Thermodynamic limit and twisted boundary energy of the XXZ spin chain with antiperiodic boundary condition

We investigate the thermodynamic limit of the inhomogeneous T-Q relation of the antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It is shown that the contribution of the inhomogeneous term at the ground state can be neglected when the system-size N tends to infinity, which enables us to reduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones. Then the quantum numbers at the ground states are obtained, by which the system with arbitrary size can be studied. We also calculate the twisted boundary energy of the system.


Introduction
The XXZ spin chain with the antiperiodic boundary condition (or the twist boundary condition) is a very interesting quantum system [1,2,3,4]. By using the Jordan-Wigner transformation, the model can describe a p-wave Josephson junction embedded in a spinless Luttinger liquid [5,6,7]. Although there exists a twisted bound at the boundary which breaks the usual U (1)-symetry of the bulk system (or the closed chain case) [8], it can be proved that the system is still integrable. By using the off-diagonal Bethe ansatz (ODBA) method [9,10,11], the exact solution of the model was obtained [9], which is described by an inhomogeneous T − Q relation (c.f. the ordinary homogeneous T − Q one [12,13]). Such an inhomogeneous T − Q relation has played a universal role to describe the eigenvalue of the transfer matrix for quantum integrable systems [8]. However, due to the fact that Bethe roots should satisfy the inhomogeneous Bethe ansatz equations (BAEs), it is hard to study the thermodynamic properties [14] of the corresponding systems [15,16,17].
Based on an intelligent trick, the thermodynamic limit of the spin-1 2 XXZ chain with the generic offdiagonal boundary terms in the gapless region (i.e., the isotropic parameter η in (2.1) below being an imaginary number) was succeeded in obtaining [18]. The most important observation in the paper is that the contribution of the inhomogeneous term at the ground state, in the gapless region, can be neglected when the system-size N tends to infinity. Such a fact has been confirmed recently by the studies of other integrable models [19,20,21,22] whose eigenvalue of the transfer matrix is given in terms of the inhomogeneous T − Q relation.
In this paper, we propose a method to study the thermodynamic limit of the XXZ spin chain with the twist boundary condition at the antiferromagnetic region (i.e., η being a real number). We first study the

The model and its ODBA solution
The spin- 1 2 XXZ quantum chain is described by the Hamiltonian where the anti-periodic boundary condition reads σ α N +1 = σ x 1 σ α 1 σ x 1 (α = x, y, z), and σ α j is the Pauli matrix. For such a topological boundary condition, the spin on the N th site couples with that on the first site after rotating by an angle π along the x-direction (a kink on the (N, 1) bond) and the system forms a torus in the spin space. This kink could be smoothly shifted to the (j, j + 1) bond without changing the energy spectrum. That is to say that the Hamiltonian is unchanged with the transformation Due to the fact [H, U x ] = 0, the model possesses a global Z 2 invariance. Note that the braiding occurs in the quantum space rather than in the real space. Therefore, the model describes a quantum Möbius strip.
The integrability of the model (2.1) is associated with the well-known six-vertex R-matrix where u is the spectral parameter and η is the crossing parameter (or isotropic parameter). The R-matrix satisfies the Yang-Baxter equation and possesses the properties: Initial condition: Unitarity: Crossing relation: where P 1,2 is the permutation opetator, and t i denotes the transposition in the ith space. Here and below we adopt the standard notations: for any matrix A ∈ End(C 2 ), A i is an embedding operator in the tensor space C 2 ⊗ C 2 ⊗ · · · , which acts as A on the i-th space and as identity on the other factor spaces; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones. The associated monodromy matrix is given as Because of the Z 2 -symmetry (2.8), the following relation holds which directly gives rise to the fact that [t(u), t(v)] = 0, (2.11) where the transfer matrix t(u) is defined as The first order derivative of the logarithm of the transfer matrix gives the Hamiltonian (2.1) This ensures the integrability of the model. By means of the off-diagonal Bethe ansatz method, the eigenvalues Λ(u) of the transfer matrix t(u) is given by the inhomogeneous T − Q relation [8] Λ(u) = e u a(u) , (2.14) where Q(u) is a trigonometric polynomial of the type and The N parameters {λ j } in Eq. (2.15) should satisfy the associated BAEs The eigenvalue of the Hamiltonian (2.1) is then expressed in terms of the associated Bethe roots as The numerical simulation implies that the inhomogeneous BAEs (2.20) indeed give the correct and complete spectrum of the model [8].

