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arXiv:cond-mat/9912117 · Physical review. B, Condensed matter

Some sum rules for non-Fermi liquids: Applications taking into account the mass renormalization factor

José Jesús Rodríguez-Núñez, I. Ţifrea, S. G. Magalhães

Published Aug 1, 2000• 5 citations
arXivHTMLPDF

Abstract

Restudying the non-Fermi-liquid one-particle Green functions (NFLGF) we have extended the work of Balatsky [Philos. Mag. Lett. 68, 251 (1993)] and Yin and Chakravarty [Int. J. Mod. Phys. B 10, 805 (1996)], among others. We use the moment approach of Nolting [Z. Phys. 255, 25 (1972)] to compute the unknown parameters of the NFLGF's in the framework of the Hubbard model. The zeroth-order moment requires that our one-particle Green functions describe fermionic degrees of freedom. In order to satisfy the first-order sum rule, a renormalization, $\ensuremath{\gamma}\ensuremath{\ne}1,$ of the free-electron mass is called for. The second-order sum rule or moment imposes a relation between the non-Fermi-liquid parameter, $\ensuremath{\alpha},$ the Coulomb interaction, U, and the frequency cutoff, ${\ensuremath{\omega}}_{c}.$ We have calculated the effect of the mass renormalization factor, $\ensuremath{\gamma},$ on some physical quantities, such as (i) the correlated momentum distribution function, ${n}_{c}(\stackrel{\ensuremath{\rightarrow}}{k}),$ close to the effective chemical potential, at $T=0;$ (ii) the superconducting critical temperature, ${T}_{c};$ and (iii) the superconducting critical interaction, ${\ensuremath{\lambda}}_{\mathrm{cr}},$ and compared them with analytical results found in the literature. Also, we have calculated the isotope effect, ${\ensuremath{\alpha}}^{\ensuremath{'}},$ for non-Fermi-liquid systems, which reduces to ${\ensuremath{\alpha}}^{\ensuremath{'}}=1/2$ (the BCS result) when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}0.$ As a case of non-Fermi-liquid systems, in the Appendix, we have studied two inequivalent coupled Hubbard layers for which we calculate the one-particle spectral functions on the layers and perpendicular to them. We discuss the features which appear due to the shift in the two effective chemical potentials and propose some experiments to detect the features found from our expressions.

