Crystalline topological phases as defect networks

A crystalline topological phase is a topological phase with spatial symmetries. In this work, we give a very general physical picture of such phases: a topological phase with spatial symmetry $G$ (with internal symmetry $G_{\mathrm{int}} \leq G$) is described by a *defect network*: a $G$-symmetric network of defects in a topological phase with internal symmetry $G_{\mathrm{int}}$. The defect network picture works both for symmetry-protected topological (SPT) and symmetry-enriched topological (SET) phases, in systems of either bosons or fermions. We derive this picture both by physical arguments, and by a mathematical derivation from the general framework of [Thorngren and Else, Phys. Rev. X 8, 011040 (2018)]. In the case of crystalline SPT phases, the defect network picture reduces to a previously studied dimensional reduction picture, thus establishing the equivalence of this picture with the general framework of Thorngren and Else applied to crystalline SPTs.Comment: 13 pages + 2 pages of appendices. v3 published version, with better justification of the equivalence relatio

Collisionless dynamics of general non-Fermi liquids from hydrodynamics of emergent conserved quantities

Given the considerable theoretical challenges in understanding strongly coupled metals and non-Fermi liquids, it is valuable to have a framework to understand properties of metals that are universal, in the sense that they must hold in any metal. It has previously been argued that an infinite-dimensional emergent symmetry group is such a property, at least for clean, compressible metals. In this paper, we will show that such an emergent symmetry group has very strong implications for the dynamics of the metal. Specifically, we show that consideration of the hydrodynamics of the associated infinitely many emergent conserved quantities automatically recovers the collisionless Boltzmann equation that governs the dynamics of a Fermi liquid. Therefore, the hydrodynamic prediction is that in the low-temperature, collisionless regime where the emergent conservation laws hold, the dynamics and response to external fields of a general spinless metal will be identical to a Fermi liquid. We discuss some potential limitations to this general statement, including the possibility of non-hydrodynamic modes. We also report some interesting differences in the case of spinful metals.Comment: 10 pages + 3 pages appendices. v2 Some minor improvements and clarification

Pre-thermal phases of matter protected by time-translation symmetry

In a periodically driven (Floquet) system, there is the possibility for new phases of matter, not present in stationary systems, protected by discrete time-translation symmetry. This includes topological phases protected in part by time-translation symmetry, as well as phases distinguished by the spontaneous breaking of this symmetry, dubbed "Floquet time crystals". We show that such phases of matter can exist in the pre-thermal regime of periodically-driven systems, which exists generically for sufficiently large drive frequency, thereby eliminating the need for integrability or strong quenched disorder that limited previous constructions. We prove a theorem that states that such a pre-thermal regime persists until times that are nearly exponentially-long in the ratio of certain couplings to the drive frequency. By similar techniques, we can also construct stationary systems which spontaneously break *continuous* time-translation symmetry. We argue furthermore that for driven systems coupled to a cold bath, the pre-thermal regime could potentially persist to infinite time.Comment: Published version, with new title and introductio

Fragile topological phases in interacting systems

Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbors nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed "fragile topological phases." In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a $\mathrm{U}(1)$ charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g. electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons.Comment: 14 pages. Comments welcome; v2: several discussions are improve

Prethermal Strong Zero Modes and Topological Qubits

We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This non-equilibrium dynamical phenomenon is a form of prethermalization, and occurs because of obstructions to the equilibriation of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a non-perturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes, and on trapped ion chains.Comment: 20 pages. v2: reorganized and added overview sectio