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arXiv:2211.16836 · Communications in Mathematical Physics

Adiabatic Evolution of Low-Temperature Many-Body Systems

Rafael L. Greenblatt, Markus Lange, Giovanna Marcelli, Marcello Porta

Published Mar 1, 2024• 3 citations
arXivHTMLPDF

Abstract

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

DOI
10.1007/s00220-023-04903-6

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Large scale dynamical response of interacting 1d1d1d Fermi systems
We consider the dynamics of a class of weakly interacting, gapless 1d1d1d fermionic systems, in presence of small external perturbations slowly varying in space and in time. We consider the evolution of the expectation values of the charge density and of the current density, in the thermodynamic limit and for low enough temperatures. We prove the validity and the asymptotic exactness of linear response in the limit of vanishing space-time variation of the perturbation, and we provide the explicit expression of the response of the system. The proof relies on the representation of the real time Duhamel expansion in terms of Euclidean correlation functions, for which we provide sharp estimates using rigorous renormalization group methods. The asymptotic exactness of linear response holds thanks to a cancellation for the scaling limit of the correlations that is reminiscent of bosonization, and which is derived rigorously using emergent chiral Ward identities.
Recent work by Greenblatt, Lange, Marcelli, and Porta (Commun. Math. Phys. 2024) provides the first rigorous justification of adiabatic theory for interacting quantum systems at positive temperature. Their analysis treats weakly interacting lattice fermions and combines a rigorous Wick rotation with fermionic cluster expansion techniques. Here, we study a class of weakly interacting quantum spin systems and derive response theory at positive temperature, thereby covering non-Fermi systems for the first time. Our proof identifies and exploits the locality property of local indistinguishability as a key ingredient in carrying out low-temperature adiabatic expansions. The fact that these systems satisfy local indistinguishability was recently established jointly with Adhikari (arXiv:2508.15907) using cluster expansion methods together with a probabilistic swapping trick.
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 consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable AAA at time ttt. By analyzing the convergence of the perturbative expansion in the interaction strength λλλ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime λ∣t∣2D+1=O(1)λ|t|^{2D+1}=\mathcal{O}(1)λ∣t∣2D+1=O(1), when the Hamiltonian is geometrically local on a DDD-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to λ∣t∣=O(1)λ|t|=\mathcal{O}(1)λ∣t∣=O(1) with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to λ∣t∣=O(1)λ|t|=\mathcal{O}(1)λ∣t∣=O(1), modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.
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 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.