arXiv subjects

Marius Wesle

Publications and source records attributed to Marius Wesle.

7 recordsLinked to original sources

arXiv2026

The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram

We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux $b$, we construct an open region in the three-dimensional parameter space $(b,\mu,\lambda)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane $\lambda=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2\pi\sigma^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

math-ph↗

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.

math-ph↗

arXiv2025

The generalized adiabatic theorem for extended lattice systems

We prove an adiabatic theorem for infinitely extended lattice fermion systems with gapped ground states, allowing perturbations that may close the gap. The Heisenberg dynamics on the CAR-algebra is generated by a time dependent two-parameter family of Hamiltonians $H^{\varepsilon,\eta}_t=\eta^{-1}(H_t+\varepsilon(H^1_t+V_t))$, where $H_t$ is assumed to have a gapped ground state $\omega_t$, $\eta \in (0,1]$ is the adiabatic parameter and $ \varepsilon \in [0,1]$ controls the strength of the perturbation. We construct a quasi-local dressing transformation $\beta^{\varepsilon,\eta}_t=\exp(i \mathcal{L}_{S^{\varepsilon,\eta}_t})$ that yields super-adiabatic states $\omega^{\varepsilon,\eta}_t =\omega_t \circ \beta^{\varepsilon,\eta}_t$ which, when tested against local observables, solve the corresponding time-dependent Schr\"odinger equation up to errors asymptotically smaller than any power of $\eta$ and $\varepsilon$. The construction is local in space and time, does not assume uniqueness of the ground state, and works under super-polynomial decay of the interactions $H_t$ and $H_t^1$ rather than exponential decay. If the Hamiltonian is time-independent on an interval, the dressed state is $\eta$-independent and forms a non-equilibrium almost-stationary state with lifetime of order $\varepsilon^{-\infty}$. The result provides a rigorous basis for linear response to macroscopic changes in gapped systems, including a proof of Ohm's law for macroscopic Hall currents.

math-ph↗

arXiv2025

Automorphic equivalence within gapped phases of infinitely extended fermion systems

We prove automorphic equivalence within gapped phases of infinitely extended lattice fermion systems (as well as spin systems) with super-polynomially decaying interactions. As a simple application, we prove a version of Goldstone's theorem for such systems: if an infinite volume interaction is invariant under a continuous symmetry, then any gapped ground state is also invariant under that symmetry.

math-ph↗

arXiv2025

A note on Hall conductance and Hall conductivity in interacting Fermion systems

In this note we consider lattice fermions on $\mathbb{Z}^2$ with a gapped ground state and show how to apply the NEASS approach to linear response to derive a formula for the Hall conductance in terms of the ground state expectation of a commutator of modified step functions. This formula is usually derived by a charge pumping argument going back to Laughlin. Here we show that it can also be obtained as the linear response coefficient of the microscopic current response to an adiabatic increase of the chemical potential on a half plane (or more generally on any cone-like region). Indeed, in a manner reminiscent of the bulk-boundary correspondence, we show that raising the chemical potential in any cone-like region gives rise to a current that flows along its boundary and is nearly linear in the increase in chemical potential. We also discuss the connection with the double commutator formula with modified position operators for the Hall conductivity derived in arXiv:2411.06967 as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.

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arXiv2024

Near linearity of the macroscopic Hall current response in infinitely extended gapped fermion systems

We consider an infinitely extended system of fermions on a $d$-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a locally unique gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $\varepsilon$ is of order $\mathcal{O}(\varepsilon^\infty)$, i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $\varepsilon$ up to terms of order $\mathcal{O}(\varepsilon^\infty)$. The proportionality factor $\sigma_\mathrm{H}$ is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $d=2$ the relevant observable that "measures" the Hall conductivity in experiments, the Hall conductance, not only agrees with $\sigma_{\mathrm{H}}$ in expectation up to $\mathcal{O}(\varepsilon^\infty)$, but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.

math-ph↗

arXiv2024

ftint: Calculating gradient-flow integrals with pySecDec

The program ftint is introduced which numerically evaluates dimensionally regulated integrals as they occur in the perturbative approach to the gradient-flow formalism in quantum field theory. It relies on sector decomposition in order to determine the coefficients of the individual orders in $\epsilon=(4-D)/2$, where $D$ is the space-time dimension. For that purpose, it implements an interface to the public library pySecDec. The current version works for massive and massless integrals up to three-loop level with vanishing external momenta, but the underlying method is extendable to more general cases.

hep-ph↗