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Off-diagonal Bethe ansatz and exact solution of a topological spin ring

A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with M{\"o}bius like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring is discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

Integrable models play important roles in statistical physics, quantum field theory and condensed matter physics, because those models provide some benchmarks for understanding the corresponding universal classes. Since Yang and Baxter's pioneering works [1][2][3], the Yang-Baxter relation has become a cornerstone for constructing and solving the integrable models. Especially, the T − Q relation method [2,3] and the algebraic Bethe ansatz method [4][5][6] developed from the Yang-Baxter equation have become two very popular methods for dealing with the exact solutions of the known integrable models. Generally speaking, there are two classes of integrable models. One possesses U (1) symmetry and the other does not. Three well known examples without U (1) symmetry are the XY Z spin chain [5,7], the XXZ spin chain with antiperiodic boundary condition [8][9][10][11][12][13][14] and the ones with unparallel boundary fields [15][16][17][18][19]. It has been demonstrated that the algebraic Bethe ansatz and T − Q relation can successfully diagonalize the integrable models with U (1) symmetry. However, for those without U (1) symmetry, only some very special cases such as the XY Z spin chain with even site number [5,7] and the XXZ spin chain with constrained unparallel boundary fields [15][16][17] can be dealt with because of the existence of a proper "local vacuum state" in these special cases. The main obstacle applying the algebraic Bethe ansatz and Baxter's method to general integrable models without U (1) symmetry lies in the absence of such a "local vacuum". A promising method for approaching such kind of problems is Sklyanin's separation of variables method [20,21] which has been recently applied to some integrable models [11-14, 18, 19]. However, a systematic method is still absent to derive the usual Bethe ansatz equations (BAEs) which are crucial for studying the physical properties in the thermodynamic limit.
In this letter, we develop a general method for dealing with the integrable models without U (1) symmetry. The central point lies in how to construct a T − Q relation and the usual BAEs for those models based on the connection between two basic invariants of the monodromy matrix, i.e., its trace (transfer matrix) and its quantum determinant which do not depend on the basis choice and whether there is a reference state. As a concrete example, we study the spectrum of the XXZ spin ring with Möbius like topological boundary condition, as it is tightly related to the recent study on the topological states of matter. In fact, the topological boundary problem in many body systems has been rarely touched. With the inhomogeneous XXZ topological spin ring model, we elucidate how our method works to derive the exact spectrum and the BAEs by constructing and solving a recursive functional equations. Particular attention is focused on the elementary excitations of the homogeneous XX spin ring with antiperiodic boundary condition, as it is the simplest quantum realization of the Möbius stripe. Our exact solution shows that the elementary excitations of this simple model indeed exhibit a nontrivial topological nature.
We start from the following model Hamiltonian with the anti-periodic boundary conditions σ α N +1 = σ x 1 σ α 1 σ x 1 . N is the site number of the system and σ α j (α = x, y, z) is the Pauli matrix on the site j along the α direction. With such a topological boundary condition, the spin on the N -th site connects with that on the first site after rotating π-angle along the x-direction (a kink on the (N, 1) bond) and forms a torus in the spin space. With an unitary transformation U n HU −1 n , U n = n j=1 σ x j , the kink can be shifted to the (n, n + 1) bond without changing the spectrum of the Hamiltonian. Notice here the braiding is in the quantum space rather than in the real space and therefore the present model describes a quantum Möbius stripe. We define a Z 2 operator U N = N j=1 σ x j . It can be easily checked that U N 2 = 1 and [H, U N ] = 0. Therefore, the present model possesses a global Z 2 invariance, indicating the double degeneracy of the eigenstates.
The integrability of the present model is associated with the following Lax operator , and the monodromy matrix whereλ j = λ − θ j , λ is the spectral parameter and θ j are the site inhomogeneous constants; η is the crossing parameter as usual; the index 0 indicates the auxiliary space and j indicates the quantum space. Both the Lax operator and the monodromy matrix satisfy the Yang-Baxter relation with R 12 (λ) = L 12 (λ|θ j = 0). The transfer matrix of the system is defined as where tr 0 means tracing the auxiliary space. From Eq.
