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Perturbative vs. Non-Perturbative Study of the Sine-Gordon Model

S Hariharakrishnan

Published May 5, 2022• 0 citations
OpenAlex

Abstract

Functional Renormalization Group (FRG) flow equations of Sine-Gordon model in 1, 2 and 3 dimensions are obtained by using power law regulator in additive and multiplicative approaches in the leading order and next to leading order in the Taylor expansion of the Fourier amplitude with LPA′ and single Fourier mode approximations. The 2-dimensional FRG flow equations are compared to the Perturbative Renormalization Group (PRG) flow equations of Daniel J Amit, Yadin Y Goldschmidt, and S Grinstein. Renormalisation group analysis of the phase transition in the 2d coulomb gas, sine-gordon theory and xy-model. Journal of Physics A: Mathematical and General, 13(2):585, 1980., where a universal behaviour is claimed. This thesis explored whether FRG also reproduces this universal behaviour and it is concluded that no such universal behaviour exists in the FRG framework.

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Functional Renormalization Group (FRG) flow equations of Sine-Gordon model in 1, 2 and 3 dimensions are obtained by using power law regulator in additive and multiplicative approaches in the leading order and next to leading order in the Taylor expansion of the Fourier amplitude with LPA′ and single Fourier mode approximations. The 2-dimensional FRG flow equations are compared to the Perturbative Renormalization Group (PRG) flow equations of Daniel J Amit, Yadin Y Goldschmidt, and S Grinstein. Renormalisation group analysis of the phase transition in the 2d coulomb gas, sine-gordon theory and xy-model. Journal of Physics A: Mathematical and General, 13(2):585, 1980., where a universal behaviour is claimed. This thesis explored whether FRG also reproduces this universal behaviour and it is concluded that no such universal behaviour exists in the FRG framework.
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.
The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this work we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner-Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner-Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas.
The exact renormalization group is used to study the RG flow of quantities in field theories. The basic idea is to write an evolution operator for the flow and evaluate it in perturbation theory. This is easier than directly solving the differential equation. This is illustrated by reproducing known results in four dimensional ϕ 4 field theory and the two dimensional Sine-Gordon theory. It is shown that the calculation of beta function is somewhat simplified. The technique is also used to calculate the c-function in two dimensional Sine-Gordon theory. This agrees with other prescriptions for calculating c-functions in the literature. If one extrapolates the connection between central charge of a CFT and entanglement entropy in two dimensions, to the c-function of the perturbed CFT, then one gets a value for the entanglement entropy in Sine-Gordon theory that is in exact agreement with earlier calculations (including one using holography) in arXiv:1610.04233 .
The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.