Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry

Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to"already known"topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states.


Introduction
Strongly correlated systems such as cuprate superconductors, quantum Hall systems, and quantum anti-ferromagnets (QAFM) have been offering arenas for unexpected emergent phenomena brought about by strong many-body correlation. In particular, the study of QAFM bears the longest history since Heisenberg introduced the celebrated quantum-spin model [1,2] and Bethe [3] found the first non-trivial exact solution to the quantum many-body problem, and still is providing us with attractive topics in modern physics. Generally, in the presence of many-body interaction, it is very hard to obtain exact many-body ground-state wave functions and, even if possible, it is rare that we are able to write down them in compact and physically meaningful forms. Fortunately, in the above mentioned three cases (superconductivity (SC), quantum Hall effects (QHE), and QAFM), the paradigmatic ground-state wave functions have been known and greatly contributed to our understanding of exotic physics of these systems: the Bardeen-Cooper-Schrieffer (BCS) state [4] for SC, the Laughlin wave function [5] for QHE, and for QAFM the valence bond solid (VBS) states [6,7], which are the exact ground states of a certain class of quantum-spin models called the VBS models. In the present paper, we give a pedagogical review of the VBS models and their supersymmetric (SUSY) extension, i.e. the supersymmetric valence bond solid (SVBS) models [8,9,10], with particular emphasis on their relation to fuzzy geometry. The VBS models had been originally introduced by Affleck, Kennedy, Lieb and Tasaki (AKLT) [6,7] as a class of "exactly solvable" models that exhibit the properties conjectured by Haldane [11,12], namely the qualitative difference in excitation gaps and spin correlations in one-dimensional (1D) QAFM between half-odd-integer-spin and integer-spin cases. As other quantum-disordered paramagnets, the VBS states have a finite excitation gap and exponentially decaying spin correlations. However, the VBS states are not "mere" disordered non-magnetic spin states; the spin-S VBS states necessarily have gapless modes at their edges, or more specifically, emergent spin-S/2 edge spins [13,14]. This might remind the readers of the (chiral) gapless modes at the edge of the QHE systems where excitations are gapful in the bulk (see Fig. 1). Similar features are found in topological insulators as well [15,16,17] and considered as a hallmark of the topological state of matter. Furthermore, in 1D, the VBS states are known to exhibit a non-local order called the hidden string order [18,19]. What is prominent in the VBS states is that they most typically exhibit a certain kind of topological order even in 1D, while QHE needs at least 2D to work. This is a great advantage of the investigation of the VBS states since in 1D most calculations of physical quantities can be carried out exactly by using the matrix product state (MPS) representation [20,21,22,23,24] combined with the transfer-matrix method.
Since the topological character of a system is believed to be encoded in the many-body groundstate wave functions rather than in its Hamiltonian, the relatively simple structure of the VBS wave function is suitable to investigate its topological properties by using such modern means as entanglement entropy [25,26] or entanglement spectrum [27]. For the above reasons, the VBS states, or more generally, the MPS as a class of model states that satisfy the so-called arealaw constraint [28] have attracted renewed attention as a "theoretical laboratory" in the recent study of topological states of matter. In fact, the MPS representation is now regarded as a natural and efficient way to describe quantum entangled many-body states, and for a given (1D) Hamiltonian the density matrix renormalization group (DMRG) [29] provides a powerful tool to find the optimized variational wave function in the form of MPS (see Refs. [30,31,32] for more details about the MPS method and DMRG). The key idea there is that for generic short-range interacting systems, only a small part of the entire Hilbert space is important and, depending on the problems in question, there are variety of ways to parametrize this physically relevant subspace. Although the Wigner's 3j-symbol may give a convenient description of SU (2)-invariant MPS states [33,34,35], we adopt in the present paper the Schwinger-particle formalism (see, for instance, chapters 7 and 19 of [36]) to emphasize the analogies to the lowest Landau level physics. The list of possible applications includes a convenient description of gapped quantum ground states [37,38,28] as well as its application to efficient simulations of dynamics [39,40,41,42,43,44] and variational calculations [45,46,47,48]. Due to their simple structure, the entanglement entropy of the VBS states comes only from the bond(s) cut by the entanglement bipartition [49,50,51,52].
In some 'anisotropic' MPSs, it is known that a generalized quantum phase transitions (QPT) occur as the parameters contained in the MPSs are varied [53,54,55,56]. However, a remark is in order about the interpretation of this kind of 'quantum phase transitions' in MPS. Normally, these QPTs in MPS are characterized by the divergence of spatial correlation lengths [56]. In generic lattice models, on the other hand, the diverging spatial correlation does not necessarily mean the vanishing of the excitation gap, while in the Lorentz-invariant systems, these two occur hand in hand. In fact, by the structure of MPS, the block (with size L) entanglement entropy never exceeds the L-independent value 2 ln D [57] (with D being the dimension of the MPS matrix) while in quantum critical ground systems whose low-energy physics is well described by conformal field theories, the block entanglement entropy is proportional to ln L [57,58]. Therefore, one should take the QPTs in MPS mentioned above in a wider sense. This kind of phase transitions in the VBS-type of states in 2D have been discussed in [53,54].
Finally, we would like to mention the recent application of MPS to the classification of the gapped topological phases in 1D. As is well-known, there is no true topological phase characterized by long-range entanglement [59]. However, if certain symmetries (e.g. time reversal) are imposed, topological phases characterized by short-range entanglement are possible. Since this kind of topological phases are stable only in the presence of certain protecting symmetries, they are called symmetry-protected topological (SPT) [60]. As has been mentioned above, an appropriate choice of MPS faithfully represents any given gapped (short-range entangled) ground state. Therefore, the problem of classifying all possible gapped topological phases reduces to classifying all possible MPSs by using group cohomology [59,63]. This program has been carried out for such elementary symmetries as time-reversal and link-parity in Refs. [61,62] (for the fermionic systems, see Refs. [64,65]) and for the Lie-group symmetries in Ref. [66]. Topological quantum phase transitions among these SPT phases have been discussed in, e.g., [60,67,68].
In a sense, the second important keyword of this paper, fuzzy (super)geometry obtained by replacing the ordinary (commuting) coordinates with non-commutative ones, is again closely related to Heisenberg, who made a pioneering contribution in physics when he had built quantum mechanics on the basis of non-commuting phase-space variables. Snyder first substantiated Heisenberg's idea of non-commutative coordinates in his paper "quantized space-time" [69] 1 . In fact, the VBS models have many interesting connections to QHE and fuzzy geometry. To explain the interesting relationship among them, let us begin with an analogy between the VBS states and the Laughlin wave functions of fractional quantum Hall effects (FQHE). Soon after the proposal of AKLT [6,7], Arovas, Auerbach and Haldane [71] realized that the Laughlin wave function and the VBS states (generalized to higher-spin cases) have analogous mathematical structures upon identifying the odd integer m, which characterizes the filling fraction ν = 1/m, and the spin quantum number S. Specifically, the VBS state is transformed to the Laughlin wave function of the electron system on a two-sphere, i.e. the Laughlin-Haldane wave function [72] by using the coherent-state representation of the VBS states and assigning appropriate correspondences between their physical quantities. In such translation, the external symmetry (or, SU (2)-rotation of the spatial coordinates) of the Laughlin-Haldane wave function for QHE on a two-sphere is translated into the internal symmetry (or, spin-SU (2) symmetry) of the VBS states for the integer-spin chains. This analogy can be generalized and one can readily see that the Laughlin-Haldane wave functions with some external symmetry are generally transformed to give the VBS states with the identical internal symmetry.
On the other hand, we have already seen that the Bloch spin-coherent state enables us to relate the Laughlin-Haldane wave functions to the VBS states. In correspondence with such QHE, a variety of VBS models have been constructed with the symmetries, such as SO(5), SO(2n + 1) [107,108,109], SU (N + 1) [110,111,112,113], Sp(N ) [114,115], and q-deformed SU (2) [21,22,116,117,118] [see Fig. 2]. The VBS states demonstrate manifest relevance to fuzzy geometry also, by adopting the Schwinger operator formalism (see Sec. 2.1).
One of the main goals of this review is to illustrate their relations which have been less emphasized in previous literatures. As an explicit demonstration of the correspondence, we discuss a supersymmetric (SUSY) generalization of the VBS models. Since fuzzy super-spheres have already been explored in Refs. [119,120,121,122,123] and the SUSY QHE in Refs. [124,125,126, 127] 2 , we can develop a SUSY version of the VBS models (SVBS models) based on the mutual relations [8,9,10]. For the inclusion of fermionic degrees of freedom, the SVBS states exhibit particular features not observed in the VBS states with higher dimensional bosonic groups. For instance, while the VBS states only exhibit a property of quantum magnets, the SVBS states accommodate two distinct sectors, spin sector and charge sector, due to the inclusion of fermionic degrees of freedom. Physically, the SVBS states can be interpreted as hole-doped VBS states with superconducting property. Mathematically, the SVBS states are regarded as a type of "superfield" in terminology of SUSY field theory. As the superfield unifies a various fields as its components, the SVBS states realize a variety of ordinary VBS states as the coefficients of the expansion with respect to Grassmann quantities.
By investigating topological features of the SVBS states, we address how SUSY affects the  stability of topological phase or entanglement spectrum. As concrete examples, we discuss topological feature of the SVBS states with U OSp(1|2) and U OSp(2|2) symmetries, which we call type-I and type-II SVBS states, respectively. To perform detail analyses of the SVBS states on 1D chain, we develop a SUSY version of the MPS formalism (SMPS) [23,20,24]. MPS formalism is now regarded as an appropriate formalism to treat gapped 1D quantum systems [37,28]. Since the MPS formalism naturally incorporates edge states, MPS provides a powerful formalism to discuss relations between topological order and edge state. Taking this advantage, we can obtain general lessons for SUSY effects in topological phases. Specifically, the robustness of the topological phase in the presence of SUSY is discussed in light of modern symmetry-protected topological order argument with emphasis on edge superspin picture. We also generalize the lower SUSY analyses to the case of higher SUSY; fuzzy four-superspheres and U OSp(1|4) SVBS states.
The following is the organization of the present paper. In Sec. 2, we give a brief introduction to the original SU (2) VBS states and explain its relations to fuzzy two-sphere and the 2D QHE through the Schwinger-operator representation and the Hopf map. In Sec. 3, the SUSY extensions of fuzzy two-sphere are presented in detail. The corresponding SVBS states and their basic properties are reviewed in Sec. 4. In Sec. 5, the SMPS formalism is introduced for the analysis of SVBS states. Gapped excitations of SVBS states are derived with use of SMPS formalism within single mode approximation. We investigate topological properties of the SVBS states in Sec. 6, where the entanglement spectrum and entanglement entropy are derived. (Unpublished results for U OSp(2|2) SVBS chain are also included here.) The stability of topological phases in the presence of SUSY is discussed too. In Sec. 7, we extend the discussions to the case of fuzzy foursuperspheres and the U OSp(1|4) SVBS states. Sec. 8 is devoted to summary and discussions. In Appendix A, we provide mathematical supplements for the super-Lie group, U OSp(N |2K). Fuzzy four-superspheres with arbitrary number of SUSY are described in Appendix B.

Fuzzy Geometry and Valence Bond Solid States
In this section, we give a quick introduction to fuzzy two-spheres and the (bosonic) VBS states. Their mutual relations will be discussed, too.
The original idea of quantization of two-sphere was first introduced by Berezin in '70s [89]. In the beginning of '80s, Hoppe [90] explored the algebraic structure and field theories on a quantized sphere, and subsequently in the early '90s, Madore [91] further examined their structures coining the name "fuzzy sphere". Unlike the ordinary sphere, the fuzzy sphere has a minimum scale of area, while respecting the SU (2) continuum (rotational) symmetry as the ordinary two-sphere. This is a remarkable property of the fuzzy sphere, when we consider the field theories on it. In fact, in the other regularized field theories, such as lattice field theories, the extrinsic cut-off cures the UV divergence at the cost of continuous symmetries and the resulting theories only respect the discrete space-group symmetry of the lattice on which they are defined. On the other hand, field theories defined on the fuzzy manifolds contain an intrinsic "cut-off" coming from the minimum area of the fuzzy sphere, and the non-commutative field theories constructed on it were expected to have the innate property which might soften the UV divergence to be appropriately renormalized in conventional field theories.
In the middle of '90s, Grosse et al. [92] generalized the notion of the fuzzy two-sphere to construct four-dimensional fuzzy spheres and supersymmetric fuzzy spheres, i.e. fuzzy superspheres [119,120]. In late 90s, the fuzzy geometry was rediscovered in the context of the "second revolution of string theory" and attracted a lot of attentions. Researchers began to recognize that the fuzzy geometry is closely related to the geometry that the string theory attempts to describe: The geometry of multiple D-branes is naturally described by fuzzy geometry (see Refs. [138,139,140] as reviews) and fuzzy manifolds are also known to arise as classical solutions of Matrix theory (see e.g. Refs. [93,141]). Similarly, field theories on fuzzy superspheres provide a set-up for SUSY field theories with UV regularization [119,142,143], and the fuzzy superspheres were also found to arise as classical solutions of supermatrix models [144,145]. For details and applications of the fuzzy sphere, interested readers may consult Refs. [146,147,148,149]. Non-commutative geometry and fuzzy physics also found their applications in gravity [150,151,152] and in condensed matter physics [86,87,88].

