arXiv subjects

Joscha Henheik

Publications and source records attributed to Joscha Henheik.

At least 19 recordsLinked to original sources

arXiv2026

Anomalous rate of eigenstate thermalisation at singularities of the density of states

We prove the Eigenstate Thermalisation Hypothesis (ETH), also known as Quantum Unique Ergodicity (QUE), for large $N\times N$ mean-field random matrices with general correlation structure. We identify the microcanonical ensemble and establish the optimal fluctuation scale of eigenvector overlaps around it. Our results invalidate the inverse-density scaling predicted by Feingold and Peres [Phys. Rev. A 34, 591 (1986)] (and incorporated into Srednicki's ansatz [Phys. Rev. E 50, 888-901 (1994)]) in the physics literature of quantum chaos, based upon popular semiclassical theory, and uncover the genuine mechanism which relies on multi-resolvent local laws. Although fluctuations are expected to increase as the density of states vanishes, and indeed scale as $N^{-1/2}$ in the special cusp regime, rather than $N^{-1}$ in the bulk, we find, unexpectedly, that the same $N^{-1}$ rate persists at regular spectral edges. Hence, generically, in the absence of cusps, the entire eigenbasis fluctuates on the same scale as a Haar unitary. This anomaly stems from delicate cancellations in the solution of the underlying matrix Dyson equation, which form the core of our analysis.

math-ph↗

arXiv2026

On a Rosenzweig-Porter-type model

We consider a very general Rosenzweig-Porter-type model, $H=H_0+\lambda W$, where $H_0$ is an arbitrary Hermitian matrix and $W$ is a standard Wigner matrix. We precisely trace the localization properties of the eigenvectors and the eigenstate thermalisation hypothesis (ETH) as the coupling constant $\lambda$ interpolates between the trivial $\lambda=0$ case and the fully mean field regime of large $\lambda$. Our results hold uniformly in $H_0$ and $\lambda$, substantially generalising all previous local laws on deformed Wigner matrices even in the mean field regime. Our proof precisely captures the deterministic approximation to the resolvent which exhibits a strongly inhomogeneous structure. As a byproduct, we conclude the emergence of a mobility edge and study the phenomenon of re-entrant localization.

math-ph↗

arXiv2025

Spectral rigidity of two-dimensional Liouville tori

It is a folklore conjecture that Liouville metrics are the only Riemannian metrics with integrable geodesic flow on the two-dimensional torus. In this paper, we study Laplace-isospectral deformations, within a fixed conformal class, of generic Liouville metrics and prove two rigidity results: First, any isospectral deformation that is affine in $\epsilon$ within the fixed conformal class is trivial. Second, any isospectral deformation that is analytic in the perturbation parameter and whose Taylor coefficients are trigonometric polynomials in the spatial variables remains Liouville and is obtained by componentwise rearrangement of the original metric. For the first result, the proof combines a wave-trace noncancellation result, which allows us to recover suitable length data from the Laplace spectrum, with a second-variation analysis of the energy functional along closed geodesics. For the second result, we prove that, within the Liouville class considered here, the relevant length-isospectrality condition is equivalent to componentwise rearrangement.

math.DG↗

arXiv2025

Law of fractional logarithm for random matrices

We prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an $N\times N$ Wigner matrix and its $(N-k)\times (N-k)$ minor if $k\ll N^{2/3}$. On the other hand, we use dynamical methods to show that this correlation is weak if $k\gg N^{2/3}$.

math.PR↗

arXiv2025

Decorrelation transition in the Wigner minor process

We consider the Wigner minor process, i.e. the eigenvalues of an $N\times N$ Wigner matrix $H^{(N)}$ together with the eigenvalues of all its $n\times n$ minors, $H^{(n)}$, $n\le N$. The top eigenvalues of $H^{(N)}$ and those of its immediate minor $H^{(N-1)}$ are very strongly correlated, but this correlation becomes weaker for smaller minors $H^{(N-k)}$ as $k$ increases. For the GUE minor process the critical transition regime around $k\sim N^{2/3}$ was analyzed by Forrester and Nagao (J. Stat. Mech.: Theory and Experiment, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality.

math.PR↗

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.

math-ph↗

arXiv2025

Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model

We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices [arXiv:2403.10359] and an intricate analysis of the deterministic approximation.

