Topological limit shape phase transitions: melting of Arctic Circles

Topological limit shape phase transitions: melting of Arctic Circles
Dimitri Gangardt, University of Birmingham
Dimitri Gangardt
Date and time: Mon, Feb 14, 2022 - 3:00pm
Category: Many-Body Physics Seminar
Abstract:

A limit shape phenomenon in statistical mechanics is the appearance of a most probable macroscopic state. An iconic example of this phenomenon is given by the Arctic Circle Theorem [1] of random tilings which, in the certain scaling limit, can be mapped to the imaginary time evolution of free fermions. A limit shape is usually characterized by a well-defined boundary separating frozen and liquid spatial regions. The earliest studies related to this phenomenon in the context of crystal shapes
are in works by Pokrovsky and Talapov [2].

In this talk, I will present a phase transition of limit shape, which can be visualized as merging two melted regions (Arctic circles). By mapping onto a free fermionic problem and calculating correspondent correlation functions we identify the transition as the third-order transition known in lattice QCD [3].  We make connections to algebraic geometry, stressing the topological nature of the transition and identifying universal features of the limiting shape.

[1] W. Jockusch, J. Propp and P. Shor, arXiv preprint math/9801068 (1998).
"Random domino tilings and the arctic circle theorem."

[2] V. L. Pokrovsky and A. L. Talapov, Phys. Rev. Lett. 42, 65 (1979).
"Ground State, Spectrum, and Phase Diagram of Two-Dimensional Incommensurate
Crystals."

[3] D. J. Gross and E. Witten, Phys. Rev. D, 21 (2): 446, 1980.  "Possible
third-order phase transition in the large-n lattice gauge theory.”;
S. R. Wadia, "N = ∞ phase transition in a class of exactly soluble model
lattice gauge theories", Phys. Lett. 93, 403 (1980)