While it is a common knowledge that any quantum states in d spatial dimensions can be represened as a d+1 dimensional classical statistical-mechanical model, some of important quantum states, such as the AKLT state, can be represented as a classical model in the same number of dimensions. It is also notable that many quantum states at the focus of attention, such as the Haldane state and the Kitaev spin liquid, are in close vicinity to such classically-representable quantum states; they are connected smoothly to their respective classically-representable states. A few years ago, we found [1] that the Kitaev spin liquid on a honeycomb lattice is smoothly 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. More 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)