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Physical Review

Phase Transitions in Systems of Interacting Fermions

S. Gartenhaus, G. Stranahan

Published May 31, 1965• 14 citations
Publisher

Abstract

Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.

DOI
10.1103/physrev.138.a1346

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Condensed Matter Physics and Exactly Soluble Models: Selecta of Elliott H. Lieb

Élliott H. Lieb, Bruno Nachtergaele, Jan Philip Solovej +1 more

A Survey by the Editors.- I.1 Violation of the Noncrossing Rule: The Hubbard Hamiltonian for Benzene.- I.2 Two Theorems on the Hubbard Model.- I.3 The Hubbard Model: Some Rigorous Results and Open Problems.- I.4 Flux Phase of the Half-Filled Band.- II.1 Proof of the Peierls Instability in One Dimension.- II.2 Stability of the Peierls Instability for Ring Shaped Molecules.- III.1 A Model for Crystallization: A Variation on the Hubbard Model.- III.2 Phase Separation due to Quantum Mechanical Correlations.- IV.1 Theory of Ferromagnetism and the Ordering of Electronic Energy Levels.- IV.2 Ordering Energy Levels of Interacting Spin Systems.- IV.3 Magnetic Properties of Some Itinerant-Electron Systems at T > 0.- V.1 Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors.- V.2 Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field.- V.3 Constructive Macroscopic Quantum Electrodynamics.- V.4 The Laser: A Reversible Quantum Dynamical System with Irreversible Classical Macroscopic Motion.- VI.1 A Proof of Part of Haldane's Conjecture on Spin Chains.- VI.2 Rigorous Results on Valence-Bond Ground States in Antiferromagnets.- VI.3 Valence Bond Ground States in Isotropic Quantum Antiferromagnets.- VI.4 Ground State Properties of a Fully Frustrated Quantum Spin System.- VII.1 Exact Ground State Energy of the Strong-Coupling Polaron.- VIII.1 Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas.- VIII.2 Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.- VIII.3 The Quantum-Mechanical Many-Body Problem: The Bose Gas.- Some of the Early History of Exactly Soluble Models.- IX.1 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.- IX.2 Exact Solution of the F Model of an Antiferroelectric.- IX.3 Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.- IX.4 Residual Entropy of Square Ice.- IX.5 Two-Dimensional Ferroelectric Models.- IX.6 Relations Between the 'Percolation' and 'Colouring' Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the 'Percolation' Problem.- IX.7 Analytic Properties of the Free Energy for the Ice Models.- X.1 Two-Dimensional Ising Model as a Soluble Problem of Many Fermions.- X.2 Solution of the Dimer Problem by the Transfer Matrix Method.- XI.1 Two Soluble Models of an Antiferromagnetic Chain.- XII.1 Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State.- XII.2 Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum.- XIII.1 Delta-Function Fermi Gas with Two-Spin Deviates.- XIII.2 Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model on One Dimension.- XIV.1 Exact Solution of a Many-Fermion System and Its Associated Boson Field.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.

