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arXiv:nucl-th/0502026 · Physical Review C

Long-wavelength spin- and spin-isospin correlations in nucleon matter

G. I. Lykasov, E. Olsson, C. J. Pethick

Published Aug 29, 2005• 14 citations
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Abstract

We analyze the long-wavelength response of a normal Fermi liquid using Landau theory. We consider contributions from intermediate states containing one additional quasiparticle-quasihole pair as well as those from states containing two or more additional quasiparticle-quasihole pairs. For the response of an operator corresponding to a conserved quantity, we show that the behavior of matrix elements to states with more than one additional quasiparticle-quasihole pair at low excitation energies $\ensuremath{\omega}$ varies as $1/\ensuremath{\omega}$. It is shown how rates of processes involving transitions to two quasiparticle-quasihole states may be calculated in terms of the collision integral in the Landau transport equation for quasiparticles.

DOI
10.1103/physrevc.72.025805

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We analyze the long-wavelength response of a normal Fermi liquid using Landau theory. We consider contributions from intermediate states containing one additional quasiparticle-quasihole pair as well as those from states containing two or more additional quasiparticle-quasihole pairs. For the response of an operator corresponding to a conserved quantity, we show that the behavior of matrix elements to states with more than one additional quasiparticle-quasihole pair at low excitation energies \ensuremathω\ensuremath{\omega}\ensuremathω varies as 1/\ensuremathω1/\ensuremath{\omega}1/\ensuremathω. It is shown how rates of processes involving transitions to two quasiparticle-quasihole states may be calculated in terms of the collision integral in the Landau transport equation for quasiparticles.
We use Landau's theory of a normal Fermi liquid to derive expressions for the static response of a system with a general tensor interaction that conserves the total spin and the total angular momentum of the quasiparticle-quasihole pair. The magnetic susceptibility is calculated in detail, with the inclusion of the center of mass tensor and cross vector terms in addition to the exchange tensor one. We also introduce a new parametrization of the tensor Landau parameters which significantly reduces the importance of high angular harmonic contributions. For nuclear matter and neutron matter we find that the two most important effects of the tensor interaction are to give a contribution from multipair states and to renormalize the magnetic moments. Response to a weak probe may be calculated using similar methods, replacing the magnetic moments with the matrix elements of the weak charges.
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.
We introduce a new renormalisation scheme to construct the Landau quasiparticles of Fermi fluids. The scheme introduces an energy cutoff ΛΛΛ to remove the resonant couplings, enabling the dressing of the particles into quasiparticles via a unitary transformation. The dynamics of the quasiparticles is then restricted to low-energy transitions and is fully determined by an effective Hamiltonian which unifies the Landau function fff, the pair interaction ggg responsible for superfluidity, and the collision amplitude A\mathcal{A}A responsible for transport and equilibration. Studying the flow equation that results from infinitesimal variations of the cutoff, we recover the Bethe-Salpeter relation between fff and the forward limit of A\mathcal{A}A, and we demonstrate an analogue relation between ggg and the frontal limit of A\mathcal{A}A. We show that our effective theory captures all the low-energy phenomena of Fermi liquids, from the equation of state to the transport properties, both in the normal and in the superfluid phase. We apply it to the calculation of non-Fermi liquid corrections to the quasiparticle lifetime. This publication is continued by arXiv:2607.07041 where we apply the effective theory to a Fermi fluid of ultracold atoms.
We derive the Boltzmann equation describing long-wavelength low-frequency transport and relaxation processes in a superfluid Fermi liquid close to the transition temperature Tc{T}_{c}Tc​. In the superfluid, the quasiparticle number is not conserved and therefore one has to take into account the decay of one quasiparticle into three and the inverse process, as well as the two-quasiparticle scattering process. We calculate the collision term in the Boltzmann equation to first order in the superfluid gap $\ensuremath{\Delta}$ and show that it is related rather simply to the corresponding normal-state collision term. As applications of the Boltzmann equation, we solve it exactly to calculate the shear viscosity, and the intrinsic spin relaxation rate. The shear viscosity drops as $\ensuremath{\Delta}\ensuremath{\propto}{({T}_{c}\ensuremath{-}T)}^{\frac{1}{2}}$ for temperatures just below Tc{T}_{c}Tc​, and we determine the coefficient of $\ensuremath{\Delta}$ as a function of the normal-state collision probability. Leggett and Takagi's characteristic spin-relaxation time is shown to be equal to the relaxation time of a normal-state quasiparticle at the Fermi energy at Tc{T}_{c}Tc​. The results provide one with a useful consistency check on experimental measurements which is independent of any assumption about the normal-state collision probability.
We propose a new method to calculate the particle-hole Green's function for an infinite system of fermions in the long-wavelength limit of the random-phase approximation. Contrary to the standard approach the present method does not require approximating the residual particle-hole interaction by a sum of separable terms. The method can also be applied to the Landau theory of Fermi liquids. We illustrate the method by evaluating the response function of nuclear matter in the long-wavelength limit for two different isospin channels.
An expression for the response of the system, valid up to the second order in the general potential, has been derived by developing the Green function by an equation-of-motion method. In the long-wavelength limit, contact with other theories can be made in a transparent manner. The expression for the width contains contributions from direct and exchange processes, the latter contribution being exactly half of the former. The decay of zero sound into two-particle-two-hole pair excitations is found to vary as q 2 . The dispersion relation up to the second order in the potential is an upward parabola.