Exact solution of a anisotropic $J_1-J_2$ spin chain with antiperiodic boundary condition

The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbour, next-nearest-neighbour and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the $U(1)$-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous $T-Q$ relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant $J=-1$ although the Bethe Ansatz equations are the inhomogeneous ones.


Introduction
The Heisenberg model is a typical system to describe the quantum magnetism, where the spin exchanging interaction is the nearest-neighbor (NN) one. A nontrivial generalization of the Heisenberg model is the J 1 − J 2 model, where the NN and the next-nearest-neighbor (NNN) interactions are involved [1,2,3,4,5,6,7]. Many interesting phenomena have been found in the J 1 − J 2 model. For example, at the point of J 2 /J 1 = 0.241, the J 1 − J 2 model has a topological phase transition [8,9]. At the Majumdar-Ghosh point, J 2 /J 1 = 0.5, the model Hamiltonian degenerates into a projector operator and only the ground state can be obtained exactly [10].
Although the J 1 − J 2 model can not be solved exactly, people find that the J 1 − J 2 model with some additional terms is integrable. For example, Popkov and Zvyagin proposed the integrable two-chain and multichain quantum spin model [11,12,13,14]. Frahm and Rödenbeck constructed an integrable model of two coupled Heisenberg chains by taking the derivative of the logarithm of product of two transfer matrices with different spectral parameters [15,16,17]. Using the samilar idea, Ikhlef, Jacobsen and Saleur constructed the Z 2 staggered vertex model [18,19]. These models are equivalent to the J 1 − J 2 model with some spin chirality terms, where the extra scalar chirality terms are introduced to ensure the integrability. Tavares and Ribeiro studied the thermodynamic properties of this kind of models by using the quantum transfer matrix method [20,21]. Recently, the models with chirality terms have attracted lot of interest in the context of quantum spin liquids [22,23].
The quantization condition used in the above references are the periodic boundary condition. In this paper, we study the integrable anisotropic J 1 − J 2 spin chain with antiperiodic boundary condition. We note that in this case, the U(1) symmetry of the system is broken and the traditional Bethe ansatz method does not work due to the lack of reference state.
The antiperiodic (twisted) boundary condition is tightly related to the recent study on the topological states of matter. The model Hamiltonian considered in this paper is where σ j ≡ (σ x j , σ y j , σ z j ) are the Pauli matrices at site j, a and η are the generic constants describing the coupling strengths, and the boundary condition is the antiperiodic one In the Hamiltonian (1.1), the first two terms describe an anisotropic NN interaction, the third term is an isotropic NNN interaction and the last one corresponds to an anisotropic chiral three-spin interaction. We note that the hermitian of the Hamiltonian (1.1) requires that a must be real if η is imaginary (gapped regime), and a must be imaginary if η is real (gapless regime). We use the off-diagonal Bethe Ansatz (ODBA) [24,25] to solve the model.
The paper is organized as follows. In the next section, we prove that the model (1.1) is integrable. In section 3, we derive the exact energy spectrum and the Bethe Ansatz equations. Ground state and twisted boundary energy with J = 1 are given in section 4 and the corresponding results with J = −1 are discussed in section 5. Section 6 is attributed to the concluding remarks.

Integrability
Throughout, V denotes a two-dimensional linear space and let {|m , m = 0, 1} be an orthogonal basis of it. We shall adopt the standard notations: for any matrix A ∈ End(V ), A j is an embedding operator in the tensor space V ⊗ V ⊗ · · · , which acts as A on the j-th space and as identity on the other factor spaces. For B ∈ End(V ⊗ V ), B ij is an embedding operator of B in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
where u is the spectral parameter. The R-matrix (2.1) satisfies the following relations Initial condition : R 0,j (0) = P 0,j , Crossing relation : where φ(u) = − sinh(u + η) sinh(u − η)/ sinh 2 η, t 0 (or t j ) denotes the transposition in the space V 0 (or V j ) and P 0,j is the permutation operator possessing the property 3) The R-matrix (2.1) satisfies the Yang-Baxter equation (YBE) We define the monodromy matrices as where V 0 is the auxiliary space, V 1 ⊗ V 2 ⊗ · · · ⊗ V 2N is the physical or quantum space, 2N is the number of sites and a is the inhomogeneous parameter. From the YBE (2.4) and the one can prove that the monodromy matrix T (u) satisfies the Yang-Baxter relation The transfer matrices are the trace of monodromy matrices in the auxiliary space t(u) = tr 0 T 0 (u),t(u) = tr 0T0 (u). (2.8) Using the crossing symmetry in Eq.(2.2), we obtain the relations between transfer matrices From the Yang-Baxter relation (2.7) and Eq.(2.9), one can prove that the transfer matrices t(u) [ort(u)] with different spectral parameters commute with each other. Meanwhile, the transfer matrices t(u) andt(u) also commute with each other Therefore, both t(u) andt(u) serve as the generating functions of all the conserved quantities of the system, and the transfer matrices t(u) andt(u) can be diagonalized simultaneously.
Using the initial condition of the R-matrix (2.2), we obtain Taking the derivative of transfer matrix t(u) with respect to u and consider the values at The integrable Hamiltonian can be constructed from the transfer matrices t(u) andt(u) as (2.14) Substituting the relations (2.11)-(2.13) into above expression (2.14), we obtain The derivative of the R-matrix reads The commutative relation between the permutation operators is

