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Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “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 a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed 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 a superconducting property that interpolates various VBS states, depending on the 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. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.
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 it is still providing us with attractive topics in modern physics. Generally, in the presence of a 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 them down 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 the 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 with 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 Figure 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 2 D to work. This is a great advantage for 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 [20,21,22,23,24,25] combined with the transfer-matrix method.
Since the topological character of a system is believed to be encoded in the many-body ground-state wave functions rather than in its Hamiltonian one, the relatively simple structure of the VBS wave function is suitable to investigate its topological properties by using such modern means as entanglement entropy [26,27] or the entanglement spectrum [28]. For the above reasons, the VBS states or, more generally, the MPS as a class of model states that satisfy the so-called area-law constraint [29] 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) [30] provides a powerful tool to find the optimized variational wave function in the form of MPS (see [31,32,33] for more details about the MPS method and DMRG). The key idea here 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 [34,35,36], we adopt in the present paper the Schwinger-particle formalism (see, for instance, chapters 7 and 19 of [37]) to emphasize the analogies to the lowest Landau level physics. The list of possible applications includes a convenient description of gapped quantum ground states [29,38,39], as well as its application to efficient simulations of dynamics [40,41,42,43,44,45] and variational calculations [46,47,48,49]. Due to their simple structure, the entanglement entropy of the VBS states comes only from the bond(s) cut by the entanglement bipartition [50,51,52,53].
(Color online) Physical analogies between quantum Hall effects (QHE) and valence bond solid (VBS) state. The bulk excitation on QHE is gapful, while the edge mode is a gapless (chiral) mode. Meanwhile, the bulk excitation on the VBS state is gapful while the motion of edge spins is a freely rotating gapless mode.
In some “anisotropic” MPSs, it is known that generalized quantum phase transitions (QPT) occur as the parameters contained in the MPSs are varied [54,55,56,57]. 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 [57]. 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, 2lnD [58] (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 lnL [58,59,60]. Therefore, one should take the QPTs in MPS mentioned above in a wider sense. This kind of phase transition in the VBS-type of states in 2D have been discussed in [54,55].
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 [61,62]. However, if certain symmetries (e.g., time reversal) are imposed, topological phases characterized by short-range entanglement are possible. Since this kind of topological phase is stable only in the presence of certain protecting symmetries, they are called symmetry-protected topological (SPT) [63]. 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 [61,62,64]. This program has been carried out for such elementary symmetries as time-reversal and link-parity in [65,66] (for the fermionic systems, see [67,68]) and for the Lie-group symmetries in [69,70]. Topological quantum phase transitions among these SPT phases have been discussed in, e.g., References [63,71,72].
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” [73] (see [74] for Heisenberg’s contribution to the original idea of non-commutative geometry and related historical backgrounds.) 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 [75] 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 [76], by using the coherent-state representation of the VBS states and assigning appropriate correspondences between their physical quantities. In such a 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.
In the past decade, there have been remarkable developments in higher-dimensional generalization of QHE (see [77,78] for reviews). So far, the set-up [5,76] of 2D QHE has been generalized to higher-dimensional manifolds, such as 4D [79], 8D [80], S2n [81] and CPN [82,83]. There also exist q-deformed QHE [84] and QHE on non-compact manifolds [85,86,87,88]. As is well-known, QHE is a physical realization of the non-commutative geometry [89], and non-commutative geometry brings exotic properties to QHE [90,91,92,93]. Therefore, for each higher-dimensional QHE, one can think of the underlying higher-dimensional fuzzy geometry, such as fuzzy two-sphere [94,95,96], four-sphere [97], 2n sphere [98,99,100,101,102,103], CPN [104,105,106], q-deformed sphere [107] and fuzzy hyperboloids [108,109,110,111].
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) [112,113,114], SU(N+1) [115,116,117,118], Sp(N) [119,120] and q-deformed SU(2) [22,23,121,122,123] [see Figure 2]. The VBS states demonstrate manifest relevance to fuzzy geometry also, by adopting the Schwinger operator formalism (see Section 2.1).
Close relations among fuzzy geometry, QHE and VBS. They are “transformed” to each other with appropriate translations.
One of the main goals of this review is to illustrate their relations, which have been less emphasized in previous literature. 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 [124,125,126,127,128] and the SUSY QHE in [129,130,131,132] (a variety of super Landau models with super-unitary symmetries have been constructed in [133,134,135,136,137,138,139] and fuzzy supergeometries have also been investigated in [110,140,141,142]), 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 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 UOSp(1|2) and UOSp(2|2) symmetries, which we call type-I and type-II SVBS states, respectively. To perform detailed analyses of the SVBS states on a 1D chain, we develop a SUSY version of the MPS formalism (SMPS) [21,24,25]. MPS formalism is now regarded as an appropriate formalism to treat gapped 1D quantum systems [29,38]. 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 the modern symmetry-protected topological order argument, with emphasis on the edge superspin picture. We also generalize the lower SUSY analyses to the case of higher SUSY: fuzzy four-superspheres and UOSp(1|4) SVBS states.
The following is the organization of the present paper. In Section 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 Section 3, the SUSY extensions of fuzzy two-sphere are presented in detail. The corresponding SVBS states and their basic properties are reviewed in Section 4. In Section 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 Section 6, where the entanglement spectrum and entanglement entropy are derived (unpublished results for UOSp(2|2) SVBS chain are also included here). The stability of topological phases in the presence of SUSY is discussed, too. In Section 7, we extend the discussions to the case of fuzzy four-superspheres and the UOSp(1|4) SVBS states. Section 8 is devoted to summary and discussions. In Appendix A, we provide mathematical supplements for the super-Lie group, UOSp(N|2K). Fuzzy four-superspheres with an arbitrary number of SUSY are described in Appendix B.
2. 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 the 70s [94]. In the beginning of the 80s, Hoppe [95] explored the algebraic structure and field theories on a quantized sphere, and subsequently in the early 90s, Madore [96] 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 that might soften the UV divergence to be appropriately renormalized in conventional field theories.
In the middle of 90s, Grosse et al. [97] generalized the notion of the fuzzy two-sphere to construct four-dimensional fuzzy spheres and supersymmetric fuzzy spheres, i.e., fuzzy superspheres [124,125]. In the late 90s, fuzzy geometry was rediscovered in the context of the “second revolution of string theory” and attracted a lot of attention. 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 [143,144,145] as reviews), and fuzzy manifolds are also known to arise as classical solutions of Matrix theory (see, e.g., References [98,146]). Similarly, field theories on fuzzy superspheres provide a set-up for SUSY field theories with UV regularization [124,147,148], and the fuzzy superspheres were also found to arise as classical solutions of supermatrix models [149,150]. For details and applications of the fuzzy sphere, interested readers may consult References [151,152,153,154]. Non-commutative geometry and fuzzy physics also found their applications in gravity [155,156,157] and in condensed matter physics [90,91,92,93].
