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arXiv:1208.3483 · Physical Review B

First-Matsubara-frequency rule in a Fermi liquid. I. Fermionic self-energy

Andrey V. Chubukov, Dmitrii L. Maslov

Published Oct 19, 2012• 72 citations
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Abstract

We analyze in detail the fermionic self-energy $\ensuremath{\Sigma}(\ensuremath{\omega},T)$ in a Fermi liquid (FL) at finite temperature $T$ and frequency $\ensuremath{\omega}$. We consider both canonical FLs, i.e., systems in spatial dimension $D>2$, where the leading term in the fermionic self-energy is analytic [the retarded $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)=C({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})]$, and noncanonical FLs in $1<D<2$, where the leading term in $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ scales as ${T}^{D}$ or ${\ensuremath{\omega}}^{D}$. We relate the ${\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2}$ form to a special property of the self-energy, ``the first-Matsubara-frequency rule,'' which stipulates that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ in a canonical FL contains an $\mathcal{O}(T)$ but no ${T}^{2}$ term. We show that in any $D>1$, the next term after $\mathcal{O}(T)$ in ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ is of order ${T}^{D}$ (${T}^{3}\mathrm{ln}T$ in $D=3$). This ${T}^{D}$ term comes from only forward and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the ${T}^{D}$ term vanishes in the ``local approximation,'' when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the ${T}^{D}$ term vanishes, but also that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ does not contain any terms beyond $\mathcal{O}(T)$. This rule imposes two constraints on the scaling form of the self-energy: upon replacing $\ensuremath{\omega}$ by $i\ensuremath{\pi}T$, $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must vanish and $\mathrm{Re}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must reduce to $\mathcal{O}(T)$. These two constraints should be taken into consideration in extracting scaling forms of ${\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ from experimental and numerical data.

DOI
10.1103/physrevb.86.155136

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We analyze in detail the fermionic self-energy \ensuremathΣ(\ensuremathω,T)\ensuremath{\Sigma}(\ensuremath{\omega},T)\ensuremathΣ(\ensuremathω,T) in a Fermi liquid (FL) at finite temperature TTT and frequency \ensuremathω\ensuremath{\omega}\ensuremathω. We consider both canonical FLs, i.e., systems in spatial dimension D>2D>2D>2, where the leading term in the fermionic self-energy is analytic [the retarded Im\ensuremathΣR(\ensuremathω,T)=C(\ensuremathω2+\ensuremathπ2T2)]\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)=C({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})]Im\ensuremathΣR(\ensuremathω,T)=C(\ensuremathω2+\ensuremathπ2T2)], and noncanonical FLs in 1<D<21<D<21<D<2, where the leading term in Im\ensuremathΣR(\ensuremathω,T)\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)Im\ensuremathΣR(\ensuremathω,T) scales as TD{T}^{D}TD or \ensuremathωD{\ensuremath{\omega}}^{D}\ensuremathωD. We relate the \ensuremathω2+\ensuremathπ2T2{\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2}\ensuremathω2+\ensuremathπ2T2 form to a special property of the self-energy, the first-Matsubara-frequency rule,'' which stipulates that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ in a canonical FL contains an $\mathcal{O}(T)$ but no ${T}^{2}$ term. We show that in any $D>1$, the next term after $\mathcal{O}(T)$ in ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ is of order ${T}^{D}$ (${T}^{3}\mathrm{ln}T$ in $D=3$). This ${T}^{D}$ term comes from only forward and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the ${T}^{D}$ term vanishes in the local approximation,'' when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the TD{T}^{D}TD term vanishes, but also that \ensuremathΣR(i\ensuremathπT,T){\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)\ensuremathΣR(i\ensuremathπT,T) does not contain any terms beyond O(T)\mathcal{O}(T)O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing \ensuremathω\ensuremath{\omega}\ensuremathω by i\ensuremathπTi\ensuremath{\pi}Ti\ensuremathπT, Im\ensuremathΣR(\ensuremathω,T)\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)Im\ensuremathΣR(\ensuremathω,T) must vanish and Re\ensuremathΣR(\ensuremathω,T)\mathrm{Re}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)Re\ensuremathΣR(\ensuremathω,T) must reduce to O(T)\mathcal{O}(T)O(T). These two constraints should be taken into consideration in extracting scaling forms of \ensuremathΣR(\ensuremathω,T){\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)\ensuremathΣR(\ensuremathω,T) from experimental and numerical data.
Motivated by recent optical measurements on a number of strongly correlated electron systems, we revisit the dependence of the conductivity of a Fermi liquid $\ensuremath{\sigma}(\ensuremath{\Omega},T)$ on the frequency $\ensuremath{\Omega}$ and temperature TTT. Using the Kubo formalism and taking full account of vertex corrections, we show that the Fermi-liquid form $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+4{\ensuremath{\pi}}^{2}{T}^{2}$ holds under very general conditions, namely, in any dimensionality above one, for a Fermi surface of an arbitrary shape (but away from nesting and van Hove singularities), and to any order in the electron-electron interaction. We also show that the scaling form of $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)$ is determined by the analytic properties of the conductivity along the Matsubara axis. If a system contains not only itinerant electrons but also localized degrees of freedom which scatter electrons elastically, e.g., magnetic moments or resonant levels, the scaling form changes to $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+b{\ensuremath{\pi}}^{2}{T}^{2}$, with $1\ensuremath{\le}b<\ensuremath{\infty}$. For purely elastic scattering, b=1b=1b=1. Our analysis implies that the value of $b\ensuremath{\approx}1$, reported for URu2Si2{\mathrm{URu}}_{2}{\mathrm{Si}}_{2}URu2​Si2​ and some rare-earth-based doped Mott insulators, indicates that the optical conductivity in these materials is controlled by an elastic scattering mechanism, whereas the values of $b\ensuremath{\approx}2.3$ and 5.65.65.6, reported for underdoped cuprates and organics, correspondingly, imply that both elastic and inelastic mechanisms contribute to the optical conductivity.
We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic (n=2)(n=2)(n=2) Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ for a model case of the long-range interaction in the nematic channel. Already within the random-phase approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue ZZZ and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, ZZZ, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where gc,2{g}_{c,2}gc,2​ is the n=2n=2n=2 charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but gc,2{g}_{c,2}gc,2​ Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction DDD with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
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