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Sine-Gordon/Coulomb Gas Soliton Correlation Functions and an Exact Evaluation of the Kosterlitz-Thouless Critical Exponent

Leonardo Mondaini, Eduardo C. Marino

Published Jan 1, 2004• 7 citations
OpenAlex

Abstract

We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent η = 1/4 at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at β 2 = 8π, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure. KEY WORDS: soliton correlation functions, sine-Gordon theory, XY-model, 2D Coulomb gas, Kosterlitz-Thouless transition

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Xiaoling Cui, Shi-Jian Gu, Junpeng Cao +2 more

A relation between the correlation entropy and the correlation functions for the general spin-1/2 systems is obtained. It is shown that the correlation entropy catches some characters of correlation behavior and can be used to quantify the quantum and finite-temperature phase transitions, including the infinite order or topological ones. As an example, the Kosterlitz–Thouless transition in the quantum two-dimensional XY model is investigated. The critical temperature and the critical exponents are determined from the finite-size scaling analysis of the correlation entropy.

Journal of Physics A Mathematical and Theoretical•Oct 23, 2007•8 citations
We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent η = 1/4 at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at β 2 = 8π, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure. KEY WORDS: soliton correlation functions, sine-Gordon theory, XY-model, 2D Coulomb gas, Kosterlitz-Thouless transition
A systematic renormalisation group technique for studying the 2D sine-Gordon theory (Coulomb gas, XY model) near its phase transition is presented. The new results are (a) higher order terms in the flow equations beyond those of Kosterlitz (1974) give rise to a new universal quantity; (b) this in turn gives the universal form as well as the relative coefficient of the next-to-leading term in the correlation function of the XY model; (c) the free energy (1PI vacuum sum) is calculated after the singularity at beta 2 =4 pi is treated; (d) vortices with multiple charges are shown to be irrelevant; (e) symmetry breaking fields are analysed systematically. The main ideas that the sine-Gordon theory can be defined as a double expansion in alpha (fugacity) and delta = beta 2 /8 pi -1 (distance from the critical temperature at alpha =0). Wave-function and coupling constant ( alpha ) renormalisations are necessary and sufficient, around beta 2 =8 pi where cos phi acquires dimension 2, for functions with elementary SG fields. This gives rise to renormalisation of beta . The renormalisability is proved to the order calculated in the context of the SG theory, and in general, by using the equivalence to the Thirring-Schwinger model. The renormalised beta 2 plays a role analogous to the dimension in a phi 4 theory-8 pi being the critical dimension. beta 2 >8 pi gives an infrared asymptotically free theory which leads to the well-known fixed line. The infrared properties are understood by analogy with the non-linear sigma model.
A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi n (K c (n) -K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y 0 )/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu 0 , nu < nu 0 =1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta c , of the spin-spin correlation function at the critical point is found to be eta c =1/(4+1/ pi 2 ).
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.
Abstract At low temperature, the large distance properties of the O(2) spin lattice model can be described by the O(2) non-linear σ-model. The latter model is free and massless in two dimensions. The origin of this peculiarity can be found in the local structure of the field manifold: for N = 2, the O(N) sphere reduces to a circle, which cannot be distinguished locally from a straight line. Because the physical fields are sin θ or cos θ, or equivalently e± iθ, instead of θ, a field renormalization is necessary, and temperature-dependent anomalous dimensions are generated. However, the free θ action cannot describe the long-distance properties of the lattice model for all temperatures, since a high temperature analysis of the corresponding spin model shows that the correlation length is finite at high temperature, and thus a phase transition is required. In fact, it is necessary to take into account the property that θ is a cyclic variable. This condition is irrelevant at low temperature, but when the temperature increases, classical configurations with singularities at isolated points, around which θ varies by a multiple of 2π become important. The action of these configurations (vortices) can be identified with the energy of a neutral Coulomb gas, which exhibits a transition between a low temperature of bound neutral molecules and a high temperature phase of a plasma of free charges. The Coulomb gas can be mapped onto the sine-Gordon (sG) model, mapping in which the low- and high-temperature regions of the models are exchanged. This correspondence helps to understand some properties of the famous Kosterlitz-Thouless (KT) phase transition, which separates an infinite correlation length phase without order, the low-temperature phase of the O(2) spin model, from a finite correlation length phase, the high-temperature phase of the O(2) spin model.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.
A relation between the correlation entropy and the correlation functions for the general spin-1/2 systems is obtained. It is shown that the correlation entropy catches some characters of correlation behavior and can be used to quantify the quantum and finite-temperature phase transitions, including the infinite order or topological ones. As an example, the Kosterlitz–Thouless transition in the quantum two-dimensional XY model is investigated. The critical temperature and the critical exponents are determined from the finite-size scaling analysis of the correlation entropy.