Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Heat production of noninteracting fermions subjected to electric fields

J.-B. Bru, de Siqueira Pedra, W.

Published Jan 1, 2017• 11 citations
OpenAlex

Abstract

Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength $u\\in \\mathbb{R}$ of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly 1. In particular, the convergence of perturbation series at $u=0$ of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\\"{o}lder inequalities for general non-commutative Lp-spaces derived by Araki and Masuda.

Related papers

Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems

Walter A. de S. Pedra, Manfred Salmhofer B

It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many–fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Ω (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided Ω is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.

arXiv:0707.2810•Communications in Mathematical Physics•May 5, 2008•34 citations

A rigorous Keldysh functional integral for fermions

Philipp Benjamin Aretz, Manfred Salmhofer B

Abstract We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values $$\gamma ^\textrm{c}_{m,\bar{m}}$$ γ m , m ¯ c of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound $$\delta _C$$ δ C and decay constant $$\alpha _C$$ α C of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.

arXiv:2508.01787•Journal of Statistical Physics•Feb 14, 2026•0 citations

The asymptotics of an eigenfunction-correlation determinant for Dirac-$\delta$ perturbations

Martin Gebert

We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-$\delta$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ \rho>0$, where $\varphi_j^L$ and $\psi_k^L$ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schr\odinger operators. The decay exponent is given in terms of the s-wave scattering phase shift $\zeta(E):=\delta^2(\sqrt E)/{\pi^2}$. For an attractive Dirac-$\delta$ perturbation we conclude that the decay exponent $\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}$ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.

•Mar 12, 2015•0 citations

The asymptotics of an eigenfunction-correlation determinant for Dirac-δ perturbations

Martin Gebert

We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 3-dimensional ball BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.

arXiv:1503.03654•Journal of Mathematical Physics•Jul 1, 2015•5 citations

Heat Production of Noninteracting Fermions Subjected to Electric Fields

J.-B. Bru, Walter de Siqueira Pedra, Carolin Hertling

Electric resistance in conducting media is related to heat (or entropy ) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the nonautonomous C *‐dynamical system obtained from the fermionic second quantization of a discrete Schrödinger operator with bounded static potential in the presence of an electric field that is time‐ and space‐dependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context and is studied the present paper. We use tree‐decay bounds of the n ‐point, n ∊ 2ℕ, correlations of the many‐fermion system to analyze this regime. We verify below the first law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among otheg things, technically convenient. We initially focus our study on noninteracting (or free) fermions, but our approach will be later applied to weakly interacting fermions. © 2015 Wiley Periodicals, Inc.

arXiv:1611.07706•Communications on Pure and Applied Mathematics•Jul 21, 2014•9 citations

A Simple Bound on the Error of Perturbation Theory in Quantum Mechanics

Daniel Harlow

I provide a straightforward proof that a simple harmonic oscillator perturbed by an (almost) arbitrary positive interaction has a perturbative expansion for any finite-time Euclidian transition amplitude which obeys the following result: the difference of the sum of the first N terms of the series and the exact result is bounded in absolute value by the next term in the series. The proof makes no assumptions about either the strength of the interactions or the convergence of the perturbation series. I then argue that the result generalizes immediately to a much broader class of quantum mechanical systems, including bare perturbation theory in quantum field theory. The case of renormalized perturbation theory is more subtle and remains open, as does the generalization to energy levels and connected S-matrix elements.

arXiv:0905.2466•May 15, 2009•1 citation
Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength uinmathbbRu\\in \\mathbb{R}uinmathbbR of the interparticle interaction. We provide, for large determinants of fermionic convariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly 1. In particular, the convergence of perturbation series at u=0u=0u=0 of any fermionic quantum field theory is ensured if the matrix entries, with respect to some fixed orthonormal basis, of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use H\"{o}lder inequalities for general non-commutative Lp-spaces derived by Araki and Masuda.
It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many–fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Ω (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided Ω is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.
Abstract We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values γm,mˉc\gamma ^\textrm{c}_{m,\bar{m}}γm,mˉc​ γ m , m ¯ c of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound δC\delta _CδC​ δ C and decay constant αC\alpha _CαC​ α C of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.
The asymptotics of an eigenfunction-correlation determinant for Dirac-δ\deltaδ perturbations
We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 333-dimensional ball BLB_LBL​ of radius LLL in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ\deltaδ perturbation. More precisely, we show the algebraic decay of the correlation determinant ∣det⁡(⟨φjL,ψkL⟩)j,k=1,...,N∣2=L−ζ(E)+o(1)\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}​det(⟨φjL​,ψkL​⟩)j,k=1,...,N​​2=L−ζ(E)+o(1), as N,L→∞N,L\to\inftyN,L→∞ and N/∣BL∣→ ρ>0N/|B_L|\to~ \rho>0N/∣BL​∣→ ρ>0, where φjL\varphi_j^LφjL​ and ψkL\psi_k^LψkL​ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schr\odinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E):=δ2(E)/π2\zeta(E):=\delta^2(\sqrt E)/{\pi^2}ζ(E):=δ2(E​)/π2. For an attractive Dirac-δ\deltaδ perturbation we conclude that the decay exponent 1π2∥arcsin⁡∣T(E)/2∣∥HS2\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}π21​∥arcsin∣T(E)/2∣∥HS2​ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.
We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 3-dimensional ball BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ>0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.
Electric resistance in conducting media is related to heat (or entropy ) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the nonautonomous C *‐dynamical system obtained from the fermionic second quantization of a discrete Schrödinger operator with bounded static potential in the presence of an electric field that is time‐ and space‐dependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context and is studied the present paper. We use tree‐decay bounds of the n ‐point, n ∊ 2ℕ, correlations of the many‐fermion system to analyze this regime. We verify below the first law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among otheg things, technically convenient. We initially focus our study on noninteracting (or free) fermions, but our approach will be later applied to weakly interacting fermions. © 2015 Wiley Periodicals, Inc.
I provide a straightforward proof that a simple harmonic oscillator perturbed by an (almost) arbitrary positive interaction has a perturbative expansion for any finite-time Euclidian transition amplitude which obeys the following result: the difference of the sum of the first N terms of the series and the exact result is bounded in absolute value by the next term in the series. The proof makes no assumptions about either the strength of the interactions or the convergence of the perturbation series. I then argue that the result generalizes immediately to a much broader class of quantum mechanical systems, including bare perturbation theory in quantum field theory. The case of renormalized perturbation theory is more subtle and remains open, as does the generalization to energy levels and connected S-matrix elements.