Twisted boundary energy and low energy excitation of the XXZ spin torus at the ferromagnetic region

We investigate the thermodynamic limit of the one-dimensional ferromagnetic XXZ model with twisted (or antiperiodic ) boundary condition. It is shown that the distribution of the Bethe roots of the inhomogeneous Bethe Ansatz equations (BAEs) for the ground state as well as for the low-lying excited states satisfy the string hypothesis, although the inhomogeneous BAEs are not in the standard product form which has made the study of the corresponding thermodynamic limit nontrivial. We also obtain the twisted boundary energy induced by the non-trivial twisted boundary conditions in the thermodynamic limit.


Introduction
The XXZ spin-1 2 torus model is a XXZ spin chain with Möbius like topological boundary condition (or twisted boundary condition) [1,2,3,4], and it is tightly related to the recent study on the topological edge states of matter. The integrability of this model is associated with the Z 2 -symmetry of the six-vertex R-matrix [5,6,7,8]. Due to the topological boundary condition, the U(1)-symmetry is broken, making it much different from the periodic XXZ model. For an example, the lack of an obvious reference state prevents us applying the conventional Bethe Ansatz methods [9,10,11,12,13,8,14,15] to the model.
The off-diagonal Bethe Ansatz (ODBA) method is a newly developed analytic theory to approach exact solutions of quantum integrable models, especially those with nontrivial integrable boundaries [16,17,18,19]. The exact solutions of the systems is characterized by the inhomogeneous T − Q relations where the Bethe roots should satisfy the inhomogeneous Bethe Ansatz equations (BAEs). Then, a natural question is what is the distribution of the Bethe roots of the inhomogeneous BAEs in the complex plain. It is very important because that it is the start point to study the thermodynamic properties of the system. However, due to the existence of the inhomogeneous term, it is hard to use the usual thermodynamic Bethe Ansatz method [15]. Some interesting approaches are proposed [20,21]. For examples, because the inhomogeneous BAEs can degenerate into the conventional cases when the model parameters taking some special values, then one can use the results at the degenerate points to infinitely approximate the actual values [20]. Another method is that one can study the contribution of the inhomogeneous term, then the thermodynamic limit can be obtained by using the finite-size scaling behavior [21,22]. In this paper, we directly investigate the Bethe roots from the inhomogeneous BAEs without any approximation. We use the XXZ spin-1 2 torus as the example. From the numerical analysis, we obtain the structure of the Bethe roots at the ground state as well as at the low-lying excited states in the thermodynamic limit. Based on them, the physical quantities such as the twisted boundary energy and the gap are studied.
The paper is organized as follows. Section 2 serves as an introduction of the model and its exact solution. In section 3, we discuss the distribution of Bethe roots of the ground state.
Base on them, we calculate the twisted boundary energy. In section 4, the nearly degenerate states are studied. In section 5, we discuss the elementary excitation and the energy gap. In section 6, the limiting behavior is considered, which is used to check our results. Concluding remarks are given in section 7.
2 Spin-1 2 XXZ torus The spin-1 2 XXZ torus is characterized by the Hamiltonian where N is the number of sites, η is the crossing parameter (or anisotropic parameter) and the boundary condition is the antiperiodic one, namely, It is remarked that the antiperiodic boundary condition (2.2) breaks the bulk U(1)-symmetry (c.f. the spin-1 2 XXZ chain with the periodic boundary condition: σ α N +1 = σ α 1 ). This leads to that the exact solution of the model challenged many years until the work [17].
The integrability of the model is associated with the well-known six-vertex R-matrix where u is the spectral parameter. From the R-matrix, we can define the monodromy matrix The R-matrix and the monodromy matrix satisfy the RTT relation The transfer matrix of the system is defined as From the RTT relation, one can prove that The Hamiltonian (2.1) can be exactly solved by using the ODBA method [16,17]. The eigen-energy is then expressed in terms of the Bethe roots where the Bethe roots {u j } should satisfy the inhomogeneous BAEs We note that the period of Bethe roots is π, thus we fix the real part of Bethe roots in the interval [− π 2 , π 2 ).

