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International Journal of Modern Physics B

PAIRING IN NON-FERMI LIQUID IN TERMS OF THE BETHE-SALPETER EQUATION

Y. M. Malozovsky, J. D. Fan

Published Jul 30, 2007• 10 citations
Publisher

Abstract

Pairing between two fermionic excitations in a non-Fermi liquid is considered in terms of the Bethe-Salpeter equation. We consider the pairing in a Fermi system with vanishing spectral weight. It is well known that the quasiparticle pole in the single-particle Green's function in a Non-Fermi liquid is absent or weak. The examples of such systems can be viewed as the "Marginal" Fermi liquid and Luttinger liquid. Another example that has also been considered is the Fermi system with pseudogap behavior in the spectral weight. Although the pairing between two excitations in a non-Fermi liquid is, in general, absent, nevertheless we show that the Cooper's type of pairing can occur and the conditions for such a kind of pairing are discussed. The results have also been applied to the systems with smeared or non-monotonic Fermi distribution. A system that transits from Fermi-type to Bose-type behavior can serve as an example of such systems as discussed.

DOI
10.1142/s0217979207044019

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Pairing between two fermionic excitations in a non-Fermi liquid is considered in terms of the Bethe-Salpeter equation. We consider the pairing in a Fermi system with vanishing spectral weight. It is well known that the quasiparticle pole in the single-particle Green's function in a Non-Fermi liquid is absent or weak. The examples of such systems can be viewed as the "Marginal" Fermi liquid and Luttinger liquid. Another example that has also been considered is the Fermi system with pseudogap behavior in the spectral weight. Although the pairing between two excitations in a non-Fermi liquid is, in general, absent, nevertheless we show that the Cooper's type of pairing can occur and the conditions for such a kind of pairing are discussed. The results have also been applied to the systems with smeared or non-monotonic Fermi distribution. A system that transits from Fermi-type to Bose-type behavior can serve as an example of such systems as discussed.
Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature $(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})$ and a critical interaction strength $({u}_{c\ensuremath{\infty}})$ characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\ensuremath{\omega}}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit $(\ensuremath{\beta}u\ensuremath{\gg}1)$, the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent [O(1)]\left[O(1)\right][O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to 12\frac{1}{2}21​. Weak impurity scattering $(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})$ leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.
Non-Fermi liquid behavior is found for the first time in a two-dimensional (2D) system with non-singular interactions using Haldane’s bosonization scheme. The bosonized system is solved exactly by a generalized Bogoliubov transformation. The fermion momentum distribution, calculated using a generalized Mattis-Lieb technique, exhibits a non-universal power law in the vicinity of the Fermi surface for intermediate interaction strengths. In recent years, there has been great interest in possible non-Fermi-liquid-like behavior in the ground state of two-dimensional (2D) strongly correlated fermionic systems [1]. It is well-known that in one dimension the ground state of a fermionic system with arbitrarily weak interactions is not a Fermi liquid but a state usually known as a Luttinger liquid [2]. For the 2D case, many studies have suggested that, in the limit of weak interactions, this behavior does not appear [3]. Studies of systems with a small number of particles also suggest Fermi liquid behavior in 2D [4]. Although Khveshchenko et al have found non-Fermi-liquid behavior in 2D, they worked only with a singular long-range current-current interaction [5]. The one-dimensional Luttinger liquid solution can be obtained by a bosonization
A model of two interacting one-dimensional fermion systems (``Luttinger liquids'') coupled by single-particle hopping is investigated. Bosonization allows a number of exact statements to be made. In particular, for forward scattering only, the model contains two massless boson sectors and an Ising-type critical sector. For general repulsive interactions, there is a spin excitation gap and either d-type pairing or orbital antiferromagnetic fluctuations dominate. It is argued that the same behavior is also found for strong interactions. A possible scenario for the crossover to a Fermi liquid in a many-chain system is discussed. \textcopyright{} 1996 The American Physical Society.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
We present a numerical study on the non-Fermi liquid behavior of a three-dimensional strongly correlated system. The Hubbard model in a simple cubic lattice is simulated by the dynamical cluster approximation; in particular, the quasi-particle weight is calculated at finite dopings for a range of temperatures. By fitting the quasi-particle weight to the marginal Fermi liquid form at finite doping near the putative quantum critical point, we find evidence of a separatrix between Fermi liquid and non-Fermi liquid regions. Our results suggest that a marginal Fermi liquid and possibly a quantum critical point exist in the non-symmetry broken solution of the three-dimensional interacting electron systems. We also calculate the spectral function, close to the half-filling, and we obtain evidence of pseudogap.
The effect of static fluctuations in the phase of the order parameter on the normal and superconducting properties of a 2D system with attractive four-fermion interaction has been studied. Analytic expressions for the fermion Green function, its spectral density and the density of states are derived. The resultant single-particle Green function clearly demonstrates non-Fermi liquid behavior. The results show that as the temperature increases through the 2D critical temperature the width of the quasiparticle peaks broadens significantly. At the same time one retains the gap in quasiparticle spectrum. The spectral density for the dynamical fluctuations can also be obtained. Clearly the dynamical fluctuations fill the gap giving the observed pseudogap behaviour.