2.1. Fuzzy Two-Spheres and the Lowest Landau Level Physics
The fuzzy two-sphere [94,95,96] is one of the simplest curved fuzzy manifolds (The fuzzy spheres also play a crucial role in studies of string theory; see [158,159,160] for reviews). The coordinates of the fuzzy two-sphere, Xi(i=1,2,3), are regarded as the operators that are constituted of the SU(2) generators
Xi=2RdJi
Here, R denotes the radius of the sphere, and d is the dimension of the SU(2) irreducible representation:
d=2j+1
with the SU(2) Casimir index j(j=0,12,1,32,2,⋯), and Ji(i=1,2,3) are the SU(2) matrices of the corresponding representation that satisfy:
[Ji,Jj]=iϵijkJk
and
JiJi=j(j+1)1d
where 1d denotes d×d unit matrix. The “coordinates”, Xi, defined in Equation (1), satisfy:
[Xi,Xj]=i2RdϵijkXk,XiXi=R21-1d21d
Since J3 takes the eigenvalues, j,j-1,j-2,⋯,-j, the eigenvalues of X3 are given by:
X3=R1-1d,R1-3d,R1-5d,⋯,R-1+1d
Each “latitude” of X3 corresponds to a patch of width, 2R/d, i.e., the minimum state, on the fuzzy sphere ( here, two different viewpoints are possible; one assumes, as is done here, R to be fixed and, in the large-d (or large-j) limit, the width of the patch, 2R/d, becomes zero to recover the continuous space). The other fixes the width, 2R/d, and the radius of the fuzzy two-sphere, R∼d, diverges in the large-d limit. The former is reminiscent of the scaling limit of lattice field theories upon identifying R with the physical mass. 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 [Xi,Xj]=0 and XiXi|d→∞=R2, and the discrete spectrum (6) of X3 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 [153,161] and introduce the following operator:
Φ=(a,b)t
whose components satisfy the following ordinary bosonic commutation relations:
[a,a†]=[b,b†]=1,[a,b]=[a,b†]=0
By sandwiching the Pauli matrices with the Schwinger operators, Φ, one may simply represent the coordinates Xi of the fuzzy two-sphere as:
Xi=RdΦ†σiΦ
where σi(i=1,2,3) are the Pauli matrices:
σ1=0110,σ2=0-ii0,σ3=100-1
This is the well-known construction of the SU(2) angular momentum operators introduced by Schwinger [162]. In fact, one can easily check that Ji=(1/2)Φ†σiΦ satisfies 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:
XiXi=Rd2n(n+2)1n+1
where n denotes the eigenvalues of the number operator for the Schwinger bosons, n^=Φ†Φ=a†a+b†b. Comparing this with Equation (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):
|n1,n2〉=1n1!n2!a†n1b†n2|vac〉
where n1 and n2 are non-negative integers that satisfy n1+n2=n. Physically, |n1,n2〉 stand for a finite number of basis states constituting fuzzy two-spheres, which are the eigenstates of X3 with the eigenvalues:
X3=Rd(n1-n2)=Rd(n-2k)
where k≡n2=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 X3-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, the 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 (the coherent state here is usually referred to as the Bloch spin coherent state in the literature.) 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 the two-sphere, (x1,x2,x3)(xixi=1), is defined as a two-component complex vector satisfying (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 Equation (15)).
(xi·σi)ϕ({xi})=ϕ({xi})
In general, the (normalized) coherent state is represented in terms of the Euler angles as:
ϕ({xi})=uv=12(1+x3)1+x3x1+ix2e-i2χ=cosθ2sinθ2eiφe-i2χ
where (x1,x2,x3)=(sinθcosφ,sinθsinφ,cosθ) (0≤φ<2π, 0≤θ≤π, 0≤χ<2π) and e-i2χ denotes an arbitrary U(1) phase factor. The relation between the point, (x1,x2,x3), on a two-sphere and the two-component Hopf spinor (u,v) is given by the so-called first Hopf map (for construction of higher dimensional Hopf maps, see Reference [77], for instance):
xi=ϕ†σiϕ
which maps the three-sphere, S3, onto the two-sphere, S2:
S3⟶S1S2
In fact, a space of normalized two-component complex spinors, ϕ=(u,v)t, subject to ϕ†ϕ=1 is isomorphic to S3, and Equation (17) gives the mapping onto S2:
xixi=(ϕ†ϕ)2=1
The Hopf spinor, ϕ, is regarded as the classical counterpart of the Schwinger operator, Φ [Equation (7)] that satisfies:
Xi·Φ†σi=Rd(n+2)Φ†
If we note that R(n+2)/d→R, Xi→Rxi in the limit, d→∞, the resemblance between (15) and (20) is clear. Multiplying (R/d)Φ from the right to both sides of Equation (20) and using Φ†Φ=n, we reproduce Equation (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 (xi↦Xi, ϕ↦Φ) of the Hopf map (17) [163].
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):
un1,n2=1n1!n2!un1vn2J+=u∂∂v,J-=v∂∂u,Jz=12u∂∂u-v∂∂v
with n1+n2=n, n1,n2≥0.
(Color online) One-body-level relationship among fuzzy two-sphere (upper left), Haldane sphere (upper right) and local spin states of the VBS state (lower middle). 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 a Dirac monopole at its center. The S=n/2 is the monopole charge quantized as a half-integer or an 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.
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 un1,n2 represents the wave functions of a certain kind of quantum-mechanical systems in two dimensions. In fact, the wave functions, un1,n2 (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 [76]):
H=12MR2ΛiΛi
where Λi(i=1,2,3) are the covariant angular momenta:
Λi=-iϵijkxj(∂k+iAk)
in the presence of the gauge field, Ai, generated by the Dirac monopole at the origin:
Ai=n2ϵij3xjR(R+x3)
In the fuzzy geometry side, the monopole charge, n/2, corresponds to the quantized radius of the fuzzy two-sphere (11) (see also Figure 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 fuzzy geometry and the lowest Landau level physics [77].
2.2. 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:
|↑〉=a†|vac〉,|↓〉=b†|vac〉
The fully symmetric representation (13) constructed out of n Schwinger operators automatically realizes the local spin defined on each site with the spin magnitude:
S=(12⊗12⊗⋯⊗12)︷nfullysymm.=n2
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 [6,7,75], 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 Figure 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 (since the spin-singlet state of the two spin-1/2s maximizes the entanglement entropy, the pair is sometimes called maximally entangled in modern literature.) called the valence bonds (VB) (see Figure 4). Last, we project the entire 2n-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., Reference [114]). This kind of construction (valene-bond construction) applies to any lattice in any dimensions (see Figure 4 for typical examples in 1D and 2D), and the state obtained thereby is called the VBS state or, in modern terminology, the projected entangled-pair state (PEPS) [164,165].
It is also possible to construct the VBS states for other symmetry groups (e.g., SU(N) [115,116] and SO(N) [112,113]), 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 (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.) However, if we use the matrix-product representation [21] in 1D or, in general, tensor-product representation (or, equivalently, the vertex-state representation [54,55]) 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, References [21,23,56] for 1D and References [54,55] for 2D honeycomb and square lattices.
(Color online) VBS states on 1D and 2D lattices. Filled circles denote auxiliary spin-1/2 objects, which are finally symmetrized to form S=Mz physical spins at each site. Solid lines stand for singlet valence bonds between the spin-1/2s.
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 Equation (25)), the singlet valence bond on the bond 〈ij〉 reads:
(ai†bj†-bi†aj†)
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:
(ai†bj†-bi†aj†)M
to represent M valence bonds between the sites, i and j. Thus, one sees that the original valence-bond construction [6,7] of the VBS states is equivalent to the following representation in terms of the Schwinger bosons [75]:
|VBS〉=∏〈ij〉∈N.N.(ai†bj†-bi†aj†)M|vac〉
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. From [Sia+Sja,ai†bj†-bi†aj†]=0 (a=x,y,z), it is obvious that the state Equation (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 Mz Schwinger bosons per site in the VBS state (29):
ai†ai+bi†bi=zM
and, hence, the local spin quantum number, Si=12(ai†ai+bi†bi), is given by:
Si=12zM
In particular, for the 1D (i.e., z=2) M=1 VBS state, we have:
Si=1
and the local Hilbert space is spanned by the following three basis states:
|1〉=12ai†2|vac〉,|0〉=ai†bi†|vac〉,|-1〉=12bi†2|vac〉
We were a bit sloppy in writing down Equation (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 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-Ms at the edges. The edge polynomials for M=1 are given by:
f↑(a†,b†)=a†,f↓(a†,b†)=b†
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
|VBS〉=(a1†)p(b1†)M-p∏j=1L-1(aj†bj+1†-bj†aj+1†)M(aL†)M-q(bL†)q|vac〉(0≤p,q≤M)
Since the VBS Hamiltonian is defined as the projection operator for the Hilbert space of a pair of neighboring spins (see Section 4.3 for detail), the state of the form (29) is the ground state regardless of the edge polynomials. Therefore, there appear (M+1)×(M+1) degenerate ground states for the spin-S(=M) VBS state on a finite open chain (see Figure 5) [7]. From similar considerations, it is obvious that the higher-dimensional VBS ground states (e.g., those in Figure 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 Sz 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 Equation (29) and Figure 4) and is non-degenerate (this is true for a periodic chain and the bulk in a infinite chain; on a finite open chain, the ground state may not be spin-singlet and, hence, may show degeneracy corresponding to the edge states), while the classical ground state, i.e., the Néel state, is infinitely degenerate with respect to the SU(2) rotational symmetry. This implies that though the magnitude of the local spin is Si=1, its expectation value is zero, 〈Si〉VBS=0, in the bulk (on a finite open chain, 〈Si〉VBS may take a non-zero finite value near the two boundaries and decays exponentially to zero toward the center of the system; in this sense, magnetism revives near the boundaries, and this is the manifestation of the emergent edge states), and the system is non-magnetic. In this sense, the VBS state is purely quantum-mechanical and does not have any classical counterpart.
