We derive the quantum Boltzmann equation (QBE) of composite fermions at/near the \ensuremath{\nu}=1/2 state using the nonequilibrium Green's-function technique. The lowest-order perturbative correction to the self-energy due to the strong gauge-field fluctuations suggests that there is no well-defined Landau quasiparticle. Therefore, we cannot assume the existence of the Landau quasiparticles a priori in the derivation of the QBE. Using an alternative formulation, we derive the QBE for the generalized Fermi-surface displacement which corresponds to the local variation of the chemical potential in momentum space. From this QBE, one can understand in a unified fashion the Fermi-liquid behaviors of the density-density and the current-current correlation functions at \ensuremath{\nu}=1/2 (in the long-wavelength and the low-frequency limits) and the singular behavior of the energy gap obtained from the finite-temperature activation behavior of the compressibility near \ensuremath{\nu}=1/2. Implications of these results for recent experiments are also discussed.
We derive the quantum Boltzmann equation (QBE) of composite fermions at/near the \ensuremath{\nu}=1/2 state using the nonequilibrium Green's-function technique. The lowest-order perturbative correction to the self-energy due to the strong gauge-field fluctuations suggests that there is no well-defined Landau quasiparticle. Therefore, we cannot assume the existence of the Landau quasiparticles a priori in the derivation of the QBE. Using an alternative formulation, we derive the QBE for the generalized Fermi-surface displacement which corresponds to the local variation of the chemical potential in momentum space. From this QBE, one can understand in a unified fashion the Fermi-liquid behaviors of the density-density and the current-current correlation functions at \ensuremath{\nu}=1/2 (in the long-wavelength and the low-frequency limits) and the singular behavior of the energy gap obtained from the finite-temperature activation behavior of the compressibility near \ensuremath{\nu}=1/2. Implications of these results for recent experiments are also discussed.
Quantitative theory of composite fermions in Bose-Fermi mixtures at ν=1.
Composite fermion provides a simple and unified picture to understand a vast amount of phenomenology in the quantum Hall regime. However it has remained challenging to formulate this concept properly within a single Landau level. Recently a low-energy noncommutative field theory for bosons at Landau-level filling factor ν=1 has been formulated by Dong and Senthil. In the limit of long-wavelength and small-amplitude gauge fluctuation, they found it reduces to the celebrated Halperin-Lee-Read theory of composite fermion liquid. In this work we consider a Bose-Fermi mixture at total filling factor ν=1. Different from previous work, the number density of composite fermions in the mixture and corresponding Fermi momentum can be tuned by changing the filling factor of bosons, νb=1−νf. This tunability enables us to study the dilute limit νb≪1, which allows for a controlled and asymptotically exact calculation of the energy dispersion and effective mass of composite fermions. Furthermore, the approximation of the low-energy description by a commutative field theory is manifestly justified. Most importantly, we demonstrate gauge fluctuations acquire a Higgs mass due to the presence of a composite boson condensate, as a result of which the system behaves like a genuine Landau Fermi liquid. Combined with the irrelevance of four-fermion interaction in the dilute limit, we are able to obtain asymptotically exact properties of this composite fermion Fermi liquid. In the opposite limit of νf≪1, the Higgs mass goes to zero and we find crossover between Fermi liquid and non-Fermi liquid as temperature increases. Observing these properties either experimentally or numerically provides unambiguous evidence of not only the composite fermions and the Fermi surface they form, but also the presence of emergent gauge fields and their fluctuations due to strong correlation.
The theory of composite fermion provides a simple and unified picture to understand a vast amount of phenomenology in the quantum Hall regime. However it has remained challenging to formulate this concept properly within a single Landau level, which provides the relevant degrees of freedom in the limit of strong magnetic field. Recently a low-energy noncommutative field theory for bosons at Landau-level filling factor $\ensuremath{\nu}=1$ has been formulated by Dong and Senthil [Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020)]. In the limit of long-wavelength and small-amplitude gauge fluctuation, they found it reduces to the celebrated Halperin-Lee-Read theory of a composite fermion liquid. In this work we consider a Bose-Fermi mixture at total filling factor $\ensuremath{\nu}=1$. Different from previous work, the number density of composite fermions in the mixture and corresponding Fermi momentum can be tuned by changing the filling factor of bosons, ${\ensuremath{\nu}}_{b}=1\ensuremath{-}{\ensuremath{\nu}}_{f}$. This tunability enables us to study the dilute limit ${\ensuremath{\nu}}_{b}\ensuremath{\ll}1$, which allows for a controlled and asymptotically exact calculation of the energy dispersion and effective mass of composite fermions. Furthermore, the approximation of the low-energy description by a commutative field theory is manifestly justified. Most importantly, we demonstrate that gauge fluctuations acquire a Higgs mass due to the presence of a composite boson condensate, as a result of which the system behaves like a genuine Landau- Fermi liquid. Combined with the irrelevance of four-fermion interaction in the dilute limit, we are able to obtain asymptotically exact properties of this composite fermion Fermi liquid. In the opposite limit of ${\ensuremath{\nu}}_{f}\ensuremath{\ll}1$, the Higgs mass goes to zero and we find crossover between Fermi liquid and non-Fermi liquid as temperature increases. Observing these properties either experimentally or numerically provides unambiguous evidence of not only the composite fermions and the Fermi surface they form, but also the presence of emergent gauge fields and their fluctuations due to strong correlation.
