Phase structure of XX0 spin chain and nonintersecting Brownian motion

Abstract

We study finite size and temperature XX0 Heisenberg spin chain in weak and strong coupling regimes. By using an elegant connection of the model to integrable combinatorics and probability, we explore and interpret a possible phase structure of the model in asymptotic limit: the limit of large inverse temperature and size. First, the partition function and free energy of the model are derived by using techniques and results from random matrix models and nonintersecting Brownian motion. We show that, in the asymptotic limit, partition function of the model, written in terms of matrix integral, is governed by the Tracy-Widom distribution. Second, the exact analytic results for the free energy, which is obtained by the asymptotic analysis of the Tracy-Widom distribution, indicate a completely new and sophisticated phase structure of the model. This phase structure consists of second-and third-order phase transitions. Finally, to shed light on our new results, we provide a possible new interpretation of the phase structure in terms of dynamical behaviour of magnons in the spin chain. We demonstrate distinct features of the phases with schematic spin configurations which have definite features in each region of the phase diagram.

Exact solvability and asymptotic aspects of generalized XX0 spin chains

“…In contrast, because of a correspondence between the XX0 model and Non-Intersecting Brownian Motion (NIBM) [3], this model faces a rather rare third-order phase transition [1], [9]. In our previous study, we explored the phase structure of XX0 spin chain and despite its simplicity we found a sophisticated phase structure for this model [1]. In this study, we generalize the model, in the simplest way, to a long-range interacting model and in the light of the simplicity of the model, we conjecture universality features in the phase structure of the generalized model arising from the conjectural universality of Tracy-Widom distribution (TW) [10], [11], [12].…”

“…Among the variety of classical, quantum and topological phase transitions that occur in XYZ models, the XX0 model does not experience those types of transitions. In contrast, because of a correspondence between the XX0 model and Non-Intersecting Brownian Motion (NIBM) [3], this model faces a rather rare third-order phase transition [1], [9]. In our previous study, we explored the phase structure of XX0 spin chain and despite its simplicity we found a sophisticated phase structure for this model [1].…”

“…The XX0 spin chain as one of the simplest integrable and exactly solvable one-dimensional models, in the class of half-spin XYZ Heisenberg model on d-dimensional lattice, has been studied extensively [1], [2], [3], [4], [5], [6], [7], [8]. Among the variety of classical, quantum and topological phase transitions that occur in XYZ models, the XX0 model does not experience those types of transitions.…”

“…In order to study the generalized XX0 model, we extend the methods from random matrix theory and integrable probability [3], [17], which are introduced and adopted to XX0 model in our previous study [1]. Similar to the original model, the observables i.e.…”

“…In the physics language, the asymptotic analysis is the study of the model in the thermodynamic limit, in which the phase structure of the model emerges and thus we expect that the TW determines all the information relevant to the phase structure. Applying the introduced method in [1], we compute the free energy of the finite model in the asymptotic limit from the TW and thus extract the phase structure of the model. In the case of original XX0 model, we apply this new spirit and method from TW to reproduce the known results of Gross-Witten (GW) phase transition [21], namely the free energy near the phase transition point.…”