Fuzzy two-spheres and the lowest Landau level physics
The fuzzy two-sphere [89,90,91] is one of the simplest curved fuzzy manifolds 3 . The coordinates of fuzzy two-sphere, X i (i = 1, 2, 3), are regarded as the operators that are constituted of the SU (2) generators Here, R denotes the radius of the sphere and d is the dimension of the SU (2) irreducible representation: with the SU (2) Casimir index j (j = 0, 1 2 , 1, 3 2 , 2, · · · ), and J i (i = 1, 2, 3) are the SU (2) matrices of the corresponding representation that satisfy and Since J 3 takes the eigenvalues j, j − 1, j − 2, · · · , −j, the eigenvalues of X 3 are given by Each "latitude" of X 3 corresponds to a patch of width 2R/d, i.e. the minimum state, on the fuzzy sphere 4 . If the limit of large SU (2) representation d → ∞ (with fixed radius R) is taken, the patches become invisible and the discrete nature of the fuzzy sphere is smeared off. In fact, one readily sees [X i , X j ] = 0 and X i X i | d→∞ = R 2 and the discrete spectrum (6) of X 3 becomes continuum raging between −R and R. In this sense, the fuzzy sphere reduces to the ordinary (commutative) sphere with radius R in the limit d → ∞.
As a realization of the fuzzy two-sphere, a convenient way is to utilize the Schwinger operator formalism [156,148] and introduce the following operator whose components satisfy the following ordinary bosonic commutation relations: By sandwiching the Pauli matrices with the Schwinger operators Φ, one may simply represent the coordinates X i of the fuzzy two-sphere as: where σ i (i = 1, 2, 3) are the Pauli matrices: This is the well-known construction of the SU (2) angular momentum operators introduced by Schwinger [157]. In fact, one can easily check that J i = (1/2)Φ † σ i Φ satisfy the standard SU (2) commutation relations (3) and that the Schwinger-operator representation (9) coincides with the original definition (1). The square of the radius is given by where n denotes the eigenvalues of the number operator for the Schwinger bosons,n = Φ † Φ = a † a + b † b. Comparing this with eq.(5), one sees j = n/2 and d(n) = n + 1.
Since we utilize the Schwinger operator, the irreducible representation is given by the fully symmetric representation of SU (2): where n 1 and n 2 are non-negative integers that satisfy n 1 + n 2 = n. Physically, |n 1 , n 2 stand for a finite number of basis states constituting fuzzy two-sphere which are the eigenstates of X 3 with the eigenvalues: where k ≡ n 2 = 0, 1, 2, · · · , n. The dimension of the space spanned by the basis states (13) is given by d(n). From the expression (14), it is not obvious that the definition (9) respects the SU (2) rotational symmetry of the fuzzy two-sphere. However this is an artifact of the particular choice of the X 3 -diagonal basis states (13). Any other complete sets of the basis states of the fuzzy two-sphere can be obtained by applying SU (2) transformations to the basis set (13). In this sense, whole Hilbert space of fuzzy two-sphere is "symmetric" with respect to the SU (2) transformation, and hence the fuzzy two-sphere is SU (2) (rotationally) symmetric like the ordinary (continuum) sphere.
Another description of fuzzy two-sphere is to utilize the coherent state 5 formalism, which will be useful in understanding the connection with the VBS states. The coherent state φ, or more precisely the Hopf spinor, that is labeled by a point on two-sphere ( 5 The coherent state here is usually referred to as the Bloch spin coherent state in literature. 6 The usual coherent 'state' |φ (for S = 1/2) defined as |φ ≡ Φ † ·φ|0 satisfies (2xi·Ji)|φ = |φ .
Expressing the above in the basis a † |0 and b † |0 , we obtain eq.(15).
In general, the (normalized) coherent state is represented in terms of the Euler angles as where (x 1 , x 2 , x 3 ) = (sin θ cos ϕ, sin θ sin ϕ, cos θ) (0 ≤ ϕ < 2π, 0 ≤ θ ≤ π, 0 ≤ χ < 2π), and e − i 2 χ denotes an arbitrary U (1) phase factor. The relation between the point (x 1 , x 2 , x 3 ) on a two-sphere and the two-component Hopf spinor (u, v) is given by the so-called 1st Hopf map 7 which maps the three-sphere S 3 onto the two-sphere S 2 : In fact, a space of normalized two-component complex spinors φ = (u, v) t subject to φ † φ = 1 is isomorphic to S 3 and eq.(17) gives the mapping onto S 2 : The Hopf spinor φ is regarded as the classical counterpart of the Schwinger operator Φ [eq. (7)] that satisfies If we note that R(n + 2)/d → R, X i → Rx i in the limit d → ∞, the resemblance between (15) and (20) is clear. Multiplying (R/d)Φ from the right to both sides of Eq.(20) and using Φ † Φ = n, we reproduce Eq. (11). By replacing the Schwinger operator with the Hopf spinor, one sees that the Schwinger operator construction (9) of fuzzy coordinates is an operator-generalization (x i → X i , φ → Φ) of the Hopf map (17) [158]. The fully symmetric SU (2) representation in terms of the Hopf spinor (u, v) is obtained if the Schwinger operator in (13) is replaced with the Hopf spinor (a † , b † ) → (u, v) with n 1 + n 2 = n, n 1 , n 2 ≥ 0. So far, (u, v) are just the two auxiliary variables needed to realize the fuzzy two-sphere. However, if we regard them as the physical coordinates of the usual two-dimensional sphere [via the Hopf map (18)], we may think that u n 1 ,n 2 represents the wave functions of a certain kind of quantum-mechanical systems in two dimensions. In fact, the wave functions u n 1 ,n 2 (21) coincide with those of the lowest eigenstates of the Landau Hamiltonian on a two-sphere (with radius R) in the Dirac monopole background (i.e. the so-called Haldane's sphere [72]): 7 For construction of higher dimensional Hopf maps, see Ref. [73] for instance. . The fuzzy two-sphere consists of a finite number of patches, i.e., the basis states, with width 2R/(n + 1). The Haldane sphere is a two-sphere with Dirac monopole at its center. The S = n/2 is the monopole charge quantized as half-integer or integer by the Dirac quantization condition. In the local spin state of the VBS state (lower center), each blob denotes spin-1/2 degrees of freedom, and n blobs amount to S = n/2 local spin by a large Hund coupling on each site.
where Λ i (i = 1, 2, 3) are the covariant angular momenta in the presence of the gauge field A i generated by the Dirac monopole at the origin: In the fuzzy geometry side, the monopole charge n/2 corresponds to the quantized radius of the fuzzy two-sphere (11) (see also Fig. 3). Thus, the lowest Landau level eigenfunctions (on a two-sphere) can be "derived" from the fuzzy two-sphere by switching from the Schwinger operator formalism to the coherent state formalism (or, the Weyl representation). This is the key observation of correspondence between the fuzzy geometry and the lowest Landau level physics [73].

Valence bond solid states
In order to translate the above features to those of QAFM, we first express the spin 1/2 state in terms of the Schwinger bosons: The fully symmetric representation (13) constructed out of n Schwinger operators automatically realizes the local spin defined on each site with the spin magnitude Physically, this bosonic construction realizes the ferromagnetic (Hund) coupling among n spin-1/2s to yield the maximal spin, S = n/2, at each site. We have already seen that the spin wave function, written in terms of the Hopf spinor (u, v), coincides with those of the non-interacting electrons moving on a two-sphere in the presence of the monopole magnetic field. Now we demonstrate that a similar analogy exists even when we switch on strong interactions. Specifically, the exact ground-state wave functions of a class of interacting spin models called the valence-bond solid (VBS) model [6,7,71] closely resemble the Laughlin wave functions on a two-sphere. The ground states of the VBS models (dubbed the VBS states) on a lattice with coordination number z (see Fig. 4) are constructed as follows. As the first step of the construction, we prepare n (= z) local S = 1/2 spins (auxiliary spins) on each vertex (site) of the lattice. Next, we connect every pair of two spin-1/2s on the nearest neighbor sites by spin-singlet bonds 8 called the valence bonds (VB) (see Fig. 4). Last, we project the entire 2 n -dimensional Hilbert space of n spin-1/2s onto the subspace with the desired value of the (physical) spin S (S ≤ n/2) on each site to obtain the spin-S many-body singlet state. Depending on which irreducible representation we use to realize the physical spin, we obtain different states even for the same lattice structure (some examples may be found in, e.g., Ref. [109]). This kind of construction (valene-bond construction) applies to any lattice in any dimensions (see Fig. 4 for typical examples in 1D and 2D) and the state obtained thereby is called the VBS state or, in a modern terminology, the projected entangled-pair state (PEPS) [159,160].
It is also possible to construct the VBS states for other symmetry groups (e.g., SU (N ) [110,111] and SO(N ) [107,108]) with due extension of the notion of the singlet valence bond. In these examples, the VBS states thus constructed are constrained by specific symmetry groups and normally they do not have any tunable parameter 9 . However, if we use the matrix-product representation [20] in 1D or, in general, tensor-product representation (or, equivalently, the vertexstate representation [53,54]) in higher dimensions to express the VBS states and consider their "anisotropic" extensions of the tensors, it is possible to obtain parameter-dependent states. These generalized VBS states are known to have interesting properties. See, for instance, Refs. [20,22,55] for 1D and Refs. [53,54] for 2D honeycomb-and square lattices. Now let us come back to the SU(2) VBS states. If we represent the up and down degrees of freedom of auxiliary spin-1/2 by the Schwinger bosons a † and b † (see eq.(25)), the singlet valence bond on the bond ij reads 8 Since the spin-singlet state of the two spin-1/2s maximizes the entanglement entropy, the pair is sometimes called maximally entangled in modern literatures. 9 However, the parent Hamiltonians (i.e. the VBS models) may have tunable parameters. For instance, the parent Hamiltonian of the spin-2 VBS state (in 1D) allows one free parameter to tune even after we fix the overall energy scale.  Physically, the spin singlet bond denotes the state with no specific spin polarization made of two spin 1/2 states, and hence the valence bond represents a non-magnetic spin pairing between the two neighboring sites. Of course, we can promote the power of the singlet bond from one to an arbitrary integer M to represent M valence bonds between the sites i and j. Thus, one sees that the original valencebond construction [6,7] of the VBS states is equivalent to the following representation in terms of the Schwinger bosons [71] where ij ∈ N.N. implies that the product is taken over all nearest-neighbor bonds ij and |vac denotes the vacuum of the Schwinger bosons.
, it is obvious that the state eq.(29) is spin-singlet. As z bonds emanate from each site of the lattice (for instance, z = 2 is for 1D chain and z = 2D for D-dimensional hypercubic lattice), we have M z Schwinger bosons per site in the VBS state (29) and hence the local spin quantum number In particular, for the 1D (i.e., z = 2) M = 1 VBS state, we have and the local Hilbert space is spanned by the following three basis states: We were a bit sloppy in writing down eq. (29). When considering a finite open chain (with length L), we should be careful in dealing with the edges of the system, while, for the VBS states on a circle, the expression (29) is correct without any modification. In fact, in (29), the number of the Schwinger bosons at the sites 1 and L is M (half of that of the other sites) and we have to add the extra edge degrees of freedom represented by the M -th order homogeneous polynomials in a † and b † to recover the correct spin-M s at the edges. The edge polynomials for M = 1 are given by This representation naturally incorporates the physical emergent edge spins [13,14] localized around the two edges. For general M , the precise form of the VBS states on an open chain reads as Since the VBS Hamiltonian is defined as the projection operator for the Hilbert space of a pair of neighboring spins (see Sec  [7]. From similar considerations, it is obvious that the higher-dimensional VBS ground states (e.g., those in Fig. 4) have degeneracy exponentially large in the size of the boundary. In the following, unless otherwise stated, we implicitly assume that the VBS state is defined in 1D and will focus on the spin-1 case.
Let us look at some key features of the VBS states in more detail. Classically, AFM on a bipartite lattice assumes the Néel-ordered state, where any pair of neighboring spins point the opposite directions. To be specific, in the Néel state of (classical) spin-1 AFM, the z-component S z of local spins at the site i takes either +1 or −1 depending on to which sublattice the site belongs. (Without loss of generality, we may assume that ordered spins are parallel to the z-axis.) The spin configuration described by the VBS state is totally different from that of the classical Néel state described above. First of all, we note that the VBS state is SU (2)-invariant (being a product of singlet bonds between pairs of adjacent sites; see eq.(29) and Fig. 4) and is non-degenerate, 10 while the classical ground state, i.e. Néel state, is infinitely degenerate with respect to the SU (2) rotational symmetry. This implies that though the magnitude of the local spin is S i = 1, its expectation value is zero S i VBS = 0 in the bulk 11 and the system is non-magnetic. In this sense, the VBS state is purely quantum-mechanical and does not have any classical counterpart.
For better understanding of the VBS state, let us expand the spin-1 VBS state in the S z -basis: = | · · · 000 · · · + | · · · 00 − · · · + | · · · 0 − + · · · + | · · · 0 − 0 · · · + | · · · 0 + 0 · · · + | · · · − + − · · · + | · · · − 0 + · · · + | · · · − +0 · · · + | · · · + 00 · · · + | · · · + 0 − · · · + | · · · + − + · · · + | · · · + −0 · · · + | · · · 0 + 0 · · · + | · · · 0 + − · · · + | · · · 00 + · · · + | · · · 000 · · · , where the coefficient in front of each term on the right-hand side is omitted for simplicity. What is remarkable with the above VBS state is that all the states appearing on the right-hand side have a very special feature; the states S z = +1 and −1 appear in an alternating manner with intervening S z = 0 states. Namely, the ground state exhibits an analogue of the classical Néel order called the string order [18,19] albeit "diluted" by randomly inserted zeros (see Sec. 6.1 for further detail). Unlike in the case of the classical Néel order, by the SU(2) symmetry of the state, the existence of the string order does not rely on the particular choice of the quantization axis (z-axis here). For the sake of later discussions, we introduce here a concise representation of the VBS state and point out remarkable similarity [71] to the Laughlin-Haldane wave function [72] for FQHE on a two-sphere. First, we note that by using the SU (2) U Sp(2) invariant matrix given by the 2×2 antisymmetric matrix (see Appendix A) the VBS states (29) on generic lattices can be rewritten compactly as where Φ i denotes the Schwinger operator on the site i   Then, we rewrite the VBS state (38) into the form of the wave function on a two-sphere. Specifically, by replacing the Schwinger operator with the Hopf spinor we obtain the coherent-state (or, Weyl) representation of the VBS state 12 : Now the formal similarity to the Laughlin-Haldane wave function of QHE on a two-sphere [71] is clear. 13 Though the physical interpretation of the quantities appearing in these wave functions are different (Table 1), mathematical similarities between the VBS model and QHE may be manifest from the above constructions. Similarities between topological properties of VBS and QHE have also been discussed in Refs. [161,162]. 12 Precisely, if we use the coherent-state basis for the local spin-M z/2 states and expand the VBS state in these basis, we obtain the 'wave function' {Ωj}|VBS ∝ ΦVBS({u * i , v * i }). 13 By the stereographic projection from a two-sphere to a complex plane: ΦLH (42) is, in the thermodynamic limit, reduced to the celebrated Laughlin wave function, 3 Fuzzy Two-supersphere Next, we proceed to the SUSY version of fuzzy two-sphere 14 and generalized VBS states. We mostly focus on the cases of the SUSY numbers N = 1 and 2.
The fundamental representation (j = 1/2) matrices of the U OSp(1|2) generators are expressed by the following 3×3 matrices: where τ α (α = 1, 2) are Eq.(48) may be regarded as a SUSY extension of the Pauli matrices. They are "Hermitian" in the sense where ‡ signifies the super-adjoint defined by 14 Fuzzy superspheres are also realized as a classical solution of supermatrix models [144,145].
Similar to the case of fuzzy sphere (9), the coordinates of fuzzy two-supersphere [119,120] are constructed by the graded version of the Schwinger operator formalism [122,123,124]. The graded Schwinger operator consists of two bosonic components, a and b, and one fermion component f , with satisfying With use of Ψ, the coordinates of fuzzy two-supersphere, S 2|2 f (2 on the left to | denotes the bosonic degrees of freedom, while 2 on the right to | does the fermionic degrees of freedom), are constructed as which satisfy Square of the radius of fuzzy supersphere is given by the U OSp(1|2) Casimir Notice the zero-point energy in (56) reflects the difference between the bosonic and fermionic degrees of freedom. The basis states on fuzzy supersphere consist of the graded fully symmetric representation specified by the superspin j = n/2: where n 1 , n 2 , m 1 and m 2 are non-negative integers that satisfy n 1 + n 2 = m 1 + m 2 + 1 = n. |m 1 , m 2 ) is the fermionic counterpart of |n 1 , n 2 , and thus |n 1 , n 2 and |m 1 , m 2 ) exhibit N = 1 SUSY. The bosonic and fermionic basis states are the eigenstates of the fermion parity (−1) n F with the eigenvalues +1 and −1, respectively. Their degrees of freedom are also respectively given by and the total degrees of freedom is Figure 6: (Color online) N = 1 fuzzy supersphere is a "compound" of two fuzzy two-spheres with radii, R d n and R d (n − 1). This figure corresponds to n = 2.
The eigenvalues of X 3 for (57) read as where k = 0, 1, 2, · · · , 2n. Even k (k = 2n 2 ) correspond to the bosonic states (57a), while odd k (k = 2m 2 + 1) the fermionic states (57b). Compare the X 3 eigenvalues of fuzzy supersphere (60) with those of the fuzzy (bosonic) sphere (14): The degrees of freedom of fuzzy supersphere for even k are accounted for by those of fuzzy sphere with radius n, while those for odd k are by fuzzy sphere with radius n − 1. In this sense, the fuzzy two-supersphere S 2|2 f (n) of radius n can be regarded as a "compound" of two fuzzy spheres with different radii, R d n and R d (n − 1) [ Fig. 6]: The first S 2 f (n) accounts for the bosonic degrees of freedom (57a), while the second S 2 f (n − 1) does for the fermionic degrees of freedom (57b).
Similar to the 1st Hopf map, we can construct a graded version of the 1st Hopf map [166,167] ψ where l i and l α are the U OSp(1|2) matrices of fundamental representation (48), and ψ denotes a normalized U OSp(1|2) superspinor Here, the super-adjoint of the superspinor is defined by ψ ‡ ≡ (u * , v * , −η * ) [see also eq.(51)] and * represents the pseudo-conjugation. 15 The first two components of ψ are Grassmann-even and 15 The pseudo-conjugation is defined as (η * ) * = −η and (η1η2) * = η * 1 η * 2 for Grassmann odd quantities. See Ref. [168] for instance. the third component is Grassmann-odd. ψ subject to the normalization (64) can be regarded as a coordinate of the manifold S 3|2 . From (62), we find that x i and θ α satisfy the condition of S 2|2 : Consequently, the map (62) represents The bosonic part of (66) exactly corresponds to the 1st Hopf map. Note that ψ satisfies the super-coherent state equation and ψ is referred to as the super-coherent state or spin-hole coherent state [36] in literature. x i are Grassmann even but not usual c-numbers, since the square of x i is not a c-number, x i x i = 1 − αβ θ α θ β , as seen from (65). Instead of x i , one can introduce c-number quantities {y i } as which denote coordinates on S 2 , the "body" of S 2|2 , as confirmed from y i y i = 1. With use of the coordinates on S 2|2 , ψ can be expressed as [124] ψ = 1 where e iχ stands for the arbitrary U (1) phase factor. The last expression on the right-hand side manifests the graded Hopf fibration, S 3|2 ∼ S 2|2 ⊗ S 1 (here, ∼ denotes local equivalence): The S 1 ( U (1))-fibre, e iχ , is canceled in the graded Hopf map (62), and the other quantities, x i and θ α in (69), correspond to the coordinates on S 2|2 .
To specify a fuzzy manifold, an appropriate choice of irreducible representation is crucial as well. The irreducible representation of U OSp(2|2) is classified into two categories; typical representation and atypical representation [164,165]. Since the quadratic Casimir (72) is identically zero for atypical representation, we adopt typical representation for the construction of S 2|4 f [123]. The U OSp(2|2) matrices for typical representation for minimal dimension are represented by the following 4 × 4 matrices: Applying the Schwinger construction, we introduce the coordinates of S 2|4 f as where Ψ signifies the four-component U OSp(2|2) Schwinger operator Here a and b are bosonic operators, while f and g are fermionic operators that satisfy It is straightforward to evaluate the square of the radius of S 2|4 f : With a given n, the basis states of S 2|4 f are constituted of the graded fully symmetric representation of the U OSp(2|2): where n 1 , n 2 , m 1 , m 2 , m 1 , m 2 , l 1 and l 2 denote non-negative integers that satisfy n 1 + n 2 = m 1 + m 2 + 1 = m 1 + m 2 + 1 = l 1 + l 2 + 2 = n. The dimension of bosonic basis states, |n 1 , n 2 and |l 1 , l 2 , and that of fermionic basis states, |m 1 , m 2 and |m 1 , m 2 , are found to be equal: and the total degrees of freedom amount to Square of the radius of S 2|4 f (77) does not have the zero-point "energy", since the bosonic and fermionic degrees of freedom are canceled exactly 16 . The four sets of the basis states (78) forms the N = 2 SUSY multiplet, and suggest the following geometrical structure of S 2|4 f , The latitudes for the the basis states (78) are given by with k = 0, 1, 2, · · · , 2n. The even ks (k = 2n 2 , k = 2l 2 + 2) correspond to the bosonic basis states, (78a) and (78b), while odd ks (k = 2m 2 + 1, k = 2m 2 + 1) the fermionic basis states, (78c) and (78d) [ Fig. 7]. Except for non-degenerate states at the north and south poles X 3 = ±n, all the other eigenvalues of X 3 (82) are doubly-degenerate.