Finite-size effects
In this paper, we consider the massive region with a real η. In order to study the contribution of the inhomogeneous term [the last term in Eq. (2.14)] to the ground state energy, we first introduce a homogeneous T − Q relation as It should be remarked that the number of Bethe roots in Eq.
The singular analyzing of the T − Q relation (3.1) gives rise to homogeneous BAEs .

(3.4)
Taking the logarithm of Eq. (3.4), we have (3.6) Here the notation [ ] represents the Gauss Mark, and the quantum number {I j } are certain integers (half odd integers) for N − M even (N − M odd). Corresponding to Eq. (2.19), we define Now, we define the contribution of the inhomogeneous term to the ground state energy as where E g is the ground state energy of the Hamiltonian (2.1) obtained from Eq. (2. 19) and E g hom is the minimal energy calculated by Eqs.(3.7) and (3.5).
Because we consider the massive region, the thermodynamic limit of the system with even N and that with odd N are different. We first study the contribution of inhomogeneous term E g inh with even N . In this case, M = N 2 , and all the Bethe roots in Eq. (3.5) are real and are determined completely by the quantum number Substituting the values of Bethe roots into Eq.(3.7), we obtain the value of E g hom . From Eq. (3.8), the contribution of the inhomogeneous term can be calculated and the results are shown in Fig. 1. From the fitting, we find that E g inh and N satisfy the power law Due to the fact that b 1 < 0, the value of E g inh tends to zero when the system-size N tends to infinity, which means that the contribution of the inhomogeneous term at the ground state can be neglected in the thermodynamic limit.
The contributions of the inhomogeneous term are shown in Fig. 2. From the fitting, we find that E g inh and N satisfy the exponential law Again, due to the fact that b 2 < 0, the value of E g inh tends to zero when N → ∞. Therefore, the contribution of the inhomogeneous term at the ground state can be neglected in the thermodynamic limit.
Through above finite-size scaling analysis, we conclude that the contribution of the inhomogeneous term at the ground state energy can be neglected when N → ∞. The similar results have also been obtained [22]. Therefore, the reduced BAEs (3.4) and the Eq. (3.7) can be use to calculate the ground state energy of the system (2.1) in the thermodynamic limit.
From Figs. 1 and 2, we also find that E g inh > 0 for the even N case and E g inh < 0 for the odd N case. Which means that E hom is larger than the actual value for the even N case while E hom is smaller than the actual value for the odd N case.

The thermodynamic limit
Now, we consider the thermodynamic limit of the system. For convenience, we define the counting function In the thermodynamic limit, N → ∞, M → ∞ and N/M takes the finite value. Taking the derivative of Eq. (4.1) with respect to x, we obtain and a m (x) = 1 2π where Q is chosen as π/η, ρ(x) and ρ h (x) are the densities of particles and holes, respectively. For the arbitrary periodic function f (x), x ∈ [−Q, Q], we introduce the Fourier transformatioñ Taking the Fourier transformation of Eq. (4.2), we obtaiñ whereã m (k) = e −mη|k| . (4.7) Then we haveρ (4.8) In the thermodynamic limit, the eigenvalue (3.7) can be expressed by the density of particles as For the even N case, we have M = N 2 at the ground state. Thus the following equation must hold (4.10) From Eqs. (4.8) and (4.10), we find that at the ground state, there exists one hole at x 0 ∈ [− π η , π η ]. The density of holes is given by With the Fourier transformation, we haveρ Thus the solution of (4.2) can be derived as At the ground state, the position of hole should be put at x 0 = π η to minimize the energy. Thus the ground state energy in the thermodynamic limit can be written as (4.20) with e 0 defined as (4.15). From the above calculation, we find that the ground state energy of the XXZ spin torus with even N and that with odd N are different. This is consistent with the fact that we consider the antiferromagnetic coupling and the massive region of the model (2.1), i.e., ∆ = cosh η ≥ 1 with real η. The values of e 0 and e h (π/η) have the same order. In the thermodynamic limit, the most contributions come from e 0 N and the e h (π/η) can be neglected. Thus the thermodynamic quantities calculated by the density of ground state energy e 0 do no depend on the even or odd of N . However, in this paper, we focus on the effects induced by the boundary degree of freedom, thus the contribution of e h (π/η) can not be neglected. If η → 0, then e h (π/η) → 0.

