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We construct, using fermionic functional integrals, thermodynamic Green's functions for a weakly coupled fermion gas whose Fermi energy lies in a gap. Estimates on the Green's functions are obtained that are characteristic of the size of the gap. This prepares the way for the analysis of single scale renormalization group maps for a system of fermions at temperature zero without a gap.

The first renormalization group map arising from the momentum space decomposition of a weakly coupled system of fermions at temperature zero differs from all subsequent maps. Namely, the component of momentum dual to temperature may be arbitrarily large — there is no ultraviolet cutoff. The methods of Part 1 are supplemented to control this special case.

The generic renormalization group map associated to a weakly coupled system of fermions at temperature zero is treated by supplementing the methods of Part 1. The interplay between position and momentum space is captured by "sectors". It is shown that the difference between the complete four-legged vertex and its "ladder" part is irrelevant for the sequence of renormalization group maps.

For a two-dimensional, weakly coupled system of fermions at temperature zero, one principal ingredient used to control the composition of the associated renormalization group maps is the careful counting of the number of quartets of sectors that are consistent with conservation of momentum. A similar counting argument is made to show that particle–particle ladders are irrelevant in the case of an asymmetric Fermi curve.

The momentum distribution n_k of itinerant electrons in the one-dimensional Falicov–Kimball model is calculated for various ground-state phases. In particular, we examine the periodic phases with period two, three and four (that are ground-states for all Coulomb interactions) as well as the phase separated states (that are ground states for small Coulomb interactions). For all periodic phases examined the momentum distribution is a smooth function of k with no sign of any discontinuity or singular behavior at the Fermi surface k=k_F. An unusual behavior of n_k (a local maximum) is found at k=3k_F for electron concentrations outside half-filling. For the phase separated ground states the momentum distribution n_k exhibits discontinuity at k=k₀<k_F. This behavior is interpreted in terms of a Fermi liquid.

We study an effective field theory describing cold fermionic atoms near a Feshbach resonance. The effective theory gives a precise description of the dynamics in the limit that the energy of the Feshbach resonance is tuned to be twice that of the Fermi surface. We compute the zero temperature superfluid condensate in this limit, and obtain a critical temperature T_C≃0.43 T_F.

We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.

In terms of an exact equation for the thermodynamic potential due to interaction between two particles and based on Green's function method; we have derived the Landau expansion of the thermodynamic potentials in terms of the variation of the quasiparticle distribution function. We have also derived the expansion of the thermodynamic potential in terms of the variation of an exact single particle (not quasiparticles), this derivations lead to the relationship between the interaction function for two quasiparticles and the interaction energy between two particles as shown. Further, in terms of the four-point vertex part we are led to the Pauli exclusion principle.

In this paper, we will apply the derived equation of interaction for two quasiparticles in the other paper of ours (J. D. Fan and Y. M. Malozovsky, Int. J. Mod. Phys. B, Vol. 27 No. 15, 2013)¹ to the model of a Fermi liquid with a weak contact attractive interaction between ³He particles. We will show that in terms of the perturbative approach and the random phase approximation (RPA), the evaluated Fermi-liquid interaction parameters are in remarkably good agreement with experimental data on liquid ³He. Thus, our simple model, in contrast to very complicated and numerically correlated basis function model exploited by Finberg and co-authors [E. Finberg, Theory of Quantum Fluids (Academic Press, New York, 1969)], enables to reproduce the Fermi liquid parameters for liquid ³He. In addition, using the result we obtained in Ref. 1, we have qualitatively explained superfluidity of ³He under pressure.

The main misconception regarding the interaction between quasiparticles stems from the assertion that the interaction energy between two quasiparticles is exactly identical to that of the renormalized interaction between two particles due to interparticle interaction in the Fermi system. If the main concept regarding the definition of quasiparticle as a weakly excited state of the Fermi system with conservation of charge and spin is paramount (except for the charge and spin separation models), the concept of the interaction between quasiparticles is very different from the assumption in the common sense. In this paper, we will prove a general theorem that the interaction between two quasiparticles is very much different from the renormalized interaction between two particles. The major difference lies in two places: the interaction between two quasiparticles is just negative to the renormalized interaction between two particles, and the interaction energy between the two particles is proportional to the product of two Fermi liquid renormalization factors. The result shed light on the reinterpretation of Cooper's pairing without invoking electron-photon interaction.

Special aspects of few cycle pulses propagation in semi-holographic Fermi liquid with impurities are considered in this paper. Green’s function poles which are in charge of excitation states dispersion law of the liquid under consideration were given according to the ADS/CFT correspondence. The impact of both Fermi liquid parameters and its impurities on the few cycle pulse shape was defined.

We suggest the diffuse approach to the relaxation processes within the kinetic theory for the Wigner distribution function. The diffusion and drift coefficients are evaluated taking into consideration the interparticle collisions on the distorted Fermi surface. Using the finite range interaction, we show that the momentum dependence of the diffuse coefficient D_p(p) has a maximum at Fermi momentum p = p_F whereas the drift coefficient K_p(p) is negative and reaches a minimum at p ≈ p_F. For a cold Fermi system the diffusion coefficient takes the nonzero value which is caused by the relaxation on the distorted Fermi surface at temperature T = 0. The numerical solution of the diffusion equation was performed for the particle-hole excitation in a nucleus with A = 16. The evaluated relaxation time τ_r ≈ 8.3 ⋅ 10^-23s is close to the corresponding result in a nuclear Fermi-liquid obtained within the kinetic theory.

Microscopic current fluctuations and conductance are inseparable. We present a unified kinetic theory of both quantized conductance and nonequilibrium noise in one-dimensional ballistic wires. We show that high-current ballistic fluctuations depend, directly and robustly, on carrier statistics. This is dramatically evident in the large noise peaks predicted to coincide with the quantized steps in the two-probe conductance. The outstanding features of nonlinear ballistic fluctuations are: (i) their observability by standard experimental techniques; (ii) their strong suppression by electron degenercy; (iii) their divergence in any model that ignores the explicit dynamics of dissipative scattering in the macroscopic leads; (iv) their numerical dominance over the noise predictions of the Landauer-Büttiker model. The nonequilibrium noise of hot ballistic carriers is strikingly sensitive to their inelastic energy loss on entering the leads. Uniquely and quantitatively, it characterizes the observed nonideality of quantized conductance.

A possibility and properties of spontaneous magnetization in quark matter are investigated. Magnetic susceptibility is evaluated within Fermi liquid theory, taking into account of screening effect of gluons. Spin wave in the polarized quark matter, as the Nambu-Goldstone mode, is formulated by way of the coherent-state path integral.

We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.