Supersymmetric Valence Bond Solid States
In this section, we review the basic properties of the SVBS states [8,9] and discuss its intriguing connection to the SUSY QH wave function and BCS function. is a "compound" made of four fuzzy two-spheres that are considered as N = 2 superpartners. The above picture corresponds to n = 2.

Construction of SVBS states
Here, we consider the SVBS states with U OSp(1|2) SUSY (N = 1), which we shall call the type-I SVBS states. We apply the mathematical procedure of the constructions of the VBS states described in Sec.2.2. The first we prepare is the U OSp(1|2) invariant matrix (see Appendix A) With the parameter dependent Schwinger operator the type-I SVBS state is constructed as The operators a i , b i and f i are components of the graded Schwinger operator defined on each site i and satisfy the commutation relations, Physically, the three fundamental states, a † |vac , b † |vac and f † |vac , are interpreted as spin ↑, ↓ and spinless hole states (see Table 2) 17 . Since the fermions always appear in pairs of the form f † i f † j (i, j are adjacent), the SVBS states can be regarded as hole-pair doped VBS states, and r stands for a hole doping parameter.
Here, some comments are added. The U OSp(1|2) specific feature does not enter the local Hilbert space on each site. For instance, (57) can also be regarded as an irreducible representation of SU (2|1). Meanwhile, in the construction of the type-I SVBS states (85), the U OSp(1|2) invariant matrix was utilized, and then the U OSp(1|2) structure explicitly enters in the manybody states. This implies that (super)spin interaction between adjacent sites reduces the SU (2|1) symmetry on each site to the lower symmetry U OSp(1|2) in many-body physics.
In the type-I SVB states (85), the total particle number of the Schwinger particles at each site is given by Schwinger operator SU (2) quantum number Since the fermion number f † f takes either 0 or 1, the following two eigenvalues are possible for the local spin quantum number In particular, for M = 1, each site can take two spin values and the local Hilbert space is spanned by the five (4M +1, in general) basis states These constitute N =1 SUSY multiplet with the U OSp(1|2) superspin S = 1. Similarly, the edge states consist of N = 1 SUSY multiplet with the U OSp(1|2) superspin S = 1/2: As we will see in Sec.5, the ground state of a finite open chain is nine-fold degenerate (corresponding to 3 × 3 matrix-components of the M = 1 type-I SVBS states). Intriguingly, the M = 1 type-I SVBS chain interpolates two VBS states in the two extreme limits of the hole doping: In the limit r → 0, |SVBS-I reproduces the original spin-1 VBS state |VBS , while in the limit r → ∞, |SVBS-I reduces to the Majumdar-Ghosh (MG) dimer state [170,171] |MG , where |MG is either of the two dimerized states of the MG model 18 [ Fig. 8]: 18 The open boundary condition has been implicitly assumed here; if the periodic boundary condition had been used, the two states would have been summed up with a minus sign due to the anti-commutating property of the holes. For larger M , |MG should be replaced with the inhomogeneous VBS states [71] where the number of valence bonds alternates from bond to bond.
In the super-coherent state (63) representation 19 , the SVBS state is expressed as [8] From the Grassmann (odd) property of η, Ψ SVBS-I (94) can be rewritten as where Φ VBS is the coherent state representation of the original VBS state (41). We can deduce a nice physical interpretation of the SVBS states from this expression. Since the SVBS states are written as a "product" of the exponential and the original VBS wave function Φ VBS , all physics inherent in the SVBS must be included in this exponential factor: Since every time when the factor acts to the VBS wave function, the VB between the adjacent sites i and j is replaced with a hole-pair (see Fig. 9): one sees that the SVBS wave function (95) may be expanded as Thus, the SVBS chain is expressed as the superposition of many-body states on the right-hand side (r.h.s.) of (99). The first term on the r.h.s. is the original VBS chain (which is consistent with (91)). The second term is the VBS chain with one hole-pair doped, and the third term is the VBS chain with two hole-pairs doped. In general, nth term represents the VBS chain with (n − 1) hole pairs doped. As the last term, partially dimerized chains, i.e., the VB chains whose half sites are occupied with hole-pairs, are realized (see Fig. 10) 20 . For M = 1, the last term gives rise to the Majumdar-Ghosh dimer states,  As in the original correspondence between VBS and QHE, the super-coherent state representation of SVBS (94) shows striking analogies to the SUSY Laughlin-Haldane wave function [125], Also for one-particle mechanics, there exist apparent relations between the SVBS and the SUSY Landau problem. Indeed, the super-coherent state representation of the basis states (57), gives the lowest Landau level eigenstates of a SUSY Landau Hamiltonian [124]. Here, n 1 , n 2 , m 1 and m 2 are non-negative integers that satisfy n 1 + n 2 = m 1 + m 2 + 1 = I. By the stereographic projection 20 For general M and arbitrary lattice coordination number z, the last term of the expansion realizes a resonating valence bond (RVB) state [172]. For instance, on 2D square lattice, M = 2 SVBS state with N = 3 SUSY gives the Rokhsar-Kivelson RVB state [173] as the last term. By expanding the polynomial part (z i − z j − rξ i ξ j ) m in the Grassmann odd quantities, we have where N is the total number of particles (which we assumed to be an even integer) and Φ L coincides with the Laughlin wave function on a 2D plane (up to the Grassmann factor e − i ξ i ξ * i ): Interestingly, the Pfaffian wave function Pf i for the ground state of the non-Abelian QHE [174] appears in the last term of the expansion (104) at m = 2.

N = 2
The N = 2 SVBS states, which we call the type-II SVBS states, are constructed as and U OSp(2|2) Schwinger operator we introduce the type-II SVBS states: The inclusion of two species of holes, f and g, allows us to write down a wave function more symmetric with respect to the bosonic and fermionic degrees of freedom. The new fermionic degrees of freedom g i are interpreted as another species of (spinless) hole, and satisfy the standard anti-commutation relations {g i , g † j } = δ ij , {f i , g j } = 0 and etc. In the type-II VBS states, there appear the local sites such as f † i g † i |vac with spin-0, which are not realized in the type-I SVBS states.
We have two species of fermions, and the total particle number of the Schwinger particles at each site i reads as Since the eigenvalues of n f (i)=f † i f i and n g (i)=g † i g i can take either 0 or 1, in the type-II SVBS chain (z = 2), the following four eigenvalues are allowed for the local spin quantum number In particular, for the M = 1 SVBS chain, the local spin values are given by and the local Hilbert space is spanned by the following nine basis states, The edge states are now given by and correspondingly, there appear 4 × 4 = 16 degenerate ground states for the M = 1 type-II SVBS chain. The M = 1 type-II SVBS chain has the following properties. As in the type-I SVBS state, the type-II SVBS state reproduces the original VBS state for r → 0: On the other hand, when r → ∞, it reduces to the totally uncorrelated fermionic (F) state filled with holes: Ref. [9].] Here we have assumed the open boundary condition 21 . Note that, unlike type-I, type-II SVBS states have no spin degrees of freedom for r → ∞. The super-coherent state representation of |SVBS-II is given by By expanding the exponentials in terms of r, the type-II SVBS states can be expressed as a superposition of the hole-pair doped VBS states as shown in Fig. 11. 21 If the periodic boundary condition is used, we have zero state for odd-length chains. (The sign factor depends on both the parity of the system size and the edge states).

Superconducting Properties
In both Ψ SVBS-I and Ψ SVBS-II , the fermions always appear in pairs and the wave functions can be expressed by a superposition of fermion pairs. We can point out interesting similarities between the SVBS states and BCS state of SC [175]: with electron operator c k and coherence factor g k . |BCS can be expressed as This expansion may remind us of the hole-pair expansion of the SVBS state (99). Furthermore, in the limit g k → 0, the BCS state reduces to the electron vacuum (no fermions), and for g k → ∞, it coincides with the filled Fermi sphere |Fermi : These also show apparent similarities with the asymptotic behaviors of the type-II SVBS chain, (114) and (115), under the correspondence (120b) Figure 12: The SVBS states exhibit superconducting property in the charge sector with finite r in addition to quantum AFM property in the spin sector.
From the analogies to the BCS state, the SVBS states are expected to exhibit SC property in the charge sector by the immersion of hole-pairs to (insulating) VBS states [see Fig. 12]. This is quite similar to the mechanism of the Anderson's RVB picture of high Tc SC [172]: A finite amount of hole-doping transforms the insulator of resonating valence bond state to high Tc SC state. In the following, we explore qualitative arguments for the SC aspect of the SVBS states.

