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Journal of Experimental and Theoretical Physics Letters

Berezinskii-Kosterlitz-Thouless phase transition in systems with exotic symmetries

S. A. Bulgadaev

Published May 1, 1996• 2 citations
Publisher

Abstract

A Berezinksii-Kosterlitz-Thouless phase transition in systems with the exceptional symmetry groups G=E 6,7,8, G 2, and F 2 is studied. The critical exponents and the exponents of the logarithmic corrections to the correlation functions at the transition point are found by the renormalization-group method. It is shown that for G=A, D, and E the critical exponents can be expressed in terms of the Coxeter numbers h G (or the values of the Casimir operator in the adjoint representation K 2 G ).

DOI
10.1134/1.567089

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A Berezinksii-Kosterlitz-Thouless phase transition in systems with the exceptional symmetry groups G=E 6,7,8, G 2, and F 2 is studied. The critical exponents and the exponents of the logarithmic corrections to the correlation functions at the transition point are found by the renormalization-group method. It is shown that for G=A, D, and E the critical exponents can be expressed in terms of the Coxeter numbers h G (or the values of the Casimir operator in the adjoint representation K 2 G ).
We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\ensuremath{\beta}$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in the condition of lowest temperatures, when thermal fluctuations give place to quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near the quantum critical point is considered. The approach based on the Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as well as the method of taking into account the crossover from the thermal fluctuation regime to the quantum one in the renormalization group technique. For the system with low vortex concentration, it is shown that the BKT transition unavoidably occurs at a temperature above the crossover temperature from the thermal fluctuation regime to quantum one. As a result at small temperatures the critical exponent of the BKT transition, η=1/4η=1/4η=1/4, remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.
Combining the ideas of quantum Fisher information and quantum renormalization group method, the Berezinskii-Kosterlitz-Thouless quantum phase transition of spin- 1/2 XXZ chain is investigated. Quantum Fisher informations of the whole N sites and the partial N/3 sites are studied. They display very similar behaviors, even though their mathematical formulas are very different from each other. The universally critical exponent of quantum Fisher information is obtained as beta=0.47, which is consistent with the results obtained by the renormalized concurrence or discord. We also discuss the relationship between quantum Fisher information and entanglement
We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
Using analytical and discretized Migdal-Kadanoff renormalization-group methods it is shown that similar to the standard XY model the two-component uniform multipolar systems undergo a Kosterlitz-Thouless transition in two dimensions. The critical behavior of the multipole-multipole correlation function is governed by a set of critical exponents that depend on the rank t of multipole moments. At the Kosterlitz-Thouless transition they become universal for all systems. An exactly renormalizable multipolar model is introduced. The behavior of this model on fractal and hierarchical lattices is studied. The T=0 scaling exponent of the two-component multipolar systems on a Sierpinski carpet is shown to be negative, so no Kosterlitz-Thouless transition can be expected to take place in this case.