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Hajo Leschke

Publications and source records attributed to Hajo Leschke.

At least 19 recordsLinked to original sources
arXiv2026

The Quantum Formalism Revisited

For the simple system of a point-like particle confined to a straight line, I compile, initially in a concise table, the structural elements of quantum mechanics and contrast them with those of classical (statistical) mechanics. Despite many similarities, there are the well-known fundamental differences, resulting from the algebraic non-commutativity in the quantal structure. The latter was discovered by Werner Heisenberg (1901-1976) in June 1925 on the small island of Helgoland in the North Sea, as a consequence of understanding atomic spectral data within a matrix scheme consistent with energy conservation. I discuss the differences and exemplify their quantifications by the variance and entropic indeterminacy inequalities, by (pseudo-)classical bounds on quantum canonical partition functions, and by the correlation inequalities of John Bell (1928-1990) and others.

quant-ph↗
arXiv2023

Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature

We study the local and (bipartite) entanglement Rényi entropies of the free Fermi gas in multi-dimensional Euclidean space $\mathbb{R}^d$ in thermal equilibrium. We prove positivity of the entanglement entropies with Rényi index $γ\leq1$ for all temperatures $T>0$. Furthermore, for general $γ>0$ we establish the asymptotics of the entropies for large $T$ and large scaling parameter $α>0$ for two different regimes $-$ for fixed chemical potential $μ\in\mathbb{R}$ and also for fixed particle density $ρ>0$. In particular, we thereby provide the last remaining building block for a complete proof of our low- and high-temperature results presented (for $γ=1$) in J. Phys. A: Math. Theor. $\textbf{49}$, 30LT04 (2016) [Corrigendum: $\textbf{50}$, 129501 (2017)], but being supported there only by the basic proof ideas.

math-ph↗
arXiv2023

Asymptotic growth of the local ground-state entropy of the ideal Fermi gas in a constant magnetic field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $\mathbb R^2$ perpendicular to an external constant magnetic field of strength $B>0$. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $μ\ge B$ (in suitable physical units). For this (pure) state we define its local entropy $S(Λ)$ associated with a bounded (sub)region $Λ\subset \mathbb R^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $Λ$ of finite area $|Λ|$. In this setting we prove that the leading asymptotic growth of $S(LΛ)$, as the dimensionless scaling parameter $L>0$ tends to infinity, has the form $L\sqrt{B}|\partialΛ|$ up to a precisely given (positive multiplicative) coefficient which is independent of $Λ$ and dependent on $B$ and $μ$ only through the integer part of $(μ/B-1)/2$. Here we have assumed the boundary curve $\partialΛ$ of $Λ$ to be sufficiently smooth which, in particular, ensures that its arc length $|\partialΛ|$ is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln(L)$ is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $\text L^2(\mathbb R^2)$ to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies.

math-ph↗
arXiv2021

Existence of replica-symmetry breaking in quantum glasses

By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a transverse magnetic field if the temperature and the field are sufficiently low. The argument also applies to the generalization of the model with multi-spin interactions, sometimes dubbed as transverse $p$-spin model.

cond-mat.dis-nn↗
arXiv2021

The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington-Kirkpatrick spin-glass model without external magnetic field to the quantum case with a "transverse field" of strength $b$. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $v>0$ is smaller than the temperature $1/β$, then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $b/v\geq0$. The macroscopic annealed free energy (times $β$) turns out to be non-trivial and given, for any $βv>0$, by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $βv<1$ we determine this minimum up to the order $(βv)^4$ with the Taylor coefficients explicitly given as functions of $βb$ and with a remainder not exceeding $(βv)^6/16$. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong $βb$-dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann-Gibbs operator by a Feynman-Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate $βb$. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

math-ph↗
arXiv2016

Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature

The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T=0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit the leading large-scale term of the thermal EE considerably simplifies and displays a ln(1/T)-singularity which one may identify with the known logarithmic enhancement at T=0 of the so-called area-law scaling.

quant-ph↗
arXiv2016

Trace formulas for Wiener--Hopf operators with applications to entropies of free fermionic equilibrium states

We consider non-smooth functions of (truncated) Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. Our main results are uniform estimates for trace norms ($d\ge 1$) and quasiclassical asymptotic formulas for traces of the resulting operators ($d=1$). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space $\mathbb R^d$ ($d\ge 1$) at positive temperature, $T>0$. In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small $T$ and large scaling parameter $α>0$ provided that the product $Tα$ remains bounded from below. Here $α$ is the reciprocal quasiclassical parameter. For $d=1$ we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at $T=0$), previously established by the authors for $d\ge 1$.