N = 1
Since the naive SC order parameter such as f † k f † k+1 vanishes due to the violation of particle number conservation at each site, as the SC order parameter of the type-I VBS state we adopt the following quantity: which is calculated as It takes the maximum value In particular for M = 1, |r| = 3 2 1+ √ 5 6

The expectation values for the boson and fermion
As expected, with increase of the hole doping |r|, n b monotonically decreases, while n f monotonically increases. The fluctuations for the boson and fermion numbers, δn 2 b = n 2 b − n b 2 and δn 2 f = n 2 f − n f 2 , are given by and their maximum δn b = δn f = 1 2 is met at .
The behaviors of such quantities are depicted in Fig. 13. While the BCS state (117) has the particle-hole symmetry 22 , the SVBS state does not show an exact particle-hole symmetry, r ↔ 1/r [see the SC order parameter (122) for instance]. This is because of the unequal properties between boson and fermion operators,

N = 2
We define the order parameter for the type-II SVBS state (108) as follows: In Fig. 14, we plotted its expectation value the hole-number n f , and hole fluctuation δn hole , The SC order parameter O SC takes its maximal value at |r| 1.05 for M = 1. Comparing Fig. 13 and Fig. 14, one may find that the behavior of the order parameter of the type-II SVBS chain is more symmetric with respect to |r| = 1 than that of the type-I SVBS chain. This is because of the almost compensation of the contributions from the equal number of the boson and fermion species in the type-II SVBS states.
calculated as ∆ k and δn k are symmetric under g k ↔ 1/g * k due to the particle-hole symmetry.

Parent Hamiltonians
The VBS state is the exact and unique ground state of a many-body Hamiltonian which we call the parent Hamiltonian [6,7]. The relation between the VBS state and its parent Hamiltonian is quite unique. Usually in quantum mechanics, Hamiltonian is firstly given, and then we solve the eigenvalue problem of the given Hamiltonian. In most cases, particularly in the presence of many-body interaction, it is formidable to exactly solve the eigenvalue problem, and so we need to rely on some appropriate approximation method. Interestingly in VBS models, the procedure is completely inverse: the many-body state (VBS state) is firstly given, and next the parent Hamiltonian is constructed such that its ground state is exactly given by the VBS state.
We briefly review the procedure for the construction of the parent Hamiltonian. Consider the VBS chain with bulk spin 1. The SU (2) decomposition rule of two spin 1 gives the total spin J = 0, 1, 2: However, the values for the bond spin of the VBS chain does not take J = 2, since in that case all four 1/2 spins are aligned to a same direction and do not form the spin-singlet bond between neighboring sites [see Fig. 15]. Hence, the VBS state satisfies the condition where P J denotes a projection operator to the bond spin J = 2. Since the eigenvalue of the projection operator is either 0 or 1 for arbitrary adjacent sites, the minimum eigenvalue of the "many-body operator", i P J=2 (i, i + 1), is zero. This simple fact is the key observation for the construction of the parent Hamiltonian. We then construct the parent Hamiltonian for the VBS chain as where V J=2 denotes a positive coefficient 23 . It is obvious that the eigenvalues of the parent Hamiltonian (134) are semi-positive definite and its ground-state energy is zero. Also remember that the VBS state vanishes by the projection operators in the parent Hamiltonian, and hence Therefore, the VBS chain is a zero-energy ground state of the parent Hamiltonian (134). Uniqueness of the ground state 24 for the parent Hamiltonian (up to the degeneracy coming from the edge degrees of freedom) can also be proven [7]. The projection operator is explicitly derived as In the present case S = 1, P J=2 (i, i + 1) is given by After all, for the S = 1 VBS chain the parent Hamiltonian takes the following form: where the overall proportional factor 1/2 has been absorbed in the coefficient V . The parameter V simply determines the energy scale of the system, and is not important in the dynamical 23 VJ=2 can depend on the lattice site i, but here we postulate lattice translation symmetry and drop the lattice index i. 24 On a finite-size chain, the ground state is, by construction, either unique (for a periodic chain) or degenerate with respect to the edge states (for an open chain). In order to prove the uniqueness in the infinite-size system, one has to define the infinite-size ground state carefully [7]. behaviors of the system. The construction of the parent Hamiltonian is rather technical, but the resulting parent Hamiltonian (138) appears to be "physical": The spin-spin interaction represents the Heisenberg AFM, though it contains the quadratic term whose coefficient is one-third of the first Heisenberg AFM term. It may be obvious from the above discussion that the VBS state is the exact ground state of the parent Hamiltonian, but if the Hamiltonian (138) was firstly given, one might think that it is almost impossible to derive its exact energy spectra and corresponding many-body states. However at least for the ground-state, we know the exact form of the wave function and its energy. Taking this advantage, we can develop precise discussions about physical quantities relevant to the ground-state, such as entanglement spectrum. For excitations, we need to rely on some approximation technique to extract useful information from the parent Hamiltonian (see Sec.5.3.2). It should also be emphasized that the present construction can be generalized to higher spin VBS states on arbitrary lattice in any dimensions. Generally, the parent Hamiltonian is given by where V J denotes a positive coefficient and P J (i, j) is given by In QHE, the parent Hamiltonian for the Laughlin-Haldane wave function is called the Haldane's pseudo-potential Hamiltonian [72]. As discussed in Sec.2.2, the mathematical structure of the VBS state and the QHE is similar to each other, and the pseudo-potential Hamiltonian for QHE can also be obtained by applying the translations between QHE and VBS (Table 1). Indeed, the resultant pseudo-potential Hamiltonian for Laughlin-Haldane wave function takes the form similar to (139): where P J denotes the projection operator to two-body angular momentum J.

N = 1
By replacing the SU (2) operators with U OSp(1|2) ones, we readily construct the parent Hamiltonian for the type-I SVBS states which are invariant under U OSp(1|2) transformations generated by the SU (2) bosonic generators L i , and the parameter-dependent fermionic operators K α , where Ψ(r) is defined by (84). Notice that L i and K α satisfy the U OSp(1|2) algebra (45). With use of the U OSp(1|2) generators, it is straightforward to construct the parent Hamiltonian for the SVBS states with arbitrary value of the parameter r. First, we need to derive the projection operators to the subspaces of bond-superspin J. The decomposition rule for the two superspins S is given by Note that the U OSp(1|2) decomposition rule is similar to that of the SU (2) except for the bondsuperspin decreasing by 1/2. The SVBS state does not contain any U OSp(1|2) bond superspins larger than J max = (z − 1)M , and is the exact zero-energy ground state of the parent Hamiltonian where V J are positive coefficients, and P J (i, j) are the projection operator onto the superspin-J representation of U OSp(1|2) written in terms of the Casimir operator as: which projects to the two-site subspace of the bond superspin J. Here, 2 ). Since the projection operators (146) are U OSp(1|2) invariant, the parent Hamiltonian is (145) as well. Following similar discussions about the uniqueness of the VBS state in Ref. [7], we can prove that the SVBS state is the unique zero-energy ground-state of the parent Hamiltonian (145).
For concreteness, we demonstrate the derivation of the parent Hamiltonian for the S = 1 SVBS chain (z = 2, M = 1 and then J max = 1). From (144), we obtain the parent Hamiltonian (145) for type-I SVBS chain as Here, we add several comments. Since the Casimir operator (K A (i) + K A (j)) 2 contains paircreation terms of fermions, such as , the Hamiltonian (147) does not preserve the total fermion number N f = i f † i f i (though the total particle number, i.e. the sum of boson number and fermion number, is conserved). This fermion number non-conserving term is a particular structure of the BCS Hamiltonian, and this is in agreement with the SC property of the SVBS model (see Sec.4.2).
The fermionic generators K α are non-Hermitian, and then the type-I parent Hamiltonian (145) is non-Hermitian as well. As a reasonable Hermitian extension of the type-I Hamiltonian, one may adopt The definition (148) is a natural generalization of the original type-I parent Hamiltonian, since, if the projection operators were Hermitian, from the property P 2 J = P J Eq.(148) would be reduced to the original one (145).

N = 2
The irreducible representation of the U OSp(2|2) group is specified by (j, b), which signify the indices of the largest bosonic subalgebra, SU (2) ⊕ U (1) 25 . The irreducible representation is classified into two categories; the typical representation and atypical representation. For b = ±j, the irreducible representation is called the typical representation with dimension 8j, while for b = ±j the irreducible representation becomes to the atypical representation with dimension 4j + 1. The eigenvalues of the U OSp(2|2) quadratic Casimir operator (72) for (j, b) representation are generally given by Then for the atypical representation (b = j), the Casimir eigenvalues identically vanish. Since the Casimir corresponds to the square of the radius of fuzzy manifold, it may not be probable to explore fuzzy geometry based on the atypical representation. Meanwhile, the typical representation consists of four SU (2) representations, |b, j, j 3 , |b + 1/2, j − 1 2 , j 3 , |b − 1/2, j − 1 2 , j 3 , |b, j − 1, j 3 , each of which carries the SU (2) spin index S as 25 We denote the eigenvalues of the SU (2) Casimir operator S 2 as j(j + 1) (j = 0, 1/2, 1, · · · .), and arbitrary complex number eigenvalues of the U (1) operator B as b.
For a two-site system, the U OSp(2|2) bond superspin operators are constructed as , and the quadratic Casimir operator is expressed as where C(i) = C(j) = S 2 (S = 0, 1/2, 1, 3/2, · · · ) 27 and L(i) · L(j) is defined as Tensor product of two identical typical representations is decomposed as For instance, 28 indices, (j, b) = (S, 0). 28 On the right-hand sides of (157) and (156) Similar to the U OSp(1|2) decomposition (144), the U OSp(2|2) bond superspins decrease by 1/2 [see the right-hand side of (157)]. As also observed in (157), B is specified by J: For integer J, B = 0, while for half-integer J, B = 1/2 or −1/2. Hence, the square of B is uniquely determined as a function of J: Invoking the usual arguments of constructing the parent Hamiltonian, we derive the type-II parent Hamiltonian as where P J stand for the projection operators of the bond superspin J: .
Here, B(J) 2 and B(J ) 2 are given by (158). For L = 1 type-II SVBS chain, the parent Hamiltonian is given by where the projection operators are With use of C i,i+1 = 2L(i) · L(i + 1) + 2, the type-II parent Hamiltonian (161) can be rewritten as It is noticed that, unlike the type-I parent Hamiltonians (145), the type-II parent Hamiltonians (159) themselves are Hermitian, since the U OSp(2|2) Casimir itself (154) is given by a Hermitian operator.

Supersymmetric Matrix Product State Formalism
This section reviews the MPS formalism and its supersymmetric version, SMPS. In the formalism, the edge degrees of freedom are naturally incorporated. Practically, the MPS formalism provides a powerful method to calculate physical quantities such as excitation gap, string order and entanglement spectrum.

Bosonic Matrix Product State Formalism
As we have discussed above, the VBS state is expressed as a product of valence bonds defined on two adjacent sites. In 1D, the VBS state (29) can be rewritten as a product of matrices defined on local site [20,23]: where the "state-valued" matrix A i is given by It is clear, by the Schwinger-boson construction, that the row and the column of the matrix A i correspond respectively to the valence bonds going from the site i to its adjacent left and right sites. Sometimes, it is convenient to write (165) in a slightly different way: In the first representation, the c-number matrices A(m) are given by while, in the second, A (a) is given by the Pauli matrices σ a (a = x, y, z). Using these representation, we can recast (164) into the form where the c-number coefficients and the basis part are separated explicitly: The state (164) or (168) represents a collection of the D 2 states (with D being the matrix size) specified by the matrix indices (α, β). In the above case, (α, β) have a clear physical meaning that they specify the states of the two emergent edge spins.
It is convenient to represent A(m) [and A * (m)] by the following simple tripod diagrams: , where the thick-and the thin lines respectively denote the d-dimensional physical Hilbert space (here, d = 3 spin-1 states labelled by m = −1, 0, 1 or a = x, y, z) and the D-dimensional auxiliary space (D = 2-dimensional space spanned by the spinors a † and b † , here); the matrix multiplication amounts to connecting open thin lines on the adjacent sites. Then, the bra and the ket vectors may be depicted by strings of these tripods (Fig. 16). Quantum states which can be written in the form of (164) or (168) are in general called matrix-product states (MPS). As has been mentioned in Sec. 1, any gapped gapped short-range states fall into this category. We refer the readers to recent readable reviews [31,32] for more details and the applications.
We would like to comment on an interesting property of MPS (168). When SU (2) rotation acts on the state in (166b) as |a → R ba |b , the local MPS matrix A i transforms like where R is a three-dimensional rotation matrix and U is the corresponding spinor representation. Namely, the original SU (2) symmetry for the local spin-1 objects "fractionalizes" into that for the two spin-1/2 objects (spinors). From this, it is evident that the spin-1 VBS state on a finite open chain [represented by the MPS (164)] transforms, under SU (2) rotation, as if there were two spin-1/2 objects ("quark" and "anti-quark") at the ends of the chain. The above is the simplest example of more general symmetry fractionalization property of MPS, which will be extensively used in Sec. 6.4. We can generalize the strategy to construct the parent Hamiltonian for the VBS state in Sec. 4.3 to any MPS. The idea is to prepare a cluster Hamiltonian and tune the parameters so that the Hamiltonian annihilates all the D 2 states (i.e. matrix elements) of the MPS on that cluster. In fact, it can be shown that for any given MPS there exists the parent Hamiltonian for which the MPS (164) or (168) gives the (degenerate) ground states [24]. By construction, the degree of degeneracy is equal to the number of matrix elements (D 2 ). For example, the ground state of the parent Hamiltonian of the spin-S VBS state (the VBS model) is shown to have (S+1)×(S+1)-fold degeneracy, when the model is defined on a finite open chain [6,7]. This exact degeneracy on a finite chain is peculiar to the VBS model and, if we slightly deviate from the solvable VBS point, the emergent edge spins (S/2) start interacting with each other with a coupling constant exponentially small in system size to (partially) resolve the degeneracy. For a periodic chain, on the other hand, we have to take the trace over the matrix indices and hence the ground state is not degenerate.

SMPS Formalism and Edge States
The Schwinger boson construction described in the previous section can be generalized to SUSY cases by using the Schwinger operator which contains both bosons and fermion(s).

