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arXiv:1612.01505 · Physical Review Letters

Adiabatic Theorem for Quantum Spin Systems

Sven Bachmann, Wojciech De Roeck, Martin Fraas

Published Aug 11, 2017• 69 citations
arXivHTMLPDF

Abstract

The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ϵ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.

DOI
10.1103/physrevlett.119.060201

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The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ϵ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.
We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalised super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem holds also for certain perturbations of gapped ground states that close the spectral gap (so it is an adiabatic theorem also for resonances and in this sense `generalised'), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called `super-adiabatic'). In addition to existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations also for infinite systems. While we consider the result and its proof as new and interesting in itself, they also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
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 the initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [ 10 ], 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.
We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.
The quantum adiabatic theorem is a fundamental result in quantum mechanics, with a multitude of applications, both theoretical and practical. Here, we investigate the dynamics of adiabatic processes for quantum many-body systems by analyzing the properties of observable-free intensive quantities. In particular, we study the adiabatic rate function f ( T , Δ λ ) in dependence of the ramp time T , which gives us a complete characterization of the many-body adiabatic fidelity as a function of T and the strength of the parameter displacement Δ λ . f ( T , Δ λ ) quantifies the deviation from adiabaticity for a given process and therefore allows us to control and define the notion of adiabaticity in many-body systems. First we study f ( T , Δ λ ) for the one-dimensional transverse field Ising model and the Luttinger liquid, both of which are quadratic systems and therefore allow us to look at the thermodynamic limit. For ramps across gapped phases, we relate f ( T , Δ λ ) to the transition probability of the system and for ramps across a gapless point, or gapless phase we relate it to the excitation density of the relevant quasiparticles. Then we investigate the XXZ model, which allows us to see the qualitative features that survive when interactions are turned on. Several key results in the literature regarding the interplay of the thermodynamic and the adiabatic limit are obtained as inferences from the properties of f ( T , Δ λ ) in the large- T limit.
Establishing quantitative adiabaticity criteria at finite temperature remains substantially less developed than in the pure-state setting, even though realistic quantum systems are never at absolute zero. Here, by combining a mixed-state quantum speed limit with mixed-state fidelity susceptibility within the Liouville-space formulation of quantum mechanics, we derive rigorous bounds on the Hilbert-Schmidt fidelity between mixed states. Focusing on protocols that drive an initial Gibbs state toward a quasi-Gibbs target, these bounds yield an explicit threshold driving rate for the onset of nonadiabaticity. For a broad class of local Hamiltonians in gapped phases, we show that, in the thermodynamic limit, the threshold driving rate factorizes into a system-size contribution that recovers the zero-temperature scaling and a universal temperature-dependent factor. The latter is exponentially close to unity at low temperature, whereas at high temperature it is linear in temperature. We verify the predicted scaling in several spin-1/2 chains by obtaining closed-form expressions for the threshold driving rate. Our results provide a practical and largely model-independent criterion for finite-temperature adiabaticity in driven many-body systems under closed-system unitary evolution.