Exact Solutions and Degenerate Properties of Spin Chains with Reducible Hamiltonians

The Jordan--Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of $N$ spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the $N-1$ bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan--Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model ($\sigma_x\sigma_y-\sigma_x\sigma_y$) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.

Introduction-The Jordan-Wigner (JW) transformation establishes a connection between spin-1/2 operators and spinless fermion operators, 1 and it has become a powerful tool of solving the one-dimensional spin models and the two-dimensional Ising model. 2,3][6][7][8][9][10] Despite of successful applications of the JW transformation, a common problem in the JW transformation people always encounter with is that the periodic boundary condition introduces redundancy.
To remove the redundancy, much effort is spent in doing projections, and it is no longer convenient to solve the statistical mechanics.Or an approximate result has to be adopted.Therefore the boundary problem is a long-standing defect of the JW transformation, nevertheless little progress has been made.In this paper, we find a class of systems in which the JW transformation does not introduce redundancy.Those subtle systems may inspire the method of perfecting the JW transformation.In addition, special degenerate properties are discussed for those systems.We define the holistic degeneracy for such systems, and find that the holistic degeneracy must be even-fold.Later, we present that one-dimensional (1D) compass model is an appropriate case.Proposal -For completeness, we first introduce the redundancy problems in the JW transformation, then discuss the solutions of the reducible systems, and further those systems are classified by the new-defined holistic degeneracy.
Considering a 1D spin-1/2 system with N sites labeled by 1, 2, 3, • • • , N , its Hamiltonian is H s .We assume that H s does not change the parity of the number of spinup states or spin-down states, and H s includes merely nearest-neighbor interactions. 11Let the parity be P s = (−1) Nup where N up is the number of spin-up states, we have Eq. ( 1) indicates that the eigenstates of H s can be divided into two sets according to different eigenvalues of P s .In one set P s = 1, and in another set Basic vectors in H s 's Hilbert space M are described as where α is a spin-up or -down state.An eigenstate ϕ of H s is a linear superposition of the basic vectors, which can be expressed by where ρ is the corresponding coefficient.Let P s act on ϕ, we have Since P s ϕ = ±ϕ, we have Hence ν's with nonzero ρ's have the same parity with ϕ.For zero ρ i , the parity of ν i is not certain, yet does not make sense.Accordingly, we divide M into two parts: s and H e s , 2 N −1 eigenvalues will be obtained for either one.Now we apply the JW transformation to H s .The Pauli matrices in H s are transformed by σ ± , i.e., arXiv:1804.00147v1 [cond-mat.str-el]31 Mar 2018 where σ ± = σ x ± iσ y , and the subscripts l is the number of sites.The JW transformation is as follows A one-to-one mapping between the spin-up (-down) state and the occupation (non-occupation) state of a fermion has been built, and meanwhile the communication relation of spin operators and anticommunication relation of fermion operators are preserved.We use the fermion operators to substitute for σ ± in the spin Hamiltonian, and obtaine H f .The purpose of implementing the JW transformation is to take the advantage of diagonalizing the quadratic form of fermion Hamiltonian H f .A problem here used to encounter with is the boundary condition.H s is considered to have the periodic boundary condition, i.e., σ may not be valid, in other words, it is relevant with the parity of the number of occupation states.For instance, utilizing Eq. ( 7), we have σ , namely, the anti-periodic boundary condition (APBC); for odd n, a † N +1 = a † 1 , namely, the periodic boundary condition (PBC).
Being the same with H s , H f does not change the parity of occupancies, we are able to divide the Hilbert space of H f into two subspaces: e .The dimensions of each space are To obtain exact results, we need to treat the Hamiltonian with APBC (PBC) within the physical subspace M f e (M f o ).Commonly, the dimensions of the Hamiltonian with a fixed boundary condition are two times as large as the subspace, that is to say, D( , and consequently redundancy is inevitable.In order to remove the redundancy, further projections are necessary.
It turns to the thing we focus on in this paper.We consider a kind of systems in which the fermion Hamiltonians are reducible when the JW transformation is implemented.