N = 1
Now let us consider the MPS representation for the type-I VBS chain [9]. The SVBS chain (85) is written as a string of 3×3 matrices (α, β = 1, 2, 3): where From the expression (172), it is clear that the nine-fold degenerate ground states correspond to different possible choices of the edge states: The row index specifies the left edge states and the column one the right. On the left (right) edge, the matrix indices {1, 2, 3} correspond respectively to {↓, ↑, hole} ({↑, ↓, hole}). By looking at the form of A i (173), one sees that the matrix has a block structure where B(m, n) and F (m, n) respectively denote "bosonic" and "fermionic" (i.e. anti-commuting) matrices of the dimension m×n. Therefore, it is convenient to regard A i as a supermatrix. Thus, the SVBS chain can be expressed in the form of supermatrix-product state (SMPS), and the matrix size of the SMPS is directly related to the number of edge degrees of freedom. The (S)VBS states with different edge states have finite overlaps with each other, which are exponentially decreasing as the system size L. That is, two (S)VBS states with different edge states are orthogonal to each other only in the infinite-size limit.
In constructing the SVBS state on a periodic chain, one has to treat the fermion sign carefully and one sees that the trace operation used in the standard MPS representation (171) should be replaced with the supertrace: where the supertrace is defined as From these A-matrices, we obtain the following 9×9 T -matrices (transfer matrix): Here, A * is obtained from A by |· → ·| and complex conjugation. Using the matrices A(m) and the diagrammatic representation introduced in Sec. 5.1, the transfer matrix may be expressed as The transfer matrix naturally appears in the calculations of the MPS formalism. For instance, by using the diagrammatic representation Fig. 16 and the orthogonality of the local basis states m|n i = δ mn , one can show that the overlap of the two (S)MPSs with edge states {ᾱ L ,ᾱ R } and {α L , α R } can be written as 29 (see Fig. 17(a)) where the matrix multiplication is taken over the tensor index (ᾱ, α). In the periodic case, the above expression is modified: where sgn(α) = 1 for α = 1, 2 29 For the bosonic MPS, this is straightforward. For the SMPS, one has to treat the fermion sign carefully but, in the end of the day, we can check that the final result is the same.
Notice that the transfer matrix directly appears in the right-hand side of (180), and hence the calculation is boiled down to that of the power of the transfer matrix. The eigenvalues of the transfer matrix (177) are computed as and plotted in Fig. 18. The largest eigenvalue of the transfer matrix, 1 2 3 + √ 8r 2 + 9 , will be relevant in the thermodynamic limit.

N = 2
Similar to the type-I SVBS chain, we can express the type-II SVBS chain (M = 1) in the form of SMPS: where As in the type-I SVBS state, the supertrace is necessary for the periodic system: where STr(M ) ≡ M 11 + M 22 − M 33 − M 44 . The corresponding transfer matrix is a 16×16 matrix and has seven different eigenvalues: where f (r) ≡ √ r 4 + 10r 2 + 9. Regardless of the value of r, the largest eigenvalue is 1 2 r 2 + 3 + f (r) .

Excitations
In this section, we delve into dynamical properties, i.e., low-lying excitations on the SVBS chain. In Sec. 4.3, we have already obtained the parent Hamiltonian from the VBS state. Given the form of the Hamiltonian, we can, in principle, investigate dynamical properties of the system. Unfortunately, however, even though we have the exact ground state in hand, only limited (exact or rigorous) information about the excitations is available [176,177]. Nevertheless, when the explicit form of the (whether exact or approximate) ground-state wave function is known, the single-mode approximation (SMA) gives reasonably good results [87,71]: The SMA not only provides us with a simple transparent way of calculating (approximate) excitation spectrum but also sets a rigorous upper bound for the true spectrum.

Fixing parent Hamiltonian
Since the type-I Hamiltonian (147) contains one extra parameter up to the overall factor, i.e. the ratio of V 3/2 to V 2 , we begin with fixing the form of the parent Hamiltonian. One way to fix the remaining coupling is to require that the SUSY parent Hamiltonian (147) should reduce to the original SU (2) VBS Hamiltonian (138) in the limit r → 0. This naturally fixes the two coupling constants in the type-I parent Hamiltonian (147) as and we have Some of the matrix elements in the fermionic sector have a factor 1/r, and in the limit r → 0 they are divergent. However, they are harmless in the limit, since the ground states contain no fermion in the r → 0 limit. Consequently, the type-I parent Hamiltonian projected onto the bosonic sector coincides with the spin-1 VBS Hamiltonian (138).

Crackion Excitation
Now we are ready to derive the excitation spectra by using SMA. The paradigmatic picture of the low-lying excitations in the half-odd-integer-spin chains is provided by the so-called Lieb-Schultz-Mattis construction [180], where we apply a slow twist along one of the symmetry axis (say, z-axis) of the spin Hamiltonian. Physically, this boosts the quasi-particles in the system and thereby creates low-lying excitations [with energies of the order of ∼ (chain length) −1 ] at a special momentum determined solely by the total magnetization. Unfortunately, this construction does not work in the usual VBS state [23]. Instead, as we will see, an excited triplet bond (crackion [181], i.e., a "crack" created in a "solid" of valence bonds; See Fig. 19) in the VBS gives, to good approximation, a physical low-lying excitation. Due to the simple structure of the VBS states, the excitations considered in the SMA essentially coincide with the crackions [181,23]. In the Schwinger-boson construction of the VBS states (29), the crackion excitation is obtained by replacing one of the singlet valence bonds (a . Since a single Schwinger boson a † or b † describes a spin-1/2 spinon, we may thought of the triplet crackion as the confined triplet pair of two spinons. The notion of crackion excitations can be generalized in other VBS-type models (states) with higher symmetries [e.g., SU (N )], where an intriguing picture on the relation between spinon confinement and the existence of 'Haldane gaps' has been proposed [111,113] Now let us consider the crackions in the SVBS chains [9]. Since we are dealing with a SUSY system, we consider two different types of excitations (see Fig. 19) that may be regarded as super-partners of each other: • Spin excitation: Spin triplet excitation (S = 1) created by U OSp(1|2) bosonic operators • Spinon-hole excitation: Spin doublet excitation (S = 1/2) paired with a hole created by U OSp(1|2) fermionic operators They are schematically represented in Fig. 19. In the SMA, the (unnormalized) excited-state wave function is assumed to be given (for the spin excitation) by where S a (k) denotes the Fourier transform of the local spin operator S a j . Then, the excitation spectrum (or, the Bijl-Feynman frequency [178,179]) is obtained by calculating the following quantity: where H is given by (189) and E 0 is the ground-state energy. We can consider other types of excitations by changing S a (k) to other operators.

Spin Excitation
Let us start by investigating the action of local spin operators Figure 19: (Color online) Action of bosonic (spin) operator S z (a) and fermionic generator K 1 (b) onto the SVBS state. The local operators S a (i) (a = x, y, z) and K 1,2 (i) respectively create a triplet bond and a spinon-hole pair on either of the two adjacent bonds (i − 1, i) and (i, i + 1).
[ Figure and Caption are taken from Ref. [9].] on the SVBS state. A little algebra shows that these spin operators create triplet bonds around the site i [see Fig. 19]: where |ψ ), respectively. By taking the Fourier transform of eqs.(193a) and (193b), one immediately sees that the triplon-crackion equivalence (except for the momentum-dependent form factor) found in the ordinary VBS states [181,23] holds in the SVBS case as well. By a simple algebra, it is easy to show the following bound for the true spin-excitation spectrum ω s,a true (k): The last inequality is proven by noting that the left-hand side can be rewritten as the following average ∞ 0 dω ωS aa (k, ω) ∞ 0 dωS aa (k, ω) (195) and using the spectral decomposition of the dynamical structure factor S aa (k, ω). The spinexcitation spectrum obtained [9] in this way is shown in Fig. 20. At r = 0, the dispersion reduces to the well-known result of the original VBS chain [71]: ω s,a SMA (k) = 10 27 (5 + 3 cos k) (a = x, y, z) .
In the limit r ∞, on the other hand, the spin excitation loses its dispersion. This is easily understood by noticing that the ground-state reduces to the Majumdar-Ghosh dimer states on which excitations cannot move.

Spinon-Hole Excitations
The dynamics of doped holes in the spin-gapped background is in its own right interesting [182]. In the context of the VBS models, some (both exact and approximate) results have been obtained. For instance, in Ref. [183], the motion of spin-0 holes in the spin-1 VBS background is considered and the exact one-hole spectrum is obtained. Motivated by the experiments carried out for holedoped spin-1 compound, Ref. [184] introduced a realistic model and investigated the motion of a single spin-1/2 hole immersed in the gapped spin-1 VBS background.
A similar strategy can be used to obtain the spectrum of the charged (hole, f † ) excitation which is always paired with the S = 1/2 spinon (a † or b † ). These excitations are created by applying the two fermionic generators of U OSp(1|2) to the VBS ground state. By using the explicit form of the ground-state wave function, it is easy to show where the crackion state |ψ Fig. 19(b)]. The excited state K 2 |SVBS is defined similarly with a † in the above expression replaced with b † . For r = 0, the spectrum is given by The behavior of the spectrum as a function of r [9] is plotted in Fig. 21. We add some comments about distinctions between ω s SMA (k) and ω h SMA (k). Since SUSY relates the bosonic generators S and the two fermionic generators K α , one might naively expect the same spectra for their corresponding excitations. However, this expectation relies on the existence of a 'unitary' transformation which linearly transforms the set of the SUSY generators onto themselves. Since no such transformation exists in the present SUSY, the spectra for the spin and charge sectors indeed exhibit different behaviors.

Topological Order
As has been discussed in Sec. 1, no true topological order is possible in 1D systems [59]. However, if we impose a certain kind of symmetries, there can be topologically non-trivial phases protected by the symmetries dubbed symmetry-protected topological phases. One of the typical examples would be a non-trivial topological phase with the Majorana edge mode in 1D interacting fermions [185].
We already know that there exists an analogous "topological" phase with (almost) free edge spins (S = 1/2) at the edges in 1D spin systems as well. In this section, we give detailed discussions about topological properties of the SVBS states [9,10]. In particular, we investigate the string order and the entanglement spectrum of type-I and type-II VBS states. Then we generalize the MPS argument of the symmetry-protected topological order [61,62] to SUSY cases to understand the degeneracy structure in the entanglement spectrum.

Hidden Antiferromagnetic Order and String Order Parameter
Before proceeding to the SUSY cases, we briefly recapitulate the hidden non-local order in 1D spin systems. The concept of hidden order is an isotropic generalization of the Néel order. As we have seen in Sec. 2.2, the Néel order for S = 1 antiferromagnetic spin chains looks like (if we assume that the ordering occurs in the z-axis) Here, + stands for S z = +1, and − for S z = −1. Clearly, the spin-spin correlation depends on the parity of the number n of sites between i and j. A simple way of turning this position-dependent (or, alternating) correlation into the smooth ferromagnetic one would be to insert a phase factor (−1) n between the two spins S z j S z j+n → S z j (−1) n S z j+n (202) in order to cancel the sign factor coming from the alternating +1 and −1 between the sites j and j + n. Namely, the Néel antiferromagnetic correlation translates to the ferromagnetic correlation in S z j (−1) n S z j+n . On the other hand, as is seen in the expansion of the VBS state [see (36)], a typical S z sequence appearing in the state reads as As has been pointed out already in Sec. 2.2, by removing zeros in the sequence, we can reproduce the usual Néel order. In this sense, there still exists a certain kind of Néel order, though "disordered" by randomly inserted 0s, called the hidden string order 30 [18,19]. However, the trick used above does not work since, due to the intervening 0s, the positions of +1 and −1 are random (though they still appear in an alternating way) and the phase (−1) n can not cancel the sign factor. Nevertheless, a little thought tells that the following choice will do the job: exp (iπS z tot (j, j + n)) = (−1) # of ±1 between j and j + n , where stands for the partial sum of S z k between an arbitrary pair of sites i and j. Therefore, it is suggested that we should use, instead of eq.(202), the following pair of non-local order parameters known as the string order parameters [18,19] in order to characterize the non-trivial spin order in the (S = 1) VBS state. In the first line, S x tot (j + 1, j + n) is defined similarly to S z tot (j, j + n − 1) [see eq.(205)].
The above expressions have been guided by a simple physical intuition. However, as has been pointed out in Refs. [186,187], the string order parameters have in fact a deeper meaning than we expect from the above simple argument. To see this, first we note that the string correlation functions can be recasted in a suggestive way: where S x and S z are defined as The physical meaning of these operators may be best understood by considering the 'height plot' of the VBS spin configurations [18,23], where the value S z tot (1, j) is represented by the "height" between the sites j and j+1 (hence S z j itself is a "step" at the site j). Fig. 23(a) is a plot of a typical S z -configuration of the spin-1 VBS state when the left edge state is ↑. One can clearly see that the meandering steps are always confined between the heights 0 and +1. A similar analysis in the case with the left edge state ↓ shows that the heights are either 0 or −1. From these observations, it is concluded that the string exp{iπS z tot (1, j − 1)} attached to the left of the operator S z j somehow suppresses the strong fluctuations in S z and that S z takes either 0 or +1 (0 or −1) when the left edge state is ↑ (↓). Similarly, one can show that S x becomes weakly ferromagnetic depending on the right edge states. In short, non-zero string order parameters translate to the existence of a certain kind of weakly ferromagnetic order in the x and the z directions (i.e. S a = 0 for a = x, z).
Remarkably, the following unitary transformation [188] relating the two operators S a and S a as transforms the original [SU(2)-invariant] Hamiltonian into the one U KT H U −1 KT which is invariant only under the dihedral group D 2 (or Z 2 ×Z 2 ) consisting of two π rotations with respect to the x and the z axes [186,187]. Therefore, we can interpret the existence of the string order (both in x and z) in the original system as a consequence of the spontaneous breakdown of the Z 2 ×Z 2symmetry in the transformed system U KT H U −1 KT and the resulting (weak) ferromagnetic order in S x,z .
The idea of non-local hidden order and edge states has been to some extent generalized [162,188,23,189] to other values of integer-spin-S although the hidden Z 2 ×Z 2 -symmetry is never broken [188] in the case of even-S (see Fig. 22) 31 . In the course of these studies, it has been recognized that there are some differences [188,23] in the ground-state properties according to the parity of S. Nevertheless, by analogy with the quantum-Hall systems [161], the ground state of generic integer-spin antiferromagnetic chains, including the original VBS state and its higher-spin generalizations [71], characterized by certain kinds of non-local correlations and emergent edge states have been called 'topological' in a rough sense.
To illustrate distinct behaviors depending on the parity of spin S, we may introduce the following generalized (angle-dependent) string order parameter [188]: where the parameter θ has been introduced for convenience 32 . When θ = 0, it reduces to the usual spin-spin correlation and, when θ = π, it coincides with the string order parameter discussed above. The behaviors of the generalized string order parameter (211) are shown for several values of bulk spin S in Fig. 22 [23]. The string order parameters are symmetric with respect to θ = π and generally have S peaks for the spin-S VBS state. As is expected from that the ground state is magnetically disordered, the infinite-distance limit of the usual spin-spin correlation function (θ = 0) vanishes regardless of the value of S. On the other hand, Fig. 22 demonstrates that O z,∞ string takes finite values for the odd-S VBS states while it vanishes for even-S. Therefore, in the sense of the Z 2 ×Z 2 -symmetry argument [186,187] mentioned above, this hidden symmetry is never broken [188] in the even-S VBS models. 31 This does not mean that Z2×Z2-symmetry never breaks down in any even-S chains. In fact, even when S =even, one can construct the model ground states which have non-vanishing string order parameters. 32 The introduction of the θ parameter is mainly motivated by the idea that the intermediate string might somehow cancel the fluctuations between the two spins S z (i) and S z (i + n). For θ = π, this works perfectly in the S = 1 VBS state. Except for θ = 0 (ordinary spin-spin correlation) and θ = π (string correlation), no symmetry-related reason has been found so far.

