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arXiv:1907.10098 · Physical review. B./Physical review. B

Topological and conventional phases of a three-dimensional electronic glass

Prateek Mukati, Adhip Agarwala, Subhro Bhattacharjee

Published Jan 27, 2020• 25 citations
arXivHTMLPDF

Abstract

We investigate a symmetry-protected ${Z}_{2}$ topological electron glass, a glassy equivalent of the ${Z}_{2}$ topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the ${Z}_{2}$ glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with $\frac{1}{2}$ electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the ${Z}_{2}$ glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.

DOI
10.1103/physrevb.101.035142

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We investigate a symmetry-protected Z2{Z}_{2}Z2​ topological electron glass, a glassy equivalent of the Z2{Z}_{2}Z2​ topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the Z2{Z}_{2}Z2​ glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with 12\frac{1}{2}21​ electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the Z2{Z}_{2}Z2​ glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.
Recent theoretical studies predict that structural disorder, serving as a bridge connecting a crystalline material to an amorphous material, can induce a topological insulator from a trivial phase. However, to experimentally observe such a topological phase transition is very challenging due to the difficulty in controlling structural disorder in a quantum material. Given experimental realization of randomly positioned Rydberg atoms, such a system is naturally suited to studying structural disorder induced topological phase transitions and topological amorphous phases. Motivated by the development, we study topological phases in an experimentally accessible one-dimensional amorphous Rydberg atom chain with random atom configurations. In the single-particle level, we find symmetry-protected topological amorphous insulators and a structural disorder induced topological phase transition, indicating that Rydberg atoms provide an ideal platform to experimentally observe the phenomenon using state-of-the-art technologies. Furthermore, we predict the existence of a gapless symmetry-protected topological phase of interacting bosons in the experimentally accessible system. The resultant many-body topological amorphous phase is characterized by a Z_{2} invariant.
Band topology is traditionally analyzed in terms of gauge-invariant observables associated with crystalline Bloch wave functions. Recent work has demonstrated that many of the free fermion topological characteristics survive even in an amorphous setting. In this Letter, we extend these studies to incorporate the effect of strong repulsive interactions on the fate of topology and other correlation induced phenomena. Using a parton-based mean-field approach, we obtain the interacting phase diagram for an electronic two-orbital model with tunable topology in a two-dimensional amorphous network. In addition to the (non-)topological phases that are adiabatically connected to the free fermion limit, we find a number of strongly interacting amorphous analogs of crystalline Mott insulating phases with nontrivial chiral neutral edge modes, and a fractionalized Anderson insulating phase. The amorphous networks thus provide a new playground for studying a plethora of exotic states of matter, and their glassy dynamics, due to the combined effects of nontrivial topology, disorder, and strong interactions.
We introduce a novel class of interaction-enabled topological crystalline insulators in two- and three-dimensional electronic systems, which we call ``topological crystalline magnet.'' It is protected by the product of the time-reversal symmetry T\mathcal{T}T and a mirror symmetry or a rotation symmetry R\mathcal{R}R. A topological crystalline magnet exhibits two intriguing features: (i) it cannot be adiabatically connected to any Slater insulator and (ii) the edge state is robust against coupling electrons to the edge. These features are protected by the anomalous symmetry transformation property ${(\mathcal{R}\mathcal{T})}^{2}=\ensuremath{-}1$ of the edge state. An anisotropic response to the external magnetic field can be an experimental signature.
We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z8{Z}_{8}Z8​. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.
The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.