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Register a new userTQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D
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Ryan Thorngren
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We derive a canonical form for 2-group gauge theory in 3+1D which shows they are either equivalent to Dijkgraaf-Witten theory or to the so-called EF1 topological order of Lan-Wen. According to that classification, recently argued from a different point of view by Johnson-Freyd, this amounts to a very large class of all 3+1D TQFTs. We use this canonical form to compute all possible anomalies of 2-group gauge theory which may occur without spontaneous symmetry breaking, providing a converse of the recent symmetry-enforced-gaplessness constraints of Cordova-Ohmori and also uncovering some possible new examples. On the other hand, in cases where the anomaly is matched by a TQFT, we try to provide the simplest possible such TQFT. For example, with anomalies involving time reversal, $mathbb{Z}_2$ gauge theory almost always works.
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We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.
Compact quantum electrodynamics (CQED$_3$) with Dirac fermionic matter provides an adequate framework for elucidating the universal low-energy physics of a wide variety of (2+1)D strongly correlated systems. Fractionalized states of matter correspond to its deconfined phases, where the gauge field is effectively noncompact, while conventional broken-symmetry phases are associated with confinement triggered by the proliferation of monopole-instantons. While much attention has been devoted lately to the symmetry classification of monopole operators in massless CQED$_3$ and related 3D conformal field theories, explicit derivations of instanton dynamics in parton gauge theories with fermions have been lacking. In this work, we use semiclassical methods analogous to those used by t Hooft in the solution of the $U(1)$ problem in 4D quantum chromodynamics (QCD) to explicitly demonstrate the symmetry-breaking effect of instantons in CQED$_3$ with massive fermions, motivated by a fermionic parton description of hard-core bosons on a lattice. By contrast with the massless case studied by Marston, we find that massive fermions possess Euclidean zero modes exponentially localized to the center of the instanton. Such Euclidean zero modes produce in turn an effective four-fermion interaction -- known as the t Hooft vertex in QCD -- which naturally leads to two possible superfluid phases for the original microscopic bosons: a conventional single-particle condensate or an exotic boson pair condensate without single-particle condensation.
We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin$_pm$ case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin$_-$ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the $mathbb{Z}_8$ classification of (1+1)d topological superconductors. We also compute the indicator formula of $mathbb{Z}_{16}$ valued time-reversal anomaly for (2+1)d pin$_+$ TQFT based on our construction.
Introducing both Berry curvature and chiral anomaly into Landaus Fermi-liquid theory, we construct a topological Fermi-liquid theory, applicable to interacting Weyl metals in the absence of time reversal symmetry. Following the Landaus Fermi-liquid theory, we obtain an effective free-energy functional in terms of the density field of chiral fermions. The density field of chiral fermions is determined by a self-consistent equation, minimizing the effective free-energy functional with respect to the order-parameter field. Beyond these thermodynamic properties, we construct Boltzmann transport theory to encode both the Berry curvature and the chiral anomaly in the presence of forward scattering of a Fermi-liquid state, essential for understanding dynamic correlations in interacting Weyl metals. This generalizes the Boltzmann transport theory for the Landaus Fermi-liquid state in the respect of incorporating the topological structure and extends that for noninteracting Weyl metals in the sense of introducing the forward scattering. Finally, we justify this topological Fermi-liquid theory, generalizing the first-quantization description for noninteracting Weyl metals into the second-quantization representation for interacting Weyl metals. First, we derive a topological Fermi-gas theory, integrating over high-energy electronic degrees of freedom deep inside a pair of chiral Fermi surfaces. As a result, we reproduce a topological Drude model with both the Berry curvature and the chiral anomaly. Second, we take into account interactions between such low-energy chiral fermions on the pair of chiral Fermi surfaces. We perform the renormalization group analysis, and find that only forward scattering turns out to be marginal above possible superconducting transition temperatures, justifying the topological Fermi-liquid theory of interacting Weyl metals with time reversal symmetry breaking.
We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group $G$. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: $rho$, a map from $G$ to the duality symmetry group of this $mathrm{U}(1)$ gauge theory which physically encodes how the symmetry permutes the fractional excitations, $ uinmathcal{H}^2_{rho}[G, mathrm{U}_mathsf{T}(1)]$, the symmetry actions on the electric charge, $pinmathcal{H}^1[G, mathbb{Z}_mathsf{T}]$, indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor $n$ over $mathcal{H}^3_{rho}[G, mathbb{Z}]$, the symmetry actions on the magnetic monopole. However, certain choices of $(rho, u, p, n)$ are not physically realizable, i.e. they are anomalous. We find that there are two levels of anomalies. The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly. States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state. If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar t Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion. States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state. We apply these results to some interesting physical examples.
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