Generalized Hidden String Order in SVBS Chain
Unlike the original VBS chains, S z = 1/2 and −1/2 generally appear in the S z sequence of the SVBS chain, and a typical S z sequence of the SVBS chains is given by 33 From the sequence, one can "derive" the ordinary hidden oder. First, we search the spin-half sites from the left, and whenever we encounter a pair of spin-half sites we sum the two S z values to have the effective S z (= +, 0, −) (e.g. ↓ ↓ → −): Next we remove the zeros in the sequence to recover the standard Néel pattern: This observation leads to the existence of (generalized) hidden order of the SVBS chain. Fig. 23 shows a typical height configuration corresponding to the usual VBS chain and its SUSY counterpart. Reflecting the existence of hidden order, the height configuration is always meandering between the height 0 and the height 1 34 . It should also be noted that the height-configuration 33 It may be worthwhile to give some comments on the relation to the ferrimagnetic spin chains that also consist of alternating spin 1 and spin 1/2 [46]. Though both the ferrimagnetic chains and the present SVBS chains contain spin-1 and spin-1/2 degrees of freedom, in the SVBS chains the spin 1 and spin 1/2 are not necessarily alternating [see (212) and Fig. 10]. More importantly, while ferrimagnetic spin chain can exhibit a long-range magnetic order as the order parameter commutes with the Hamiltonian, the ground state of the SVBS chain itself is spin-singlet and the SU (2)-symmetry is never broken (spontaneously). 34 The same reasoning applies to the general spin-S VBS cases and the height configurations are confined within a region of width S [23].
To substantiate the existence of the hidden order, we explicitly calculate the string parameter for the SVBS chains. The behaviors of the string order are depicted in Fig. 24 with respect to the hole doping parameter.  Fig. 24, one can find distinct behaviors of the string parameter in terms of the parity of bulk superspin. The string parameters of odd superspin SVBS chains generally decrease with increase of the hole doping, while those of the even superspin SVBS chains increase. Since the hole-doping simply reduces the spin degrees of freedom on the spin chain, the decrease of the string order of the odd superspin SVBS chains may be naturally understood. On the other hand, the string order behavior of the even superspin SVBS chains is quite interesting, since the string order revives with the hole doping. Intuitive explanation may go as follows. For instance, consider S = 2 SVBS chain. At r = 0, the S = 2 SVBS chain is exactly identical to the S = 2 VBS chain which is essentially constituted of two S = 1 VBS chains, upper and lower chains. By doping the holes to the S = 2 VBS chain, the spin degrees of freedom, say, on the upper S = 1 VBS chain decrease to have deficits in the chain. Below the deficits on the upper chain, the spin degrees of freedom on the lower S = 1 VBS chain emerge and come into effect. In this way, the spin degrees of freedom of the lower S = 1 VBS chain contribute to generate the finite string order with increase of hole doping to the S = 2 VBS chain. This intuitive explanation can be applicable to the revival of the string orders of general even superspin SVBS chains. Consequently, the SVBS states bear finite string order with a finite amount of hole-doping regardless of the parity of bulk superspin. This is the salient SUSY effect to the topological stability of quantum spin chains. We revisit this effect in the context of the symmetry protected topological order in Sec.6.3 and 6.4.

N = 2
For the S = 1 type-II SVBS infinite chain, the string correlation is computed as which is plotted in Fig. 25. The string order (216) takes the value of the S = 1 VBS chain, 4/9, at r = 0 like the S = 1 type-I chain, while approaches to zero in the r ∞ limit unlike the S = 1 type-I chain. Since spin degrees of freedom completely disappear from the type-II chain in the r ∞ limit [Fig. 11], the string order of the type-I and II chains shows qualitatively different behaviors.

Entanglement Spectrum and Edge States
As discussed above, the string order of type-I SVBS chains with even bulk-superspin S revives upon hole doping. This suggests that, in contrast to their bosonic counterpart, even the SVBS states with even superspin can host the stable topological phase. Though the string order parameter is appealing in its similarity to the order parameter in FQHE [161] and its relation to the hidden Z 2 ×Z 2 -symmetry, its fragility under perturbations has also been discussed recently [190,60] and the alternative "order parameter" has been sought for.
Recent development in quantum-information-theoretic approaches to quantum many-body problems enables us to extract information on the bulk topological order from the entanglement properties of the ground-state wave function [191,192,27]. The topological states in onedimensional (1D) spin systems have been reconsidered [60,61,62] from the modern point of view and the precise meaning of the topological Haldane phase has been clarified. In these studies, the string order parameters and the edge states, which in general are not robust against small perturbations, are replaced by more robust objects (i.e. the structure of the entanglement spectrum or the structure of tensor-network). In particular, it has been shown in Ref. [61,62] that the existence of (at least one of) the discrete symmetries (time-reversal, link-inversion and Z 2 × Z 2 symmetry) divides all states of matter in 1D into two categories: topologically-non-trivial ones and the rest. Generic odd-integer-S spin chains belong to the former while even-S chains to the latter. The hallmark of the topological phase protected by the above discrete symmetries is that all entanglement levels are even-fold degenerate. In this formulation, the difference between the odd-S VBS states and the even-S ones is naturally understood in terms of the entanglement structure; the degenerate structure exists only for odd-S cases 35 . It should also be mentioned that the topological phases of one-dimensional gapped spin systems have been classified by group cohomology, [59,63] and the detailed analyses based on the Lie group symmetries are reported in Ref. [66].
Following the proposal of Li and Haldane 36 [27], we use the structure of the entanglement spectrum (e.g. degeneracy of the entanglement levels) as the fingerprint of topological phases. Then, the problem of the topological stability of the SVBS chains translates to the stability of the degenerate structure of the entanglement spectrum. Before proceeding to the details, we briefly introduce characteristic features of the entanglement entropy of the original SU (2)-invariant VBS states. It has been reported that the entanglement entropy of the SU (2) spin-S VBS state on an infinite chain is given by a constant 37 determined essentially by the degrees of freedom of the edge spins S/2 [50,51]: Corresponding to the parity of the bulk spin of the VBS chains, there appear either integer or half-integer spin at the edge. Meanwhile, in the presence of SUSY there necessarily appear both integer and half-integer spins at the edge, since SUSY relates integer and half-integer spin degrees of freedom [ Fig. 26]. Such particular feature of the edge spins is crucial in understanding of the salient structures of the entanglement spectrum of the SVBS chains.

Schmidt Decomposition and Canonical Form of MPS
Before going into the detailed discussion, it would also be worthwhile here to give the derivation of the entanglement spectrum using MPS. Suppose we divide a system into the two parts A and 35 This does not mean that all the odd-S spin chains have the degenerate entanglement spectrum. We can construct an odd-S spin state without the even-fold degeneracy. 36 Li and Haldane proposed to take the degeneracy of entanglement spectrum as the hallmark of topological phases, which can be applicable to general topological phases beyond QAFM. 37 On the other hand, for gapless spin chains, the entanglement entropy diverges as log(L) with L the length of a subsystem for which entanglement entropy is defined [57].
B (see Fig. 27). Then, we can express any wave function (state) |Ψ as [25] with non-zero coefficients λ α (≥ 0). Here, are orthonormal basis states in the subspaces A and B, respectively: The expansion (218) called the Schmidt decomposition defines the Schmidt coefficients λ α and describes the entanglement between the two subsystems. The Schmidt number χ is the number of non-zero Schmidt coefficients and never exceeds the minimum of the dimensions of the Hilbert subspaces. From the normalization of |Ψ , the Schmidt coefficients satisfy χ α=1 λ 2 α = 1. The "spectrum" of the entanglement energy α defined by is called the entanglement spectrum [27]. In terms of the Schmidt coefficients, the (von Neumann) entanglement entropy is given by [25] It is interesting to observe that S E.E. may be viewed as the ordinary thermodynamic entropy if we introduce a fictitious "temperature" T and "partition function" as and define entanglement cut "edge" Figure 27: The VBS state is divided into two parts and the edge degrees of freedom emerge at the cut.
In the MPS formulation, the derivation of the Schmidt coefficients are rather straightforward. Since the MPS is represented as the product of matrices A j defined on each site, one can easily write down an analogue of the Schmidt decomposition (218) (see Sec. 5.2 for the physical meaning of the matrix indices): where D is the size of the matrix A j and each state-valued matrix A is expanded explicitly in terms of the c-numbered matrices A(m) and the orthogonal local basis states |m j as (d: dimension of local physical Hilbert space). For simplicity of argument, we assume that the system is uniform [A j (m) = A(m)] and defined on a open chain. One may think that eq.(224) already completes the Schmidt decomposition (218) with the identification However, the orthonormality condition (219) are not always satisfied for the above choice. Normally, we use the "gauge ambiguity" [24,31] to find an appropriate set of edge states in such a way that the following overlap matrices may equal to identity: This procedure can be carried out both for finite systems and for infinite-size systems by using the singular-value decomposition (SVD) [39,40,44]. As a result, we obtain, instead of a single matrix A(m), two different MPS matrices ΛΓ(m) for the left subsystem A and Γ(m)Λ for the right subsystem B. The (diagonal) matrix elements of the D×D diagonal matrix Λ coincide with the Schmidt coefficients: The MPS characterized by the set of matrices {Λ, Γ(m)} (m = 1, . . . , d) is called canonical [24] and automatically completes the Schmidt decomposition (218) as [39,40,44] (for a rather comprehensive account of the use of SVD in MPS, see, for instance, [32]). In infinite-size systems, the overlap calculation (227) simplifies a lot since L (R) reduces essentially to the left (right) eigenvector V R of T L and T R , respectively (see Fig. 28): If our infinite-size MPS (iMPS) assumes the canonical form, [L]ᾱ β ∝ δᾱ β = (1 D )ᾱ β and [R]ᾱ β ∝ δᾱ β = (1 D )ᾱ β by definition (with 1 D being the D-dimensional identity matrix), and 38 When the MPS is not of the canonical form, we can take Γ(m) = A(m) and Λ = 1 (and hence TL = TR). therefore the two transfer matrices satisfy 39 [20,193] ᾱ,α Fig. 29). Notice that these equations can also be regarded as the eigenvalue equations for the transfer matrices: 1 D is the left (right) eigenstate of the transfer matrix T L (T R ) with the eigenvalue 1. Eq.(232) may be thought of as the conditions in order that the iMPS is of the canonical form. Figure 29: (Color online) Graphical representation of the two conditions, (232a) and (232b), for canonical iMPS. In Ref. [193], the action of the type (a) ((b)) is denoted by E * (Λ 2 ) = Λ 2 (E(1) = 1).
On the other hand, when the MPS in question satisfies 40 in the infinite-size limit, the Schmidt decomposition for the infinite chain is obtained very easily just by rescaling the MPSs for the subsystems. The normalized state |Ψ is constructed as 39 By the assumption of pure MPS, the largest eigenvalue is unique [20,193]. (MPS is called a pure MPS when transfer matrix has non-degenerate maximal eigenvalue. In the infinite-size limit, the pure MPS is reduced to a pure state, and hence the name pure MPS.) When the value of the largest eigenvalue is not 1, we can rescale the matrices A(m) so that it may be 1. 40 This is the case for all the (S)MPSs treated in this paper.
where A(α, β) is the magnitude of |Ψ(α, β) : With these normalization constants, the normalized MPS is written as Comparing this expansion with (218), one may read off the Schmidt coefficients as In the infinite limit, λ α is not relevant to the polarization of the edge spins, α L and α R . Therefore for the infinite (S)VBS chain, we only need to evaluate the magnitude of the matrix product to obtain the Schmidt coefficients, and thus the derivation of Schmidt coefficients is boiled down to the computation of the normalization constants, A(α, β) (235).

N = 1
To substantiate the topological stability of the SVBS chain, we investigate two type-I chains with distinct bulk superspins, S = 1 and S = 2.
(i) Superspin S = 1 From the formula (237), the entanglement spectrum of the S = 1 type-I state (on an infinite chain) is derived as They are shown in Fig. 30 (left-figure) with the entanglement entropy (right- figure). At r = 0, the SVBS state reduces to the S = 1 VBS state and reproduces both entanglement spectra and entanglement entropy of the S = 1 VBS chain, i.e., λ B 2 → 1/2 and S EE → ln2, which should be compared with (217) for S = 1. Similarly, in the limit r → ∞, the SVBS chain reduces to the MG chains, and the entanglement entropy of SVBS chain also reproduces the finite entanglement entropy of the MG chain (the right-figure of Fig. 30). As in the entanglement spectra of the left of Fig. 30, we have two distinct entanglement spectra, one of which is doubly degenerate spectrum (blue curve) for the "bosonic" Schmidt coefficients corresponding to those of the original S = 1 VBS chain (238a) and the other is the non-degenerate spectrum (red curve) for the "fermionic" Schmidt coefficient (238b). The existence of such two types of entanglement spectra is a salient feature of the SUSY state and can be readily understood based on the following edge state picture. For S = 1 SVBS chain, its edge superspin states are given by the U OSp(1|2) multiplet with superspin S edge = 1/2 that consists of the ordinary SU (2) states with spin, 1/2 ⊕ 0 ( Fig. 31) : The S = 1/2 SU (2) edge spin states generate the double degeneracy in the entanglement spectra, while the S = 0 SU (2) edge spin state gives the non-degenerate one. Due to the existence of 1/2 spin edge degrees of freedom, the double degeneracy is guaranteed (we will give a detail discussion in Sec.6.4), and hence we find that the type-I SVBS chain is in the topological phase. To highlight the effect of SUSY, we consider the non-SUSY (i.e. purely bosonic) cases and replace the fermion operator f i with the boson operator c i ([c i , c † j ] = δ ij ) to define the following boson-pair VBS chain, As the insets of Fig. 30, we depicted the entanglement spectra and entanglement entropy. The crucial difference to the SUSY case will be apparent in the limit r → ∞, |Φ b.p. , is reduced to a simple product state, and the entanglement entropy vanishes (the inset of the right figure of Fig. 30).
(ii) Superspin S = 2 Next, we examine the entanglement spectrum of the S = 2 type-I SVBS chain. Though the topological phase of its bosonic counterpart, S = 2 VBS chain, is fragile under perturbation [190], the S = 2 type-I SVBS chain itself is topologically stable with finite amount of hole doping. As we shall see below, SUSY plays a crucial role for the stability of the topological phase. At r = 0, S = 2 SVBS chain is reduced to the S = 2 VBS chain, while in the limit r → ∞ the SVBS chain is reduced to the partially dimerized chain. We have the following five Schmidt coefficients for the S = 2 SVBS chain: Thus, the five Schmidt coefficients are split into the triply degenerate (243a) and doubly degenerate (243b) spectra showing distinct behaviors in Fig. 32. Again, such splitting of the Schmidt coefficients are readily understood by the edge state picture for the SUSY chain. For the S = 2 type-I SVBS chain, the edge superspin is given by S edge = 1 that consists of the SU (2) edge spin S edge = 1 and S edge = 1/2 [ Fig. 33]: (S edge = 1) = (S edge = 1) ⊕ (S edge = 1/2) (244) The S edge = 1 degrees of freedom generate the triple degeneracy in the entanglement spectra, while the S edge = 1/2 degrees of freedom give the double degeneracy. Due to the existence of SUSY, the S = 2 SVBS chain necessarily contains the S edge = 1/2 edge spin degrees of freedom that do not originally exist in the S = 2 VBS chain [ Fig. 33], and they guarantee the double degeneracy in the entanglement spectra, i.e. the stability of the topological phase [61,62]. From the above demonstrations for S = 1 and S = 2 type I SVBS chains, one may see that regardless of the parity of bulk superspin, the SUSY introduces the half-integer edge-spin states that necessitate at least double degeneracy in the entanglement spectrum.