math.SP↗
arXiv2015

Scaling of Rényi entanglement entropies of the free Fermi-gas ground state: a rigorous proof

In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev and Israel Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bi-partite von-Neumann entanglement entropy of non-interacting fermions in multi-dimensional Euclidean space at zero temperature. Based on recent progress by one of us (A.V.S.) in semi-classical functional calculus for pseudo-differential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders $α>0$. The special case $α=1/2$ is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas, exhibiting a "logarithmically enhanced area law", have been used already in many publications.

math-ph↗
arXiv2005

Energetic and dynamic properties of a quantum particle in a spatially random magnetic field with constant correlations along one direction

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a ``unidirectionally constant'' magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schroedinger operator (without scalar potential) by analysing its ``energy-band structure''. In particular, for an ergodic random UMF we provide conditions which ensure that the operator's entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localisation. The conditions are verified, for example, for Gaussian and Poissonian random UMF's with non-zero mean-values. These results may be viewed as ``random analogues'' of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J. E. Mueller [Phys. Rev. Lett. 68 (1992) 385].

math-ph↗
arXiv2005

Ballistic transport in random magnetic fields with anisotropic long-ranged correlations

We present exact theoretical results about energetic and dynamic properties of a spinless charged quantum particle on the Euclidean plane subjected to a perpendicular random magnetic field of Gaussian type with non-zero mean. Our results refer to the simplifying but remarkably illuminating limiting case of an infinite correlation length along one direction and a finite but strictly positive correlation length along the perpendicular direction in the plane. They are therefore ``random analogs'' of results first obtained by A. Iwatsuka in 1985 and by J. E. Müller in 1992, which are greatly esteemed, in particular for providing a basic understanding of transport properties in certain quasi-two-dimensional semiconductor heterostructures subjected to non-random inhomogeneous magnetic fields.

cond-mat.dis-nn↗
arXiv2004

Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of the generating Schroedinger operator also act as Carleman operators with continuous integral kernels. Applications to Schroedinger operators with rather general random scalar potentials include a rigorous justification of an integral-kernel representation of their integrated density of states - a relation frequently used in the physics literature on disordered solids.

math-ph↗
arXiv2003

A survey of rigorous results on random Schroedinger operators for amorphous solids

Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schroedinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.

cond-mat.dis-nn↗
arXiv2003

Quantum-classical transitions in Lifshits tails with magnetic fields

We consider Lifshits' model of a quantum particle subject to a repulsive Poissonian random potential and address various issues related to the influence of a constant magnetic field on the leading low-energy tail of the integrated density of states. In particular, we propose the magnetic analog of a $ 40 $-year-old landmark result of Lifshits for short-ranged single-impurity potentials $ U $. The Lifshits tail is shown to change its character from purely quantum, through quantum-classical, to purely classical with increasing range of $ U $. This systematics is explained by the increasing importance of the classical fluctuations of the particle's potential energy in comparison to the quantum fluctuations associated with its kinetic energy.

cond-mat.dis-nn↗
arXiv2002

Simple diamagnetic monotonicities for Schroedinger operators with inhomogeneous magnetic fields of constant direction

Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of M. Loss and B. Thaller [Commun. Math. Phys. 186 (1997) 95] and L. Erdos [J. Math. Phys. 38 (1997) 1289].

quant-ph↗
arXiv2001

Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volume limits of spatial eigenvalue concentrations of finite-volume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinite-volume operator, the integrated density of states is almost surely non-random and independent of the chosen boundary condition. Our proof of the independence of the boundary condition builds on and generalizes certain results by S. Doi, A. Iwatsuka and T. Mine [Math. Z. {\bf 237} (2001) 335-371] and S. Nakamura [J. Funct. Anal. {\bf 173} (2001) 136-152].

math-ph↗
arXiv2001

Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials

We study a non-relativistic charged particle on the Euclidean plane R^2 subject to a perpendicular constant magnetic field and an R^2-homogeneous random potential in the approximation that the corresponding random Landau Hamiltonian on the Hilbert space L^2(R^2) is restricted to the eigenspace of a single but arbitrary Landau level. For a wide class of Gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the Lebesgue measure. We construct explicit upper bounds on the resulting derivative, the restricted density of states. As a consequence, any given energy is seen to be almost surely not an eigenvalue of the restricted random Landau Hamiltonian.

math-ph↗
arXiv2001

The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{ö}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schrödinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.

math-ph↗