Exact solvability and asymptotic aspects of generalized XX0 spin chains

Abstract

Building on our earlier work [1], we introduce a generalized XX0 model. For a number of some specific examples, including a long-range interacting model, referred to as the Selberg model, we study correlation functions and phase structures. By using a matrix integral representation of the generalized XX0 model, as well as combinatorial and probabilistic methods, (from non-intersecting Brownian motion), closed-form formulas for the partition function and correlation functions, of the finite and infinite size generalized model and its specific examples, are obtained in terms of the Selberg integrals and Fredhom determinants. Applying asymptotic analysis of Fredholm determinants and Tracy-Widom distribution techniques, the phase structure of the model in different examples are determined. We find that tails of the Tracy-Widom distribution govern a finite/infinite third-order phase transition in short-and long-range interacting models and reproduce the Gross-Witten type phase transition of the original XX0 model. Based on our results, we conjecture universal features of the phase structure of the model. Finally, a real space renormalization group is proposed to explain the observed universality and to connect the generalized XX0 model to non-intersecting Brownian motion in a generalized potential.

How to Draw a Correlation Function

“…Their behavior was intensively investigated for the system in the thermodynamic limit. Connection between the XX0 chain and the low-energy QCD, as well as a possibility of third order phase transition [22] in the spin chain, are discussed in [35,36,37,38]. The asymptotics of the partition functions, as well as the phase diagrams, for XX0 chain and related models are studied in [36,37,38].…”

Universal edge scaling in random partitions

“…It is now known that the Tracy-Widom distribution is widely found in various contexts, including, the longest increasing subsequence of random permutations [8], the fluctuation in the (totally) asymmetric simple exclusion process ((T)ASEP) [9], the large N phase transition of matrix model and gauge theory (Gross-Witten-Wadia/Douglas-Kazakov phase transition [10,11,12]. See also [13,14]), integrable systems such as spin chains [15,16], and the growing turbulent interfaces [17,18,19].…”