Absence of Reentrance in the Two-Dimensional XY-Model with Positional Disorder

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.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.

arXiv:cond-mat/9501120Journal de Physique IMay 1, 199548 citations

Glassy vortex state in a two-dimensional disordered XY model

Stefan Scheidl

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 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.

arXiv:cond-mat/9601131Physical review. B, Condensed matterJan 1, 199751 citations

Quasi-long-range order in the 2D XY model with random phase shifts

Vincenzo Alba, Andrea Pelissetto, Ettore Vicari

We study the square-lattice XY model in the presence of random phase shifts. We consider two different disorder distributions with zero average shift and investigate the low-temperature quasi-long-range order phase which occurs for sufficiently low disorder. By means of Monte Carlo simulations we determine several universal quantities which are then compared with the analytic predictions of the random spin-wave theory. We observe a very good agreement which indicates that the universal long-distance behavior in the whole low-disorder low-temperature phase is fully described by the random spin-wave theory.We study the square-lattice XY model in the presence of random phase shifts. We consider two different disorder distributions with zero average shift and investigate the low-temperature quasi-long-range order phase which occurs for sufficiently low disorder. By means of Monte Carlo simulations we determine several universal quantities which are then compared with the analytic predictions of the random spin-wave theory. We observe a very good agreement which indicates that the universal long-distance behavior in the whole low-disorder low-temperature phase is fully described by the random spin-wave theory.

arXiv:0901.4682Journal of Physics A Mathematical and TheoreticalJul 6, 20096 citations

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

Thomas Garel, J. C. Niel, Henri Orland

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.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.

Europhysics Letters (EPL)Feb 15, 199013 citations

Phase transitions in the two-dimensional XY model with random phases: A Monte Carlo study

J. Maucourt, Daniel R. Grempel

We study the two-dimensional $\mathrm{XY}$ model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to $T=0$ in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.We study the two-dimensional $\mathrm{XY}$ model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to $T=0$ in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.

arXiv:cond-mat/9703109Physical review. B, Condensed matterAug 1, 199730 citations

XY models with disorder and symmetry-breaking fields in two dimensions

Stefan Scheidl, Michael Lehnen

The combined effect of disorder and symmetry-breaking fields on the two-dimensional $\mathrm{XY}$ model is examined. The study includes disorder in the interaction among spins in the form of random phase shifts as well as disorder in the local orientation of the field. The phase diagrams are determined and the properties of the various phases and phase transitions are calculated. We use a renormalization-group approach in the Coulomb gas representation of the model. Our results differ from those obtained for special cases in previous works. In particular, we find a changed topology of the phase diagram that is composed of phases with long-range order, quasi-long-range order, and short-range order. The discrepancies can be ascribed to a breakdown of the fugacity expansion in the Coulomb gas representation. Implications for physical systems such as planar Josephson junctions and the faceting of crystal surfaces are discussed.The combined effect of disorder and symmetry-breaking fields on the two-dimensional $\mathrm{XY}$ model is examined. The study includes disorder in the interaction among spins in the form of random phase shifts as well as disorder in the local orientation of the field. The phase diagrams are determined and the properties of the various phases and phase transitions are calculated. We use a renormalization-group approach in the Coulomb gas representation of the model. Our results differ from those obtained for special cases in previous works. In particular, we find a changed topology of the phase diagram that is composed of phases with long-range order, quasi-long-range order, and short-range order. The discrepancies can be ascribed to a breakdown of the fugacity expansion in the Coulomb gas representation. Implications for physical systems such as planar Josephson junctions and the faceting of crystal surfaces are discussed.

arXiv:cond-mat/9802111Physical review. B, Condensed matterOct 1, 19983 citations

We study the two-dimensional

\mathrm{XY}

model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to

T=0

in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.

The combined effect of disorder and symmetry-breaking fields on the two-dimensional

\mathrm{XY}

model is examined. The study includes disorder in the interaction among spins in the form of random phase shifts as well as disorder in the local orientation of the field. The phase diagrams are determined and the properties of the various phases and phase transitions are calculated. We use a renormalization-group approach in the Coulomb gas representation of the model. Our results differ from those obtained for special cases in previous works. In particular, we find a changed topology of the phase diagram that is composed of phases with long-range order, quasi-long-range order, and short-range order. The discrepancies can be ascribed to a breakdown of the fugacity expansion in the Coulomb gas representation. Implications for physical systems such as planar Josephson junctions and the faceting of crystal surfaces are discussed.