Rational indices for quantum ground state sectors

Abstract

We consider charge transport for interacting many-body systems with a gapped ground state subspace which is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of $1/p$, where $p$ is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb-Schultz-Mattis theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron-Dana-Zak relation between Hall conductance and the filling factor.

v3: Lemma 4.1 corrected. v2: New clustering result, Proposition 3.1

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Cited by in corpus (9)

• Vanishing Hall conductance for commuting Hamiltonians
• The spectral gap of a fractional quantum Hall system on a thin torus
• Exactness of linear response in the quantum Hall effect
• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Dynamical abelian anyons with bound states and scattering states
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Fractional index of Bargmann-Fock space and Landau levels