Bethe roots at the ground state
Here, we consider the case that η is real and solve the inhomogeneous BAEs (2.10) numerically. The values of Bethe roots at the ground states for finite system-size are listed in Table   1 and 2. From the data, we find that the real part of Bethe roots is nearly −π/2, and the difference between the imaginary part of two Bethe roots is nearly equal and the value is iη.
Thus the Bethe roots form a single string. In order to see this point clearly, we draw them in Fig.1. From it we see that the string is located at the boundary of the period (− π 2 ). We should mention that for the odd N case, there exists another N-string which locates at the imaginary axis. These two kinds of strings give the same ground state energy thus the corresponding Bethe states are degenerate.
Based on above facts, we conclude that all the Bethe roots form following string solution at the ground state  where o(N) stands for the small correction which is related with N, and i is the imaginary unit. Substituting the string hypothesis (3.1) into the energy expression (2.9) and neglecting the small correction, we obtain the ground state energy In order to check the validity of string hypothesis (3.1), we calculate the ground state energy of the system by exactly diagonalizing the Hamiltonian (2.1) up to N = 19 and compare the results with those obtained by the equation (3.2). The data is listed in Table 3. We see that the analytical and numerical results agree with each other very well. For the larger system-size, we also check the validity of (3.1) and (3.2) by the density matrix renormalization group (DMRG) method. The result is shown in Fig.2. Then, we can conclude that the Bethe roots at the ground state form the string solution in the thermodynamic limit and Eq. (3.2) gives the energy of the system. Now, we calculate the twisted boundary energy. It is well-known that the ground state energy of the XXZ spin chain with periodic boundary condition is [15] E p 0 = −N cosh η.
is the one calculated by the exact diagonalization. Despite some data with small N, the data can be fitted as ∆E ∝ e −αN , α ≈ η. We define the twisted boundary energy as is the ground state energy calculated from DMRG method. We can easily find the relative correction tends to zero rapidly when N tends to infinity.

Elementary excitation
Now, we consider the elementary excitation of the XXZ spin torus model. As shown in Fig.4, we find that the distribution of Bethe roots at the lowest excited states can be described by a (N − 1)-string plus an additional real root. Meanwhile, the string and the real root are nearly located at the interval boundary − π 2 3 . After some more precision calculation, we conclude that the Bethe roots at the lowest excited state take the form of 3 we regard π 2 as the same point with − π 2 due to the periodicity.
where o 1 (N) and o 2 (N) stand for the small deviations and i is the imaginary unit. The energy corresponding to this kind of excitation is We also check the validity of Eq. (5.2) by the exact diagonalization and the result is shown in Fig.5. Again, we see that with the increasing N, the energy difference between the analytic result (5.2) and the actual values tends to zero rapidly. Thus the string solution (5.1) is correct in the thermodynamic limit. is the lowest excitation energy calculated by the exact diagonalization. From the data we see that the energy difference tends to zero rapidly with the increasing system-size.
The energy gap of the XXZ spin torus is defined as In the thermodynamic limit, the energy gap reads which is the same as that of the XXZ spin chain with periodic boundary condition.

Limiting behavior
In order to check above results again, now, we consider some limit case of η → +∞. In

Conclusions
In this paper, we have studied the thermodynamic limit of the one-dimensional ferromagnetic XXZ model with the antiperiodic boundary condition which is described by the Hamiltonian (2.1) and (2.2). It is shown that even constrained by the three terms Bethe Ansatz equations, the Bethe roots of the ground state appear to be a string form (3.1). This fact enables us to calculate the twisted boundary energy of the model given by (3.6). By using the similar method, we further investigate the elementary excitation and obtain the energy gap of the model which is the same as that of the periodic boundary condition case.