math-ph↗

arXiv2025

How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem

Particle creation terms in quantum Hamiltonians are usually ultraviolet divergent and thus mathematically ill defined. A rather novel way of solving this problem is based on imposing so-called interior-boundary conditions on the wave function. Previous papers showed that this approach works in the non-relativistic regime, but particle creation is mostly relevant in the relativistic case after all. In flat relativistic space-time (that is, neglecting gravity), the approach was previously found to work only for certain somewhat artificial cases. Here, as a way of taking gravity into account, we consider curved space-time, specifically the super-critical Reissner-Nordström space-time, which features a naked timelike singularity. We find that the interior-boundary approach works fully in this setting; in particular, we prove rigorously the existence of well-defined, self-adjoint Hamiltonians with particle creation at the singularity, based on interior-boundary conditions. We also non-rigorously analyze the asymptotic behavior of the Bohmian trajectories and construct the corresponding Bohm-Bell process of particle creation, motion, and annihilation. The upshot is that in quantum physics, a naked space-time singularity need not lead to a breakdown of physical laws, but on the contrary allows for boundary conditions governing what comes out of the singularity and thereby removing the ultraviolet divergence.

math-ph↗

arXiv2024

Eigenstate thermalisation at the edge for Wigner matrices

We prove the Eigenstate Thermalisation Hypothesis for Wigner matrices uniformly in the entire spectrum, in particular near the spectral edges, with a bound on the fluctuation that is optimal for any observable. This complements earlier works of Cipolloni et. al. (Comm. Math. Phys. 388, 2021; Forum Math., Sigma 10, 2022) and Benigni et. al. (Comm. Math. Phys. 391, 2022; arXiv: 2303.11142) that were restricted either to the bulk of the spectrum or to special observables. As a main ingredient, we prove a new multi-resolvent local law that optimally accounts for the edge scaling.

math.PR↗

arXiv2024

Cusp Universality for Correlated Random Matrices

For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality [arXiv:1804.07744], while cusp universality was proven only for Wigner-type matrices with independent entries [arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus also re-establishing universality of the local eigenvalue statistics in the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744] regimes.

math.PR↗

arXiv2024

Multi-band superconductors have enhanced critical temperatures

We introduce a multi-band BCS free energy functional and prove that for a multi-band superconductor the effect of inter-band coupling can only increase the critical temperature, irrespective of its attractive or repulsive nature and its strength. Further, for weak coupling and weaker inter-band coupling, we prove that the dependence of the increase in critical temperature on the inter-band coupling is (1) linear, if there are two or more equally strongly superconducting bands, or (2) quadratic, if there is only one dominating band.

math-ph↗

arXiv2024

Eigenvector decorrelation for random matrices

We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.

math.PR↗

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 $ρ_0$ of a Hamiltonian $H_0$ is locally gapped in $Λ^{\mathrm{gap}} \subset Λ$, whenever the Liouvillian $- \mathrm{i} \, [H_0, \, \cdot \, ]$ is almost invertible on local observables supported in $Λ^{\mathrm{gap}}$ when tested in $ρ_0$. 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 $εV$ acts in a region which is further than $|\log ε|$ away from the non-gapped region $Λ\setminus Λ^{\mathrm{gap}}$.

math-ph↗

arXiv2024

Loschmidt Echo for Deformed Wigner Matrices

We consider two Hamiltonians that are close to each other, $H_1 \approx H_2 $, and analyze the time-decay of the corresponding Loschmidt echo $\mathfrak{M}(t) := |\langle ψ_0, \mathrm{e}^{\mathrm{i} t H_2} \mathrm{e}^{-\mathrm{i} t H_1} ψ_0 \rangle|^2$ that expresses the effect of an imperfect time reversal on the initial state $ψ_0$. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such $H_1$ and $H_2$.

math-ph↗

arXiv2024

Out-of-time-ordered correlators for Wigner matrices

We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables $A$ and $B$ in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Elect. J. Prob. 27, 1-38, 2022]) or the Hilbert-Schmidt norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Forum Math., Sigma 10, E96, 2022]).

math-ph↗

arXiv2024

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.

math-ph↗

arXiv2024

Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices

We consider large non-Hermitian $N\times N$ matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance $1/N$ completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by [Deutsch 1991] and proven for Wigner matrices in [Cipolloni, Erdős, Schröder 2020], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [arXiv:2005.08930] and [arXiv:2005.08908]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

math.PR↗

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↗