The reducible Hamiltonian means that the fermion Hamiltonian can be reduced to lower dimensions by appropriate methods, and such reductions always implies some symmetries in those systems.For those reducible systems, we denote the reduced Hamiltonian by H r , and H r is represented by fermion operators.We further assume that H r has Q quasiparticle states when it is diagonalized, which indicates that D(H r ) = 2 Q .Q is definitely less than N .Let H r be limited to a fixed boundary condition APBC or PBC which explicitly does not change the dimensions of H r , then we have , we have two possible situations about d: , hence H r can be exactly diagonalized in the physical subspaces, and redundancy is forbidden.Therefore H r with two boundary conditions just exactly gives all solutions of H s , and N and Q satisfy the following relation Otherwise d > 1, we find D(H A r ) < D(M f e ) and D(H P r ) < D(M f o ).Since H r is equal to H f except for the distinction of dimensions, a reasonable deduction is that those systems own some kinds of symmetries, such that a subspace can be divided into several smaller spaces that are all equal for H r .Thereby, we have r , where d is referred as the multiplicity in the group theory.It deserves to be mentioned that the degeneracy of the ground state of H s is already determined simply through the dimension of H r .Now that the subspace is divided, regarding each smaller space as an element, we are able to find some quantum numbers like q 1 , q 2 , q 3 , etc, and construct a complete set {H r , q s} to describe each smaller space. 12Clearly, by the deduction the latter case d > 1 indicates that the degeneracy of all the eigenvalues given by H r is exactly d-fold (the degeneracy with H r is not concerned here).Since d describes the degeneracy of a group of energy levels, it should be more convenient and appropriate to call such degeneracy as the holistic degeneracy.Similarly, we have Eq. ( 9) can be considered as a special case of Eq. (10).From Eq. ( 10), we easily find the relation between d, N , and i.e., the holistic degeneracy must be folds like 2-fold, 4-fold, 8-fold, etc. Furthre, it can be known that the total degeneracy for each energy level must be even-fold, and at least d-fold (here the degeneracy within H r is included).
Example-In this part, we show an excellent example of our proposal.Generally speaking, in the light of the degeneracy of those systems, it is in spin systems with special symmetries that our proposal is most possibly realized.At present, the compass model is known to own various symmetries, 13 hence we naturally search a case in the compass model.Indeed, the 1D compass model 14 , which is also referred as the reduced Kitaev model 15 , is found to be a case of our proposal.The 1D compass model has the following Hamiltonian in which J x and J y are interacting parameters for odd and even bonds, respectively.We treat Eq. ( 12) with the JW transformation, and then substitute Majorana fermions for normal fermions.The Hamiltonian in the Majorana representation is and the diagonalization of Eq. ( 13) has already been given in Appendix A.
Considering PBC in Eq. ( 12 11), we are able to find that d = 2 N/2−1 .For the special way of describing the degeneracy d is called the holistic degeneracy of H mf in this paper, and at the same time it represents the least degeneracy of the given system.
The rough statement above is not convincing enough.In the following, utilizing the method in Ref. 16, we elaborate all details to evidently show the correctness of our discussion above.Furthermore relevant symmetries in the 1D compass model will be analyzed.
As is shown in Appendix B, the Hamiltonian of the 1D compass model is decoupled in the momentum space, For W (k), H 0 , and H π , they have degenerate eigenvalues corresponding to different parity in their independent spaces.Eigenvalues of H are obtained through the combination of eigenvalues of independent terms, and different combinations possibly lead to the even or odd parity of the whole eigenstate.Considering both the degeneracy in each independent terms and equal combinations, by rigorous algebra, we are able to prove that the holistic degeneracy of the 1D compass model is 2 N/2−1 .In addition, it is straightforward to find that the ground state of the 1D compass model is 2 N/2−1 -fold degenerate, which is identical with the result obtained by the reflection positivity technique. 17o analyze the symmetric characters in the real space, we introduce some details in Ref. 14. Rotating Eq. ( 12) about the x-axis through π/2, i.e., J y → J z , and σ y → σ z , then the Hamiltonian in Ref. 14 is obtained, Using z-axis as the quantization axis, obviously the quantization axis is currently parallel with one interacting direction.Note that although the Hamiltonians of Eqs. ( 12) and ( 15) are equal, the method in Ref. 14 is out of our formalism. 18Transforming this Hamiltonian to the dual space by dividing the N -site system into N/2 odd pairs, i.e., sites 2l − 1 and 2l constitute a unit.There are four states for each pair: }, s l corresponds to the lth pair, and s l = 1 for parallel states, while s l = 0 for antiparallel states.Now the Hilbert space can be divided equally into 2 N/2 subspaces by giving the set with different values.Further, transforming Eq. ( 15) to a quantum Ising model, it can be solved in the dual subspace, although redundancy exists.A point here is that the Hamiltonian in each subspace has the same solution when l s l has identical parity.Hence undoubtedly we obtain that the holistic degeneracy is 2 N/2−1 .Now we consider symmetries in the dual space.First, for a certain l s l , the set owns a permutation symmetry.For instance, when l s l = 1, the Hamiltonians are same when s = 1 is placed in any position.Second, Hamiltonians have no difference when l s l has the same parity, thus l s l is conserved modulo 2, which can be considered as a kind of symmetry here.In a word, both the permutation and the modulo-2 symmetries in the dual space result in the holistic degeneracy of the 1D compass model.Besides, we stress that both two symmetries are inherent, thus it is irrelevant with the outside form of the Hamiltonian.
