Hall conductivities are important characterizations of phases of matter. It is known that nonzero Hall conductivities are difficult to realize in local commuting projector lattice models due to no-go theorems in ( 2 + 1 ) D . In this work we construct local commuting projector models in ( 2 + 1 ) D and ( 3 + 1 ) D with nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries on tensor product Hilbert space of finite local dimension. The model is given by a standard Z N toric code, but the symmetries do not admit expression in terms of on-site charge operators. The symmetry do not have local charges or currents on the lattice in the absence of boundaries, but there is still a notion of Hall conductivities that coincide with the continuum field theories. We construct protected gapless boundaries of the lattice models using modified Villain formalism. The generalized Hall conductivities are computed by surface currents as well as bulk flux insertion and many-body Chern number.
Hall conductivities are important characterizations of phases of matter. It is known that nonzero Hall conductivities are difficult to realize in local commuting projector lattice models due to no-go theorems in ( 2 + 1 ) D . In this work we construct local commuting projector models in ( 2 + 1 ) D and ( 3 + 1 ) D with nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries on tensor product Hilbert space of finite local dimension. The model is given by a standard Z N toric code, but the symmetries do not admit expression in terms of on-site charge operators. The symmetry do not have local charges or currents on the lattice in the absence of boundaries, but there is still a notion of Hall conductivities that coincide with the continuum field theories. We construct protected gapless boundaries of the lattice models using modified Villain formalism. The generalized Hall conductivities are computed by surface currents as well as bulk flux insertion and many-body Chern number.
By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create an exactly soluble path integral on spacetime lattice for Uκ(1) Symmetry Protected Topological (SPT) phases in 2+1 dimensions with a non-zero Hall conductance. We then convert the path integral on a 2+1d spacetime lattice into a 2d Hamiltonian lattice model, and show that the Hamiltonian consists of mutually commuting local projectors. We confirm the non-zero Hall conductance by calculating the Chern number of the exact ground state. It has recently been suggested that no commuting projector model can host a nonzero Hall conductance. We evade this no-go theorem by considering a rotor model, with a countably infinite number of states per site.
A bstract The Kapustin-Fidkowski no-go theorem forbids U(1) symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with U(1) global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d U(1) symmetric abelian topological orders of such kind, we can construct a lattice Hamiltonian that is controllably solvable at low energies, even though not “exactly” solvable; hence, this no-go theorem does not lead to significant difficulty in the lattice study of these topological orders. Moreover, for the fermionic topological orders in our construction, we introduce the lattice notion of spin-c structure — a concept important in the continuum that has previously not been adequately introduced in the lattice context.
We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases, which allow gapped boundaries without breaking the symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations.
It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of (2+1)-dimensional [(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2 symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the Z2 symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the (2+1)D topological superconductor protected by time-reversal symmetry T with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
1d lattice models for the boundary of 2d "Majorana" fermion SPTs: Kramers-Wannier duality as an exact Z2 symmetry
It is well known that symmetry protected topological (SPT) phases host non-trivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite symmetry action. This raises the question: can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of 2+1D fermion SPTs. We focus on the root SPT phase for the symmetry group G=Z2×Z2f. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective 1d lattice model for the boundary. Crucially, the Hilbert space of this 1d model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2 symmetry on this Hilbert space and find a simple 3-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising CFT where the Z2 symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the 2+1D topological superconductor protected by time-reversal symmetry T with T2=(−1)F.
We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group ${G}_{f}=G\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where G is finite and Z2f is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface M on which our model is defined, namely a ``Kasteleyn'' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the Z8 classification of 2D fermionic SPTs with G=Z2.