Loops in classical and quantum statistical-mechanical models
e-mail: kawashima@issp.u-tokyo.ac.jp
Various statistical-mechanical systems can be represented in terms of loop graphs. A classical example is the low-temperature expansion of the Ising model in two dimensions, which is identical to the simplest (i.e., mono-color) loop gas model. Quantum many-body systems may also be represented by a loop gas. As is well-known, the D-dimensional XXZ quantum spin model, for example, can be mapped, through the path integral representation, to a (D+1)-dimensional classical loop gas model, which is the basis of qnautm Monte Carlo method. A few years ago, we found [1] that the Kitaev spin liquid on a honeycomb lattice is adiabatically connected to a quantum state represented by a 2D (not 3D) classical mono-color loop gas. This is consistent with the Ising CFT characterizing the 2D Dirac electrons. Recently, we found [2] another 2D quantum system that has a representation as a 2D classical loop gas model. Every loop in this classical system has two colors. Accordingly, this system exhibits the KT transition.
[1] Hyun-Yong Lee, Ryui Kaneko, Tsuyoshi Okubo, Naoki Kawashima: Phys. Rev. Lett. 123, 087203 (2019).[2] Hosho Katsura, Naoki Kawashima, Satoshi Morita, Akinori Tanaka and Hal Tasaki: Phys. Rev. Res. 3, 033190 (2021)