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Thermodynamic limit of the off-diagonal Bethe ansatz solvable models

Yuanyuan Li, Jun Cao, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Jan 14, 2014• 1 citation
OpenAlex

Abstract

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.

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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.
In two previous papers [26], [27], the exact solutions of the spin-12 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 η=ηm, 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 η case with O(N−2) corrections in the thermodynamic limit N→∞. 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.
Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
Thermodynamic limit of the spin-12 \frac{1}{2} 21​ XYZ spin chain with the antiperiodic boundary condition
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The XXX spin- \frac{1}{2} 1 2 Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of U(1) U ( 1 ) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For U(1) U ( 1 ) symmetric spin- \frac{1}{2} 1 2 chains such NLIEs involve two functions a(x) a ( x ) and \bar{a}(x) a ‾ ( x ) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size N N . In the case considered here the U(1) U ( 1 ) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function c(x) c ( x ) , which captures the inhomogeneous contributions to the T T - Q Q relation in the ODBA. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \log(1+a(x)) log ( 1 + a ( x ) ) and \log(1+\bar a(x)) log ( 1 + a ‾ ( x ) )
In this study, we explore the precise physical quantities in the thermodynamic limit of the one-dimensional Hubbard model with nonparallel boundary magnetic fields based on the off-diagonal Bethe ansatz solution. A particular emphasis is placed on the half-filling condition to investigate the distinct patterns of Bethe roots in the reduced Bethe ansatz equations for different boundary parameters. The ground state of the system can be divided into five regions according to the distribution of Bethe roots. By analyzing these patterns, we calculate the densities of states, ground-state energy density, and surface energy. The results reveal the existence of stable-boundary bound states, which are dependent on specific constraints regarding the boundary magnetic fields.