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Europhysics Letters (EPL)

Disorder Lines and Nonmonotonous Renormalization Group Flows: Application to the Two-Dimensional XY Model

Thomas Garel, J. C. Niel, Henri Orland

Published Feb 15, 1990• 13 citations
Publisher

Abstract

We show analytically how the crossing of a disorder line in a simple one-dimensional model coincides with a qualitative change of behaviour (monotonous to nonmonotonous) in the renormalization group flow. Such a change is found in Migdal-Kadanoff recursion relations of a family of ferromagnetic two-dimensional XY models. We therefore infer the existence of a disorder line in these models. For the usual XY model, the disorder temperature corresponds to the onset of short-distance pairing of vortices. It may coincide with the dynamical transition temperature recently found by Golinelli and Derrida.

DOI
10.1209/0295-5075/11/4/010

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We show analytically how the crossing of a disorder line in a simple one-dimensional model coincides with a qualitative change of behaviour (monotonous to nonmonotonous) in the renormalization group flow. Such a change is found in Migdal-Kadanoff recursion relations of a family of ferromagnetic two-dimensional XY models. We therefore infer the existence of a disorder line in these models. For the usual XY model, the disorder temperature corresponds to the onset of short-distance pairing of vortices. It may coincide with the dynamical transition temperature recently found by Golinelli and Derrida.
Using one-dimensional models, the authors characterize the existence of a disorder line (point) in a renormalization group procedure. This procedure is defined in terms of the eigenvalues of the transfer matrix. There are various possibilities for the crossing of subdominant eigenvalues; this study therefore encompasses usual disorder lines (where the spin-spin correlation function changes), as well as other types of disorder lines (where, for instance, the energy-energy correlation function changes). In the usual case, the disorder line shows up as a separatrix between two regions of the flow diagram attracted to different paramagnetic sinks. For this case, they also consider, in the framework of the random phase approximation, the dynamical signature of a disorder line: they show that it can be experimentally detected, especially for disorder lines (points) of the second kind.
We show, that the 2D XY-model with random phase shifts exhibits for low temperature and small disorder a phase with quasi-long-range order, and that the transition to the disordered phase is {\it not} reentrant. These results are obtained by heuristic arguments, an analytical renormalization group calculation, and a numerical Migdal-Kadanoff renormalization group treatment. Previous predictions of reentrance are found to fail due to an overestimation of the vortex pair density as a consequence of independent dipole approximations. At positions, where vortex pairs are energetically favored by disorder, their statistics becomes effectively fermionic. The results may have implications for a large number of related models.
The two-dimensional XY model with random phase shifts on bonds is studied. The analysis is based on a renormalization group for the replicated system. The model is shown to have an ordered phase with quasi-long-range order. This ordered phase consists of a glasslike region at lower temperatures and of a nonglassy region at higher temperatures. The transition from the disordered phase into the ordered phase is not reentrant and is of a new universality class at zero temperature. In contrast to previous approaches the disorder strength is found to be renormalized to larger values. Several correlation functions are calculated for the ordered phase. They allow one to identify not only the transition into the glassy phase but also an additional crossover line, where the disconnected vortex correlation changes its behavior on large scales nonanalytically. The renormalization group approach yields the glassy features without a breaking of replica symmetry.
The XY\mathrm{XY}XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.
We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T , and the deformation parameter, Δ , and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024