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arXiv:1506.00592

Protection of topological order by symmetry and many-body localization

Andrew C. Potter, Ashvin Vishwanath

Published Jun 1, 2015• 37 citations
arXivHTMLPDF

Abstract

In closed quantum systems, strong randomness can localize many-body excitations, preventing ergodicity. An interesting consequence is that high energy excited states can exhibit quantum coherent properties, such as symmetry protected topological (SPT) order, that otherwise only occur in equilibrium ground states. Here, we ask: which types of SPT orders can be realized in highly excited states of a many-body (MB) localized system? We argue that this question is equivalent to whether an SPT order can be realized in an exactly solvable lattice model of commuting projectors. This perspective enables a sharp definition of MB localizability. Using this criterion, it is straightforward to establish that whereas all bosonic SPTs in spatial dimensions $d=1,\,2,\,3$ are MB localizable, chiral phases (e.g. quantum Hall fluids) are not. We also show that free fermion SPTs in $d >1$ (including topological insulators and superconductors) cannot be localized if interactions are weak. A key question is whether strong interactions can render them MB localizable, which we study in the context of a class of $d=2$ topological superconductors. Using a decorated domain wall (DDW) approach we show that some phases in this class are MB localizable, when they correspond to bosonic SPT orders. However, a similar DDW approach faces a fatal obstruction to realizing certain intrinsically fermionic SPT orders, an issue we argue may persist beyond this specific construction.

DOI
10.48550/arxiv.1506.00592

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In closed quantum systems, strong randomness can localize many-body excitations, preventing ergodicity. An interesting consequence is that high energy excited states can exhibit quantum coherent properties, such as symmetry protected topological (SPT) order, that otherwise only occur in equilibrium ground states. Here, we ask: which types of SPT orders can be realized in highly excited states of a many-body (MB) localized system? We argue that this question is equivalent to whether an SPT order can be realized in an exactly solvable lattice model of commuting projectors. This perspective enables a sharp definition of MB localizability. Using this criterion, it is straightforward to establish that whereas all bosonic SPTs in spatial dimensions d=1, 2, 3d=1,\,2,\,3d=1,2,3 are MB localizable, chiral phases (e.g. quantum Hall fluids) are not. We also show that free fermion SPTs in d>1d >1d>1 (including topological insulators and superconductors) cannot be localized if interactions are weak. A key question is whether strong interactions can render them MB localizable, which we study in the context of a class of d=2d=2d=2 topological superconductors. Using a decorated domain wall (DDW) approach we show that some phases in this class are MB localizable, when they correspond to bosonic SPT orders. However, a similar DDW approach faces a fatal obstruction to realizing certain intrinsically fermionic SPT orders, an issue we argue may persist beyond this specific construction.
Using the decorated domain wall procedure, we construct finite-depth local unitaries that realize fermionic symmetry-protected topological (SPT) phases. This results in explicit full commuting projector Hamiltonians, where ``full'' implies the fact that the ground state as well as all excited states of these Hamiltonians realize the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by the symmetry group $G={\mathbb{Z}}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$, which also has a free fermion realization in class BDI, and by the symmetry group $G={\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}^{F}$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of p+ipp+ipp+ip superconductors combined with their $p\ensuremath{-}ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
Symmetry protected topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper, we describe the classification and properties of such phases for three-dimensional (3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free-fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases, we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1)U(1)U(1) subgroup we show that this classification is complete. We describe the nontrivial surface and bulk properties of these states. In particular, we discuss interesting correlated surface states that are not captured in a free-fermion description. We show that in many, but not all, cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.
We study the excited states of interacting fermions in one dimension with particle-hole symmetric disorder (equivalently, random-bond XXZ chains) using a combination of renormalization group methods and exact diagonalization. Absent interactions, the entire many-body spectrum exhibits infinite-randomness quantum critical behavior with highly degenerate excited states. We show that though interactions are an irrelevant perturbation in the ground state, they drastically affect the structure of excited states: Even arbitrarily weak interactions split the degeneracies in favor of thermalization (weak disorder) or spontaneously broken particle-hole symmetry, driving the system into a many-body localized spin glass phase (strong disorder). In both cases, the quantum critical properties of the noninteracting model are destroyed, either by thermal decoherence or spontaneous symmetry breaking. This system then has the interesting and counterintuitive property that edges of the many-body spectrum are less localized than the center of the spectrum. We argue that our results rule out the existence of certain excited state symmetry-protected topological orders.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.