Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field - Leschke, Hajo; Sobolev, Alexander; Spitzer, Wolfgang (Communications in Mathematical Physics, 2020)
We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2 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 mu = B (in suitable physical units). For this (pure) state we define its local entropy S(Lambda) associated with a bounded (sub)region Lambda subset of R-2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Lambda of finite area vertical bar Lambda vertical bar. In this setting we prove that the leading asymptotic growth of S(L Lambda), as the dimensionless scaling parameter L > 0 tends to infinity, has the form L root B vertical bar partial derivative Lambda vertical bar up to a precisely given (positive multiplicative) coefficient which is independent of Lambda and dependent on B and mu only through the integer part of (mu/B-1)/2. Here we have assumed the boundary curve partial derivative Lambda of Lambda to be sufficiently smooth which, in particular, ensures that its arc length vertical bar partial derivative Lambda vertical bar 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 L-2(R-2) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Renyi entropies. As opposed to the case B = 0, the corresponding asymptotic coefficients depend on the Renyi index in a non-trivial way.
CRIS-Team WoS Importer:2021-01-15

Existence of Replica-Symmetry Breaking in Quantum Glasses - Leschke, Hajo; Manai, Chokri; Ruder, Rainer; Warzel, Simone (Physical Review Letters, 2021)
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 multispin interactions, sometimes dubbed as the transverse p-spin model.
CRIS-Team Scopus Importer:2021-11-26

Quantum annealing and computation: challenges and perspectives - Chakrabarti, Bikas K.; Leschke, Hajo; Ray, Purusattam; Shirai, Tatsuhiko; Tanaka, Shu (Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 2023)
In the introductory article of this theme issue, we provide an overview of quantum annealing and computation with a very brief summary of the individual contributions to this issue made by experts as well as a few young researchers. We hope the readers will get the touch of the excitement as well as the perspectives in this unusually active field and important developments there. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
CRIS-Team Scopus Importer:2022-12-16
Keywords: NP-hard problems; quantum Ising spin glass; quantum computation; quantum tunnelling; quantum annealing

Rényi Entropies of the Free Fermi Gas in Multi-Dimensional Space at High Temperature - Leschke, Hajo; Sobolev, Alexander V.; Spitzer, Wolfgang (Springer Science and Business Media Deutschland GmbH, 2022)
We study the local and (bipartite) entanglement Rényi entropies of the free Fermi gas in multi-dimensional Euclidean space in thermal equilibrium. We prove positivity of the entanglement entropies with Rényi index γ ≤ 1 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 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. 49, 30LT04 (2016); [Corrigendum. 50, 129501 (2017)], but being supported there only by the basic proof ideas.
CRIS-Team Scopus Importer:2024-12-16
Keywords: Asymptotic trace formulas; Non-smooth functions of Wiener–Hopf operators; Rényi (entanglement) entropy of fermionic equilibrium states

The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder - Leschke, Hajo; Rothlauf, Sebastian; Ruder, Rainer; Spitzer, Wolfgang (Journal of Statistical Physics, 2021)
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≥ 0. The macroscopic annealed free energy 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.
CRIS-Team Scopus Importer:2021-03-19
Keywords: Annealed free energy; Feynman–Kac formula; Quantum spin-glas model; Varadhan large-deviation principle