Abstract These lecture notes deal with aspects of out-of-equilibrium behaviour in topologically ordered systems, broadly interpreted as systems that do not develop a local order parameter at low temperature (e.g. symmetry breaking) and yet exhibit non-trivial global properties. The specific cases of classical spin ice and the quantum toric code are discussed. Within classical systems, it is found that substantial progress can be made in understanding the dynamics through effective modelling of the vacuum and its emergent excitations. However, at the quantum level, modelling is limited to that of leading energy barriers and asymptotic behavior, although it turns out that an intriguing parallel can be drawn between the toric code Hamiltonian and a class of lattice systems known as kinetically constrained models.
Abstract These lecture notes deal with aspects of out-of-equilibrium behaviour in topologically ordered systems, broadly interpreted as systems that do not develop a local order parameter at low temperature (e.g. symmetry breaking) and yet exhibit non-trivial global properties. The specific cases of classical spin ice and the quantum toric code are discussed. Within classical systems, it is found that substantial progress can be made in understanding the dynamics through effective modelling of the vacuum and its emergent excitations. However, at the quantum level, modelling is limited to that of leading energy barriers and asymptotic behavior, although it turns out that an intriguing parallel can be drawn between the toric code Hamiltonian and a class of lattice systems known as kinetically constrained models.
Topological ordered states are exotic quantum states of matter that defy the usual description in terms of symmetry breaking and local order parameters. The type or order they feature is of non-local, topological nature, and it allows such systems to present unusual and interesting behaviours, like robust degeneracy, and anyonic excitations. Here we focus on the behaviour of such systems away from equilibrium. After a sudden change of the Hamiltonian - the so called quantum quench - quantum many body systems show thermalization, unless the quench is integrable. We show how in topologically ordered systems out of equilibrium behaviour do not follow the usual thermalization behaviour.
We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.
Topological systems are an exciting frontier of condensed matter physics, as well as having deep relationships to quantum computing. Progress continues to be made both towards building practical quantum computers and experimentally identifying topological features in real materials. However, these attempts have to contend with the messiness of the real world including the effects of finite temperatures. In quantum computing thermal errors are studied through the framework of quantum error correction. This allows a characterisation of how damaging these errors are, and has provided many general results on the limits of how well given quantum codes can endure thermal errors. In the first part of this thesis we study the typical response of one of the best known quantum codes (the two-dimensional toric code) when subjected to thermal noise. This typical behaviour is compared to the limiting bounds and we note differences that arise due to entropic effects. Topological features are typically properties of the ground state of a system. In some cases this means finite temperatures can be damaging, washing out the topological properties with excitations above the ground state. But in other cases it has been shown that topological systems can have their own unique finite temperature behaviours, which can be used to identify them. In the second part of the thesis we study the topological model Kitaev's honeycomb. Fixing certain conserved quantities restricts the model to special sectors, some of which carry topological phases. We begin by investigating an intrinsically finite temperature property of these topological phases, thermal edge currents. At too high temperatures, close to the bulk gap, the topological behaviour of these currents breaks down. However, we show that at lower temperatures they can be identified and shown to have the expected properties such as robustness to perturbations. Following this we investigate the thermal states of the full model. Our approach to this allows us to probe the topological order of the model in a way that has not been done before and we identify a number of interesting new features of the finite temperature physics.
