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Modern Physics Letters B

NON-FERMI LIQUID BEHAVIOR OF A SYSTEM WITH SADDLE POINTS IN THE ENERGY

M. Crişan, Cătălin Paşcu Moca

Published Nov 20, 1995• 4 citations
Publisher

Abstract

We present a simple demonstration that the two-dimensional fermionic system with the energy ε(k)=kxky has a non-Fermi behavior. The calculation of the wave function renormalization constant Z will be performed using the “poor man’s renormalization” method and we will show that Z→0 with the infrared cutoff Λ as Z(Λ)=Λζ where ζ is a constant.

DOI
10.1142/s0217984995001777

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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.

Cambridge University Press eBooks•Aug 27, 2015•0 citations
We present a simple demonstration that the two-dimensional fermionic system with the energy ε(k)=kxky has a non-Fermi behavior. The calculation of the wave function renormalization constant Z will be performed using the “poor man’s renormalization” method and we will show that Z→0 with the infrared cutoff Λ as Z(Λ)=Λζ where ζ is a constant.
(1) The temperature dependence of the specific heat for a marginal Fermi liquid has been calculated. (2) We calculated the self-energy at T=0 for a two dimensional fermionic system with hyperbolic dispersion. The existence of the saddle points in the energy gives rise to a marginal behavior. (3) We showed that the two-dimensional fermionic system with the energy εk=kxkyε_{\mathbf{k}}=k_xk_yεk​=kx​ky​ has a non-Fermi behavior. (4) The electronic self-energy due to the electron-spin interaction is calculated for a two dimensional system. (5) We study the influence of the amplitude fluctuations of a non-Fermi superconductor on the energy spectrum of the two-dimensional Anderson non-Fermi system. (6) Using the field-theoretical methods we studied the evolution from BCS theory to Bose-Einstein condensation (BEC) for a non-Fermi system. (7) Using the renormalization group approach proposed by Millis we calculated the specific heat coefficient γ(T)γ(T)γ(T) for the magnetic fluctuations with susceptibility χ−1∼δα+∣ω∣α+f(q)χ^{-1}\sim δ^α+| ω| ^α+f(q)χ−1∼δα+∣ω∣α+f(q) near a Lifshitz point.
Using the method of continuous constructive renormalization group around the Fermi surface, we prove that a jellium two-dimensional interacting system of fermions at low temperature T is analytic in the coupling constant lambda for |lambda| |logT|</=K, where K is some numerical constant and T is the temperature. In that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior seen in Fermi liquids but, in particular, not in Luttinger liquids. Our results also prove that in 2D any transition temperature must be nonperturbative in the coupling constant, a result expected on physical grounds. The proof exploits specific momentum conservation rules in two dimensions.
In a recent paper by Schmeltzer [Phys. Rev. B 52, 7939 (1995)] it is claimed that a non-Fermi liquid behavior arises in a d-dimensional system of fermions interacting through a long-range potential of the form V(q)=g/|q${\mathrm{|}}^{\mathrm{d}+\mathrm{\ensuremath{\epsilon}}\ensuremath{-}|\mathrm{A}\mathrm{x}\mathrm{\ensuremath{-}}1}$ for 0\ensuremath{\leqslant}1. Schmeltzer arrives at this result using both renormalization group (RG) and bosonization. In this Comment we show that a proper application of the bosonization technique to this problem leads instead to a non-Fermi liquid behavior only for \ensuremath{\epsilon}-hat\ensuremath{\geqslant}d-1, in agreement with several previous analyses. We also give a direct physical interpretation of this result. Finally we comment on the application of RG to this problem.
Abstract. We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.
We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.
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.