Berezinskii–Kosterlitz–Thouless transitions in the six-state clock model

Abstract

Classical 2D clock model is known to have a critical phase with Berezinskii-Kosterlitz-Thouless(BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed the method called level spectroscopy, which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D 6-clock model which can be mapped to the quantum self-dual 1D 6-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view.

Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy

“…In the original papers [6][7][8][9][10] on Level Spectroscopy, the degeneracy of V s 1 and W ±1 was discussed in terms of the correspondence between the sine-Gordon model and the SU(2) Thirring model [49,50]. Here, we present the derivation of Level Spectroscopy in more intuitive manner.…”

“…On the other hand, many 1D quantum systems can be also described by the same effective theory and thus also exhibit the BKT transition. Interestingly, a powerful numerical finite-size scaling method called "Level Spectrscopy" was developed specifically for those 1D quantum systems by Okamoto and Nomura [6][7][8][9][10]. Based on the Conformal Field Theory (CFT) results on the finitesize energy spectrum [11,12], they found that the BKT transition can be identified with a level crossing between a certain pair of the energy levels, cancelling the logarithmic corrections.…”

“…This is nothing but the Level Spectroscopy method proposed and developed in Refs. [6][7][8][9][10][28][29][30]. Thanks to the emergent SU(2) symmetry, the transition point determined by the Level Spectroscopy is exact in all orders of the marginal coupling g, as long as the irrelevant perturbation H is negligible.…”

Quantum clock models with infinite-range interactions

“…the explicit breaking of charge conjugation symmetry [2,14,15,16], which is not present in the Z 2 case. Very little is known on the phase diagrams and the phase transitions of clock models with q > 3: for example, for q ≥ 5 the self-dual clock models exhibit phase transitions of the Kosterlitz-Thouless universality class [17,3,7]. In general, characterizing the phase transitions of clock models has required a considerable theoretical effort and the application of advanced numerical techniques [4,5,6,18,19].…”

Finite-$m$ scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model

“…We also address the scaling analysis for the entanglement spectra determined by the cornertransfer-matrix (CTM) in the intermediate phase, assuming the boundary CFT with the c = 1 Gaussian universality. We then show that the Tomonaga-Luttinger (TL) parameter extracted from the entanglement spectra is [16,24]. The organization of the rest of this paper is as follows.…”