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Symmetry and Topology in Quantum Matter

28 Apr 2020-
TL;DR: In this article, the authors proposed a sufficient criterion that can efficiently determine whether a three-dimensional crystal is a topological semimetal or not, based on the compatibility of different expressions for a special topological invariant, the quantized bulk average value of the effective axion field.
Abstract: Quantum states of matter at zero temperature are called quantum phases, which are characterized by their symmetries and topology. If two quantum phases cannot be related by symmetry-preserving continuous transformations, they are defined to be topologically distinct. The zero-temperature transitions among quantum phases are quantum phase transitions. Quantum phase transitions that happen among topologically distinct phases are called topological quantum phase transitions. The study of topological quantum phases and topological quantum phase transitions is a central topic in condensed matter physics.My research on quantum phases has been focused on the theoretical study of topologically nontrivial quantum phases in the crystals governed by non-interacting Hamiltonians, including topological insulator phases, topological semimetal phases, topological superconductor phases, and so on. First, we proposed a sufficient criterion that can efficiently determine whether a three-dimensional crystal is a topological semimetal or not, based on the compatibility of different expressions for a special topological invariant, the quantized bulk average value of the effective axion field. Second, we predicted the existence of various topological insulator phases and topological semimetal phases in half-Heusler materials. Third, we proposed the magnetic-resonance-induced current as a feasible experimental probe of a special kind of topological insulators, called the axion insulators. Fourth, we proposed a new pairing mixing mechanism, the singlet-quintet mixing, for superconductors with spin-3/2 fermions, demonstrated the topological superconductor phase induced by it, and studied various properties of it, including spin-susceptibility, upper critical field, stability against the disorder, and surface Majorana flat bands.Besides the topological quantum phases, I have also worked on quantum phase transitions. First, we constructed the first theoretical model for the emergent supersymmetry at a discontinuous quantum phase transition and proposed to realize it on the surface of a topological superconductor. Second, we proposed the discontinuous change of piezoelectricity as a probe of two-dimensional topological quantum phase transitions between insulating phases, through a systematic study of all the relevant gap closing cases. In this dissertation, I first briefly introduce the basic concepts in the field of topological quantum phases and topological quantum phase transitions with a focus on the crystals governed by noninteracting Hamiltonians. Then, I review in detail all my first-authored research works mentioned above. My other research works are briefly mentioned.
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Posted Content
TL;DR: In this paper, a higher-categorical generalization of the Karoubi envelope construction from ordinary category theory is presented, where the idempotents in the ordinary version are replaced with a notion that they call "condensations".
Abstract: We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions.

50 citations

Journal ArticleDOI
TL;DR: Wang et al. as discussed by the authors introduced a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator, and explicitly showed that both the gapped and gapless boundaries of their model are consistent with those of band-theoretic, weakly interacting topological supercondulators.
Abstract: Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy $\mathcal{CP}$ symmetry, leading to intuitive interpretations of the bulk invariant for a $\mathcal{CP}$-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.

5 citations

Journal ArticleDOI
TL;DR: In this article, the boundary supersymmetry of one-dimensional fermionic phases beyond SPT phases was investigated using the connection between Majorana edge modes and real supercharges.
Abstract: It has recently been demonstrated that protected supersymmetry emerges on the boundaries of one-dimensional intrinsically fermionic symmetry protected trivial (SPT) phases. Here we investigate the boundary supersymmetry of one-dimensional fermionic phases beyond SPT phases. Using the connection between Majorana edge modes and real supercharges, we compute, in terms of the bulk phase invariants, the number of protected boundary supercharges.

5 citations

Posted Content
09 Dec 2020
TL;DR: In this article, it was shown that boundary supersymmetry is neither protected nor forbidden by the bulk SRE phase, and hence amounts to a choice of boundary condition, and characterized this choice as a group cohomology class.
Abstract: It has recently been demonstrated that protected supersymmetry emerges on the boundaries of one-dimensional intrinsically fermionic symmetry protected trivial (SPT) phases. Here we investigate the case of one-dimensional fermionic short-range entangled (SRE) phases beyond SPT phases. We show that for SRE phases that are not SPT phases, boundary supersymmetry is neither protected nor forbidden by the bulk SRE phase, and hence amounts to a choice of boundary condition. We characterize this choice as a group cohomology class and study the behavior of supersymmetric boundary conditions under the stacking of the bulk SRE phases.

4 citations


Cites background from "Symmetry and Topology in Quantum Ma..."

  • ...Simplest among these phases are short-range entangled (SRE) phases, which are essentially trivial in their bulks but host topologically protected phenomena on their boundaries [1]....

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  • ...SRE phases may be stacked together, an operation that makes them into an abelian group (hence, their alternative name “invertible phases”) [1, 4]....

    [...]

Posted Content
TL;DR: In this paper, the authors constructed infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies.
Abstract: We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary $\mathbb{Z}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $\pi$ statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.