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

arXiv:1403.2834 · Reviews in Mathematical Physics

A rigorous proof of the Bohr–van Leeuwen theorem in the semiclassical limit

Baptiste Savoie

Published Sep 1, 2015• 8 citations
arXivHTMLPDF

Abstract

The original formulation of the Bohr–van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude–Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from -∞ to ∞ that makes the free energy density magnetic-field-independent. When considering a classical (Maxwell–Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems, one expects the BvL theorem to hold provided the thermodynamic limit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this paper is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.

DOI
10.1142/s0129055x15500191

Related papers

Magnetism in matter before the discovery of quantum spin: Bohr's less well-known contribution to the transition from classical to quantum physics.

Jean-François S. Van Huele

How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!

Bulletin of the American Physical Society•Mar 12, 2008•0 citations

Thermodynamic extension of density-functional theory. II. Finite-temperature ensemble spin-density functional theory

Robert Balawender, A. Holas

The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.

arXiv:0904.3990•Apr 25, 2009•5 citations

On Magnetic Models in Wavefunction Ensembles

Leonardo De Carlo, W. David Wick

In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable “spins” there are no finite-temperature phase transitions because of high dimensionality of the phase space. Then we focus on the simplest case, namely the mean-field (Curie–Weiss) model, in order to discover whether phase transitions are even possible in this model class. This strategy at least diminishes the dimensionality of the problem. We found that, even assuming exchange symmetry in the wavefunctions, no finite-temperature phase transitions appear when the Hamiltonian is given by the usual energy expression of quantum mechanics (in this case the analytical argument is not totally satisfactory and we relied partly on a computer analysis). However, a variant model with additional “wavefunction energy” does have a phase transition to a magnetized state. (With respect to dynamics, which we do not consider here, wavefunction energy induces a non-linearity which nevertheless preserves norm and energy. This non-linearity becomes significant only at the macroscopic level.) The three results together suggest that magnetization in large wavefunction spin chains appears if and only if we consider indistinguishable particles and block macroscopic dispersion (i.e., macroscopic superpositions) by energy conservation. Our principle technique involves transforming the problem to one in probability theory, then applying results from large deviations, particularly the Gärtner–Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems.

arXiv:2208.07688•Entropy•Mar 25, 2023•0 citations

Diamagnetism of a free particle in classical electron theory with classical electromagnetic zero-point radiation

Timothy H. Boyer

Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the $z$ axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.

Physical Review A•Jan 1, 1980•33 citations

Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein

Nicolas Rougerie

These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schrödinger effective energy functionals.

arXiv:1409.1182•Sep 3, 2014•7 citations

A Proof of the Bloch Theorem for Lattice Models

Haruki Watanabe

Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.

arXiv:1904.02700•Journal of Statistical Physics•Sep 17, 2019•40 citations
The original formulation of the Bohr–van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude–Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from -∞ to ∞ that makes the free energy density magnetic-field-independent. When considering a classical (Maxwell–Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems, one expects the BvL theorem to hold provided the thermodynamic limit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this paper is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.
How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable “spins” there are no finite-temperature phase transitions because of high dimensionality of the phase space. Then we focus on the simplest case, namely the mean-field (Curie–Weiss) model, in order to discover whether phase transitions are even possible in this model class. This strategy at least diminishes the dimensionality of the problem. We found that, even assuming exchange symmetry in the wavefunctions, no finite-temperature phase transitions appear when the Hamiltonian is given by the usual energy expression of quantum mechanics (in this case the analytical argument is not totally satisfactory and we relied partly on a computer analysis). However, a variant model with additional “wavefunction energy” does have a phase transition to a magnetized state. (With respect to dynamics, which we do not consider here, wavefunction energy induces a non-linearity which nevertheless preserves norm and energy. This non-linearity becomes significant only at the macroscopic level.) The three results together suggest that magnetization in large wavefunction spin chains appears if and only if we consider indistinguishable particles and block macroscopic dispersion (i.e., macroscopic superpositions) by energy conservation. Our principle technique involves transforming the problem to one in probability theory, then applying results from large deviations, particularly the Gärtner–Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems.
Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the zzz axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.
These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schrödinger effective energy functionals.
Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.