(Color online) For the S=1 VBS state on a finite open chain, there exist spin-1/2 degrees of freedom at each edge. By construction, the VBS state is the ground state of the VBS Hamiltonian, regardless of the spin states at the edges.
For better understanding of the VBS state, let us expand the spin-1 VBS state in the Sz-basis:
|VBS〉=∏i(ai†bi+1†-bi†ai+1†)|vac〉=|⋯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, Sz=+1 and -1, appear in an alternating manner with intervening Sz=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 Section 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 [75] to the Laughlin-Haldane wave function [76] for FQHE on a two-sphere. First, we note that by using the SU(2)≃USp(2) invariant matrix given by the 2 × 2 antisymmetric matrix (see Appendix A):
R2=iσ2=01-10
the VBS states (29) on generic lattices can be rewritten compactly as:
|VBS〉=∏〈ij〉∈N.N.(Φi†R2Φj*)M|vac〉
where Φi denotes the Schwinger operator on the site i:
Φi*≡(ai†,bi†)t
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:
Φ*=(a†,b†)t→ϕ=(u,v)t
we obtain the coherent-state (or Weyl) representation of the VBS state (precisely, if we use the coherent-state basis
|Ωi〉=1(Mz)!(uiai†+vibi†)Mz|vac〉
for the local spin-Mz/2 states and expand the VBS state in these basis, we obtain the “wave function” 〈{Ωj}|VBS〉∝ΦVBS({ui*,vi*})):
ΦVBS({ui,vi})=∏〈ij〉∈N.N.(ϕitR2ϕj)M=∏〈ij〉∈N.N.(uivj-viuj)M
Now, the formal similarity to the Laughlin-Haldane wave function of QHE:
ΦLH=∏i<j(ϕitR2ϕj)m=∏i<j(uivj-viuj)m
on a two-sphere [75] is clear. By the stereographic projection from a two-sphere to a complex plane:
ϕ=uv⟶z=v/uΦLH (42) is, in the thermodynamic limit, reduced to the celebrated Laughlin wave function:
ΦL=∏i<j(zi-zj)me-∑izizi*
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 References [166,167].
symmetry-05-00119-t001_Table 1
Correspondences between physical quantities of many-body wavefunctions of QHE and quantum anti-ferromagnets (QAFM).
QHE
QAFM
Many-body state
Laughlin-Haldane wave function
VBS state
ΦLH=∏i<jN(uivj-viuj)m
|Φ〉=∏〈ij〉z(ai†bj†-bi†aj†)M|vac〉
Power
m: inverse of filling factor
M: number of VBs between neighboring sites
Charge
S=mN/2: monopole charge
S=Mz/2: local spin magnitude
3. Fuzzy Two-Supersphere
Next, we proceed to the SUSY version of fuzzy two-sphere (fuzzy superspheres are also realized as a classical solution of supermatrix models [149,150].) and generalized VBS states. We mostly focus on the cases of the SUSY numbers, N=1 and 2.
First, we introduce N=1 SUSY algebra, UOSp(1|2) [168,169,170], which contains the SU(2) algebra as its maximal bosonic subalgebra. The UOSp(1|2) algebra consists of the five generators, three of which are SU(2) (bosonic) generators, Li(i=1,2,3), and the remaining two are SU(2) fermionic spinors, Lα(α=θ1,θ2), which amount to satisfy:
[Li,Lj]=iϵijkLk,[Li,Lα]=12(σi)βαLβ,{Lα,Lβ}=12(iσ2σi)αβLi
The UOSp(1|2) Casimir is constructed as:
K=LiLi+ϵαβLαLβ
and its eigenvalues are given by j(j+1/2): j is referred to as the superspin taking the non-negative integer or half-integer values, j=0,1,12,1,32,⋯. The UOSp(1|2) irreducible representation with the superspin index, j, consists of the SU(2)j and j-1/2 spin representations, and hence, the dimension of the UOSp(1|2) representation of superspin j is given by:
(2j+1)+(2j)=4j+1
The fundamental representation (j=1/2) matrices of the UOSp(1|2) generators are expressed by the following 3 × 3 matrices:
li=12σi000,lα=120τα-(iσ2τα)0
where τα (α=1,2) are:
τ1=10,τ2=01
Equation (48) may be regarded as a SUSY extension of the Pauli matrices. They are “Hermitian” in the sense:
li‡=li,lα‡=ϵαβlβ
where ‡ signifies the super-adjoint defined by:
ABCD‡≡A†C†-B†D†
Similar to the case of fuzzy sphere (9), the coordinates of fuzzy two-superspheres [124,125] are constructed by the graded version of the Schwinger operator formalism [127,128,129]. The graded Schwinger operator consists of two bosonic components, a and b, and one fermion component, f:
Ψ=(a,b,f)t
with satisfying:
[a,a†]=[b,b†]={f,f†}=1[a,b]=[a,f]=[b,f]=[a,b†]=[a,f†]=[b,f†]=0
With the use of Ψ, the coordinates of a fuzzy two-supersphere, Sf2|2 (two on the left to | denotes the bosonic degrees of freedom, while two on the right to | denotes the fermionic degrees of freedom), are constructed as:
Xi=2RdΨ†liΨ,Θα=2RdΨ†lαΨ
which satisfy:
[Xi,Xj]=i2RdϵijkXk,[Xi,Θα]=Rd(σi)βαΘβ,{Θα,Θβ}=Rd(iσ2σi)αβXi
where d=n+1 with n=Ψ†Ψ=nB+nF=a†a+b†b+f†f. The square of the radius of fuzzy supersphere is given by the UOSp(1|2) Casimir:
XiXi+ϵαβΘαΘβ=Rd2n(n+1)12n+1
Notice that 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:
|n1,n2〉=1n1!n2!a†n1b†n2|vac〉|m1,m2)=1m1!m2!a†m1b†m2f†|vac〉
where n1,n2,m1 and m2 are non-negative integers that satisfy n1+n2=m1+m2+1=n. |m1,m2) is the fermionic counterpart of |n1,n2〉, and thus, |n1,n2〉 and |m1,m2) exhibit N=1 SUSY. The bosonic and fermionic basis states are the eigenstates of the fermion parity, (-1)nF, with the eigenvalues, +1 and -1, respectively. Their degrees of freedom are also respectively given by:
dB=d(n)=n+1,dF=d(n-1)=n
and the total degrees of freedom is:
dT=dB+dF=2n+1
The eigenvalues of X3 for (57) read as:
X3=Rd(n-k)
where k=0,1,2,⋯,2n. Even k (k=2n2) correspond to the bosonic states (57a), while odd k (k=2m2+1) to the the fermionic states (57b). Compare the X3 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 a fuzzy sphere with radius, n, while those for odd k are by a fuzzy sphere with radius n-1. In this sense, the fuzzy two-supersphere, Sf2|2(n), of radius, n, can be regarded as a “compound” of two fuzzy spheres with different radii, Rdn and Rd(n-1) [Figure 6]:
Sf2|2(n)≃Sf2(n)⊕Sf2(n-1)
The first Sf2(n) accounts for the bosonic degrees of freedom (57a), while the second Sf2(n-1) does so for the fermionic degrees of freedom (57b).
