Thermodynamic limit and surface energy of the XXZ spin chain with arbitrary boundary fields

In two previous papers [26, 27], the exact solutions of the spin-1/2 chains with arbitrary boundary fields were constructed via the off-diagonal Bethe ansatz (ODBA). Here we introduce a method to approach the thermodynamic limit of those models. The key point is that at a sequence of degenerate points of the crossing parameter \eta=\eta_m, the off-diagonal Bethe ansatz equations (BAEs) can be reduced to the conventional ones. This allows us to extrapolate the formulae derived from the reduced BAEs to arbitrary \eta case with O(N^{-2}) corrections in the thermodynamic limit N\to\infty. As an example, the surface energy of the XXZ spin chain model with arbitrary boundary magnetic fields is derived exactly. This approach can be generalized to all the ODBA solvable models.


I. INTRODUCTION
The integrable models have played very important roles in statistical physics [1], quantum field theory [2] and low-dimensional condensed matter physics [3,4]. In the recent years, new applications have been found on cold atom systems and AdS/CFT correspondence. For examples, the Lieb-liniger model [5,6], Yang model [7] and the one-dimensional Hubbard model [8] have provided important benchmarks for the one-dimensional cold atom systems and even fitted experimental data with incredibly high accuracy [9]. On the other hand, the anomalous dimensions of single-trace operators of N = 4 super-symmetric Yang-Mills (SYM) field theory can be given by the eigenvalues of certain closed integrable spin chains [10,11] while the anomalous dimensions of the determinant-like operators of N = 4 SYM [12,13] can be mapped to the eigenvalue problem of certain open integrable spin chains with boundary fields [11,14,15]. By AdS/CFT correspondence the boundaries correspond to open strings attached to maximal giant gravitons [13,16]. Sometimes those boundaries may even break the U (1) symmetry.
Indeed, among the family of quantum integrable models, there exists a large class of models which do not possess U (1) symmetry and make the conventional Bethe ansatz methods such as coordinate Bethe ansatz [17,18], algebraic Bethe ansatz [19,20] and T −Q relation [21,22] inapplicable because of the lack of a proper reference state. Some famous examples are the XY Z spin chain with odd number of sites [23], the anisotropic spin torus [24,25] and the quantum spin chains with non-diagonal boundary fields [26][27][28][29]. Those models have been realized possessing important applications in non-equilibrium statistical physics (e.g., stochastic processes [30][31][32]), in con- * Corresponding author: yupeng@iphy.ac.cn densed matter physics (e.g., a Josephson junction embedded in a Luttinger liquid [33], spin-orbit coupling systems, one-dimensional cold atoms coupled with a BEC reservoir etc.) and in high energy physics (e.g., open strings and coupled D-Branes). Solving those models had been a longstanding problem for several decades. Very recently, a systematic method for solving the integrable models without U (1) symmetry, i.e., the so-called off-diagonal Bethe ansatz method (ODBA) was proposed [25][26][27] and several long-standing models were solved exactly [25][26][27][34][35][36][37]. However, the Bethe ansatz equations (BAEs) of those models are quite unusual and are still hard to be used in the thermodynamic limit because of the presence of the off-diagonal terms, which makes the distributions of the Bethe roots quite opaque.
In this paper, we propose that the thermodynamic limit of the ODBA solvable models for arbitrary crossing parameter η can be derived from those at a sequence of degenerate points η = η m up to the order O(N −2 ). At these special points, the ODBA equations are reduced to the usual BAEs which allow us to use the usual tools to derive the thermodynamic quantities. As η m+1 − η m = 2iπ/N , those degenerate points become dense in the thermodynamic limit N → ∞. In the following text, we take the XXZ spin chain model with arbitrary boundary fields as an example to elucidate how the method works.
The paper is organized as follows: In the next section, the Hamiltonian and the associated ODBA equations are introduced. Sec.III is attributed to the calculation of the surface energy at the degenerate points η = η m . The analysis about arbitrary η case is given in Sec.IV. Concluding remarks and discussions are given in Sec.V.

II. THE MODEL AND ITS ODBA SOLUTIONS
Let us consider a typical ODBA solvable model, i.e., the XXZ spin chain with arbitrary boundary fields. The

