Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond

Abstract

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gaugegravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gaugegravity actions for U(1) SPTs in ð4 þ 1ÞD via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context. [3][4][5]. However, even without symmetry breaking and without topological order, gapped systems can still be nontrivial if there is certain global symmetry protection, known as symmetry-protected topological states (SPTs) [6][7][8][9]. Their nontrivialness can be found in the gapless-topological boundary modes protected by a global symmetry, which shows gauge or gravitational anomalies . More precisely, they are short-range entangled states which can be deformed to a trivial product state by local unitary transformation [31][32][33] if the deformation breaks the global symmetry. Examples of SPTs are Haldane spin-1 chains protected by spin rotational symmetry [34,35] and the topological insulators [36][37][38] protected by fermion number conservation and time reversal symmetry.While some classes of topological orders can be described by topological quantum field theories (TQFT) [39][40][41][42], it is less clear how to systematically construct field theory with a global symmetry to classify or characterize SPTs for any dimension. This challenge originates from the fact that SPTs are naturally defined on a discretized spatial lattice or on a discretized spacetime path integral by a group cohomology construction [6,43] instead of continuous fields. Group cohomology construction of SPTs also reveals a duality between some SPTs and the Dijkgraaf-Witten topological gauge theory [43,44].Some important progress has been recently made to tackle the above question. For example, there are the ð2 þ 1ÞD[45] Chern-Simons theory [46][47][48][49][50], nonlinear sigma models [51,52], and an orbifolding approach implementing modular invariance on 1D edge modes [25,28]. The above approaches have their own benefits, but they may be either limited to certain dimensions, ...

Construction and field theory of bosonic-symmetry-protected topological states beyond group cohomology

“…We also note that Ref. 17 proposed the SPT states beyond group cohomology should have mixed gauge-gravity responses, which is also consistent with the formalism used in our current work.…”

Generalized global symmetries and holography

“…In particular, it would be interesting to understand if the tools of holography can be helpful in recent efforts to understand topological phases of matter (see e.g. [54]) and the phase structure of non-Abelian gauge theories from the point of view of generalized symmetry. More concretely, the existence of the higher-dimension tensor operator alluded to above is somewhat mysterious from the field-theoretical point of view.…”

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

“…1, 2 in dimensions higher than one all involve high order multiple spin interactions, and are thus unlikely to exist in realistic materials. Up to now, all approaches to classifying and characterizing BSPT states [1,2,[13][14][15][16] rely on mathematical or effective field theory descriptions, which shed little light on how to identify a realistic candidate BSPT state.…”