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.
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 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.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
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.
This thesis develops a rigorous large-scale response theory for gapless lattice Fermi systems in low dimensions, focusing on the mathematical validity of linear response -- the Kubo formula -- for transport in gapless systems, where the absence of a spectral gap prevents the direct use of standard adiabatic theorems. After reviewing fundamental results on many-body lattice fermions, we first analyse the transport properties of non-interacting, gapless, one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators. We prove the validity of the Kubo formula, in the zero-temperature and infinite-volume limit, for perturbations that are weak and slowly varying in space and time. The proof relies on expressing the real-time Duhamel series in imaginary time, which ensures its convergence uniformly in the scaling parameter and system size, at low temperatures. This representation also reveals a key cancellation in the scaling limit, linked to the emergent anomalous chiral symmetry of relativistic one-dimensional fermions. As a consequence, in the combined low-temperature and scaling limit, the linear response provides the full physical response. In particular, the method yields a dynamical proof of the quantisation of edge conductance in two-dimensional quantum Hall systems. The analysis is then extended to weakly interacting, gapless one-dimensional fermionic systems, for which we obtain explicit expressions for the response functions in terms of renormalised Fermi velocities. Here, sharp estimates for interacting Euclidean correlation functions are established via rigorous renormalisation-group methods. The asymptotic exactness of linear response persists due to a cancellation in the scaling limit of the correlations -- reminiscent of bosonisation -- derived rigorously from emergent chiral Ward identities.
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 A at time t. By analyzing the convergence of the perturbative expansion in the interaction strength λ 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), when the Hamiltonian is geometrically local on a D-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) 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), 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.