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Topology of Strongly Correlated Systems

TOPOLOGICAL EXCITATIONS AND SECOND ORDER TRANSITIONS IN 3D O(N) MODELS

Luís M. A. Bettencourt

Published Aug 1, 2001• 0 citations
Publisher

Abstract

I discuss several examples of critical phenomena in O(N) models where topological excitations play an important role at criticality. I focus particular attention on the O(2) model in 3D, where recent measurements of the vortex string length distribution in equilibrium suggest the existence of a quantitative picture of the critical behavior in terms of defects. The compatibility of this perspective with renormalization group predictions is examined.

DOI
10.1142/9789812811455_0003

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I discuss several examples of critical phenomena in O(N) models where topological excitations play an important role at criticality. I focus particular attention on the O(2) model in 3D, where recent measurements of the vortex string length distribution in equilibrium suggest the existence of a quantitative picture of the critical behavior in terms of defects. The compatibility of this perspective with renormalization group predictions is examined.
We present a study of the 3d O(2) non-linear σ\sigmaσ-model on the lattice, which exhibits topological defects in the form of vortices. They tend to organize into vortex lines that bear close analogies with global cosmic strings. Therefore, this model serves as a testbed for studying the dynamics of topological defects. It undergoes a second order phase transition, hence it is appropriate for investigating the Kibble-Zurek mechanism. In this regard, we explore the persistence of topological defects when the temperature is rapidly reduced from above to below the critical temperature; this cooling (or “quenching”) process takes the system out of thermal equilibrium. We probe a wide range of inverse cooling rates τQ\tau_\mathrm{Q}τQ​ and final temperatures, employing distinct Monte Carlo algorithms. The results consistently show that the density of persisting topological defects follows a power-law in τQ\tau_\mathrm{Q}τQ​, in agreement with Zurek’s conjecture. On the other hand, at this point, our results do not confirm Zurek’s prediction for the exponent in this power-law, but its final test is still under investigation.
We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.
The critical behavior of vortices in layered systems is studied. The vortex interactions that we use approximate those of vortices in a layered superconductor. We do a mathematically rigorous, real-space renormalization-group study on our model to derive the recursion relations. Terms not found by other studies on layered systems have been derived. It is found that one of the new terms contributes to layer decoupling just above the transition temperature, behavior which is consistent with Monte Carlo studies of this system. By analylzing the correlation length, we are able to study the dependence of the transition temperature and of the size of the three-dimensional (3D) critical region on the strength of the interlayer coupling. The size of the 3D critical region above the transition is found to depend on the interlayer coupling in a way which is different than that predicted by others. We study our results in the context of the high-temperature superconductors and find that they are in good accord with experiments on these materials.
We study a class of three-dimensional (3D) exactly solvable models of topological matter first put forward by Walker and Wang [Front. Phys. 7(2) 150 (2012)]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work, we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground-state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $\ensuremath{\nu}=\frac{1}{2}$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of bFbFbF theories. Our special example of the 3D semion model captures much of the behavior of more general ``confined Walker-Wang models.'' We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk, and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum Hall effect. We find that in these models a fermion is always deconfined in the three-dimensional bulk.
We determine the detailed thermodynamic behavior of vortices in the O(2) scalar model in two dimensions (2D) and of global monopoles in the O(3) model in 3D. We construct numerical techniques, based on cluster decomposition algorithms, to analyze the point defect configurations. We find that these criteria produce results for the Kosterlitz-Thouless temperature in agreement with a topological transition between a polarizable insulator and a conductor, at which free topological charges appear in the system. For global monopoles we find no pair unbinding transition. Instead a transition to a dense state where pairs are no longer distinguishable occurs at T<T(c), without leading to long-range disorder. We produce both extensive numerical evidence of this behavior as well as a semianalytic treatment of the partition function for defects. General expectations for N=D>3 are drawn, based on the observed behavior.
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.