DOI
10.1103/physrevb.62.4026

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Restudying the non-Fermi-liquid one-particle Green functions (NFLGF) we have extended the work of Balatsky [Philos. Mag. Lett. 68, 251 (1993)] and Yin and Chakravarty [Int. J. Mod. Phys. B 10, 805 (1996)], among others. We use the moment approach of Nolting [Z. Phys. 255, 25 (1972)] to compute the unknown parameters of the NFLGF's in the framework of the Hubbard model. The zeroth-order moment requires that our one-particle Green functions describe fermionic degrees of freedom. In order to satisfy the first-order sum rule, a renormalization, \ensuremathγ\ensuremath≠1,\ensuremath{\gamma}\ensuremath{\ne}1,\ensuremathγ\ensuremath=1, of the free-electron mass is called for. The second-order sum rule or moment imposes a relation between the non-Fermi-liquid parameter, \ensuremathα,\ensuremath{\alpha},\ensuremathα, the Coulomb interaction, U, and the frequency cutoff, \ensuremathωc.{\ensuremath{\omega}}_{c}.\ensuremathωc​. We have calculated the effect of the mass renormalization factor, \ensuremathγ,\ensuremath{\gamma},\ensuremathγ, on some physical quantities, such as (i) the correlated momentum distribution function, nc(k\ensuremath→),{n}_{c}(\stackrel{\ensuremath{\rightarrow}}{k}),nc​(k\ensuremath→​), close to the effective chemical potential, at T=0;T=0;T=0; (ii) the superconducting critical temperature, Tc;{T}_{c};Tc​; and (iii) the superconducting critical interaction, \ensuremathλcr,{\ensuremath{\lambda}}_{\mathrm{cr}},\ensuremathλcr​, and compared them with analytical results found in the literature. Also, we have calculated the isotope effect, \ensuremathα\ensuremath′,{\ensuremath{\alpha}}^{\ensuremath{'}},\ensuremathα\ensuremath′, for non-Fermi-liquid systems, which reduces to \ensuremathα\ensuremath′=1/2{\ensuremath{\alpha}}^{\ensuremath{'}}=1/2\ensuremathα\ensuremath′=1/2 (the BCS result) when \ensuremathα\ensuremath→0.\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}0.\ensuremathα\ensuremath→​0. As a case of non-Fermi-liquid systems, in the Appendix, we have studied two inequivalent coupled Hubbard layers for which we calculate the one-particle spectral functions on the layers and perpendicular to them. We discuss the features which appear due to the shift in the two effective chemical potentials and propose some experiments to detect the features found from our expressions.
We evaluate the one-particle and double-occupied Green functions for the Hubbard model at half-filling using the moment approach of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. Our starting point is a self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ which has a single pole, $\ensuremath{\Omega}(\stackrel{\ensuremath{\rightarrow}}{k}),$ with spectral weight, $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{k}),$ and quasiparticle lifetime, $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{k})$ [J. J. Rodr\'{\i}guez-N\'u\~nez and S. Schafroth, J. Phys. Condens. Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)]. In our approach, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes the central feature of the many-body problem and due to three unknown $\stackrel{\ensuremath{\rightarrow}}{k}$ parameters we have to satisfy only the first three sum rules instead of four as in the canonical formulation of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. This self-energy choice forces our system to be a non-Fermi liquid for any value of the interaction, since it does not vanish at zero frequency. The one-particle Green function, $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ shows the fingerprint of a strongly correlated system, i.e., a double peak structure in the one-particle spectral density, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ vs $\ensuremath{\omega}$ for intermediate values of the interaction. Close to the Mott insulator-transition, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes a wide single peak, signaling the absence of quasiparticles. Similar behavior is observed for the real and imaginary parts of the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}).$ The double-occupied Green function, ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega}),$ has been obtained from $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ by means of the equation of motion. The relation between ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ and the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ is formally established and numerical results for the spectral function of ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ ${\ensuremath{\chi}}^{(2)}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})\ensuremath{\equiv}\ensuremath{-}(1/\ensuremath{\pi}){limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}{0}^{+}}\mathrm{Im}[{G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})],$ are given. Our approach represents the simplest way to include (1) lifetime effects in the moment approach of Nolting, as shown in the paper, and (2) Fermi or/and marginal Fermi liquid features as we discuss in the conclusions.
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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.
We present the detailed formalism of the extremely correlated Fermi liquid theory, developed for treating the physics of the $t\ensuremath{-}J$ model. We start from the exact Schwinger equation of motion for the Green's function for projected electrons, and develop a systematic expansion in a parameter $\ensuremath{\lambda}$, relating to the double occupancy. The resulting Green's function has a canonical part arising from an effective Hamiltonian of the auxiliary electrons, and a caparison part playing the role of a frequency-dependent adaptive spectral weight. This adaptive weight balances the requirement at low $\ensuremath{\omega}$ of the invariance of the Fermi volume, and at high $\ensuremath{\omega}$ of decaying as $\frac{{c}_{0}}{i\ensuremath{\omega}}$, with a correlation-depleted c0<1{c}_{0}<1c0​<1. The effective Hamiltonian Heff{H}_{\mathrm{eff}}Heff​ describing the auxiliary fermions is given a natural interpretation with an effective interaction Veff{V}_{\mathrm{eff}}Veff​ containing both the exchange Jij{J}_{ij}Jij​ and the hopping parameters tij{t}_{ij}tij​. It is made Hermitian by adding suitable terms that ultimately vanish, in the symmetrized theory developed in this paper. Simple but important shift invariances of the $t\ensuremath{-}J$ model are noted with respect to translating its parameters uniformly. These play a crucial role in constraining the form of Veff{V}_{\mathrm{eff}}Veff​ and also provide checks for further approximations. The auxiliary and physical Green's function satisfy two sum rules, and the Lagrange multipliers for these are identified. A complete set of expressions for the Green's functions to second order in $\ensuremath{\lambda}$ is given, satisfying various invariances. A systematic iterative procedure for higher order approximations is detailed. A superconducting instability of the theory is noted at the simplest level with a high transition temperature.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.