Thermodynamics of the Spin-1/2 Heisenberg–Ising Chain at High Temperatures: a Rigorous Approach

Abstract

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg-Ising (or XXZ) spin-1/2 chain at finite temperature T . Within the quantum inverse scattering method the physically pertinent observables at finite T , such as the per-site free energy or the correlation length, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. We rigorously prove all these conjectures for temperatures large enough. As a by product of our analysis, we obtain the large-T asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths. This result was never obtained previously, not even on heuristic grounds. 1

Report on scipost_202004_00045v1

“…[15]) and by considering the XX chain (this is recommended as an exercise, for some information see [45]). In addition we would like to recommend the work [12], where the case of high but finite temperature was treated with full mathematical rigour.…”

Report on scipost_202004_00045v1

“…(ii) For the special case of the XXZ chain a rigorous proof was recently provided for high enough finite temperatures [12]. Since the proof uses basically only the locality of the interaction of the model, it is expected to be generalizable at least to all fundamental integrable models.…”

“…. , M , into the Bethe Ansatz equations (5.122), performing the limit T → +∞ and using (6.156), we obtain a set of equations that determine the x j , In [12] the high-T limit was worked out on more rigorous grounds, starting from the nonlinear integral equations for the excited states.…”

Correlation functions of one-dimensional strongly interacting two-component gases

“…In practice, the complexity of the Bethe ansatz wave functions means that such calculations are extremely difficult and new mathematical methods need to be devised. Initial attempts focused on the case of systems equivalent with free fermions [5][6][7][8][15][16][17][18][19][20][21][22][23][24][25][26][27] but very recently results were obtained for the Lieb-Liniger [28][29][30][31][32][33][34][35][36][37][38] and the XXZ spin-chain [39][40][41][42][43][44][45][46][47][48][49][50][51] models away from the free fermion point. Analytical derivations of the low-energy asymptotics of correlation functions are important because they can be compared with the predictions of TLL but they can also provide insight and identify systems not described by TLL theory.…”