(Color online) N=1 fuzzy supersphere is a “compound” of two fuzzy two-spheres with radii, Rdn and Rd(n-1). This figure corresponds to n=2.
Similar to the first Hopf map, we can construct a graded version of the first Hopf map [171,172]:
ψ→xi=2ψ‡Liψ,θα=2ψ‡Lαψ
where li and lα are the UOSp(1|2) matrices of fundamental representation (48) and ψ denotes a normalized UOSp(1|2) superspinor:
ψ=uvη
with:
ψ‡ψ=u*u+v*v-η*η=1
Here, the super-adjoint of the superspinor is defined by ψ‡≡(u*,v*,-η*) [see, also, Equation (51)], and * represents the pseudo-conjugation (the pseudo-conjugation is defined as (η*)*=-η and (η1η2)*=η1*η2* for Grassmann odd quantities; see Refernence [173], for instance). The first two components of ψ are Grassmann-even. and the third component is Grassmann-odd. The ψ subject to the normalization (64) can be regarded as a coordinate of the manifold S3|2. From (62), we find that xi and θα satisfy the condition of S2|2:
xixi+ϵαβθαθβ=(ψ‡ψ)2=1
Consequently, the map (62) represents:
S3|2⟶S1S2|2
The bosonic part of (66) exactly corresponds to the first Hopf map. Note that ψ satisfies the super-coherent state equation:
xi·liψ+ϵαβlαψ·θβ=12ψ
and ψ is referred to as the super-coherent state or spin-hole coherent state [37] in the literature. xi are Grassmann even, but not usual c-numbers, since the square of xi is not a c-number, xixi=1-ϵαβθαθβ, as seen from (65). Instead of xi, one can introduce c-number quantities, {yi}, as:
yi=11-ϵαβθαθβxi
which denoting coordinates on S2, the “body” of S2|2, as confirmed from yiyi=1. With the use of the coordinates on S2|2, ψ can be expressed as [129]:
ψ=12(1+y3)(1+θ1θ2)1+y3y1+iy2(1+y3)θ1+(y1+iy2)θ2eiχ=12(1+x3)(1+x3)(1-12(1+x3)θ1θ2)(x1+ix2)(1+12(1+x3)θ1θ2)(1+x3)θ1+(x1+ix2)θ2eiχ
where eiχ stands for the arbitrary U(1) phase factor. The last expression on the right-hand side manifests the graded Hopf fibration, S3|2∼S2|2⊗S1 (here, ∼ denotes local equivalence): the S1(≃U(1))-fiber, eiχ, is canceled in the graded Hopf map (62), and the other quantities, xi and θα in (69), correspond to the coordinates on S2|2.
As the geometric structure of Sf2|2 is determined by the UOSp(1|2) algebra, the N=2 fuzzy supersphere, Sf2|4, is formulated by the UOSp(2|2) algebra. The UOSp(2|2) algebra contains the bosonic subalgebra, usp(2)⊕o(2)≃su(2)⊕u(1), and is isomorphic to SU(2|1). The dimension is given by:
dim[uosp(2|2)]=dim[su(2|1)]=4|4=8
Denoting the four bosonic generators as Li(i=1,2,3) and Γ and the four fermionic generators as Lα and Lα′(α=θ1,θ2), we can express the UOSp(2|2) algebra as:
[Li,Lj]=iϵijkLk,[Li,Lασ]=12(σi)βαLβσ[Γ,Li]=0,[Γ,Lασ]=12ϵτσLατ{Lασ,Lβτ}=12δστ(iσ2σi)αβLi+12ϵστϵαβΓ
where Lασ=(Lα,Lα′). Li and Lα form the UOSp(1|2) subalgebra. There are two sets of fermionic generators, Lα and Lα′, which bring N=2 SUSY. The UOSp(2|2) algebra has two Casimirs, quadratic:
K=LiLi+ϵαβLαLβ+ϵαβLα′Lβ′+ΓΓ
and cubic ones [169].
To specify a fuzzy manifold, an appropriate choice of irreducible representation is crucial as well. The irreducible representation of UOSp(2|2) is classified into two categories: typical representation and atypical representation [169,170]. Since the quadratic Casimir (72) is identically zero for atypical representation, we adopt typical representation for the construction of Sf2|4 [128]. The UOSp(2|2) matrices for typical representation for minimal dimension are represented by the following 4×4 matrices:
li=12σi020202,lα=1202τα0-(iσ2τα)t00000lα′=12020τα000-(iσ2τα)t00,γ=12020202iσ2
Applying the Schwinger construction, we introduce the coordinates of Sf2|4 as:
Xi=2RdΨ†LiΨ,Θα=2RdΨ†LαΨ,Θα′=2RdΨ†Lα′Ψ,G=2RdΨ†ΓΨ
where Ψ signifies the four-component, UOSp(2|2), Schwinger operator:
Ψ=(a,b,f,g)t
and d=n+1 with n=Ψ†Ψ=a†a+b†b+f†f+g†g. Here, a and b are bosonic operators, while f and g are fermionic operators that satisfy:
[a,a†]=[b,b†]={f,f†}={g,g†}=1[a,b]=[a,f]=[a,g]=⋯={f,g}={f,g†}=0
It is straightforward to evaluate the square of the radius of Sf2|4:
XiXi+ϵαβΘαΘβ+ϵαβΘα′Θβ′+GG=Rd2n2
With a given n, the basis states of Sf2|4 are constituted of the graded fully symmetric representation of the UOSp(2|2):
|n1,n2〉=1n1!n2!a†n1b†n2|vac〉|m1,m2)=1m1!m2!a†m1b†m2f†|vac〉|m1′,m2′)=1m1′!m2′!a†m1′b†m2′g†|vac〉|l1,l2〉=1l1!l2!a†l1b†l2f†g†|vac〉
where n1, n2, m1, m2, m1′, m2′, l1 and l2 denote non-negative integers that satisfy n1+n2=m1+m2+1=m1′+m2′+1=l1+l2+2=n. The dimension of bosonic basis states, |n1,n2〉 and |l1,l2〉, and that of fermionic basis states, |m1,m2〉 and |m1′,m2′〉, are found to be equal:
dB=d(n)+d(n-2)=2ndF=2×d(n-1)=2n
and the total degrees of freedom amount to:
dT=dB+dF=4n
The square of the radius of Sf2|4 (77) does not have the zero-point, “energy”, since the bosonic and fermionic degrees of freedom are canceled exactly (in the cases of fuzzy superspheres based on the UOSp(2|N) algebra for N>2, the dimension of fermionic basis states is larger than that of the bosonic basis states, and the zero-point “energy” becomes minus; in other words, the radius of fuzzy supersphere would become minus for the UOSp(2|N) representations with sufficiently small dimensions; thus, we only deal with the fuzzy superspheres for N=1,2). The four sets of the basis states (78) forms the N=2 SUSY multiplet and suggests the following geometrical structure of Sf2|4:
Sf2|4(n)≃Sf2(n)⊕Sf2(n-1)⊕Sf2(n-1)⊕Sf2(n-2)
The latitudes for the the basis states (78) are given by:
X3=Rd(n-k)
with k=0,1,2,⋯,2n. The even ks (k=2n2, k=2l2+2) correspond to the bosonic basis states, (78a) and (78b), while odd ks (k=2m2+1, k=2m2′+1) to the fermionic basis states, (78c) and (78d) (Figure 7). Except for non-degenerate states at the north and south poles X3=±n, all the other eigenvalues of X3 (82) are doubly-degenerate.
(Color online) Sf2|4 is a “compound” made of four fuzzy two-spheres that are considered as N=2 superpartners. The above picture corresponds to n=2.
4. 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.