N = 2
It is also straightforward to calculate the Schmidt coefficients for the type-II SVBS chain: The four Schmidt coefficients are split into two groups showing distinct behaviors [ Fig. 34] according to the SU (2) decomposition of the U OSp(1|2) edge superspin state: The first 2 on the right-hand side of (248) corresponds to the doubly degenerate blue curve in Fig. 34, while the remaining 1 ⊕ 1(= 2) represents the doubly degenerate red curve.

Supersymmetry-Protected Topological Order
In this section, we clarify the relation between the structure of the entanglement spectra discussed in the previous sections and the symmetry-protected topological order [60]. To this end, we use the MPS representation established in Sec. 5. Because of its simplicity in the entanglement structure and the wide applicability to gapped states in 1D, the MPS approach provides us with a powerful tool in investigating the topological phases in 1D [61,62,64,65,63,59].

Symmetry Operation and MPS
For later convenience, we give a quick summary of some tools used in the MPS approach. As has been discussed in Sec. 5 and 6.3.1, any MPS on an open chain may be written as or in terms of the c-number matrices A(m) [A j = d m=1 A j (m)|m j , d being the dimension of the local physical Hilbert space], as As has been mentioned in Sec.  [193,194]: The local unitary operationû acts on A in a site-wise manner: where we have used the completeness relation on each site, m |m i m| i = 1. If the symmetry operation is anti-unitary (like time-reversal), the complex-conjugation Γ(m) → Γ * (m) should be taken in eq.(252). Then, it can be shown that the above unitary invariance is equivalent to the existence of the following unitary operator U acting on Γ(m) [193]: where θ u denotes aû-dependent phase. Once the unitaryû is given, U is uniquely determined [193] up to an overall phase.
To be more precise, if we define a generalized transfer matrix the D-dimensional unitary matrix U is given essentially by its left eigenvector with the largest eigenvalue 41 e iθu (see Eq.(270) and the explanations around it). Since U leaves the MPS |Ψ MPS invariant, it is natural to assume that U does not change the physical entanglement spectrum Λ, i.e.
Physically, the above relation (253) implies that the original symmetry operation (acting on the physical Hilbert space at each site) "fractionalizes" into the ones (U and U † ) which act on the edge states on both ends of the system. The equation (253) plays a crucial role in the following discussions.

Case of SMPS
Now we extend the arguments developed by Pollmann et al. [62,61] for the bosonic MPS to the SUSY case. First remember that degeneracy in energy spectra of quantum mechanical Hamiltonian can be attributed to some symmetry of the Hamiltonian. Meanwhile, since the entanglement spectrum is solely determined by a ground-state wave function, the degeneracy of entanglement spectrum is expected to stem from some symmetry of the ground-state wave function. Indeed, several discrete symmetries are identified to guarantee degeneracy in the entanglement spectrum.
Since the degeneracy of the entanglement spectrum is the hallmark of the topological order, it is said that the topological order is protected by the symmetry of the ground-state wave function, and hence the name symmetry protected topological order. In the following, we will show that the SUSY guarantees the existence of at least two-fold degeneracy of entanglement spectrum regardless of the parity of superspin, provided that at least one of the three symmetries, inversion, time-reversal and Z 2 × Z 2 , is present. We will also find that the (S)MPS formalism plays a crucial role in the discussions.
As has been discussed in Sec.5.2, the SMPS is generally represented as [9] where A i (i = 1, 2, · · · , L) are supermatrices defined on the sites i. Then, we follow the same steps as in Sec. 6 In fact, the local symmetry operation need not be unitary (as will be in the following sections). In this case, the left-hand side may be replaced with the general form Γ (m). Therefore, the most general form reads as where the phase θ U depends on the symmetry operation considered. In the above equations, m labels both bosonic and fermionic states, i.e. m = i, α, and Γ(m) are given, for U OSp(1|2), by where M 1 , M 2 , N 1 and N 2 are c-number matrices.
Here, a remark is in order about the form of U . The (c-number) unitary matrix U in (257) may be postulated as: where U 1 and U 2 are unitary matrices that act on the two bosonic subspaces having different fermion numbers. The reason for choosing the above form may be seen as follows. First we note that eq.(257) implies that the MPS on a periodic chain transforms like where the supertrace STr is defined for general supermatrices as [see also eq.(176b)] We expect that for a periodic chain, which does not have edges, is fully invariant under the unitary operation, i.e., the expression (261) coincides with the original MPS (up to an overall phase). While in the case of bosonic MPS, this, combined with Tr(AB) = Tr(BA), immediately implies ⊗û|Ψ ∝ |Ψ , the relation STr(AB) = STr(BA) holds only when A and B are super-matrices (that contain the Grassmann-odd blocks in their off-diagonal parts): In fact, if an arbitrary pair of two super matrices A and B were merely c-number matrices, A and B, in general, would not commute inside STr(·): STr(AB) = STr(BA). In order to satisfy STr(AB) = STr(BA) only with c-number matrices, either A or B is forbidden to have c-number components in the off-diagonal blocks.
For later convenience, we derive a useful property of pure canonical MPSs [24,44]. In the following sections, we assume that the MPS in question is defined on an infinite-size system. The equation (258) is the most general statement about how a given symmetry of MPS is realized by a projective representation. However, sometimes it happens that Γ (m) = Γ(m) for a certain symmetry operation and in these cases we can draw an interesting conclusion about the properties of U [61,62,10]. Suppose that we have a pure MPS whose canonical form is characterized by the MPS data [24,44] {Λ, Γ} and that it satisfies the following relation for some unitary matrix U : Since the MPS is canonical, the following holds [see eq.(232a)]: Physically, it states that the is the dominant left-eigenvector of the left transfer matrix (see Fig. 29) Plugging Γ † (m) = e −iθ U U † Γ † (m)U into (265), we obtain: This implies that the D×D unitary matrix when viewed as a D 2 -dimensional vector, is the left-eigenvector of T L with the eigenvalue e iθ U : Since, by assumption of canonical MPS, 1 D is the only left-eigenvector having the eigenvalue |λ| = 1, we conclude Since, in deriving the above, we have only assumed that the (infinite-system) MPS in question is pure and takes the canonical form, (271) holds for any MPS (including SMPS) satisfying the assumption.

Inversion Symmetry
Now let us look at what inversion symmetry I with respect to a given link implies about the structure of the entanglement spectrum. It is convenient to consider an inversion transformation on a circle: The inversion on a given link transforms the SMPS chain as where 'st' stands for supertranposition defined by It is easy to verify that applying the supertrace twice does not return a supermatrix to the original one: where the matrix has been defined in such a way that its adjoint action P (·)P multiples the fermionic blocks by a factor (−1). [1 1 and 1 2 correspond to the unit matrices of two bosonic subspaces and should not be confused with the D × D unit matrix 1 D .] Therefore, at the level of the local MPS matrix, the inversion I is realized as Finally, from eq.(258), one sees that when Ψ has the inversion symmetry, the A-matrix should satisfy the relation [10] Note that at this stage, we can not apply the argument in Sec. 6.4.
This is of the form of eq.(264) and then (271) immediately implies that After multiplying U t I from the right and making transposition, we deduce [note P U I = U I P by eqs. (260) and (277)]. Therefore, we obtain [10] U t I = ±P U I .
Eq. (283) states that when U 1 is symmetric (anti-symmetric), U 2 is anti-symmetric (symmetric). It should be noted that unlike in the bosonic case [61,62], the symmetry constraint is imposed on each of the spin-S "bosonic" sector U 1 and S − 1/2 "fermionic" sector U 2 in the case of SUSY. When U 1 is anti-symmetric, for instance, the sector with (−1) F = +1 must have a special structure in its entanglement spectrum. In fact, by computing the determinant of U 1 one can immediately see that d 1 should be even (the same argument applies to U 2 as well). From this, one can conclude that either fermionic (when the sign + occurs) or bosonic (−) sector has even-fold degeneracy in each entanglement level, which we can use as the fingerprint [60,61,62] of the SUSY-protected topological order [10].

Time-Reversal Symmetry
We can draw a similar conclusion for the time-reversal symmetry T . Let us recall the time-reversal operation for the superspin, S i and S α with complex conjugation operator K. The set of equations (285) implies that, for integer superspins S, T = Ke −iπSy satisfies 43 where the d-dimensional matrix P, which acts on the physical Hilbert space and multiples a minus sign when the state is fermionic (i.e. when the fermion number F = odd), is analogous to the D-dimensional matrix P [see eq.(277)] acting on the auxiliary space. From (285), one sees that, in terms of the canonical matrix, the time reversal transformation is represented as Γ(m) where m, n = i, α and R y (π) = e iπSy . For instance, R y (π) is given, for superspin S = 1, by Again by eq.(258), the time-reversal invariance implies that there exists a unitary matrix U T that satisfies Γ(m) Applying this twice and using T 2 = P, we obtain (290) 43 Recall that, in the SU (2) case, T 2 = +1 for integer S. When superspin S is half-odd-integer, the time reversaloperator satisfies T 2 = −P . (This is indeed a generalization of T 2 = −1 for the SU (2) half-integer spin case.) Using the relation (P) ll Γ(l) = P Γ(l)P , we may rewrite (290) into the following form to which we can apply the argument presented in Sec. 6.4.2 [see eq.(264)]. Thus we obtain (the value of θ T can not be determined by this method). This is of exactly the same form as (281) and we deduce the same conclusion: Therefore, as in the previous case [see eq.(283)], we see that time-reversal symmetry guarantees the existence of at least double degeneracy in the entanglement spectrum of (either of the bosonic or the fermionic sector).

Z 2 × Z 2 Symmetry
Lastly, we consider the two independent π rotations around the x [û x (π)] and the z axes [û z (π)]. 44 As will be seen below,û x (π) orû z (π) alone does not lead to any significant conclusion. However, the combination of the two [61,62] leads to a similar conclusion about the entanglement spectrum. Under the π rotation around the x (z) axisû x (π) (û z (π)), SMPS transforms as When the SMPS respects such a symmetry, we have where (R a (π)) 2 = P for integer superspin S. Again we follow the same steps as before to show Note that the phase factor e iΦa can be absorbed in the definition of U a and hereafter we assume U † a = P U a (a = x, z). Unlike in the previous cases, this relation alone does not give any useful information about U a .
Since the phases of U x and U z have been already fixed, the phase of U x U z cannot be arbitrary and has a definite physical meaning. By multiplying U z P and U x from the left and the right of (298), respectively, and using U † a = P U a = U a P repeatedly, one can show which is combined with (298) to give e iΦxz = ±1. To summarize, the two unitary matrices U x and U z obey the following commutation relation: In terms of the block components U a,1 and U a,2 , eq.(300) reads as As in the previous cases, (301) immediately implies at least double degeneracy of each entanglement levels in the sector (1 or 2) taking the minus sign in the above equation.

