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Low temperature analysis of two dimensional Fermi systems with symmetric Fermi surface

G. Benfatto, Alessandro Giuliani, Vieri Mastropietro

Published Jan 1, 2002• 12 citations
OpenAlex

Abstract

We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivity.

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The physical system modelled is a periodic lattice that provides a background for interacting fermions. An effective model of the interacting fermions at zero temperature may be studied using standard techniques of many-body quantum mechanics. All the physical information of such a system is encoded in its Schwinger's functions that tell us how states evolve in time and what the matrix elements of all operators are. It is natural to do calculations on the interacting system by treating it as a perturbation of the corresponding noninteracting system, as the eigenstates and energy levels of the latter are known. All Schwinger functions may be conveniently cast in the path integral formalism, and combined into a generating functional. The Schwinger functions are shown to be sums of Feynman diagrams, where the electron propagators blow-up at the Fermi surface. Power counting is described and used to reveal subdiagram divergences. Renormalization via the renormalized Gallavotti-Nicolo Tree Expansion is implemented in an attempt to circumvent these subdiagram divergences and obtain finite values for the Schwinger functions. It is shown that if the Fermi surface has at worst quadratic singularities, the perturbation coefficients of the renormalizing counterterms are finite and remain so as the infrared cutoff is removed.

•Jan 1, 2009•0 citations
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.
We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential μ=2−μ0μ=2-μ_0μ=2−μ0​, μ0=10−10μ_0=10^{-10}μ0​=10−10 fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant łłł in the domain $ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}$ for any fixed temperature T>0T>0T>0, suggesting that there is a phase transition with critical temperature $T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}$. Here C1,C2C_1, C_2C1​,C2​ are positive constants independent of TTT and łłł. We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
We present a renormalization-group analysis of two-dimensional interacting fermion systems with a closed and partially flat Fermi surface. Numerical solutions of the one-loop flow equations show that for a bare local repulsion, the system evolves through three different regimes as the typical energy is lowered: a high-energy transient with a strong competition between particle-particle and particle-hole channels, an intermediate regime with dominant spin-density wave correlations, and finally a ``hot spot'' regime exhibiting d-wave superconductivity. We study, mostly by analytical methods, how this flow pattern depends on the number N of Fermi-surface patches used in the numerical solution. This clearly indicates that the final regime requires vanishingly small microscopic interactions, for the one-loop approximation to be valid, as N is going to infinity.
The physical system modelled is a periodic lattice that provides a background for interacting fermions. An effective model of the interacting fermions at zero temperature may be studied using standard techniques of many-body quantum mechanics. All the physical information of such a system is encoded in its Schwinger's functions that tell us how states evolve in time and what the matrix elements of all operators are. It is natural to do calculations on the interacting system by treating it as a perturbation of the corresponding noninteracting system, as the eigenstates and energy levels of the latter are known. All Schwinger functions may be conveniently cast in the path integral formalism, and combined into a generating functional. The Schwinger functions are shown to be sums of Feynman diagrams, where the electron propagators blow-up at the Fermi surface. Power counting is described and used to reveal subdiagram divergences. Renormalization via the renormalized Gallavotti-Nicolo Tree Expansion is implemented in an attempt to circumvent these subdiagram divergences and obtain finite values for the Schwinger functions. It is shown that if the Fermi surface has at worst quadratic singularities, the perturbation coefficients of the renormalizing counterterms are finite and remain so as the infrared cutoff is removed.