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

Physical Review

Statistical Mechanics of an Assembly of Quasiparticles

Tohru Morita, Tomoyasu Tanaka

Published May 17, 1965• 7 citations
Publisher

Abstract

Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., $n$ quasiparticles are of $O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})$, respectively, where $V$ is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.

DOI
10.1103/physrev.138.a1088

Related papers

The Fermi liquid theory with fractional exclusion statistics

Dragoş-Victor Anghel

The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations, is temperature dependent, then the Landau's quasiparticle gas cannot describe accurately the thermodynamics of the system. A general solution to this problem is presented in this paper, in which the quasiparticle energies are redefined in such a way that the total energy of the system is identical to the sum of the energies of the quasiparticles. This implies also that the thermodynamic properties of the system and those of the quasiparticle gas are identical. By choosing a perspective in which the quasiparticle energies are fixed while the density of states along the quasiparticle axis vary, we transform our quasiparticle system into an ideal gas which obey fractional exclusion statistics.

arXiv:1204.4064•Apr 18, 2012•1 citation

Luttinger many-fermion model

Ladislav Šamaj, Zoltán Bajnok

Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.

Cambridge University Press eBooks•May 16, 2013•0 citations

MANY-BODY ASPECTS OF THE THEORY OF FERMI LIQUIDS NEAR BOUNDARIES Dierk

Rainer Physikalisches

Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.

•Jan 1, 2011•0 citations

Notes on Fermi and Bose Statistics

Morikazu Toda, Fumihiko Takano

By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become \(p_{0}/\sqrt{3}m\), where p 0 is the momentum at the Fermi surface.

Journal of the Physical Society of Japan•Jan 1, 1954•5 citations

Theory of Strongly Interacting Quantum Liquids. I. Formulation

W. L. McMillan

We make use of our understanding of the ground state ${\ensuremath{\psi}}_{0}$ and elementary excitations ${\ensuremath{\psi}}_{q}$ of the strongly interacting Bose gas to construct a theory of strongly interacting mixed and Fermi gases. We consider the overcomplete basis set ${\ensuremath{\varphi}}^{k}{\ensuremath{\psi}}_{q}$, where ${\ensuremath{\varphi}}^{k}$ is a determinant of plane waves, and interpret the off-diagonal part of the hamiltonian as a three-point vertex, coupling the Fermi quasiparticles to the collective modes. This vertex dresses each fermion with its backflow and induces a two-fermion scattering via the interaction of the backflows. There is no residual (screened) two-body interaction in the problem. Finally we discuss the expansion parameter for strong interactions.

Physical Review•Nov 5, 1968•20 citations

Chemical Potential and Low-Lying Excitation in the Many-Body Bose System

K. Sawada, Ludwig W. Bruch

For a Bose system interacting through a two-body potential there is a large occupation number in the zero-momentum state with small fluctuations. Thus, we can express the chemical potential and low-lying excitation energy in terms of the density fluctuation only. The formulas derived are shown to coincide with well-established results in the low-density Bose gas. The use of experimental neutron scattering data is suggested to improve the numerical answer. A particular application is made to the ${\mathrm{He}}^{4}$ case. If we utilize the Feynman variational excited state as the (approximate) true state vector, which is correct experimentally for low-momentum excitations, we get an explicit phonon excitation for actual ${\mathrm{He}}^{4}$.

Physical Review•Aug 1, 1963•2 citations
Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., nnn quasiparticles are of O(V\ensuremath−1),O(V\ensuremath−2),\ensuremath⋯,O(V\ensuremath−n+1)O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})O(V\ensuremath−1),O(V\ensuremath−2),\ensuremath⋯,O(V\ensuremath−n+1), respectively, where VVV is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.
The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations, is temperature dependent, then the Landau's quasiparticle gas cannot describe accurately the thermodynamics of the system. A general solution to this problem is presented in this paper, in which the quasiparticle energies are redefined in such a way that the total energy of the system is identical to the sum of the energies of the quasiparticles. This implies also that the thermodynamic properties of the system and those of the quasiparticle gas are identical. By choosing a perspective in which the quasiparticle energies are fixed while the density of states along the quasiparticle axis vary, we transform our quasiparticle system into an ideal gas which obey fractional exclusion statistics.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
We make use of our understanding of the ground state ${\ensuremath{\psi}}_{0}$ and elementary excitations ${\ensuremath{\psi}}_{q}$ of the strongly interacting Bose gas to construct a theory of strongly interacting mixed and Fermi gases. We consider the overcomplete basis set ${\ensuremath{\varphi}}^{k}{\ensuremath{\psi}}_{q}$, where ${\ensuremath{\varphi}}^{k}$ is a determinant of plane waves, and interpret the off-diagonal part of the hamiltonian as a three-point vertex, coupling the Fermi quasiparticles to the collective modes. This vertex dresses each fermion with its backflow and induces a two-fermion scattering via the interaction of the backflows. There is no residual (screened) two-body interaction in the problem. Finally we discuss the expansion parameter for strong interactions.
For a Bose system interacting through a two-body potential there is a large occupation number in the zero-momentum state with small fluctuations. Thus, we can express the chemical potential and low-lying excitation energy in terms of the density fluctuation only. The formulas derived are shown to coincide with well-established results in the low-density Bose gas. The use of experimental neutron scattering data is suggested to improve the numerical answer. A particular application is made to the He4{\mathrm{He}}^{4}He4 case. If we utilize the Feynman variational excited state as the (approximate) true state vector, which is correct experimentally for low-momentum excitations, we get an explicit phonon excitation for actual He4{\mathrm{He}}^{4}He4.