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arXiv:cond-mat/9504074 · Journal of Physics A Mathematical and General

Correlation functions of the 2D sine-Gordon model

K. Nomura

Published Oct 7, 1995• 104 citations
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Abstract

A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.

DOI
10.1088/0305-4470/28/19/003

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A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.
The Berezinskii-Kosterlitz-Thouless (BKT) transition, the transition of the two-dimensional sine-Gordon model, plays an important role in low-dimensional physics. We relate the operator content of the BKT transition to that of the SU (2) Wess-Zumino-Witten model, using twisted boundary conditions. With this method, in order k - 1 to determine the BKT critical point, we can use the level crossing of the lower excitations instead of those for the periodic boundary case, thus the convergence to the transition point is highly improved. We verify the efficiency of this method by applying it to the S = 1, 2 spin chains.
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.
Abstract We introduce and study the properties of a periodic model interpolating between the sine-and the sinh-Gordon theories in 1 + 1 dimensions. This model shows the peculiarities, due to the preservation of the functional form of their potential across RG flows, of the two limiting cases: the sine-Gordon, not having conventional order/magnetization at finite temperature, but exhibiting Berezinskii–Kosterlitz–Thouless (BKT) transition; and the sinh-Gordon, not having a phase transition, but being integrable. The considered interpolation, which we term as sn-Gordon model, is performed with potentials written in terms of Jacobi functions. The critical properties of the sn-Gordon theory are discussed by a renormalization-group approach. The critical points, except the sinh-Gordon one, are found to be of BKT type. Explicit expressions for the critical coupling as a function of the elliptic modulus are given.
It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.
A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.