4.1. Construction of SVBS States4.1.1. <inline-formula>
<mml:math id="mm424" display="block">
<mml:mrow>
<mml:mi mathvariant="script">N</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>
Here, we consider the SVBS states with UOSp(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 Section 2.2. The first we prepare is the UOSp(1|2) invariant matrix (see Appendix A):
R1|2=010-10000-1
with the parameter dependent Schwinger operator:
Ψ(r)=(a,b,rf)t
the type-I SVBS state is constructed as:
|SVBS-I〉=∏〈ij〉(Ψi†(r)R1|2Ψj*(r))M|vac〉=∏〈ij〉(ai†bj†-bi†aj†-rfi†fj†)M|vac〉
The operators ,ai, bi and fi, are components of the graded Schwinger operator defined on each site, i, and satisfy the commutation relations, [ai,aj†]=[bi,bj†]=δij and {fi,fj†}=δij. Physically, the three fundamental states, a†|vac〉, b†|vac〉and f†|vac〉, are interpreted as spin ↑, ↓ and spinless hole states (see Table 2) (in Refernece [174], such a UOSp(1|2) triplet is dubbed the superqubit). Since the fermions always appear in pairs of the form fi†fj† (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 UOSp(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 UOSp(1|2) invariant matrix was utilized, and then the UOSp(1|2) structure explicitly enters in the many-body states. This implies that (super)spin interaction between adjacent sites reduces the SU(2|1) symmetry on each site to the lower symmetry UOSp(1|2) in many-body physics.
symmetry-05-00119-t002_Table 2
The physical interpretation of the local states made by the Schwinger operators. f† denotes the hole degrees of freedom.
Schwinger operator
SU(2) quantum number
Spin state
a†
1/2
|↑〉=a†|vac〉
b†
-1/2
|↓〉=b†|vac〉
f†
0
|h〉=f†|vac〉
In the type-I SVB states (85), the total particle number of the Schwinger particles at each site is given by:
zM=ai†ai+bi†bi+fi†fi
Since the fermion number f†f takes either 0 or 1, the following two eigenvalues are possible for the local spin quantum number Si=12(ai†ai+bi†bi):
Si=M,M-12
In particular, for M=1, each site can take two spin values:
Si=1,12
and the local Hilbert space is spanned by the five (4M+1, in general) basis states:
|1〉=12ai†2|vac〉,|0〉=ai†bi†|vac〉,|-1〉=12bi†2|vac〉|↑〉=ai†fi†|vac〉,|↓〉=bi†fi†|vac〉
These constitute a N=1 SUSY multiplet with the UOSp(1|2) superspin S=1. Similarly, the edge states consist of N=1 SUSY multiplet with the UOSp(1|2) superspin S=1/2:
|↑〉〉=a†|vac〉,|↓〉〉=b†|vac〉,|0〉〉=f†|vac〉
As we will see in Section 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〉:
|SVBS-I〉→|VBS〉=∏i(ai†bi+1†-bi†ai+1)|vac〉
while in the limit, r→∞, |SVBS-I〉 reduces to the Majumdar-Ghosh (MG) dimer state [175,176] |MG〉,
|SVBS-I〉→∏ifi†|MG〉
where |MG〉 is either of the two dimerized states of the MG model (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) [Figure 8]:
|MG〉=|ΦA〉=∏i:even(ai†bi+1†-bi†ai+1†)|vac〉|ΦB〉=∏i:odd(ai†bi+1†-bi†ai+1†)|vac〉
For larger M, |MG〉 should be replaced with the inhomogeneous VBS states [75], where the number of valence bonds alternates from bond to bond.
(Color online) Two MG dimer states related by translation.
In the super-coherent state (63) representation (in the following discussion, the explicit form of the superspinor is not important; only the Grassmann property of the components matters), the SVBS state is expressed as [8]:
ΨSVBS-I=∏〈ij〉(uivj-viuj-rηiηj)M
From the Grassmann (odd) property of η, ΨSVBS-I (94) can be rewritten as:
ΨSVBS-I=exp-Mr∑〈ij〉ηiηjuivj-viuj·ΦVBS({ui,vi})
where ΦVBS 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 ΦVBS, all physics inherent in the SVBS must be included in this exponential factor:
exp-Mr∑〈ij〉ηiηjuivj-viuj=∏〈ij〉1-Mrηiηjuivj-viuj
Since every time when the factor:
ηiηjuivj-viuj
acts to the VBS wave function, the VB between the adjacent sites, i and j, is replaced with a hole-pair (see Figure 9):
uivj-viuj→ηiηjuivj-viujηiηj
one sees that the SVBS wave function (95) may be expanded as:
ΨSVBS-I=ΦVBS-Mr∑iηiηi+1uivi+1-viui+1·ΦVBS+12(Mr)2∑iηiηjuivi+1-viui+12+⋯+(-Mr)L/2∏iηi·(∏i:even-∏i:odd)1uivi+1-viui+1·ΦVBS
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, the 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 Figure 10) (for general M and the arbitrary lattice coordination number, z, the last term of the expansion realizes a resonating valence bond (RVB) state [177]; for instance, on a 2D square lattice, the M=2 SVBS state with N=3 SUSY gives the Rokhsar-Kivelson RVB state [178] as the last term). For M=1, the last term gives rise to the Majumdar-Ghosh dimer states:
(∏i:even-∏i:odd)1uivi+1-viui+1·ΦVBS=(∏i:odd-∏i:even)(uivi+1-viui+1)=-ΦA+ΦB
where ΦA and ΦB are the coherent state representation of |ΦA〉 and |ΦB〉 (93). Now, the physical meaning of the SVBS states is transparent: the SVBS states signify a superposed state by all possible hole-pair doped VBS states, which can be viewed as a generalization of the resonating valence bond state [177] (see Section 4.2 for more details).
(Color online) When the exponential factor (97) acts on the VBS state, the factor breaks the VB between i and i+1 sites and creates a hole-pair instead. The figure and caption are taken from [8].
(Color online) The type-I SVBS is a superposed state of hole-pair-doped VBS states. With the finite hole-doping parameter, r, all of the hole-pair-doped VBS states are superposed to form the SVBS state, and the SVBS state exhibits the SC property. At r=0, the SVBS state is reduced to the original VBS state (depicted as the first chain), while for r→∞, the SVBS state is reduced to the MG dimer state (depicted as the last two chains). The figure and caption are taken from [9].
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 [130]:
ΦSLH=∏i<jN(ψi(r)tR1|2ψj(r))M=∏i<jN(uivj-viuj-rηiηj)m
Furthermore, for one-particle mechanics, there exists apparent relations between the SVBS and the SUSY Landau problem. Indeed, the super-coherent state representation of the basis states (57):
φn1,n2(B)=1n1!n2!un1vn2φm1,m2(F)=1m1!m2!um1vm2η
gives the lowest Landau level eigenstates of a SUSY Landau Hamiltonian [129]. Here, n1, n2, m1 and m2 are non-negative integers that satisfy n1+n2=m1+m2+1=I. By the stereographic projection, zi=vi/ui and ξi=ηi/ui, the SUSY Laughlin-Haldane wave function, ΦSLH (101), is transformed to the SUSY Laughlin wave function defined by:
ΦSL≡∏i<jN(zi-zj-rξiξj)me-∑i(zizi*+ξiξi*)
By expanding the polynomial part (zi-zj-rξiξj)m in the Grassmann odd quantities, we have:
ΦSL=ΦL-mr∑i<jξiξjzi-zjΦL+⋯+1(N/2)!mN2(-r)Nξ1,ξ2⋯ξN·Pf1zi-zjΦL
where N is the total number of particles (which we assumed to be an even integer) and ΦL coincides with the Laughlin wave function on a 2D plane (up to the Grassmann factor e-∑iξiξi*):
ΦL=∏i<jN(zi-zj)me-∑i(zizi*+ξiξi*)
Interestingly, the Pfaffian wave function, Pf1zi-zj∏i<jN(zi-zj)me-∑izizi*, for the ground state of the non-Abelian QHE [179] appears in the last term of the expansion (104) at m=2.