Frustration in many-body spin systems is a situation where the spins cannot orient themselves to fully minimize the interactions with their neighbors. It is caused, in general, by competing interactions or by the lattice structure, as in the antiferromagnets on non-bipartite lattices. Over the years frustrated systems have been studied substantially, since frustration can destabilize the antiferromagnetic order, lead to different phases of matter and induce various exotic properties. Frustration can be caused also by a choice of boundary conditions, i.e. the lattice topology. For instance, a ring with an odd number of sites is non-bipartite, which makes antiferromagnets in this setting frustrated. The subject of this thesis are zero-temperature one-dimensional quantum systems exactly in this setting. The major content is discussing how such a simple setting can influence the antiferromagnetic order, by studying spin-1/2 chains with discrete symmetries, whose breaking is related to the onset of the order. After reviewing the known results on topological frustration, dealing with non-thermodynamic quantities, we develop an approach for studying symmetry breaking, that is both suitable for exact analytical computations and meaningful for discussing the order also in a finite system. It consists of the realization that many spin-1/2 chains without external magnetic fields, such as the quantum XY chain, possess anticommuting global symmetries when the number of lattice sites is an odd number, implying the ground-state degeneracy already in a finite system. In this framework, we discover that topological frustration, despite being induced only by the choice of boundary conditions and even/odd system size choice, can affect the magnetization in the thermodynamic limit and system's quantum phase transitions. We find that topological frustration can destroy local order, create a site-dependent magnetization that varies in space with an incommensurate pattern, induce a first-order quantum phase transition that is not present without frustration and modify the nature of a second-order transition, by destroying local order at both sides of the transition and preserving only non-local string order parameters. All these results indicate the incompleteness of the approach to quantum many-body systems, based on the Ginzburg-Landau theory, that tries to capture the properties of the system by taking the system size to infinity before computing the observables and neglects the influence of the chain length as a relevant scale. We find that topological frustration can affect also non-equilibrium properties, by considering a local quantum quench setup. Namely, we study the Loschmidt echo and find it displays qualitatively and quantitatively different behavior for antiferromagnetic rings with an even and odd number of sites. Most of all, the differences become clearer for large system sizes, thus allowing to distinguish in a simple out-of-equilibrium experiment a system made by a certain, large, number of spins from the one with a single additional spin. The thesis also contains a mathematical part. In studying order parameters of models mappable to free fermions, Toeplitz determinants with symbols that possess a part proportional to the delta function arise. We derive asymptotic formulas for this type of determinants.
Quantum tunneling often allows pathways to relaxation past energy barriers which are otherwise hard to overcome classically at low temperatures. However, this is not always the case. In this paper we provide exactly solvable examples where the barriers each system encounters on its approach to lower and lower energy states become increasingly wide and eventually scale with the system size. If the environment couples locally to the physical degrees of freedom in the system, tunneling under these barriers requires processes whose order in perturbation theory is proportional to the width of the barrier. This results in quantum relaxation rates that are exponentially suppressed in system size: For these quantum systems, no physical bath can provide a mechanism for relaxation that is not dynamically arrested at low temperatures. The examples discussed here are drawn from three-dimensional generalizations of Kitaev's toric code, originally devised in the context of topological quantum computing. They are devoid of any local order parameters or symmetry breaking and are examples of topological quantum glasses. We construct systems that have slow dynamics similar to either strong or fragile glasses. The example with fragile-like relaxation is interesting in that the topological defects are neither open strings nor regular open membranes, but fractal objects with dimension d* = ln3/ln2.
A fascinating feature of strongly correlated systems is that excitations may carry nontrivial quantum numbers under emergent symmetries which, in some cases, may strongly constrain their motion. We show that such constrained excitations can play a central role in driving symmetry-breaking long-range order through a kinetic mechanism: coupling to a local order-parameter field activates the mobility of otherwise constrained excitations, and condensation of the order parameter then enables a large gain in kinetic energy, thereby stabilizing a phase with long-range order. We illustrate this mechanism in two settings: (i) a dimerization instability in a spin chain with a magnetization dipole conservation law, and (ii) itinerant magnetism of electrons in the Kitaev-Kondo model, where charge carriers couple to local moments that form a spin liquid. This mechanism becomes operative whenever a finite density of mobility-constrained excitations is present, either through non-equilibrium preparation or thermal activation. In the latter case, we argue that increasing temperature can, counterintuitively, promote order by enhancing the kinetic-energy gain of thermally activated excitations.