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arXiv:1905.05790 · Physical review. B./Physical review. B

UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases

Nick Bultinck

Published Oct 18, 2019• 18 citations
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Abstract

Symmetry-protected phases are gapped phases of matter which are distinguished only in the presence of a global symmetry $G$. These quantum phases lack any symmetry-breaking or topological order and have short-range entangled ground states. Based on this short-range entanglement property, we give a general argument for the existence of an emergent antiunitary (and sometimes also a unitary) ${\mathbb{Z}}_{2}$ symmetry at a critical point separating two different bosonic symmetry-protected phases in any dimension. Often, the emergent symmetry group at criticality has a mixed global anomaly. For those phases classified by group cohomology, we identify a criterion for when such a mixed global anomaly is present, and write down representative cocycles for the corresponding anomaly class. We illustrate our results with a series of examples and make connections to recent results on $(2+1)$-dimensional beyond-Landau critical points.

DOI
10.1103/physrevb.100.165132

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Symmetry-protected phases are gapped phases of matter which are distinguished only in the presence of a global symmetry GGG. These quantum phases lack any symmetry-breaking or topological order and have short-range entangled ground states. Based on this short-range entanglement property, we give a general argument for the existence of an emergent antiunitary (and sometimes also a unitary) Z2{\mathbb{Z}}_{2}Z2​ symmetry at a critical point separating two different bosonic symmetry-protected phases in any dimension. Often, the emergent symmetry group at criticality has a mixed global anomaly. For those phases classified by group cohomology, we identify a criterion for when such a mixed global anomaly is present, and write down representative cocycles for the corresponding anomaly class. We illustrate our results with a series of examples and make connections to recent results on (2+1)(2+1)(2+1)-dimensional beyond-Landau critical points.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+12+12+1 dimensions, the quantum phases in the presence of a global symmetry group SGSGSG, and with topological order described by a finite gauge group GGGGGG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1d+1d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
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Edge theories of symmetry-protected topological phases are well known to possess global symmetry anomalies. In this Letter we focus on two-dimensional bosonic phases protected by an on-site symmetry and analyze the corresponding edge anomalies in more detail. Physical interpretations of the anomaly in terms of an obstruction to orbifolding and constructing symmetry-preserving boundaries are connected to the cohomology classification of symmetry-protected phases in two dimensions. Using the tensor network and matrix product state formalism we numerically illustrate our arguments and discuss computational detection schemes to identify symmetry-protected order in a ground state wave function.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG. 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)\mathit{SO}(3)SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1)U(1)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 ddd-dimensional bosonic SPT phases with on-site symmetry GGG (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H1+d[G,UT(1)]{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]H1+d[G,UT​(1)], the Borel (1+d)(1+d)(1+d)-group-cohomology classes of GGG over the GGG module UT(1){U}_{T}(1)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 $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry GGG 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 ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal Z2T{Z}_{2}^{T}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 H1+d[Z2T,UT(1)]{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]H1+d[Z2T​,UT​(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 ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and U(1)U(1)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: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where GH{G}_{H}GH​ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
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 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 phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. 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 anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. 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_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.