arXiv subjects

Tom Wessel

Publications and source records attributed to Tom Wessel.

10 recordsLinked to original sources

arXiv2025

Dynamics generated by spatially growing derivations on quasi-local algebras

We prove global existence and uniqueness of dynamics on the quasi-local algebra $\mathcal{A}$ of a quantum lattice system for spatially growing derivations $\mathcal{L}_\Phi = \sum_x [ \Phi_x , \cdot ]$. Existing results assume that the local terms $\Phi_x\in\mathcal{A}$ of the generator are uniformly bounded in space with respect to appropriate weighted norms $\lVert \Phi_x \rVert_{G,x}$. Analogous to the global existence result for first order ODEs, we show that global existence and uniqueness persist if the size of the local terms $\lVert \Phi_x \rVert_{G,x}$ grows at most linearly in space. This considerably enlarges the class of derivations known to have well-defined dynamics. Moreover, we obtain Lieb-Robinson bounds with exponential light cones for such dynamics. For the proof, we assume Lieb-Robinson bounds with linear light cones for dynamics, whose generators have uniformly bounded local terms. Such bounds are known to hold, for example, if the local terms are of finite range or exponentially localized.

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arXiv2025

Uniform-in-temperature locality estimates for weakly interacting quantum systems

The locality of thermal quantum states has emerged as a key input for applications to thermalization, response theory, and efficient simulability. Locality is either captured by the decay of correlations or by local indistinguishability, which allows to approximate local expectation values by those of local thermal states. Most techniques for deriving locality bounds deteriorate at small temperature, a physically highly relevant regime and so it is of interest to identify conditions for uniform-in-temperature bounds. Here we prove that a class of weakly interacting quantum Hamiltonians satisfies exponential decay of correlations and local indistinguishability uniformly in the temperature. The proof uses a low-temperature cluster expansion and a quantum version of a probabilistic swapping trick developed by the first author and Cao (Ann. Probab. 53, 2025) in the context of lattice gauge theories.

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arXiv2025

Gaussian filters in quantum lattice systems: Applications to spectral flow, local perturbations, clustering, and the quantum Hall effect

We consider the locality and spectral properties of the smearing \[ \tau_f(A) = \int_{-\infty}^\infty dt \, f(t) \, \tau_t(A) \] when applied to the dynamics $\tau_t$ of quantum spin systems. While recent applications of this map have used superpolynomially but not exponentially decaying functions $f$ to ensure exact spectral properties, we use here Gaussian filters. This improves the locality at the expense of errors on the spectral side. We propose a number of concrete applications, from quasi-adiabatic continuation to correlation decay, and exponential stability away from impurities. Finally, we discuss an application to the quantum Hall effect.

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arXiv2025

Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions

We prove a Lieb-Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb-Robinson bound is based on the work of Else et al. (2020), and our results also apply to spin systems. We explain why some newer Lieb-Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.

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arXiv2024

Enhanced Lieb-Robinson bounds for commuting long-range interactions

Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In $D$ spatial dimensions, interactions decaying as a power-law $r^{-\alpha}$ with $\alpha > 2 D+1$ exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold $2D +1$ is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form $t \, r^{-\alpha}$ for any $\alpha>0$, and this scaling is sharp. In particular, the linear light cone occurs at $\alpha = 1$ in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.

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arXiv2024

Response theory for locally gapped systems

We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state $\rho_0$ of a Hamiltonian $H_0$ is locally gapped in $\Lambda^{\mathrm{gap}} \subset \Lambda$, whenever the Liouvillian $- \mathrm{i} \, [H_0, \, \cdot \, ]$ is almost invertible on local observables supported in $\Lambda^{\mathrm{gap}}$ when tested in $\rho_0$. To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation $\epsilon V$ acts in a region which is further than $|\log \epsilon|$ away from the non-gapped region $\Lambda \setminus \Lambda^{\mathrm{gap}}$.

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arXiv2024

Equality of magnetization and edge current for interacting lattice fermions at positive temperature

We prove that the magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch's theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.

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arXiv2023

From decay of correlations to locality and stability of the Gibbs state

We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb-Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.

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arXiv2022

On adiabatic theory for extended fermionic lattice systems

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite as well as infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this note is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.

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arXiv2021

Local stability of ground states in locally gapped and weakly interacting quantum spin systems

Based on a result by Yarotsky (J. Stat. Phys. 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.

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