Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1609.02431 · Physical review. E

Quench dynamics of the three-dimensional U(1) complex field theory: Geometric and scaling characterizations of the vortex tangle

Michikazu Kobayashi, Leticia F. Cugliandolo

Published Dec 29, 2016• 16 citations
arXivHTMLPDF

Abstract

We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II superconductors, magnetic materials, and aspects of cosmology. We characterize the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls off algebraically and we associate it to a geometric percolation transition that we characterize in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermodynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach toward a state that, within numerical accuracy, shares many features with the one at the percolation threshold; a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops; and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.

DOI
10.1103/physreve.94.062146

Related papers

Thermal quenches in the stochastic Gross-Pitaevskii equation: Morphology of the vortex network

Michikazu Kobayashi, Leticia F. Cugliandolo

We study the evolution of 3d weakly interacting bosons at finite chemical potential with the stochastic Gross-Pitaevskii equation. We fully characterise the vortex network in an out of equilibrium. At high temperature the filament statistics are the ones of fully-packed loop models. The vortex tangle undergoes a geometric percolation transition within the thermodynamically ordered phase. After infinitely fast quenches across the thermodynamic critical point deep into the ordered phase, we identify a first approach towards the critical percolation state, a later coarsening process that does not alter the fractal properties of the long vortex loops, and a final approach to equilibrium. Our results are also relevant to the statistics of linear defects in type II superconductors, magnetic materials and cosmological models.

arXiv:1606.03262•Europhysics Letters (EPL)•Jul 1, 2016•15 citations

Vortex-line percolation in the three-dimensional complex ∣ψ∣4 model

Elmar Bittner, Axel Krinner, Wolfhard Janke E

In discussing the phase transition of the three-dimensional complex $\ensuremath{\mid}\ensuremath{\psi}{\ensuremath{\mid}}^{4}$ theory, we study the geometrically defined vortex-loop network as well as the magnetic properties of the system in the vicinity of the critical point. Using high-precision Monte Carlo techniques we investigate if both of them exhibit the same critical behavior leading to the same critical exponents and hence to a consistent description of the phase transition. Different percolation observables are taken into account and compared with each other. We find that different connectivity definitions for constructing the vortex-loop network lead to different results in the thermodynamic limit, and the percolation thresholds do not coincide with the thermodynamic phase transition point.

arXiv:cond-mat/0510347•Physical Review B•Sep 20, 2005•13 citations

Quench dynamics of the 2dXYmodel

Asja Jelić, Leticia F. Cugliandolo

We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii–Kosterlitz–Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long-range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble–Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results for a host of physical systems with vortex excitations, including planar ferromagnets and liquid crystals.

arXiv:1012.0417•Journal of Statistical Mechanics Theory and Experiment•Feb 18, 2011•89 citations

Critical quench dynamics of Wegner’s Z 2 gauge model: a geometric perspective

Ramgopal Agrawal, Leticia Cugliandolo, Marco Picco

Abstract Wegner’s Z 2 gauge model is the earliest formulation of pure lattice gauge theory and predicts the topological nature of the confinement–deconfinement transition. In three dimensions ( D = 3 ), the equilibrium critical behavior of the model is understood in terms of geometrically defined objects, namely loop excitations and Fortuin–Kasteleyn clusters. This work investigates the critical quench dynamics of this model from a geometric perspective, following quenches from both a high-temperature percolation phase and the zero-temperature ground state. Using time-dependent finite-size scaling analysis, we find that the critical non-equilibrium relaxation of the percolation order parameter is governed by a dynamical exponent z p ∼ 2.6 , consistent with that associated with the energy density, z c . Importantly, the value of z p is robust with respect to the initial quench condition and the choice of geometrical objects. Furthermore, we provide a detailed characterization of the kinetics of different geometrical objects during the evolution from the percolation phase. Notably, we observe that the quench dynamics obeys dynamic scaling in terms of a growing lengthscale, ξ p ( t ) ∼ t 1 / z p , despite the absence of a local order parameter.

arXiv:2605.15841•Journal of Statistical Mechanics Theory and Experiment•Oct 1, 2026•0 citations

Evolution of Vortex Strings after a Thermal Quench in a Holographic Superfluid

Chuan-Yin Xia, András Grabarits, Hua-Bi Zeng +1 more

The formation of topological defects during continuous phase transitions exhibits nonequilibrium universality. While the Kibble-Zurek mechanism (KZM) predicts universal scaling of point-like defect numbers under slow driving, the statistical properties of extended defects remain largely unexplored across both slow and fast protocols. We investigate vortex string formation in a three-dimensional holographic superfluid. For slow quenches, the vortex string number follows KZM scaling, while for rapid quenches, it exhibits complementary universal scaling governed by the final temperature. Beyond the vortex string number, the loop-length distribution reveals a richer structure: individual loops follow the first-return statistics of three-dimensional random walks, $P(\ell) \sim \ell^{-5/2}$. While the total vortex length distribution remains Gaussian, its cumulants obey universal scaling laws with varying power-law exponents, and thus differ markedly from those observed in point-defect systems, indicating distinct statistical features of extended topological defects.