String Order Parameters and Entanglement Spectrum
In the previous sections, we have shown that, as in the purely bosonic cases [60,61,62], such elementary symmetries as inversion and time-reversal protect the Haldane phase from collapsing into trivial gapped phases on the basis of the assumption that the even-fold degeneracy in the entanglement spectrum either in the fermionic or in the bosonic sector is the entanglement fingerprint of the topological "Haldane phase" in SUSY systems. On the other hand, the string order parameters [see Sec. 6.1] have been used traditionally to characterize the Haldane phase [18,19]. Then, a natural question arises: is there any connection between the description by the string order parameters and the modern characterization in terms of the entanglement spectrum? In this section, we give the answer to this question. Let us first consider the structure of the string order parameters O x,∞ string and O z,∞ string [eq.(206)] from the MPS viewpoint [23,193]. When we evaluate them using MPSs, we encounter the following matrices [23]: as well as the usual transfer matrix T . By using these matrices, the MPS expression of the string order parameter O z string is given as where we have omitted the denominator necessary to normalize the MPS. The parts T N L and T N R are easy; for the canonical MPS, they reduces, in the infinite-size limit, to (see Sec. 6.3.1): The boundary dependent factors δᾱ L ,α L and δβ R ,β R are cancelled by those coming from the denominator. Therefore, the quantity which we have to compute is Since we are interested in the long-distance limit |i − j| ∞, we need know the asymptotic behavior of the string (T string ) |i−j| . First we note that T string may be thought of as the overlap MPS|û ⊗û ⊗ · · · ⊗û|Ψ MPS (306) withû = exp(iπS a ) (a = x, z) [see eq.(251)]. Then, it can be shown [193] that in order for the string (T string ) |i−j| not to vanish in the long-distance limit, the MPS should be invariant under both of the π-rotationsû x (π) andû z (π). Then, from the discussion in Sec.6.4.5 (we just replace P → 1 to obtain the results for the purely bosonic case), we see that there exists a pair of unitary matrices satisfying (295) and [see eq.(300)]. Unfortunately, we cannot tell whether the even-fold degeneracy in the entanglement spectrum, which is the fingerprint of the topological Haldane phase, happens or not since we do not know which of the ± signs is chosen. Now we show that when the string order parameters are non-vanishing O z,x string = 0, the minus sign in fact realizes (i.e. U x and U z anti-commute) in eq.(307) and the entanglement spectrum exhibits the degenerate structure. To this end, we investigate eq.(305). Since the largest (right)eigenvalue of T string is e iθa [see Sec. 6.4.1], (T string ) |i−j| reduces essentially to a phase (e iθa ) |i−j| = (±1) |i−j| . The price to pay is the following boundary factors appearing at the two end points of the string correlators (see Fig.35): where V In fact, the eigenvalues of T string precisely coincides with those of T for all the spin-S VBS states and the different behaviors in the string order for the odd-S and the even-S chains comes only from these boundary factors.
To see whether the boundary factors are non-vanishing or not, we consider the right-boundary factor (1 {1⊗U z } T z ) of O z string . First we rewrite it by using (see the second figure of Fig.36): The unitary operatorsû † x andû x appearing on both sides of −S z can be absorbed into the MPS matrices by using eq.(295) (the third figure of Fig.36). By re-arranging the unitary matrices U x U z with the help of eq.(307) (the fourth figure of Fig.36), we arrive at the expression: Therefore, we see that the boundary factor (1 {1⊗U z } T z ) vanishes (and so does the string order parameter O z,∞ string ) when U x and U z are commuting (i.e. when the minus sign in the last expression is chosen). By the explicit construction of U x and U z , we can easily see that for the even-S VBS state, U x U z = +U z U x holds. Therefore, it immediately results that the string order parameters identically vanish for the even-S VBS state solely for a symmetry reason. In short, the existence of non-vanishing string order parameters O x,∞ string = 0 and O z,∞ string = 0 implies that the existence of the two anti-commuting unitary matrices U x and U z , and thereby guarantees the even-fold degeneracy in the entanglement spectrum. To put it another way, the string order parameters work as the sufficient condition for the topological Haldane phase. It is crucial that both of the string order parameters are finite. For instance, if we "deform" the original SU (2)-invariant VBS state by using the quantum group U q (su(2)), we obtain a VBS state with uniaxial anisotropy [22], where one of the string order parameters (O x,∞ string = 0) vanishes while When U x and U z are anti-commuting, the minus sign coming fromû x S zû † x = −S z is cancelled and an overall plus sign is recovered. (Figure has been adapted from Ref. [10].) the other is still finite [116]. In this case, the unitary U x does not exist and the existence of the degenerate structure in the entanglement spectrum is no longer guaranteed. In fact, explicit calculation shows that the two-fold degenerate entanglement levels in the S = 1 VBS state split into two non-degenerate levels in the deformed state.
It is straightforward to generalize [10] this to SUSY cases by taking into account the appearance of the P -matrix. Again by usingû x (π)S zû † x (π) = −S z and U x U z = ±P U z U x , the second factor of the right-hand side of eq.(308) can be recasted as: Since this implies we see that one of the two components (bosonic or fermionic) vanishes just by symmetry: α∈B when e iΦxz = −1 .
In conclusion, we have established the connection between the string order parameters, which have been commonly used [195,162,196,197,198] to characterize the Haldane phase, and the entanglement spectrum [27] which is a modern tool to look at the topological properties in the bulk. As has been shown in Secs. 6.4.3-6.4.5, SUSY guarantees, regardless of the parity of the bulk superspin S, the existence of the even-fold degeneracy in the entanglement spectrum of either of the bosonic or the fermionic sector, provided at least one of the three symmetries (inversion, time-reversal and Z 2 ×Z 2 ) is present. Therefore, in contrast to the usual bosonic (i.e. SU (2)) VBS states, in which the Haldane phase is stable only for odd-integer spins, in the SVBS states, SUSY plays the crucial role in protecting the topological Haldane phase. The "revival" of the string order parameters upon doping the system (discussed in Sec. 6.2) may be naturally understood by the above argument.
As has been seen above, that both of the two string order parameters are non-vanishing is the sufficient condition for the symmetry-protected Haldane phase. However, one may seek more faithful order parameters and, in fact, some results have been obtained in this direction. See, for instance, Refs. [199] and [200] for recent attempts at finding better order parameters.

Higher Symmetric Generalizations
In this section, we extend the previous SUSY formulation to the higher symmetric U OSp(1|4) case. We begin with the construction of fuzzy four-supersphere based on the U OSp(1|4) algebra [123]. As the coordinates of fuzzy four-sphere correspond to the SO(5) gamma matrices [92], the coordinates of the fuzzy four-supersphere are constructed from "super gamma matrices" of the U OSp(1|4) algebra. Next, based on the U OSp(1|4) structure, we derive the U OSp(1|4) SVBS state [10] whose bosonic counterpart is the SO(5) VBS state [107,108,109] 45 . We also develop the SMPS formalism for the U OSp(1|4) SVBS state and investigate the topological properties.

Fuzzy Four-supersphere
Here, we utilize U OSp(1|4) algebra to construct fuzzy four-supersphere with N = 1 SUSY. It may be worthwhile to first point out a nice correspondence between algebras and fuzzy spheres. For fuzzy two-sphere, we have and for fuzzy four-sphere, The U OSp(1|4) algebra is constituted of fourteen generators: 45 There still exist the correspondences among fuzzy geometry, QHE, and VBS in the higher dimension case. The SO(5) VBS corresponds to the Laughlin-Haldane wave function in 4D QHE [75,201], and 4D QHE realizes the fuzzy geometry of fuzzy four-sphere.
ten of which are the bosonic generators Γ ab = −Γ ba (a, b = 1, 2, · · · , 5) (the SO(5) generators) and the remaining four are the fermionic ones Γ α (α = 1, 2, 3, 4) (the SO(5) spinor). They amount to satisfy the following algebra: where C is the SO(5)( U Sp (4)) charge conjugation matrix and γ ab are the SO(5) matrices. In the following discussion, we take the SO(5) matrices as The U OSp(1|4) quadratic Casimir is given by The fundamental 5-dimensional representation matrices of uosp(1|4) are constructed as follows. From the (bosonic) "gamma matrices" of U OSp(1|4) algebra with we can derive the SO(5) generators The fermionic "gamma matrices" are also constructed as where They satisfy the "hermiticity" condition where the super-adjoint ‡ is defined in (50). It is straightforward to check that (324) where Ψ is the U OSp(1|4) Schwinger operator: Here, b α (α = 1, 2, 3, 4) and f respectively denote four bosonic and one fermionic operators that satisfy Square of the radius of fuzzy four-supersphere is also readily derived as In the Schwinger formalism, the Casimir (321) is represented as where Notice that in the Schwinger operator formalism the Casimir (332) is equivalent to (331) up to 1/2 coefficient on the r.h.s. The basis states on fuzzy four-supersphere are given by the graded fully symmetric representation: where n 1 , n 2 , · · · , m 4 are all non-negative integers satisfying the constraint, n 1 + n 2 + n 3 + n 4 = m 1 + m 2 + m 3 + m 4 + 1 = n ≡ Ψ † Ψ. Therefore, the dimensions of bosonic states (334a) and fermionic states (334b) are respectively given by and the total dimension is The bosonic and fermionic degrees of freedom (335) are respectively accounted for by the SO(5) symmetric basis states with the Casimir indices, n and n − 1, and hence the N = 1 fuzzy foursupersphere can be regarded as a "compound" of two fuzzy four-spheres with different radii n and n − 1: The eigenvalues of X 5 take the following values, with k = 0, 1, 2, · · · , 2n. These eigenvalues are equal to those of X 3 of fuzzy two-supersphere (60) except for the degeneracy of the basis states at each latitude. The degeneracies at the latitude (338) for even k = 2l and for odd k = 2l + 1 are respectively given by which reproduce the total dimensions of bosonic and fermionic basis states (335) as From (331), one may find that the condition for fuzzy four-supersphere is invariant under the SU (4|1) rotation of the Schwinger operator Ψ, which is larger than the original U OSp(1|4) symmetry. It may be pedagogical to demonstrate how such "hidden" SU (4|1) structure is embedded in the algebra of fuzzy four-supersphere. First notice that the fuzzy four-supersphere coordinates, X a and Θ α , do not satisfy a closed algebra by themselves, On the right-hand sides of (341), there appear "new" operators: where Γ ab and D α are respectively given by (324) and X ab and Θ α respectively act as SO(5) antisymmetric 2-rank tensor and spinor. Commutation relations for these new operators can be derived as where the last equation further yields another new operator with Commutation relations with Z are obtained as Consequently, for the closure of the algebra of the fuzzy coordinates X a and Θ α , we have introduced the fifteen new "coordinates", X ab , Θ α and Z. In total, with the original coordinates they amount to twenty four operators that satisfy the SU (4|1) algebra. The basic concept of the noncommutative geometry is the algebraic construction of geometry, and the SU (4|1) structure, the symmetry of the basis states on fuzzy four-supersphere, has indeed appeared as the fundamental algebra of the fuzzy four-supersphere.

U OSp(1|4) SVBS States
Similar to the U OSp(1|2) SVBS case, the U OSp(1|4) SVBS states 46 can be constructed as where Ψ(r) denotes the parameter-dependent U OSp(1|4) Schwinger operator The SO(5) spin magnitude on site i reads as In the following, we focus on the M = 1 U OSp(1|4) SVBS chain and its corresponding SMPS representation: where A is a supermatrix given by with the matrix elements The basis states |α, α and |α, β α =β are SO(5) 10-dimensional adjoint representation while |α are SO(5) 4-dimensional spinor. In total, the components of A consist of U OSp(1|4) 14-dimensional representation of the graded fully symmetric representation (334) for n = 2.
46 To be precise, there exist two types of U OSp(1|4) VBS states, one of which is the tensor type (348) and the other is the vector type [10]. Here, we focus on the tensor type. For the U OSp(1|4) SVBS infinite chain, the Schmidt coefficients are computed as The bosonic Schmidt coefficients are quadratically degenerate while the fermionic one is nondegenerate. The behaviors of the Schmidt coefficients and the entanglement entropy, S E.E. = −4λ 2 B log 2 λ 2 B − λ 2 F log 2 λ 2 F , are plotted in Fig. 37. The qualitative behaviors of the entanglement spectra of the U OSp(1|4) VBS chain are quite similar to those of the type-I SVBS chain [see Fig.  30] except for the quadratical degeneracy in the blue curve.
The degeneracy of the entanglement spectra of the U OSp(1|4) SVBS chain can be understood based on the arguments of the symmetry protected topological order. Before proceeding to the case of U OSp(1|4), we introduce the original arguments for its bosonic counterpart, the SO(5) VBS state [107,108,109]. Each of inversion symmetry and time reversal symmetry guarantees at least double degeneracy of the entanglement spectra of the SO(5) VBS states as proven by similar manner to the SU (2) VBS states. This is simply because each of the inversion symmetry and time reversal symmetry is a realization of Z 2 symmetry. The crucial difference to the SU (2) case is the existence of (Z 2 × Z 2 ) 2 symmetry of the SO(5) VBS states originating from the SO(5) spin degrees of freedom at edge [202]. The SO(5) rotation represents a rotation in five-dimensional space, and we "divide" the five-dimensional coordinates (1,2,3,4,5) to two three-dimensional subsectors, (1,2,5) and (3,4,5). The π-rotational symmetry around x and z axes in each sector generates the Z 2 × Z 2 symemtry and in total two independent sets of such discrete rotations give rise to (Z 2 × Z 2 ) 2 symmetry in the five-dimensional space. The group elements of (Z 2 × Z 2 ) 2 consist of 16 bases: where u ab are the SO(5) group elements of π rotation generated by the SO(5) generator σ ab : u ab (π) = e iπσ ab .
As the Z 2 × Z 2 symmetry generates at least doubly degeneracy in the entanglement spectra, the (Z 2 ×Z 2 ) 2 symmetry guarantees the four-fold degeneracy of the entanglement spectra of the SO(5) VBS chain. For the SUSY case, just as we have discussed in the U OSp(1|2) SVBS case, symmetry transformations are attributed to those of the bosonic and fermionic sectors, Similar to the discussions about double degeneracy in the U OSp(1|2) case, either of the bosonic and fermionic sectors brings the quadruple degeneracy to the entanglement spectra of the U OSp(1|4) SVBS state in the presence of (Z 2 × Z 2 ) 2 symmetry.

Summary and Discussions
We reviewed the constructions and basic properties of the fuzzy superspheres and SVBS models. Particularly, fuzzy superspheres and SVBS models with U OSp(1|2), U OSp(2|2) and U OSp(1|4) symmetries were discussed in detail. We clarified the mutual relations among the fuzzy spheres, QHE and VBS states were also emphasized based on the Schwinger operator formalism. It was illustrated that, though the SVBS states incorporate fermionic degrees of freedom, they "inherit" all the nice properties of the VBS: • Solvable parent Hamiltonian • Gapped bulk and gapless edge excitations • Generalized hidden order.
We explicitly derived the spectra for gapped excitations (magnon and hole excitations) on 1D SVBS chain within SMA. Physical properties of the SVBS models are qualitatively different from those of the other generalized VBS models based on bosonic Lie groups, in the sense that the SVBS states accommodate the charge sector in addition to the spin sector. In each sector, the SVBS states exhibit the following properties: • In charge sector, the SVBS states have the superconducting property (SSB).
• In spin sector, the SVBS states show non-trivial topological order of QAFM (no SSB).
We also established the SMPS formalism for the SVBS states, which naturally incorporates the edge degrees of freedom to provide a powerful tool in investigating topological property. For practical use, the (S)MPS formalism greatly simplifies the calculations of physical quantities, such as the string order and entanglement spectrum. The SVBS states bear a finite string order regardless of the parity of the bulk superspin unlike the original VBS states. From the explicit calculations of the entanglement spectra of the SVBS chains, we demonstrated that there exists a hallmark of topological phase, the double degeneracy in SUSY entanglement spectrum. The degeneracy is naturally understood by invoking particular edge states in SUSY chain: Since SUSY relates the integer and half-integer edge-spin states, the SVBS chains always accommodate half-integer edge spin (as well as integer edge spin) that guarantees at least double degeneracy of the entanglement spectra. Consequently, the topological order is stabilized regardless of the parity of bulk-spin in the presence of SUSY. Though we focused on the SVBS states, the arguments of the present SUSY protected topological order are applicable to general boson-fermion systems. By reformulating boson-fermion system, such as boson-fermion mixture cold atom system [203] with SUSY, we may apply the present results to discuss the stability for their topological phases. The idea of topological insulator has begun to be applied to particle theory model [204]. It may also be interesting to apply the present arguments to other SUSY models that are not directly related to condensed matter physics.