Hamiltonian reads
where σ α j (α = x, y, z) are the Pauli matrices as usual and h ± = (h x ± , h y ± , h z ± ) are the boundary magnetic fields. For convenience, we adopt the notations in Ref. [27] to parameterize the boundary fields as The eigenvalues of the Hamiltonian thus read where the Bethe roots µ j are determined by the ODBA equationsc with n a non-negative even (odd) integer for even (odd) N [38] and M = N + n. Interestingly, when the boundary parameters and the crossing parameter η satisfy the following constraint condition [27,39] there does exist a solution to (4)- (5) such that the parameterc = 0 and hence the Bethe roots are classified into two types of pairs with M 1 the number of the first pairs and m an arbitrary integer.
Let us focus on the gapless region, i.e., imaginary η and θ ± case. Not losing generality, we put α ± imaginary and β ± real to ensure the boundary fields being real. Let us examine the solutions at the degenerate points η = η m (corresponding to the case ofc = 0) and β ± = ±β, In this case, M 1 = N and the reduced BAEs give a complete set of solutions as verified numerically [40]. For convenience, let us take With these parameters, the reduced BAEs for η = η m become where j = 1, . . . , N . The corresponding eigenenergy is given by III. THE SURFACE ENERGY FOR η = ηm Let us consider the ground state energy at the degenerate crossing parameter points given by (7). Since real λ j contributes negative energy, the Bethe roots should fill the real axis as long as possible. However, the maximum number of Bethe roots accommodated by the real axis is only N/2, some of the roots must be repelled to the complex plane and form a string [41]. Suppose there is a k string in the ground state configuration with where λ r is the position of the string on the real axis and δ is a positive number to account for the small deviation. Substituting (10) into (8) and omitting the exponentially small corrections we obtain We consider the a ± ∈ ( π 2 , π) case. Taking the logarithm of (11) we have where I j is an integer and For convenience, let us put λ l = −λ −l and define the counting function Z(λ) as Obviously, Z(λ j ) = I j /(2N ) coincides with Eq.(12). In the thermodynamic limit N → ∞, the density of the real roots ρ(λ) is with a m (λ) = 1 2π where the δ(λ) term accounts for the hole at λ = 0 which is a solution of the BAEs but can never be occupied in any case. With the Fourier transformationŝ whereρ 0 (ω) =â with δ m ≡ mθ 2π − ⌊ mθ 2π ⌋ denoting the fraction part of mθ 2π . For ρ(λ) is the density of the real roots and M 1 = N , the following equation must hold which gives the length of the string k, Obviously, k has the order of N/2. In the ground state, λ r → ∞ to minimize the energy. The ground state energy in the thermodynamic limit can be easily derived as and with e 0 the ground state energy density of the periodic chain and e b the surface energy, where cos(βω)b(ω)dω. (28) Some remarks are in order: (1)The extra string in the ground state configuration contributes nothing to the energy in the thermodynamic limit. However, for a finite N , the string may induce exponentially small corrections. We note that for the present model, there is another type of strings centered at i[θ/π] [41]. Different choices of the bulk string in the ground state configuration induce the same effect as λ r → ∞.
(2)Above we considered only the parameter region a ± ∈ (π/2, π). For the boundary parameters out of this region, stable boundary bound states exist in the ground state [42,43]. However, the energy is indeed a smooth function about the boundary parameters as demonstrated in the diagonal boundary field case [44,45]. (3)An interesting fact is that the contributions of a + , a − , β to the energy are completely separated and the surface energy does not depend on θ ± at all, which indicate that the two boundary fields behave independently in the thermodynamic limit. Similar phenomenon often occurs in the dilute impurity systems. We note the surface energy does depend on the relative directions of the boundary fields to the z-axis because of the anisotropy of the bulk. (4)In the above calculations, we put the integral limits to infinity which is reasonable to the surface energy. To account for the finite size corrections of order 1/N (Casimir effect or central charge term), one should keep a finite cutoff for the integrals. Calculations can also be performed by the standard finite size correction and Wiener-Hopf methods [42,43,46,47]. The correlations between the two boundaries exist in this order [43]. (5)The thermodynamic equations at the degenerate points η = η m can also be derived by following the standard method [41]. (6)When β = 0, the boundary magnetic fields lie in the x − y plane. Taking the limit η → 0 of Eq. (27) we obtain the surface energy of the XXX spin chain with arbitrary boundary fields, which obviously does not depend on the angles θ ± . The θ ±dependence of the ground state energy only occurs in the order of 1/N as verified by the numerical simulations [48]. Now let us turn to arbitrary β ± case. In this case, the degenerate points of η takes complex values and the above derivations are invalid. However, we can deduce the surface energy with the following argument. In principle, for N → ∞ the surface energy takes the form because the two boundaries decouple completely as long as the bulk is not long-range ordered. Here the second and the third terms are the contributions of the boundary fields. For arbitrary real β ± , supposē When β ± = ±β, from Eqs. (27) - (29) we havē which indicates thatĪ(α ± , β ± , θ ± ) does not depend on α ± and θ ± . In addition, for α − = iπ/2, the boundary field is an even function of β − . SinceĪ(α − , β − , θ − ) is independent of α − , θ − , it must be an even function of β − . The same conclusion holds for β + . Therefore we conclude that ǫ b = ǫ 0 b + I 1 (a + ) + I 1 (a − ) + I 2 (β + ) + I 2 (β − ). (32) The above formula is valid for arbitrary boundary fields and η in the thermodynamic limit N → ∞ since η m become dense.

IV. PHYSICAL QUANTITIES FOR LARGE N AND GENERIC η
With the reduced BAEs at the degenerate η points, most of the physical quantities as functions of η m can be derived up to the order of 1/N with the conventional methods, i.e., Let us treat f n (η) (n = 0, 1, 2) as known functions. For a generic iη m+1 ≤ iη ≤ iη m , we suppose that the corresponding quantities aref n (η) which are initially unknown functions. We suppose further both f n (η) and f n (η) are smooth functions about η. Obviously, andf 0 (η) = f 0 (η) because f 0 is boundary-field independent and is the same calculated from the corresponding periodic system. Let us make the following Taylor expansions around η m and η m+1 (n = 1, 2) we readily havē which means that the unknown functionf n (η) can be replaced by the known function f n (η) up to the order of O(N −2 ).

V. CONCLUDING REMARKS
In conclusion, a systematic method is proposed for approaching the thermodynamic limit of the ODBA solv-able models with the open XXZ spin chain as an example. The central idea of this method lies in that at a sequence of degenerate crossing parameter points, the ODBA equations can be reduced to the conventional BAEs, which allows us to derive the thermodynamic quantities with the well developed methods. We remark that there are no degenerate points for the isotropic Heisenberg spin chain model [26] and the XXZ spin torus [25]. However, the thermodynamic quantities can be observed from their anisotropic correspondences. For the Heisenberg chain, we may take the limit η → 0 of the XXZ chain and for the XXZ torus, we may take a proper limit of the XY Z torus. In fact, for most of the rational integrable models, their trigonometric and elliptic counterparts exist. The latter ones normally possess degenerate points and thus the present method works.