The N=2 SVBS states, which we call the type-II SVBS states, are constructed as UOSp(2|2) invariant VBS states. With the UOSp(2|2) invariant matrix:
R2|2=0100-100000-10000-1
and UOSp(2|2) Schwinger operator:
Ψ(r)≡(a,b,rf,rg)t
we introduce the type-II SVBS states:
|SVBS-II〉=∏〈ij〉(Ψi†(r)R2|2Ψj*(r))M|vac〉=∏〈ij〉(ai†bj†-bi†aj†-rfi†fj†-rgi†gj†)M|vac〉
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, gi, are interpreted as another species of a (spinless) hole, and satisfy the standard anti-commutation relations, {gi,gj†}=δij, {fi,gj}=0, etc. In the type-II VBS states, there appear local sites, such as fi†gi†|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:
zM=ai†ai+bi†bi+fi†fi+gi†gi
Since the eigenvalues of nf(i)=fi†fi and ng(i)=gi†gi 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, Si=12(ai†ai+bi†bi):
Si=M,M-12,M-12,M-1
In particular, for the M=1 SVBS chain, the local spin values are given by:
Si=1,12,12,0
and the local Hilbert space is spanned by the following nine basis states:
|1〉=12ai†2|vac〉,|0〉=ai†bi†|vac〉,|-1〉=12bi†2|vac〉|↑〉=ai†fi†|vac〉,|↓〉=bi†fi†|vac〉|↑′〉=ai†gi†|vac〉,|↓′〉=bi†gi†|vac〉|0′〉=gi†fi†|vac〉
The edge states are now given by
|↑〉〉=a†|vac〉,|↓〉〉=b†|vac〉,|0〉〉=f†|vac〉,|0′〉〉=g†|vac〉
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:
|SVBS-II〉→|VBS〉=∏i(ai†bi+1†-bi†ai+1†)|vac〉
On the other hand, when r→∞, it reduces to the totally uncorrelated fermionic (F) state filled with holes:
|SVBS-II〉→|F-VBS〉≡±∏ifi†gi†|vac〉
Here, we have assumed the open boundary condition (if the periodic boundary condition is used, we have a zero state for odd-length chains; the sign factor depends on both the parity of the system size and the edge states). 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:
ΨSVBS-II=∏<ij>(uivj-viuj-rηiηj-rηi′ηj′)M=exp-Mr∑<ij>ηiηj+ηi′ηj′uivj-viuj·exp-Mr2∑<ij>ηiηiηj′ηj′(uivj-viuj)2·ΦVBS
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 Figure 11.
4.2. Superconducting Properties
In both ΨSVBS-I and ΨSVBS-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 [180]:
|BCS〉=∏k≥011+|gk|2(1+gkck†c-k†)|vac〉
with electron operator, ck, and coherence factor, gk. |BCS〉 can be expressed as:
|BCS〉∝∏k≥0(1+gkck†c-k†)|vac〉=exp(∑k≥0gkck†c-k†)|vac〉=|vac〉+(∑k≥0gkck†c-k†)|vac〉+12(∑k≥0gkck†c-k†)2|vac〉+⋯+∏k≥0gkck†c-k†|vac〉
This expansion may remind us of the hole-pair expansion of the SVBS state (99). Furthermore, in the limit, gk→0, the BCS state reduces to the electron vacuum (no fermions), and for gk→∞, it coincides with the filled Fermi sphere |Fermi〉:
|BCS〉⟶gk→0|vac〉|BCS〉⟶gk→∞|Fermi〉≡∏k≥0ck†c-k†|vac〉
These also show apparent similarities with the asymptotic behaviors of the type-II SVBS chain, (114) and (115), under the correspondence:
|vac〉↔|VBS〉ck†c-k†↔fi†gi†
From the analogies to the BCS state, the SVBS states are expected to exhibit a SC property in the charge sector by the immersion of hole-pairs to (insulating) VBS states (see Figure 12). This is quite similar to the mechanism of the Anderson’s RVB picture of high Tc SC [177]: a finite amount of hole-doping transforms the insulator of resonating valence bond state to a high Tc SC state. In the following, we explore qualitative arguments for the SC aspect of the SVBS states.
(Color online) Like the type-I SVBS chain, the type-II SVBS chain is also expressed as a superposition of the hole-pair doped VBS chains. What is different to the type-I SVBS chain is the appearance of the spinless sites, depicted by the large white circles with double holes. The MG states are realized in the “middle” of the sequence. The original VBS state and the hole-VBS state are respectively realized in the first and last lines. The figure and caption are taken from [9].
The SVBS states exhibit a superconducting property in the charge sector with finite r in addition to a quantum AFM property in the spin sector.
Since the naive SC order parameter, such as 〈fk†fk+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:
Δ=(ak†bk+1†-bk†ak+1†)fk†fk+1†
which is calculated as:
〈Δ〉=2M(M+12)2rM(M+1)(1+|r|2)+14+12(M+12)2-14(M+12)2
It takes the maximum value:
|Δmax|=(5-2)2M(1+5)M+1
at:
|r|=M+121+52M(M+1)
In particular, for M=1, |r|=321+56≃1.10. The expectation values for the boson and fermion numbers, nb(i)=ai†ai+bi†bi and nf(i)=fi†fi, are respectively calculated as:
〈nb〉=2M-1+2M+14M(M+1)(1+|r|2)+1〈nf〉=1-2M+14M(M+1)(1+|r|2)+1
As expected, with an increase of the hole doping, |r|, 〈nb〉 monotonically decreases, while 〈nf〉 monotonically increases. The fluctuations for the boson and fermion numbers, δnb2=〈nb2〉-〈nb〉2 and δnf2=〈nf2〉-〈nf〉2, are given by:
δnb2=δnf2=2M+14M(M+1)(1+|r|2)+1-(2M+1)24M(M+1)(1+|r|2)+1
and their maximum δnb=δnf=12 is met at
|r|=31+14M(M+1)
The behaviors of such quantities are depicted in Figure 13.
(Color online) Plot of Osc=〈Δ〉, the hole density, 〈nf〉, and the hole-number fluctuation, δnf2, as a function of r. Here,the bulk values are plotted. Inset: pProfile of the hole density (r=0.5) for a finite system (L=20) with different left edge states (↑, ↓ and “hole”). Only the left edge state is changed with the right one fixed to sR=↑. The hole density approaches exponentially to the bulk value as we move away from the edge. The figure and caption are taken from [9].
While the BCS state (117) has the particle-hole symmetry (for (117), the order parameter, Δk=〈ck†c-k†〉, the electron number, nk=ck†ck, and its fluctuation, δnk, are calculated as:
Δk=gk*1+|gk|2=1gk+gk*-1〈nk〉=|gk|21+|gk|2δnk2=〈(nk-〈nk〉)2〉=|gk|2(1+|gk|2)2=1(gk+gk*-1)(gk*+gk-1)Δk and δnk are symmetric under gk↔1/gk*, due to the particle-hole symmetry. 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, (aibj-biaj)↔fifj.
We define the order parameter for the type-II SVBS state (108) as follows:
Δi≡(aibi+1-biai+1)(fi†fi+1†+gi†gi+1†)
In Figure 14, we plotted its expectation value:
Osc=〈Δi〉
the hole-number 〈nf〉, and hole fluctuation, δnhole:
〈nf〉=〈fi†fi〉=〈gi†gi〉=〈ng〉δnhole2=〈nhole2〉-〈nhole〉2(nhole≡nf+ng)
The SC order parameter, OSC takes its maximal value at |r|≃1.05 for M=1. Comparing Figure 13 and Figure 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.
(Color online) Plot of Osc=〈Δi〉, the hole density, 〈nhole(i)〉=〈fi†fi〉, and the hole-number fluctuation, δnhole2, as a function of r. Inset: profile of the hole density (r=0.5) for a finite system (L=20). Only the left edge state is changed with the right one fixed to sR=↑. The figure and caption are taken from [9].
4.3. 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:
1⊗1=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 Figure 15).
(Color online) The bond spin, J=2, cannot be realized in the S=1 VBS chain.