Obviously, the conjectured Λ(λ) satisfies Eq. (14) automatically. The BAEs determined by the regularity of Λ(λ) (which ensures Λ(λ) to be a trigonometrical polynomial of degree N − 1) read The BAEs for the homogeneous model are exactly the above equations by putting all θ j = 0. The eigenvalues of Hamiltonian (1) take the following form For odd N , we put M = (N + 1)/2 and where φ 1 and φ 2 take the same form as Eq. (19) with M = (N + 1)/2 and θ j = 0. In this case, the BAEs and the eigenvalue of the Hamiltonian are still given by Eq. (20) and Eq.(21), respectively. We note the BAEs about the roots {z j } can also be derived from the recursive equations Eq. (9). To show the physical effect of the topological boundary clearly, let us focus on the η = iπ/2 case, i.e., the topological XX spin ring or equivalently the topological free fermion ring (via a Jordan-Wigner transformation). In the homogeneous case, the roots {z j } satisfy the equation a 2 (z j ) = d 2 (z j ) (see below). This equation has 2(N − 1) solutions. However, we need only N − 1 roots to form a solution set. That means a selection rule is needed. From the n = N case of Eq.(9) we obtain that where µ 1 , · · · , µ N −1 = {µ l } are free parameters; z is one of the roots of the eigenvalue Λ(λ); From Eqs.(23) and (24), we have The singular point of Eq.(25) is µ j = z ± iπ/2. Since F N −1 is a polynomial for all the variables µ j , the residue of the right hand side of Eq.(25) must be zero at the singular point. Notice the fact that M j n (z, {µ j }) = 0 if µ j = z ± iπ/2 andM kl n (z, {µ j }) = 0 as long as µ k or µ l = z ± iπ/2. We readily have the conclusion that F N −1 (z, {µ l }) is proportional to F 2 (z, z ±iπ/2) when one of µ l 's is equal to z ± iπ/2, which must be zero according to the above analysis. This gives the constrain condition of the root z as a(z)d(z + iπ/2) = d(z)a(z + iπ/2). Therefore, the roots z n of Λ(λ) satisfy the following Bethe ansatz equation Equivalently, we have coth(z n ) = e iπn N ≡ e ikn , n = ±1, · · · , ±(N − 1). (27) The N −1 pair solutions {z j , z j + π 2 i} mod(iπ) are located on two lines with imaginary part ±iπ/4. The root sets are formed by choosing one and only one in each pair. This selection rule comes from that the poles of the righthand-side of Eq.(25) do not enter into the set of roots {z j } because the poles and the zeros satisfy the same equation (26). Therefore, there are 2 N −1 possible choices to form a solution of Λ(λ). With the Z 2 symmetry of the system, we demonstrate that the solutions are complete.
The ground state is formed by filling all roots along the −iπ/4 line (as shown in Fig.1(a)) and the ground state energy reads E g = −2 cot π 2N which is slightly different from that of the periodic boundary condition case. The elementary excitations of the system can be constructed by digging some holes in the lower solution line and putting the same number of particles in the upper solution line (as shown in Fig.1(b)). However, the positions of the holes and the particles are not arbitrary but obey the selection rules of z j = z k ± iπ/2. That means if there is a hole at −k, there must be a particle at k (as shown in Fig.2). The energy of a particle-hole excitation is thus ǫ(k) = 4 sin |k|. Such an excitation character is quite unlike to that in the usual Luttinger liquids, where both the forward scattering and backward scattering are allowed and there is no constrain for the particle-hole excitations besides the Pauli principle in the charge neutral sector. In the present topological boundary case, each particle with momentum k must lock a hole with momentum −k to form a virtual bound state, indicating the topological nature of the excitations. Whether the topological boundary modifies Haldane's Luttinger liquid relation [22] is an interesting issue to be studied.
In conclusion, we developed a general method for diagonalizing the integrable models without U (1) symmetry. As an example, we constructed the exact solution of the XXZ spin-ring with topological boundary condition. We remark that the present method could be used to other integrable models without U (1) symmetry. For those models, some off-diagonal elements of the monodromy matrix enter into the transfer matrix expression τ (λ). With the commutation relations derived from the corresponding Yang-Baxter equation, similar relation of Λ(θ j )Λ(θ j − η) ∼ ∆ q (θ j ) can be constructed based on