•Nov 29, 2004•13 citations
Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.
A study is made of a certain model Hamiltonian which describes a system of fermions. Making use of the variational results of Girardeau one can obtain, in the limit of infinite volume, an explicit formula for the free energy in terms of the solution of a nonlinear integral equation. It is established that for certain values of the coupling constant the model exhibits condensation, and that qualitatively at least, the isotherms for the model reproduce those for actual physical systems. The model is applied to He3{\mathrm{He}}^{3}He3 and it is shown that by a suitable choice for the parameters of the model, reasonable agreement with the low-temperature properties of He3{\mathrm{He}}^{3}He3 is obtained. The second virial coefficient for He4{\mathrm{He}}^{4}He4 is also reproduced with the same parameter values which give the correct virial coefficient for He3{\mathrm{He}}^{3}He3.
We propose a simple model of an interacting, fully spin-polarized Fermi gas in dimensions d=2 and d=3, and derive the approximate expression for the energy spectrum and the corresponding formula for the Helmholtz free energy. We analyze the thermodynamics of the system and find the lines of first-order phase transitions between the low- and high-density phases terminating at critical points. The properties of the corresponding phase diagrams are qualitatively different for d=2 and 3, and sensitively depend on the interparticle attraction, which marks a departure from the standard van der Waals theory. The differences originate from the Pauli exclusion principle and are embedded in the fermionic nature of the system under study.
The problem of finding a microscopic theory of phase transitions across a critical point is a central unsolved problem in theoretical physics. We find a general solution to that problem and present it here for the cases of Bose–Einstein condensation (BEC) in an interacting gas and ferromagnetism in a lattice of spins, interacting via a Heisenberg or Ising Hamiltonian. For BEC, we present the exact, valid for the entire critical region, equations for the Green's functions and order parameter, that is a critical-region extension of the Beliaev–Popov and Gross–Pitaevskii equations. For the magnetic phase transition, we find an exact theory in terms of constrained bosons in a lattice and obtain similar equations for the Green's functions and order parameter. In particular, we outline an exact solution for the three-dimensional Ising model.
A general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed. Besides the well-known Ising and "ice" models, other soluble cases are also considered. After discussing some general symmetry properties of this model, we consider in detail a particular class of the soluble cases, the "free-fermion" model. The explicit expressions for all thermodynamic functions with the inclusion of an external electric field are obtained. It is shown that both the specific heat and the polarizability of the free-fermion model exhibit in general a logarithmic singularity. An inverse-square-root singularity results, however, if the free-fermion model also satisfies the ice condition. The results are illustrated with a specific example.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
A Survey by the Editors.- I.1 Violation of the Noncrossing Rule: The Hubbard Hamiltonian for Benzene.- I.2 Two Theorems on the Hubbard Model.- I.3 The Hubbard Model: Some Rigorous Results and Open Problems.- I.4 Flux Phase of the Half-Filled Band.- II.1 Proof of the Peierls Instability in One Dimension.- II.2 Stability of the Peierls Instability for Ring Shaped Molecules.- III.1 A Model for Crystallization: A Variation on the Hubbard Model.- III.2 Phase Separation due to Quantum Mechanical Correlations.- IV.1 Theory of Ferromagnetism and the Ordering of Electronic Energy Levels.- IV.2 Ordering Energy Levels of Interacting Spin Systems.- IV.3 Magnetic Properties of Some Itinerant-Electron Systems at T > 0.- V.1 Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors.- V.2 Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field.- V.3 Constructive Macroscopic Quantum Electrodynamics.- V.4 The Laser: A Reversible Quantum Dynamical System with Irreversible Classical Macroscopic Motion.- VI.1 A Proof of Part of Haldane's Conjecture on Spin Chains.- VI.2 Rigorous Results on Valence-Bond Ground States in Antiferromagnets.- VI.3 Valence Bond Ground States in Isotropic Quantum Antiferromagnets.- VI.4 Ground State Properties of a Fully Frustrated Quantum Spin System.- VII.1 Exact Ground State Energy of the Strong-Coupling Polaron.- VIII.1 Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas.- VIII.2 Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.- VIII.3 The Quantum-Mechanical Many-Body Problem: The Bose Gas.- Some of the Early History of Exactly Soluble Models.- IX.1 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.- IX.2 Exact Solution of the F Model of an Antiferroelectric.- IX.3 Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.- IX.4 Residual Entropy of Square Ice.- IX.5 Two-Dimensional Ferroelectric Models.- IX.6 Relations Between the 'Percolation' and 'Colouring' Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the 'Percolation' Problem.- IX.7 Analytic Properties of the Free Energy for the Ice Models.- X.1 Two-Dimensional Ising Model as a Soluble Problem of Many Fermions.- X.2 Solution of the Dimer Problem by the Transfer Matrix Method.- XI.1 Two Soluble Models of an Antiferromagnetic Chain.- XII.1 Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State.- XII.2 Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum.- XIII.1 Delta-Function Fermi Gas with Two-Spin Deviates.- XIII.2 Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model on One Dimension.- XIV.1 Exact Solution of a Many-Fermion System and Its Associated Boson Field.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.