Hence, the VBS state satisfies the condition:
PJ=2(i,i+1)|VBS〉=0
where PJ 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”, ∑iPJ=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:
H=∑iVJ=2PJ=2(i,i+1)
where VJ=2 denotes a positive coefficient (VJ=2 can depend on the lattice site, i, but here, we postulate lattice translation symmetry and drop the lattice index, i). It is obvious that the eigenvalues of the parent Hamiltonian (134) are semi-positive definite, and its ground-state energy is zero. Furthermore, remember that the VBS state vanishes by the projection operators in the parent Hamiltonian, and hence:
H|VBS〉=0
Therefore, the VBS chain is a zero-energy ground state of the parent Hamiltonian (134). There is uniqueness to the ground state (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].) for the parent Hamiltonian (up to the degeneracy coming from the edge degrees of freedom), which can also be proven [7]. The projection operator is explicitly derived as:
PJ(i,i+1)=∏J′≠J(S(i)+S(i+1))2-J′(J′+1)J(J+1)-J′(J′+1)=∏J′≠J2S(i)·S(i+1)+2S(S+1)-J′(J′+1)J(J+1)-J′(J′+1)
In the present case, S=1, PJ=2(i,i+1) is given by:
PJ=2(i,i+1)=∏J′=0,12S(i)·S(i+1)+4-J′(J′+1)6-J′(J′+1)=12S(i)·S(i+1)+13(S(i)·S(i+1))2+23
After all, for the S=1 VBS chain, the parent Hamiltonian takes the following form:
H=V∑iS(i)·S(i+1)+13(S(i)·S(i+1))2+23
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 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 Section 5.3.2). It should also be emphasized that the present construction can be generalized to higher spin VBS states on an arbitrary lattice in any dimensions. Generally, the parent Hamiltonian is given by:
H=∑〈i,j〉∑(z-1)M<JzMVJPJ(i,j)
where VJ denotes a positive coefficient and PJ(i,j) is given by:
PJ(i,j)=∏J′≠J(S(i)+S(j))2-J′(J′+1)J(J+1)-J′(J′+1)=∏J′≠J2S(i)·S(j)+2S(S+1)-J′(J′+1)J(J+1)-J′(J′+1)
In QHE, the parent Hamiltonian for the Laughlin-Haldane wave function is called the Haldane’s pseudo-potential Hamiltonian [76]. As discussed in Section 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 the Laughlin-Haldane wave function takes the form similar to (139):
H=∑i<j∑m(N-2)<Jm(N-1)VJPJ(i,j)
where PJ denotes the projection operator to two-body angular momentum, J.
By replacing the SU(2) operators with UOSp(1|2) ones, we readily construct the parent Hamiltonian for the type-I SVBS states, which are invariant under UOSp(1|2) transformations generated by the SU(2) bosonic generators, Li:
Li=12Ψ†(r)σi000Ψ(1/r)
and the parameter-dependent fermionic operators, Kα:
Kα=12Ψ†(r)0τα-(iσ2τα)t0Ψ(1/r)
where Ψ(r) is defined by (84). Notice that Li and Kα satisfy the UOSp(1|2) algebra (45). With use of the UOSp(1|2) generators, it is straightforward to construct the parent Hamiltonian for the SVBS states with the 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:
J=S⊗S=0⊕1/2⊕1⊕3/2⊕2⊕⋯⊕2S
Note that the UOSp(1|2) decomposition rule is similar to that of the SU(2), except for the bond-superspin decreasing by 1/2. The SVBS state does not contain any UOSp(1|2) bond superspins larger than Jmax=(z-1)M and is the exact zero-energy ground state of the parent Hamiltonian:
Htype-I=∑〈ij〉∑J=Jmax+122SVJPJ(i,j)
where VJ are positive coefficients and PJ(i,j) are the projection operator onto the superspin-J representation of UOSp(1|2), written in terms of the Casimir operator as:
PJ(i,j)=∏J′≠J2S(KA(i)+KA(j))2-J′(J′+12)J(J+12)-J′(J′+12)=∏J′≠J2S2KA(i)KA(j)+2S(S+12)-J′(J′+12)J(J+12)-J′(J′+12)
which projects to the two-site subspace of the bond superspin, J. Here, KA(i)KA(j)≡Li(i)Li(j)+ϵαβKα(i)Kβ(j) and KA(i)KA(i)≡KA(j)KA(j)=S(S+12). Since the projection operators (146) are UOSp(1|2) invariant, the parent Hamiltonian is (145), as well. Following similar discussions about the uniqueness of the VBS state in Reference [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 Jmax=1). From (144), we obtain the parent Hamiltonian (145) for the type-I SVBS chain as:
Hchain-I=∑iV32P32(i,i+1)+V2P2(i,i+1)=∑i{135(5V2+63V32)KA(i)KA(i+1)+245(9V2-7V32)(KA(i)KA(i+1))2+1645(V2-5V32)(KA(i)KA(i+1))3+32315(V2-7V32)(KA(i)KA(i+1))4+V32}
Here, we add several comments. Since the Casimir operator, (KA(i)+KA(j))2, contains pair-creation terms of fermions, such as fi†fj†(aibj-biaj), the Hamiltonian (147) does not preserve the total fermion number, Nf=∑ifi†fi (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 Section 4.2).
The fermionic generators, Kα, 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:
H′type-I=∑〈ij〉∑J=Jmax+122SVJPJ†(i,j)PJ(i,j)
The definition (148) is a natural generalization of the original type-I parent Hamiltonian, since, if the projection operators were Hermitian, from the property PJ2=PJ, Equation (148) would be reduced to the original one (145).
The irreducible representation of the UOSp(2|2) group is specified by (j,b), which signify the indices of the largest bosonic subalgebra, SU(2)⊕U(1) (we denote the eigenvalues of the SU(2) Casimir operator, S2, as j(j+1) (j=0,1/2,1,⋯.) and arbitrary complex number eigenvalues of the U(1) operator, B, as b.) 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 the atypical representation with dimension 4j+1. The eigenvalues of the UOSp(2|2) quadratic Casimir operator (72) for (j,b) representation are generally given by:
C=j2-b2
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,j3〉, |b+1/2,j-12,j3〉, |b-1/2,j-12,j3〉, |b,j-1,j3〉, each of which carries the SU(2) spin index S as:
(i)S=j⋯(2j+1)-dim(ii)S=j-1/2⋯2j-dim(iii)S=j-1/2⋯2j-dim(iv)S=j-1⋯(2j-1)-dim
For instance, an eight-dimensional typical representation, (j,b)=(1,0), is constructed with the use of the components of the UOSp(2|2) Schwinger operator (107):
(i)|+〉=12ai†2|vac〉,|0〉=ai†bi†|vac〉,|-〉=12bi†2|vac〉(ii)|↑〉=ai†fi†|vac〉,|↓〉=bi†fi†|vac〉(iii)|↑′〉=ai†gi†|vac〉,|↓′〉=bi†gi†|vac〉(iv)|0′〉=gi†fi†|vac〉
They give rise to a N=2 SUSY multiplet. The UOSp(2|2) superspin operators can also be given by:
Si=Ψ(r)†liΨ(r),Kα=Ψ†(r)lαΨ(r),Dα=Ψ†(r)lα′Ψ(r),B=Ψ†(r)γΨ(r)
with li, lα, lα′ and γ (73). Obviously, Si and -iB are the generators of the subalgebra su(2)⊕u(1). The basis states (151) carry the SU(2)⊕U(1) indices as:
(i):|b=0,j=1〉,(ii)±i(iii):|b=±1/2,j=1/2〉,(iv):|b=0,j=0〉
For a two-site system, the UOSp(2|2) bond superspin operators are constructed as Stot=S(i)+S(j), Kαtot=Kα(i)+Kα(j), Dαtot=Dα(i)+Dα(j), Btot=B(i)+B(j), and the quadratic Casimir operator is expressed as:
Ci,j=Stot·Stot+ϵαβKαtotKβtot-ϵαβDαtotDβtot-BtotBtot=C(i)+C(j)+2{S(i)·S(j)+ϵαβKα(i)Kβ(j)-ϵαβDα(i)Dβ(j)-B(i)B(j)}=C(i)+C(j)+2L(i)·L(j)=2S2+2L(i)·L(j)
where C(i)=C(j)=S2(S=0,1/2,1,3/2,⋯) (the graded fully symmetric representation made of the UOSp(2|2) Schwinger operator carries the SU(2)⊕U(1) indices, (j,b)=(S,0)), and L(i)·L(j) is defined as:
L(i)·L(j)=S(i)·S(j)+ϵαβKα(i)Kβ(j)-ϵαβDα(i)Dβ(j)-B(i)B(j)
The tensor product of two identical typical representations is decomposed as:
(J,B)=(S,0)⊗(S,0)=⊕n=02S-1(2S-n,0)⊕n=02S-1(2S-1/2-n,1/2)⊕n=02S-1(2S-1/2-n,-1/2)⊕n=12S-1(2S-n,0)
For instance, (on the right-hand sides of (157) and (156)) (1/2,1/2)⊕(1/2,-1/2) is replaced by a not-completely reducible atypical representation consisting of a semi-direct sum of atypical representations, (0,0), (1/2,-1/2), (1/2,1/2), (0,0) (for details, see Section 2.5.3 in [173]).
