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arXiv:hep-th/9506141 · Europhysics Letters (EPL)

The large- N random phase sine-Gordon model

Michel Bauer, Denis Bernard

Published Feb 1, 1996• 4 citations
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Abstract

At large distances and in the low-temperature phase, the quenched correlation functions in the 2 d random phase sine-Gordon model have been argued to be of the form: , with = ( T − T c ). However, renormalization group computations predict B ≠ 0, while variational approaches (which are supposed to be exact for models with a large number of components) give B = 0. We introduce a large- N version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient B suppressed by a factor 1/ N 3 compared to A .

DOI
10.1209/epl/i1996-00329-2

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At large distances and in the low-temperature phase, the quenched correlation functions in the 2 d random phase sine-Gordon model have been argued to be of the form: , with = ( T − T c ). However, renormalization group computations predict B ≠ 0, while variational approaches (which are supposed to be exact for models with a large number of components) give B = 0. We introduce a large- N version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient B suppressed by a factor 1/ N 3 compared to A .
We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature Tc{T}_{c}Tc​, in an expansion in small $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$, where TTT denotes the temperature. We derive renormalization group equations in cubic order in the anharmonicity, and show that they contain two universal invariants. Using them we obtain that the correlation function in the super-rough phase for temperature T<TcT<{T}_{c}T<Tc​ behaves at large distances as $\overline{\ensuremath{\langle}{[\ensuremath{\theta}(x)\ensuremath{-}\ensuremath{\theta}(0)]}^{2}\ensuremath{\rangle}}=\mathcal{A}{\mathrm{ln}}^{2}(|x|/a)+\mathcal{O}[\mathrm{ln}(|x|/a)]$, where the amplitude A\mathcal{A}A is a universal function of temperature $\mathcal{A}=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})$. This result differs at two-loop order, i.e., $\mathcal{O}({\ensuremath{\tau}}^{3})$, from the prediction based on results from the ``nearly conformal'' field theory of a related fermion model. We also obtain the correction-to-scaling exponent.
The sine-Gordon model serves as a foundational 1+11+11+1-dimensional quantum field theory with numerous applications in condensed matter physics. Despite its integrability, characterizing its finite-temperature behavior remains a significant theoretical challenge. Here we use the previously developed Method of Random Surfaces (MRS) to evaluate two-point and higher-order correlation functions. We cross-check these results with known analytical limits, demonstrating that the MRS provides reliable, non-perturbative data in intermediate regimes where traditional form-factor expansions and semiclassical methods are inapplicable. Furthermore, we derive an exact result for arbitrary NNN-point functions satisfying an appropriate selection rule, providing a direct computational method for complex multi-point observables at finite temperature. We also characterize the non-Gaussianity of correlations and demonstrate that the results align with intuitive theoretical expectations.
We compute the time evolution of correlation functions after quantum quenches in the sine-Gordon model within the semiclassical approximation, which is expected to yield accurate results for small and slow quenches producing slow quasiparticles with low density. We demonstrate this by reproducing results of a recent form-factor calculation of the relaxation of expectation values [B. Bertini, D. Schuricht, and F. H. L. Essler, J. Stat. Mech. (2014) P100351742-546810.1088/1742-5468/2014/10/P10035]. Extending these results, we find that-in the universal limit of vanishingly small quasiparticle velocities-the expectation values of most vertex operators do not decay to zero. We give analytic expressions for the relaxation of dynamic correlation functions and show that they have diffusive behavior for large timelike separation.
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 authors study here the sine-Gordon quantum field theory, relevant for many 1D gapped systems and experimental settings. Despite its integrability, computing correlation functions, especially at finite temperatures, remains challenging. Existing methods like perturbative expansions, Monte Carlo simulations, and Hamiltonian truncation have limitations. The authors develop here a numerical approach using the method of random surfaces, which is efficient, scalable, and free from the sign problem. They validate their results against established techniques and propose future applications for multipoint functions and quench dynamics.