The thermodynamic limit of the periodic XXZ spin chain
In order to study the effects induced by the twisted boundary, now we should study the thermodynamic limit of the XXZ spin chain with periodic boundary condition. The model Hamiltonian reads with the constraint σ α N +1 = σ α 1 . We consider the same case that η is real, thus the eigenvalues of the Hamiltonian (5.1) is where the M Bethe roots {x j } are determined by the BAEs [14] sin N η .

(5.3)
Taking the logarithm of Eq. (5.3), we have where {I j } are certain integers (half odd integers) for N − M odd (N − M even). For convenience, we define the counting function Obviously, Z p (x j ) = Ij N corresponds to the Eq. (5.4) and it will turn to be a continuous function in the thermodynamic limit. When N → ∞ and M → ∞, the distribution of Bethe roots are continuous, i.e., Z p (x j ) = Z p (x). Taking the derivative of Eq. (5.5) with respect to x, we obtain where ρ(x) and ρ h (x) are the densities of the particles and holes, respectively. Taking the Fourier transformation of Eq. (5.6), we obtainρ Thus the density of particles can be expressed as (5.8) In the thermodynamic limit, the energy (5.2) of the periodic XXZ spin chain is For the even N , all the Bethe roots are real at the ground state and fill the region (− π η , π η ]. Meanwhile, the number of Bethe roots M = N 2 . Thus the following equation must hold which means the magnetization at the ground state is 0 [14]. From Eqs. (5.8) and (5.10), we find that such a configuration is described by ρ h (x) = 0 and the density of particles is . where e 0 is the density of ground state energy of the system defined by Eq. (4.15). For the odd N , the ground state of the system (5.1) is described by N −1 2 real Bethe roots in the region (− π η , π η ] and one hole at x 0 ∈ [− π η , π η ]. Thus the following equation must hold In this case, the density of holes is given by Eq.(4.11). Then from Eq. (5.8), we obtain the density of particles as (5.14) With the help of Eqs.(5.9) and (5.14), we have where e h (x 0 ) is the energy carried by one hole defined by Eq. (4.16). At the ground state, x 0 = π η to minimize the energy. Thus the ground state energy in the thermodynamic limit can be expressed by Again, we find that the ground state energy of the periodic XXZ spin chain with even N and that with odd N are different. In the thermodynamic limit, comparing with e 0 N , the e h (π/η) is a small quantity and can be neglected. The thermodynamic behavior of the system with even N and those with odd N obtained by the density of ground state energy e 0 are the same.
Comparing the relations (5.12) and (4.17), (5.16) and (4.20), we find that the parity of N of the XXZ spin torus and the parity of N of the periodic XXZ spin chain are reversed. That is to say, the ground state energy of the periodic XXZ spin chain with even N equals to that of the antiperiodic XXZ spin chain with odd N . While the ground state energy of the periodic XXZ spin chain with odd N equals to that of the antiperiodic XXZ spin chain with even N . This is because of the existence of the twisted bond. 6 The twisted boundary energy 15   The twisted boundary energy is a physical quantity to measure the effects induced by twisted boundary at the ground state, which is defined as Therefore, the twisted boundary energy E even b with even N equals to the minus of twisted boundary energy The twisted boundary energies with η = 2 and η = 3 are derived as Now, we check the above results by the DMRG method. The twisted boundary energies for different system-size N obtained by DMRG are shown in Fig. 3. For the even N case, the data in Fig. 3(a)  which are also highly consistent with the analytical results (6.5). Now, we consider the degenerate case. When η = 0, the XXZ spin torus degenerates into the isotropic XXX spin chain with the anti-periodic boundary conditions. From Eq. The ground state energy of the periodic XXX spin chain is E g pXXX = (1 − 4 ln 2)N. (6.12) Therefore, the twisted boundary energy of the XXX spin torus is zero.

Conclusions
In this paper, we have studied the thermodynamic limits of the spin-1 2 XXZ chain both with the antiperiodic and the periodic boundary conditions. We find that due to the twisted bond, the ground state energy of the antiperiodic XXZ spin chain with even N equals to that of the periodic XXZ spin chain with odd N . While the ground state energy of the antiperiodic XXZ spin chain with odd N equals to that of the periodic XXZ spin chain with even N . We also find that the contribution of the inhomogeneous term in the T − Q relation of the antiperiodic XXZ spin chain at the ground state can be neglected when the system-size N tends to the infinity. By using the reduced BAEs, we study the twisted boundary energy and show that the twisted boundary energy E even b of the system with even N differs from the one E odd b with odd N by a minus sign. We check these results by the DMRG, which leads to that the analytical results and the numerical ones agree with each other very well.