arXiv subjects

Joscha Henheik

Publications and source records attributed to Joscha Henheik.

32 records · Page 2Linked to original sources

arXiv2023

Prethermalization for Deformed Wigner Matrices

We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_λ$.

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

Gaussian fluctuations in the Equipartition Principle for Wigner matrices

The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.

math-ph↗

arXiv2023

The BCS Energy Gap at High Density

We study the BCS energy gap $Ξ$ in the high-density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential $V$ on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3 (2022)) on the critical temperature $T_c$ at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.

math-ph↗

arXiv2023

The BCS Critical Temperature at High Density

We investigate the BCS critical temperature $T_c$ in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential $V$ on the Fermi-surface. Our results include a rigorous confirmation for the behavior of $T_c$ at high densities proposed by Langmann, Triola, and Balatsky (Phys. Rev. Lett. 122, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.

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↗

arXiv2023

Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results

We first review the problem of a rigorous justification of Kubo's formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo's formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.

math-ph↗

arXiv2023

Universal behavior of the BCS energy gap

We consider the BCS energy gap $Ξ(T)$ (essentially given by $Ξ(T) \approx Δ(T, \sqrtμ)$, the BCS order parameter) at all temperatures $0 \le T \le T_c$ up to the critical one, $T_c$, and show that, in the limit of weak coupling, the ratio $Ξ(T)/T_c$ is given by a universal function of the relative temperature $T/T_c$. On the one hand, this recovers a recent result by Langmann and Triola (Phys. Rev. B 108.10 (2023)) on three-dimensional $s$-wave superconductors for temperatures bounded uniformly away from $T_c$. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions $d \in \{1,2,3\}$, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to $T_c$.

math-ph↗

arXiv2023

Eigenstate Thermalisation Hypothesis for Translation Invariant Spin Systems

We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with mean field interaction. This mathematically verifies the observation made in [L.Santos and M.Rigol, Phys.Rev.E 82, 031130 (2010, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.031130)] that ETH may hold for systems with additional translation symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interactions.

cond-mat.stat-mech↗

arXiv2023

Universality in low-dimensional BCS theory

It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature $Ξ$ and the critical temperature $T_c$ is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density, and high density. The goal of this short note is to extend the universal behavior to lower dimensions $d=1,2$ and give an exemplary proof in the weak coupling limit.

math-ph↗

arXiv2022

Creation Rate of Dirac Particles at a Point Source

Only recently has it been possible to construct a self-adjoint Hamiltonian that involves the creation of Dirac particles at a point source in 3d space. Its definition makes use of an interior-boundary condition. Here, we develop for this Hamiltonian a corresponding theory of the Bohmian configuration. That is, we construct a Markov jump process $(Q_t)_{t\in\mathbb{R}}$ in the configuration space of a variable number of particles that is $|ψ_t|^2$-distributed at every time $t$ and follows Bohmian trajectories between the jumps. The jumps correspond to particle creation or annihilation events and occur either to or from a configuration with a particle located at the source. The process is the natural analog of Bell's jump process, and a central piece in its construction is the determination of the rate of particle creation. The construction requires an analysis of the asymptotic behavior of the Bohmian trajectories near the source. We find that the particle reaches the source with radial speed 0, but orbits around the source infinitely many times in finite time before absorption (or after emission).

quant-ph↗

arXiv2022

Deformational rigidity of integrable metrics on the torus

It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: We consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations the deformed metric is again Liouville. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. In order to put our results in perspective, we review existing results about integrable metrics on the torus.

math.DS↗

arXiv2022

Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions

A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, that is, for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has been done successfully already in 1 space dimension, and more generally for codimension-1 boundaries, the situation of point sources in 3 dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3d, which also correspond to a boundary condition. And indeed, we confirm this expectation here by proving that there is no self-adjoint operator on (a truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with IBC (on the boundary consisting of configurations with a particle at the origin) that are, away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.

math-ph↗

arXiv2022

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.

math-ph↗