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We show that the concept of topological order, introduced to describe ordered quantum systems which cannot be classified by broken symmetries, also applies to classical systems. Starting from a specific example, we show how to use pure state density matrices to construct corresponding thermally mixed ones that retain precisely half the original topological entropy, a result that we generalize to a whole class of quantum systems. Finally, we suggest that topological order and topological entropy may be useful in characterizing classical glassy systems.
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In this chapter we discuss aspects of the quantum critical behavior that occurs at a quantum phase transition separating a topological phase from a conventionally ordered one. We concentrate on a family of quantum lattice models, namely certain deformations of the toric code model, that exhibit continuous quantum phase transitions. One such deformation leads to a Lorentz-invariant transition in the 3D Ising universality class. An alternative deformation gives rise to a so-called conformal quantum critical point where equal-time correlations become conformally invariant and can be related to those of the 2D Ising model. We study the behavior of several physical observables, such as non-local operators and entanglement entropies, that can be used to characterize these quantum phase transitions. Finally, we briefly consider the role of thermal fluctuations and related phase transitions, before closing with a short overview of field theoretical descriptions of these quantum critical points.
As new kinds of stabilizer code models, fracton models have been promising in realizing quantum memory or quantum hard drives. However, it has been shown that the fracton topological order of 3D fracton models occurs only at zero temperature. In this Letter, we show that higher dimensional fracton models can support a fracton topological order below a nonzero critical temperature $T_c$. Focusing on a typical 4D X-cube model, we show that there is a finite critical temperature $T_c$ by analyzing its free energy from duality. We also obtained the expectation value of the t Hooft loops in the 4D X-cube model, which directly shows a confinement-deconfinement phase transition at finite temperature. This finite-temperature phase transition can be understood as spontaneously breaking the $mathbb{Z}_2$ one-form subsystem symmetry. Moreover, we propose a new no-go theorem for finite-temperature quantum fracton topological order.
We study topological order in a toric code in three spatial dimensions, or a 3+1D Z_2 gauge theory, at finite temperature. We compute exactly the topological entropy of the system, and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature Tc, dropping to zero above Tc. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes, and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature, and its quantum and classical aspects.
Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. In this paper, we develop a unified framework based on variational iPEPS for the quantitative study of both topological and conventional phase transitions through entanglement order parameters. To this end, we employ tensor networks with suitable physical and/or entanglement symmetries encoded, and allow for order parameters detecting the behavior of any of those symmetries, both physical and entanglement ones. First, this gives rise to entanglement-based order parameters for topological phases. These topological order parameters allow to quantitatively probe topological phase transitions and to identify their universal behavior. We apply our framework to the study of the Toric Code model in different magnetic fields, which in some cases maps to the (2+1)D Ising model. We identify 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we moreover find an unknown critical exponent beta=0.021. We then apply our framework of entanglement order parameters to conventional phase transitions. We construct a novel type of disorder operator (or disorder parameter), which is non-zero in the disordered phase and measures the response of the wavefunction to a symmetry twist in the entanglement. We numerically evaluate this disorder operator for the (2+1)D transverse field Ising model, where we again recover a critical exponent hitherto unknown in the model, beta=0.024, consistent with the findings for the Toric Code. This shows that entanglement order parameters can provide additional means of characterizing the universal data both at topological and conventional phase transitions, and altogether demonstrates the power of this framework to identify the universal data underlying the transition.
The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional (2D) topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping, we demonstrate that the interaction induces a Majorana stripe state, a gapped state spontaneously breaking lattice translational and rotational symmetries, as opposed to the previously conjectured topological quantum criticality. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization.
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