arXiv subjects

Stefan Teufel

Publications and source records attributed to Stefan Teufel.

At least 19 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 $(b,μ,λ)$ 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 $λ=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2πσ^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

math-ph↗

arXiv2025

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.

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 $\mathcal{L}_Φ= \sum_x [ Φ_x , \cdot ]$. Existing results assume that the local terms $Φ_x\in\mathcal{A}$ of the generator are uniformly bounded in space with respect to appropriate weighted norms $\lVert Φ_x \rVert_{G,x}$. 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 $\lVert Φ_x \rVert_{G,x}$ 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 $H^{\varepsilon,η}_t=η^{-1}(H_t+\varepsilon(H^1_t+V_t))$, where $H_t$ is assumed to have a gapped ground state $ω_t$, $η\in (0,1]$ is the adiabatic parameter and $ \varepsilon \in [0,1]$ controls the strength of the perturbation. We construct a quasi-local dressing transformation $β^{\varepsilon,η}_t=\exp(i \mathcal{L}_{S^{\varepsilon,η}_t})$ that yields super-adiabatic states $ω^{\varepsilon,η}_t =ω_t \circ β^{\varepsilon,η}_t$ which, when tested against local observables, solve the corresponding time-dependent Schrödinger equation up to errors asymptotically smaller than any power of $η$ 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 $η$-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

Adiabatically driven dissipative many-body quantum spin systems

We explore the evolution of a strongly interacting dissipative quantum Ising spin chain that is driven by a slowly varying time-dependent transverse field. This system possesses an extensive number of instantaneous (adiabatic) stationary states which are coupled through non-adiabatic transitions. We analytically calculate the generator of the ensuing slow dynamics and analyze the creation of coherences through non-adiabatic processes. For a certain choice of the transverse field shape, we show that the system solely undergoes transitions among classical basis states after each pulse. The concatenation of many of such pulses leads to an evolution of the spin chain under a many-body dynamics that features kinetic constraints. Our setting not only allows for a quantitative investigation of adiabatic theorems and non-adiabatic corrections in a many-body scenario. It also directly connects to many-body systems in the focus of current research, such as ensembles of interacting Rydberg atoms which are resonantly excited by a slowly varying laser pulse and subject to dephasing noise.

quant-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.

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

Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

We say of an isolated macroscopic quantum system in a pure state $ψ$ that it is in macroscopic thermal equilibrium (MATE) if $ψ$ lies in or close to a suitable subspace $\mathcal{H}_{eq}$ of Hilbert space. It is known that every initial state $ψ_0$ will eventually reach and stay there most of the time (``thermalize'') if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation $H_θ^{fF}$ of the Hamiltonian $H_0^{fF}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{fF}$. Here, we first point out that also for degenerate Hamiltonians all $ψ_0$ thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $H_0^{fF}$. Inspired by the fact that there is one eigenbasis of $H_0^{fF}$ for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given $H_0$ that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $H_0$ lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $H=H_0+λV$ with $λ\ll 1$, for most perturbations $V$ the perturbed Hamiltonian $H$ satisfies ETH and all states thermalize.

cond-mat.stat-mech↗

arXiv2025

Existence of Schrodinger Evolution with Absorbing Boundary Condition

Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $ψ$ expressed by a Schrödinger equation in $Ω$ together with an ``absorbing'' boundary condition on $\partial Ω$ first considered by Werner in 1987, viz., $\partial ψ/\partial n=iκψ$ with $κ>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $ψ$; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.

math-ph↗

arXiv2025

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.

math-ph↗

arXiv2025

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 $σ_\mathrm{H}$ 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 $σ_{\mathrm{H}}$ 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↗

arXiv2025

Computing the spectrum and pseudospectrum of infinite-volume operators from local patches

We show how the spectrum of normal discrete short-range infinite-volume operators can be approximated with two-sided error control using only data from finite-sized local patches. As a corollary, we prove the computability of the spectrum of such infinite-volume operators with the additional property of finite local complexity and provide an explicit algorithm. Such operators appear in many applications, e.g. as discretizations of differential operators on unbounded domains or as so-called tight-binding Hamiltonians in solid state physics. For a large class of such operators, our result allows for the first time to establish computationally also the absence of spectrum, i.e. the existence and the size of spectral gaps. We extend our results to the $\varepsilon$-pseudospectrum of non-normal operators, proving that also the pseudospectrum of such operators is computable.

math.SP↗

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.

math-ph↗

arXiv2024

From orbital magnetism to bulk-edge correspondence

By extending the gauge covariant magnetic perturbation theory to operators defined on half-planes, we prove that for $2d$ random ergodic magnetic Schrödinger operators, the zero-temperature bulk-edge correspondence can be obtained from a general bulk-edge duality at positive temperature involving the bulk magnetization and the total edge current. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a ``zero-temperature" limit.

math-ph↗

arXiv2024

Typical Macroscopic Long-Time Behavior for Random Hamiltonians

We consider a closed macroscopic quantum system in a pure state $ψ_t$ evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces $\mathcal{H}_ν$ (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of $ψ_t$ looks like macroscopically, specifically on how much of $ψ_t$ lies in each $\mathcal{H}_ν$. Previous bounds concerned the \emph{absolute} error for typical $ψ_0$ and/or $t$ and are valid for arbitrary Hamiltonians $H$; now, we provide bounds on the \emph{relative} error, which means much tighter bounds, with probability close to 1 by modeling $H$ as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of $H$ are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of $ψ_0$ from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin.

math-ph↗

arXiv2024

Canonical Typicality For Other Ensembles Than Micro-Canonical

We generalize Lévy's lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $ρ$ on a separable Hilbert space $\mathcal{H}$, GAP$(ρ)$ is the most spread out probability measure on the unit sphere of $\mathcal{H}$ that has density matrix $ρ$ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $\|ρ\|$ of $ρ$ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for ``most'' pure states $ψ$ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $ψ$-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time-evolution, for ``most'' pure states $ψ$ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $ψ_t$ is very close to a $ψ$-independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for GAP$(ρ)$, provided the density matrix $ρ$ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

math-ph↗

arXiv2023

Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk

We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding GNS-Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite volume Hamiltonians need not have a spectral gap.

math-ph↗

arXiv2023

Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap

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.

math-ph↗