Berezinskii–Kosterlitz–Thouless Transition Through the Eyes of Duality

Abstract

A new "bond-algebraic" approach to duality transformations provides a very powerful technique to analyze elementary excitations in the classical two-dimensional XY and p-clock models. By combining duality and Peierls arguments we establish the non-Abelian symmetries, phase structure and transitions of these models, unveil the nature of their topological excitations, and explicitly show the continuous U(1) symmetry that emerges when p ≥ 5. The latter is associated with the appearance of discrete vortices and Berezinskii-Kosterlitz-Thouless-type transitions. IntroductionIn this chapter we investigate, via dualities, classical two-dimensional (D = 2) systems, such as the XY and clock models, 1,2 displaying BerezinskiiKosterlitz-Thouless (BKT)-type transitions. 3-6 BKT transitions, notably characterized by essential singularities in the free energy, emerge in many physical situations including screening in Coulomb gases, 7 surface roughening, 8 melting in D = 2 solids, 4,9 and many other classical and quantum problems, such as deconfinement in D = 3 + 1 lattice gauge theories. [10][11][12][13] We study these models by invoking a bond-algebraic approach that we have recently developed. 12-15 Within our duality-based method, one relates singularities in the free energy at one temperature (or coupling constants) to those at a dual temperature (or dual coupling constants). 93 40 Years of Berezinskii-Kosterlitz-Thouless Theory Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/18/15. For personal use only.

Quantum Berezinskii–Kosterltz–Thouless transition for topological insulator

“…According to Mermin-Wagner-Hohenberg theorem [1,2], there is no possibility for the phase transition from disordered state to ordered state by spontaneous symmetry breaking for d ≤ 2 (d = dimension). However, a phase transition of a different sort is predicted by an ingenious renormalization group analysis [3,4]. This is a topological phase transition, which seperates two phases of matter containing rather different topological objects.…”

“…Motivation: Quantum BKT transition is a topological quantum phase transition in low dimensional quantum many-body system. But it has not been explored explicitly in the literature for the topological state of quantum matter [4,12]. Therefore it is also interesting to investigate the quantum BKT transition for the topological insulators, which is the main motivation of this study.…”

A study of quantum Berezinskii–Kosterlitz–Thouless transition for parity-time symmetric quantum criticality

“…At the BKT transition temperature the pairs unbind and the vortices proliferate, resulting in a state with no spin rigidity and the correlation function decays exponentially with temperature. This is the classical picture of the BKT transition [42][43][44][45] . In quantum BKT (QBKT), there are no temperature dependent transitions as it observed for the classical BKT.…”

A Study of Interaction Effects and Quantum Berezinskii- Kosterlitz-Thouless Transition in the Kitaev Chain

“…At the Berezinskii-Kosterlitz and Thouless (BKT) transition the pairs unbind and the vortices proliferate, resulting in a state with no spin rigidity and the correlation function decays exponentially. This is the classical picture of the BKT transition [30][31][32][33] . These topological defects occur in different physical systems such as vortices in superfluid helium and dislocations in a periodic crystal.…”

“…These topological defects occur in different physical systems such as vortices in superfluid helium and dislocations in a periodic crystal. These topological defects cannot be eliminated by continuous change of the order parameter [28][29][30] . In quantum BKT, there is no topological transition dependent on the temperature as we observe for classical BKT.…”