Exact solution
We first introduce the inhomogeneous monodromy matrix where the {θ j , j = 1, · · · , N} are the inhomogeneous parameters. The matrix form of monodromy matrix T g 0 (u) in the auxiliary space is where A(u), B(u), C(u) and D(u) are the operators acting in the quantum space. We denote the all spins aligning up state as the vacuum state |0 , The elements of the monodromy matrix T g 0 (u) acting on the vacuum state gives The transfer matrix t(u) defined as Suppose |Φ is the eigenstate of the transfer matrix t g (u) and the corresponding eigenvalue According to the results given in [25], we known that Λ g (u) satisfies following functional Meanwhile, Λ g (u) is a polynomial of e u with the degree 2N − 1 and satisfies the periodicity The constraints (3.7) and (3.8) show that the eigenvalue Λ g (u) can be parameterized as the where Q(u) is a trigonometric polynomial of the type andc(u) is given byc The singularity of Λ g (u) requires that the Bethe roots {λ j } in Eq.(3.9) should satisfy the Bethe ansatz equations (BAEs) Put θ 2j−1 = −a and θ 2j = a for j = 1, · · · , N, we obtain the eigenvalue Λ(u) of the transfer matrix t(u) 14) and the BAEs read  Table 1. We see that the eigenvalues obtained by solving the BAEs are exactly the same as those obtained by the exact diagonalization of the Hamiltonian (1.1). Meanwhile, the expression (3.16) gives the complete spectrum of the system.

Ground state and twisted boundary energy
In this section, we consider the case of J = 1. We first analyze the contribution of the third term in the inhomogeneous T − Q relation (3.13). For simplicity, we constrain that η is real and a is imaginary. We first introduce following homogeneous T − Q relation where Q h (u) is a trigonometric polynomial of the type The period of Bethe roots {λ j } is 2π, thus we fix the real part of Bethe roots in the interval [−π, π). For convenience, we put λ j = iu j /2 − η/2 and a = ib. The singularity of Eq.(4.1) gives where the function a n (x) is given by a n (x) = 1 2π sinh(nη) cosh(nη) − cos x , (4.5) and the Bethe roots {u j } satisfy the BAEs (4.3). Taking the logarithm of Eq.(4.3), we have From Eq.(4.7), we see that there is a hole in the real axis which can be put at the boundary x 0 = −π to minimize the energy.
Denote the true ground state energy of Hamiltonian (1.1) as E g . We use the physical to characterize the contribution of the inhomogeneous term in the T − Q relation (3.13) at the ground state. By using the density matrix renormalization group (DMRG) method, we obtain the ground state energy E g of the Hamiltonian (1.1). By solving the BAEs (4.6) and substituting the values of Bethe roots into the Eq.(4.4), we obtain the energy E g h . Substituting E g and E g h into Eq.(4.8), we obtain the values of E g inh and the results are shown as the dots in Fig.1. The fitting of the data gives that the contribution of the inhomogeneous term tends to zero when the system-size tends to infinity. Then, we conclude that the Eq. (4.4) is a suitable approximation of the energy of the system (1.1) in the thermodynamic limit.
In the following, we use E g h to quantity the ground state energy of the model (1.1).
Obviously, Z(u j ) = I j /2N corresponds to the Eq.(4.6). In the thermodynamic limit, N → ∞, M → ∞ and N/M finite, the counting function Z(u) becomes a continue function of u.
Taking the derivative of Eq.(4.9) with respect to u, we obtain where ρ(x) and ρ h (x) are the densities of particles and holes, respectively. The Fourier transformation of a n (x) is At the ground state, there is a hole at the point of x 0 = −π. Thus the density of holes reads 12) and the corresponding Fourier transformation is Taking the Fourier transformation of Eq.(4.10), we obtaiñ (4.14) The ground state energy of model (1.1) is The twisted boundary energy is defined as where E g p is the ground state energy for the Hamiltonian (1.1) with periodic boundary condition [26]

Bethe roots of inhomogeneous BAEs
In this section, we consider the case J = −1. In this case, the ground state spin configuration and the solution of Bethe roots in BAEs (3.15) are different from those with J = 1. Again, η is set as real and a is set as imaginary. For convenience, put λ j = iu j − η/2. The BAEs By careful analysis and numerical check, we find the Bethe roots in Eq.(5.1) form the 2Nstrings at the ground state where x 0 is real and o(2N) stands for a small correction which is related with 2N and i is the imaginary unit. The numerical results of Bethe roots at the ground state with 2N = 8 is shown in Fig.2. From which, the string structure of Bethe roots can been seen very clearly.
Substituting the string hypothesis (5.2) into the energy expression (3.16) and neglecting the small correction, we obtain the energy for the 2N-string states  When system size 2N is very large, the energy for the 2N-string (5.3) turns to  (2N ) . Thus, the energy difference ∆E will be zero in the thermodynamic limit.

Conclusion
In this paper, we study an integrable anisotropic J 1 − J 2 spin chain with antiperiodic boundary condition. By means of the off-diagonal Bethe Ansatz, we obtain the exact solution of the system. We show that the contribution of inhomogeneous term in the T − Q relation can be neglected when the system-size tends to infinity. Based on it, we discuss the ground state energy and the twisted boundary energy. We find the string structure of Bethe roots at