Except for the 1D compass model, it is easy to find another example of our proposal, which is as follows Since this Hamiltonian can be solved by the similar method of solving the 1D compass model, we do not elaborate the discussion on this model.Conclusion-To sum up, we have theoretically proposed that the JW transformation will not introduce the common redundancy in such spin chains whose Hamiltonians in the fermion representation can be reduced to lower dimensions.The conditions include that the Hamiltonian merely involves the nearest neighbor interactions, and that the quadratic form does not change the parity of the number of fermions.In those systems, the holistic degeneracy is defined, and it is found to be definitely even-fold.Those systems are further classified according to the folds of the holistic degeneracy, and possible high degeneracy exists in those systems.In addition, we take the 1D compass model as the example to demonstrate degenerate properties of those systems.It is significant that by our work, one is able to determine the complete energy spectra with the reduced fermion Hamiltonian, and no extra effort is needed to remove the redundancy.At last but not least, our finding perhaps sheds light on better methods on solving spin models.For instance, one notion based on this work is that, the redundancy problems can be solved completely for 1D situations if we are able to find a general operator to reduce the dimensions of the fermion Hamiltonian.Acknowledgement-S.F.thanks H. Q. Lin and J. Lee for useful discussions.S.F. acknowledges the support from H. Q. Lin.
Appendix A: Diagonalization of Eq. ( 13) The Hamiltonian for the 1D compass model is where J x is the interacting strength for the odd bonds which include only x-axis spin interactions, and J y is for another half bonds which merely include y-axis spin interactions.A toy model for this Hamiltonian is depicted in Fig. 1(a).We use the famous JW transformation to transform the spin Hamiltonian.The JW transformation is as follows where σ ± = σ x ± iσ y .Now we have Taking the advantage of the Majorana fermion operators, the Hamiltonian has a more concise form where c 2l−1 = i(a † 2l−1 − a 2l−1 ) and c 2l = a † 2l + a 2l are Majorana fermion operators.The corresponding illustration is given in Fig. 1(b).It is known that a normal fermion can be described by a pair of Majorana fermions, and accordingly each Majorana fermion has the √ 2 degrees of freedom.Interestingly, the Hamiltonian here does not include another half Majorana fermions d's which pair with c's, which will result in high degeneracy of energy levels.Details are left for discussions behind.
For a N -site system, we consider the periodic boundary condition σ N +1 = σ 1 in the spin Hamiltonian.Under this condition, we have a † N +1 = −a † 1 and a N +1 = −a 1 in Eq. (A3), c N +1 = −c 1 in Eq. (A4) for states with even normal fermions; and a † N +1 = a † 1 and a N +1 = a 1 in Eq. (A3), c N +1 = c 1 in Eq. (A4) for states with odd normal fermions.The former condition is anti-periodic boundary condition (APBC), and the latter one is periodic boundary condition (PBC) for the fermion Hamiltonian.Besides, the character that the Hamiltonian Eq. (A3) changes the number of normal fermions in pairs guarantees no interaction between even-and odd-occupation basic vectors, therefore, basic vectors with even normal fermions construct a subspace with N/2 dimensions, and so do the other basic vectors.Thus, the Hamiltonian with APBC (PBC) should be solved in the subspace with even (odd) normal fermions.Normally the dimension of the subspace does not match that of the corresponding Hamiltonian, thus redundancy is introduced.For small systems, the redundancy can be dropped off by a project operator; for large systems, the approximate result is always adopted in either APBC or PBC.
To diagonalize the Hamiltonian, we mark the odd sites as A, and the even sites B. Thus N sites are divided into N/2 cells.Eq. (A4) becomes and each part can be solved independently.For each W (k), its Hilbert space has sixteen dimensions, and the Hilbert space can be divided into two subspaces with the same dimensions.Firstly, we solve W (k) in the subspace with even parity.The subspace with even parity has the following basic vectors

FIG. 1 :
FIG. 1: (a) A toy model for the 1D compass model, and (b) the Majorana representation of this model.Jx and Jy are the interacting strength between two nearest sites, and Jx and Jy are for the x-axis and y-axis interactions, respectively.c's and d's are paired Majorana states, remarkably c's are exclusively involved in the Majorana representation.

N/ 2 l=1(− i 2 N/ 2 l=1(
J x c l,A c l,B − J y c l,B c l+1,A ) = J x c l,A c l,B − J x c l,B c l,A − J y c l,B c l+1,A + J y c l+1,A c l,B ) (A5)where l is the number of cells.Eq. (A5) can be described byΨ † H ij Ψ, (A6)whereΨ † = • • • c jA c jB • • • , Ψ = in which M o and M e consist of odd-and even-parity basic vectors, respectively.By the assumptions, we have H s = H o ), since each Majorana fermion takes merely √ 2 degrees of freedom, H mf with a certain boundary condition have Q = N/2 quasiparticle states when it is diagonalized.Therefore the dimensions of the Hamiltonian have been reduced from 2 N (H s ) to 2 N/2 (H mf ).By the definition d = H mf with APBC and PBC gives eigenvalues in two physical subspaces, respectively, and they constitute all the eigenvalues of H s with no redundancy.Otherwise d > 1, H mf with two boundaries gives a part of eigenvalues of H s , and no redundancy is introduced, instead the solutions are not complete.The approach of obtaining the complete solutions is to duplicate the solutions of H A mf and H P mf for d times.By Eq. (