(1,0)⊗(1,0)=(2,0)⊕(3/2,1/2)⊕(3/2,-1/2)⊕(1,0)⊕(1,0)⊕(1/2,1/2)⊕(1/2,-1/2)
Similar to the UOSp(1|2) decomposition (144), the UOSp(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:
B(J)2=18(1-(-1)2J)
Invoking the usual arguments of constructing the parent Hamiltonian, we derive the type-II parent Hamiltonian as:
Htype-II=∑(1-1/z)2S<J2S∑<ij>VJPJ(Ci,j)
where PJ stand for the projection operators of the bond superspin J:
PJ(Ci,j)=∏J′≠JCi,j-(J′2-B(J′)2)(J2-B(J)2)-(J′2-B(J′)2)
Here, B(J)2 and B(J′)2 are given by (158). For a L=1 type-II SVBS chain, the parent Hamiltonian is given by:
Hchain-II=∑i{V3/2P3/2(Ci,i+1)+V2P2(Ci,i+1)}
where the projection operators are:
P3/2(Ci,i+1)=∏J′=2,1,1/2,0Ci,i+1-(J′2-B(J′)2)2-(J′2-B(J′)2)=-18Ci,i+12(Ci,i+1-1)(Ci,i+1-4)P2(Ci,i+1)=∏J′=3/2,1,1/2,0Ci,i+1-(J′2-B(J′)2)4-(J′2-B(J′)2)=196Ci,i+12(Ci,i+1-1)(Ci,i+1-2)
With the use of Ci,i+1=2L(i)·L(i+1)+2, the type-II parent Hamiltonian (161) can be rewritten as:
Hchain-II=∑i{112(V2+36V3/2)(L(i)·L(i+1))+13(V2+3V3/2)(L(i)·L(i+1))2+112(5V2-36V3/2)(L(i)·L(i+1))3+16(V2-12V3/2)(L(i)·L(i+1))4+V3/2}
It is noticed that, unlike the type-I parent Hamiltonians (145), the type-II parent Hamiltonians (159) themselves are Hermitian, since the UOSp(2|2) Casimir itself (154) is given by a Hermitian operator.
5. 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.
5.1. 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 [21,24]:
|VBS〉αβ=(R2Φ1*)α∏i=1L-1(Φi†R2Φi+1*)(Φ*)Lβ|vac〉≡(R2Φ1*)αΦ1†·∏i=2L-1R2Φi*Φi†·(R2ΦL*ΦL*β)|vac〉=(A1A2⋯AL)αβ
where the “state-valued” matrix Ai is given by:
Ai=R2Φi*·Φi†|vac〉i=ai†bi†(bi†)2-(ai†)2-ai†bi†|vac〉i=|0〉i2|-1〉i-2|1〉i-|0〉i
It is clear, by the Schwinger-boson construction, that the row and the column of the matrix, Ai, 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:
Ai=∑m=-11A(m)|m〉i(representation-(i))
or
Ai=∑a=-x,y,zA′(a)|a〉i(representation-(ii))|x〉=-12|+1〉-|-1〉,|y〉=i2|+1〉+|-1〉,|z〉=|0〉
In the first representation, the c-number matrices, A(m), are given by:
A(1)=00-20,A(0)=100-1,A(-1)=0200
while, in the second, A′(a) is given by the Pauli matrices σa (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:
(A1A2⋯AL)αβ=∑{mj}A(m1)A(m2)⋯A(mL)αβ|m1〉1⊗|m2〉2⊗⋯⊗|mL〉L
The state (164) or (168) represents a collection of the D2 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 labeled 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†); 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 (Figure 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 Section 1, any gapped short-range states fall into this category. We refer the readers to recent readable reviews [32,33] for more details and the applications.
Diagrammatic representation of MPS and its dual.
We would like to comment on an interesting property of MPS (168). When SU(2) rotation acts on the state in (166b) as |a〉→Rba|b〉, the local MPS matrix, Ai, transforms, like:
Ai→SU(2)∑a,b=x,y,zRbaA′(a)|b〉i=∑a,b=x,y,zRbaσa|b〉i=∑b=x,y,zU†σbU|b〉i=U†AiU
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 Section 6.4.
We can generalize the strategy to construct the parent Hamiltonian for the VBS state in Section 4.3 to any MPS. The idea is to prepare a cluster Hamiltonian and tune the parameters, so that the Hamiltonian annihilates all the D2 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 [25]. By construction, the degree of degeneracy is equal to the number of matrix elements (D2). 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:
|MPS〉PBC=Tr⨂i=1LAi
and hence the ground state is not degenerate.
5.2. Supermatrix-Product State (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).
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):
|SVBS-I〉αβ=(R1|2Ψ1(r)*)α∏i=1L-1(Ψi(r)†R1|2Ψi+1(r)*)Ψ*(r)Lβ|vac〉≡(R1|2Ψ1(r)*)αΨ1(r)†·∏i=2L-1R1|2Ψi(r)*Ψi(r)†·(R1|2ΨL(r)*ΨL(r)†)β|vac〉=(A1A2⋯AL)αβ
where:
Ai=RIΨi(r)*·Ψi(r)†|vac〉i=ai†bi†(bi†)2rbi†fi†-(ai†)2-ai†bi†-rai†fi†-rfi†ai†-rfi†bi†0|vac〉i=|0〉i2|-1〉ir|↓〉i-2|1〉i-|0〉i-r|↑〉i-r|↑〉i-r|↓〉i0
From the expression (172), it is clear that the nine-fold degenerate ground states correspond to different possible choices of the edge states:
|SVBS-I〉open=⨂i=1LAi=|sL=↓;sR=↑〉|sL=↓;sR=↓〉|sL=↓;sR=∘〉|sL=↑;sR=↑〉|sL=↑;sR=↓〉|sL=↑;sR=∘〉|sL=∘;sR=↑〉|sL=∘;sR=↓〉|sL=∘;sR=∘〉
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 Ai (173), one sees that the matrix has a block structure:
B(2,2)F(2,1)F(1,2)B(1,1)
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 Ai as a supermatrix. Thus, the SVBS chain can be expressed in the form of a 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 exponentially decrease by 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:
|SVBS-I〉periodic=STr⨂i=1LAi
where the supertrace is defined as:
STr(M)≡M11+M22-M33
From these A-matrices, we obtain the following 9×9 T-matrices (transfer matrix):
Tα¯,α;β¯,β≡A*(α¯,β¯)A(α,β)=10002000r0-100000000000000-r0000-10000020001000r000000r0000000-r00000r000000r000r0000(α¯,α,β¯,β=1,2,3)
Here, A* is obtained from A by |·〉↦〈·| and complex conjugation.
Using the matrices, A(m), and the diagrammatic representation introduced in Section 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 in Figure 16 and the orthogonality of the local basis states, 〈m|n〉i=δmn, one can show that the overlap of the two (S)MPSs with edge states {α¯L,α¯R} and {αL,αR} can be written as (for the bosonic MPS, this is straightforward; for the SMPS, one has to treat the fermion sign carefully, but at the end of the day, we can check that the final result is the same) (see Figure 17a):
(α¯