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arXiv:2208.13785 · Journal of High Energy Physics

Fractional Hall conductivity and spin-c structure in solvable lattice Hamiltonians

Zhaoyu Han, Jingyuan Chen

Published Feb 13, 2023• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep02(2023)130

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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.
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)U^κ(1)Uκ(1) Symmetry Protected Topological (SPT) phases in 2+12+12+1 dimensions with a non-zero Hall conductance. We then convert the path integral on a 2+12+12+1d spacetime lattice into a 222d 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.
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.
A (2+1)D topological ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It is recently discovered that there are "higher" versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states, which obstructs symmetry-preserving gapped edge state beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the "partial rotation" unitary which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with ν=1/3ν=1/3ν=1/3, 1/51/51/5, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determines if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented.
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of GGG-crossed braided tensor categories (G×G\timesG×BTCs) for SETs. We consider systems with U(1)U(1)U(1) charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all 555 orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains 999 terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the G×G\timesG×BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the G×G\timesG×BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.
The Z2{Z}_{2}Z2​ topological order in Z2{Z}_{2}Z2​ spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.