We derive the quantum Boltzmann equation (QBE) by using generalized Landau-interaction parameters, obtained through the nonequilibrium Green's function technique. This is a generalization of the usual QBE formalism to non-Fermi liquid (NFL) systems, which do not have well-defined quasiparticles. We apply this framework to a controlled low-energy effective field theory for the Ising-nematic quantum critical point, in order to find the collective excitations of the critical Fermi surface in the collisionless regime. We also compute the nature of the dispersion after the addition of weak Coulomb interactions. The zero angular momentum longitudinal vibrations of the Fermi surface show a linear-in-wavenumber dispersion, which corresponds to the zero sound of Landau's Fermi liquid theory. The Coulomb interaction modifies it to a plasmon mode in the long-wavelength limit, which disperses as the square-root of the wavenumber. Remarkably, our results show that the zero sound and plasmon modes exhibit the same behaviour as in a Fermi liquid, although an NFL is fundamentally different from the former.
Field theories of the composite-fermion (CF) metal model it as a Fermi sea of composite fermions coupled to an emergent gauge field. Within a random phase approximation, these theories predict that the Landau damping of the gauge field resulting from its coupling to the low-energy, long-wavelength CF particle-hole excitations modifies the electrons' density-density correlation function related to the static structure factor S(q) at wave vector q. This produces a nonanalytic correction ∝q^{3}lnq to S(q) (with the magnetic length ℓ_{B}=1). Thanks to the recently developed quaternion formulation for Jain-Kamilla projection of CF wave functions, the evaluation of S(q) from the accurate microscopic theory of composite fermions has now become possible for systems containing as many as N=900 CFs, which enables a reliable determination of the small-q behavior of S(q). We study CF metals corresponding to electrons at Landau level filling factors ν=1/2 and 1/4, and for completeness, also of bosons at ν=1 and 1/3. In the q→0 limit, our microscopic calculation reveals a q^{3} term in S(q) of the CF metals rather than q^{3}lnq. This behavior is well predicted by a model of a noninteracting Fermi sea of dipolar CFs, which also obtains its coefficient accurately.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
Abstract The low-temperature properties of a wide range of many-fermion systems spanning metals, quantum gases and liquids to nuclear matter are well understood within the framework of Landau’s theory of Fermi liquids. The low-energy physics of these systems is governed by interacting fermionic quasiparticles with momenta and energies near a Fermi surface in momentum space. Nonequilibrium properties are described by a kinetic equation for the distribution function for quasiparticles proposed by Landau. Quasiparticle interactions with other quasiparticles, phonons, or impurities lead to internal forces acting on a distribution of nonequilibrium quasiparticles, as well as collision processes that ultimately limit the transport of mass, heat, charge, and magnetization, as well as limiting the coherence times of quasiparticles. For Fermi liquids that are close to a second-order phase transition, e.g., Fermi liquids that undergo a superfluid transition, incipient Cooper pairs—long-lived fluctuations of the ordered phase—provide a new channel for scattering quasiparticles, as well as corrections to internal forces acting on the distribution of nonequilibrium quasiparticles. We develop the theory of quasiparticle transport for Fermi liquids in the vicinity of a BCS-type superfluid transition starting from Keldysh’s field theory for nonequilibrium, strongly interacting fermions. The leading corrections to Fermi-liquid theory for nonequilibrium quasiparticle transport near a Cooper instability arise from the virtual emission and absorption of incipient Cooper pairs. Our theory is applicable to quasiparticle transport in superconductors, nuclear matter, and the low-temperature phases of liquid 3He. As an implementation of the theory we calculate the pairing-fluctuation corrections to the attenuation of zero sound in liquid 3He near the superfluid transition and demonstrate quantitative agreement with experimental results.