Symmetry protected topological orders and the group cohomology of their symmetry group
TL;DR: In this paper, it was shown that the boundary excitations of SPT phases can be described by a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models.
Abstract: Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H^(1+d)[G,UT(1)], the Borel (1+d)-group-cohomology classes of G over the G module UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ term in continuous nonlinear σ models to lattice nonlinear σ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear σ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use H^(1+d)[U(1)⋊ Z^(T)_(2),U_T(1)] to obtain interacting bosonic topological insulators (protected by time reversal Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H^(1+d)[Z^(T)_(2),U_T(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H1+d[U(1)×Z2T,UT(1)] to construct the SPT phases of integer spin systems with time-reversal and U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H,G_Ψ,H^(1+d)[G_Ψ,U_T(1)]), where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.
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TL;DR: A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. strings, membranes, etc. as discussed by the authors, which leads to Ward identities and hence to selection rules on amplitudes.
Abstract: A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a sub-group). They can also have ’t Hooft anomalies, which prevent us from gauging them, but lead to ’t Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.
952 citations
TL;DR: In this paper, the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed.
Abstract: The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in single-band systems are detailed, and the manner in which a given physical context may affect the symmetry-based topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulk-boundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.
758 citations
Cites background from "Symmetry protected topological orde..."
...(Chen et al., 2013; Chiu et al., 2016)....
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...This phenomenon defines the notion of symmetry-protected topological (SPT) phases (Chen et al., 2013; Chiu et al., 2016)....
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TL;DR: A global symmetry is a symmetry for which the charged operators are of space-time dimension $q$; e.g., strings, membranes, etc. as mentioned in this paper Theorem 1.
Abstract: A $q$-form global symmetry is a global symmetry for which the charged operators are of space-time dimension $q$; e.g. Wilson lines, surface defects, etc., and the charged excitations have $q$ spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries ($q$=0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.
616 citations
Cites background from "Symmetry protected topological orde..."
...A Symmetry Protected Topological (SPT) phase of matter is a gapped phase, which is indistinguishable from the trivial phase, if one disregards symmetry (in particular, the partition function of the low-energy TQFT is the identity on any closed manifold), but cannot be deformed to the trivial phase without undergoing a quantum phase transition and while preserving the symmetry [27]....
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TL;DR: It is shown that acoustic waves provide a fertile ground to apply the anomalous physics of Floquet topological insulators, and their relevance for a wide range of acoustic applications, including broadband acoustic isolation and topologically protected, nonreciprocal acoustic emitters.
Abstract: The unique conduction properties of condensed matter systems with topological order have recently inspired a quest for the similar effects in classical wave phenomena. Acoustic topological insulators, in particular, hold the promise to revolutionize our ability to control sound, allowing for large isolation in the bulk and broadband one-way transport along their edges, with topological immunity against structural defects and disorder. So far, these fascinating properties have been obtained relying on moving media, which may introduce noise and absorption losses, hindering the practical potential of topological acoustics. Here we overcome these limitations by modulating in time the acoustic properties of a lattice of resonators, introducing the concept of acoustic Floquet topological insulators. We show that acoustic waves provide a fertile ground to apply the anomalous physics of Floquet topological insulators, and demonstrate their relevance for a wide range of acoustic applications, including broadband acoustic isolation and topologically protected, nonreciprocal acoustic emitters.
498 citations
TL;DR: In this paper, the authors describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions, and clarify the precise meaning of the statement that in the bulk of a 3D topological insulator, the electromagnetic -angle is equal to.
Abstract: Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic -angle is equal to .
431 citations
Cites background from "Symmetry protected topological orde..."
...Symmetry-protected topological (SPT) bosonic phases of matter can be characterized by anomalies [1]....
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References
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TL;DR: Graphene is converted from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator and the spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.
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TL;DR: In this article, the effects of particle growth with momentum on information spreading near black hole horizons were investigated. But the authors only considered the earliest times of the propagation of information near the horizon.
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