Adiabatic currents for interacting fermions on a lattice

Abstract

We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [6], where Bachmann et al. proved an adiabatic theorem for interacting spin systems.As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.

Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results

“…the first order deviation from ideal adiabatic behavior. This could be done using the results of [39], which are a slight generalization of [8] in several directions: a super-adiabatic version of the theorem is formulated and proved which covers also the trace per unitvolume of extensive observables. This version is then used to derive Kubo's formula for conductance and conductivity not from adiabatic switching of a small potential, but as the adiabatic response for a Hamiltonian with time-dependent fluxes.…”

On ℤ2-indices for ground states of fermionic chains

“…Let {ω s } s∈[0,1] be any Θ-invariant continuous path of ground states connecting ω 0 and ω 1 . Then there is at least one s 0 ∈ (0, 1) such that ω s 0 cannot come from the ground state of a Θ-invariant and gapped interaction of the form (47).…”

“…Newer methods for higher-dimensional spin systems are also in development [25]. There has also been several results concerning stability of topological invariants such as the Hall conductance in interacting fermion systems [7,8,9,33,37,47].…”

Local Commuting Projector Hamiltonians and the Quantum Hall Effect

“…More precisely, the Hall conductance is defined in the thermodynamic limit L → ∞. If one assumes that the limiting ground state exists, and the spectral gap does not close in the limit L → ∞, one can indeed prove that the thermodynamic limit of the first Chern number of E exists and is equal to the Hall conductance [5,6,7]. (There is an alternative proof of the quantization of the Hall conductance which only requires H to be gapped, but does not make any assumption about the gap for nonzero β x , β y [8]).…”