hep-thCross-listedarXiv:2601.14328•Jan 21, 2026

Non-equilibrium Dynamics and Universality of 4D Quantum Vortices and Turbulence

Wei-Can Yang

The study of quantum vortices provides critical insights into non-equilibrium dynamics across diverse physical systems. While previous research has focused on point-like vortices in two dimensions and line-like vortices in three dimensions, quantum vortices in four spatial dimensions are expected to take the form of extended vortex surfaces, thereby fundamentally enriching dynamics. Here, we conduct a comprehensive numerical study of 4D quantum vortices and turbulence. Using a special visualization method, we discovered the decay of topological numbers that does not exist in low dimensions, as well as the high-dimensional counterpart of the vortex reconnection process. We further explore quench dynamics across phase transitions in four dimensions and verify the applicability of the higher-dimensional Kibble-Zurek mechanism. Our simulations provide numerical evidence of 4D quantum turbulence, characterized by universal power-law behavior. These findings reveal universal principles governing topological defects in higher dimensions, offering insights for future experimental realizations using synthetic dimensions.

arXiv:2504.04409•Apr 6, 2025•0 citations
We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II superconductors, magnetic materials, and aspects of cosmology. We characterize the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls off algebraically and we associate it to a geometric percolation transition that we characterize in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermodynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach toward a state that, within numerical accuracy, shares many features with the one at the percolation threshold; a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops; and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.
We study the evolution of 3d weakly interacting bosons at finite chemical potential with the stochastic Gross-Pitaevskii equation. We fully characterise the vortex network in an out of equilibrium. At high temperature the filament statistics are the ones of fully-packed loop models. The vortex tangle undergoes a geometric percolation transition within the thermodynamically ordered phase. After infinitely fast quenches across the thermodynamic critical point deep into the ordered phase, we identify a first approach towards the critical percolation state, a later coarsening process that does not alter the fractal properties of the long vortex loops, and a final approach to equilibrium. Our results are also relevant to the statistics of linear defects in type II superconductors, magnetic materials and cosmological models.
In discussing the phase transition of the three-dimensional complex $\ensuremath{\mid}\ensuremath{\psi}{\ensuremath{\mid}}^{4}$ theory, we study the geometrically defined vortex-loop network as well as the magnetic properties of the system in the vicinity of the critical point. Using high-precision Monte Carlo techniques we investigate if both of them exhibit the same critical behavior leading to the same critical exponents and hence to a consistent description of the phase transition. Different percolation observables are taken into account and compared with each other. We find that different connectivity definitions for constructing the vortex-loop network lead to different results in the thermodynamic limit, and the percolation thresholds do not coincide with the thermodynamic phase transition point.
We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii–Kosterlitz–Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long-range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble–Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results for a host of physical systems with vortex excitations, including planar ferromagnets and liquid crystals.
Abstract Wegner’s Z 2 gauge model is the earliest formulation of pure lattice gauge theory and predicts the topological nature of the confinement–deconfinement transition. In three dimensions ( D = 3 ), the equilibrium critical behavior of the model is understood in terms of geometrically defined objects, namely loop excitations and Fortuin–Kasteleyn clusters. This work investigates the critical quench dynamics of this model from a geometric perspective, following quenches from both a high-temperature percolation phase and the zero-temperature ground state. Using time-dependent finite-size scaling analysis, we find that the critical non-equilibrium relaxation of the percolation order parameter is governed by a dynamical exponent z p ∼ 2.6 , consistent with that associated with the energy density, z c . Importantly, the value of z p is robust with respect to the initial quench condition and the choice of geometrical objects. Furthermore, we provide a detailed characterization of the kinetics of different geometrical objects during the evolution from the percolation phase. Notably, we observe that the quench dynamics obeys dynamic scaling in terms of a growing lengthscale, ξ p ( t ) ∼ t 1 / z p , despite the absence of a local order parameter.
The formation of topological defects during continuous phase transitions exhibits nonequilibrium universality. While the Kibble-Zurek mechanism (KZM) predicts universal scaling of point-like defect numbers under slow driving, the statistical properties of extended defects remain largely unexplored across both slow and fast protocols. We investigate vortex string formation in a three-dimensional holographic superfluid. For slow quenches, the vortex string number follows KZM scaling, while for rapid quenches, it exhibits complementary universal scaling governed by the final temperature. Beyond the vortex string number, the loop-length distribution reveals a richer structure: individual loops follow the first-return statistics of three-dimensional random walks, P(ℓ)∼ℓ−5/2P(\ell) \sim \ell^{-5/2}P(ℓ)∼ℓ−5/2. While the total vortex length distribution remains Gaussian, its cumulants obey universal scaling laws with varying power-law exponents, and thus differ markedly from those observed in point-defect systems, indicating distinct statistical features of extended topological defects.
The study of quantum vortices provides critical insights into non-equilibrium dynamics across diverse physical systems. While previous research has focused on point-like vortices in two dimensions and line-like vortices in three dimensions, quantum vortices in four spatial dimensions are expected to take the form of extended vortex surfaces, thereby fundamentally enriching dynamics. Here, we conduct a comprehensive numerical study of 4D quantum vortices and turbulence. Using a special visualization method, we discovered the decay of topological numbers that does not exist in low dimensions, as well as the high-dimensional counterpart of the vortex reconnection process. We further explore quench dynamics across phase transitions in four dimensions and verify the applicability of the higher-dimensional Kibble-Zurek mechanism. Our simulations provide numerical evidence of 4D quantum turbulence, characterized by universal power-law behavior. These findings reveal universal principles governing topological defects in higher dimensions, offering insights for future experimental realizations using synthetic dimensions.