In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimensions. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of a global symmetry-breaking and thus in the absence of an order parameter. It is found that the first order phase transition undergone by this model fits into a microcanonical version of an Ehrenfest-like classification of phase transitions applied to the configurational entropy. It is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy is the effect of a deeper geometrical transition of the equipotential submanifolds of configuration space, which, in its turn, is likely to be the "shadow" of an even deeper transition of topological kind. (C) 2018 Elsevier B.V. All rights reserved.
In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimensions. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of a global symmetry-breaking and thus in the absence of an order parameter. It is found that the first order phase transition undergone by this model fits into a microcanonical version of an Ehrenfest-like classification of phase transitions applied to the configurational entropy. It is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy is the effect of a deeper geometrical transition of the equipotential submanifolds of configuration space, which, in its turn, is likely to be the "shadow" of an even deeper transition of topological kind. (C) 2018 Elsevier B.V. All rights reserved.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind. First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films. We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index. Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.
After fifty years of lattice gauge theories (LGTs), the nature of the transition between their topological phases (confinement or deconfinement) remains challenging due to the absence of a local order parameter. In this work, we conduct a percolation analysis of Wegner's three-dimensional Z_{2} lattice gauge model using intensive Monte Carlo simulations and finite-size scaling, offering fresh insights into the topological phase transitions of gauge-invariant systems. We demonstrate that, regardless of the connection rules, geometrical loops, constructed by piercing excited plaquettes percolate precisely at the thermal critical point T_{c}, with critical exponents coinciding with those of the loop representation of the dual 3D Ising model. Further, we construct Fortuin-Kasteleyn (FK) clusters in a random-cluster representation, showing that they also percolate at T_{c}, enabling access to all thermal critical exponents. Strikingly, the Binder cumulants of the percolation order parameters for both loops and FK clusters reveal a pseudo-first-order transition. This work sheds new light on the critical behavior of pure LGTs, with potential implications for condensed matter systems and quantum error correction.
Self-dual criticality in three-dimensional Z2 gauge theory with matter
The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``e'' and ``m'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the Z2 gauge-Higgs model on cubic lattices of linear size L≤96. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators A and S. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions xA and xS and dynamical exponent z. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.
It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 < n < d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.
We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with Z2 gauge symmetry. In particular, we consider the pure lattice Z2 gauge model and the lattice Z2 gauge XY model, which present various types of transitions: topological transitions without a local order parameter and transitions characterized by both gauge-invariant and non-gauge-invariant XY order parameters. We consider a standard relaxational (locally reversible) Metropolis dynamics and determine the dynamic critical exponent z that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological Z2 gauge transitions we find z=2.55(6). Therefore, the dynamics is significantly slower than in Ising systems---$z\ensuremath{\approx}2.02$ for the 3D Ising universality class---although 3D Z2 gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D Z2 gauge XY model, we find that their critical dynamics belong to the same dynamic universality class as the relaxational dynamics in ungauged XY systems independently of the gauge-invariant or non-gauge-invariant nature of the order parameter at the transition.