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arXiv:1207.6826 · Physical Review B

Extremely correlated Fermi liquids: The formalism

B. Sriram Shastry

Published Mar 15, 2013• 27 citations
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Abstract

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 ${c}_{0}<1$. The effective Hamiltonian ${H}_{\mathrm{eff}}$ describing the auxiliary fermions is given a natural interpretation with an effective interaction ${V}_{\mathrm{eff}}$ containing both the exchange ${J}_{ij}$ and the hopping parameters ${t}_{ij}$. 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 ${V}_{\mathrm{eff}}$ 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.

DOI
10.1103/physrevb.87.125124

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We present the detailed formalism of the extremely correlated Fermi liquid theory, developed for treating the physics of the t\ensuremath−Jt\ensuremath{-}Jt\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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ, 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 \ensuremathω\ensuremath{\omega}\ensuremathω of the invariance of the Fermi volume, and at high \ensuremathω\ensuremath{\omega}\ensuremathω of decaying as c0i\ensuremathω\frac{{c}_{0}}{i\ensuremath{\omega}}i\ensuremathωc0​​, 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−Jt\ensuremath{-}Jt\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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ 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 present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the ttt-JJJ model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with nnn denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high TTT, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function mk{m}_{k}mk​, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.
We present the theory of an extremely correlated Fermi liquid with $U\ensuremath{\rightarrow}\ensuremath{\infty}$. This liquid has an underlying auxiliary Fermi liquid Green's function that is further caparisoned by extreme correlations. The theory leads to two parallel hierarchies of equations that permit iterative approximations in a certain parameter. Preliminary results for the spectral functions display a broad background and a distinct TTT dependent left skew. An important energy scale $\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{k},x)$ emerges as the average inelasticity of the FL Green's function, and influences the photoemission spectra profoundly. A duality is identified wherein a loss of coherence of the ECFL results from an excessively sharp FL.
Extreme correlations arise as the limit of strong correlations, when the local interaction constant UUU goes to infinity. This singular limit transforms canonical fermions to noncanonical Hubbard-type operators with a specific graded Lie algebra replacing the standard anticommutators. We are forced to deal with a fundamentally different and more complex lattice field theory. We study the $t\text{\ensuremath{-}}J$ model, embodying such extreme correlations. We formulate the picture of an extremely correlated electron liquid, generalizing the standard Fermi liquid. This quantum liquid breaks no symmetries, and has specific signatures in various physical properties, such as the Fermi-surface volume and the narrowing of electronic bands by spin- and density-correlation functions. We use Schwinger's source field idea to generate equations for the Green's function for the Hubbard operators. A local (matrix) scale transformation in the time domain to a quasiparticle Green's function is found to be optimal. This transformation allows us to generate vertex functions that are guaranteed to reduce to the bare values for high frequencies, i.e., are ``asymptotically free.'' The quasiparticles are fractionally charged objects, and we find an exact Schwinger-Dyson equation for their Green's function, i.e., the self-energy is given explicitly in terms of the singlet and triplet particle-hole vertex functions. We find a hierarchy of equations for the vertex functions, and further we obtain Ward identities so that systematic approximations are feasible. An expansion in terms of the density of holes measured from the Mott Hubbard insulating state follows from the nature of the theory. A systematic presentation of the formalism is followed by some preliminary explicit calculations. We find a ddd-wave superconducting instability at low TTT that formally resembles that found in the resonating valence-bond theory, but with a much reduced Tc{T}_{c}Tc​.
We present the theory of an extremely correlated Fermi liquid with U→∞. This liquid has an underlying auxiliary Fermi liquid Green's function that is further caparisoned by extreme correlations. The theory leads to two parallel hierarchies of equations that permit iterative approximations in a certain parameter. Preliminary results for the spectral functions display a broad background and a distinct T dependent left skew. An important energy scale Δ(k[over →],x) emerges as the average inelasticity of the FL Green's function, and influences the photoemission spectra profoundly. A duality is identified wherein a loss of coherence of the ECFL results from an excessively sharp FL.
Low energy properties of the metallic state of the two-dimensional t - J model are presented for second neighbor hopping with hole-doping ( ) and electron-doping ( ), with various superexchange energy J . We use a closed set of equations for the Greens functions obtained from the extremely correlated Fermi liquid theory . These equations reproduce the known low energies features of the large U Hubbard model in infinite dimensions. The density and temperature dependent quasiparticle weight, decay rate and the peak spectral heights over the Brillouin zone are calculated. We also calculate the resistivity, Hall conductivity, Hall number and cotangent Hall angle. The spectral features display high thermal sensitivity at modest T for density , implying a suppression of the effective Fermi-liquid temperature by two orders of magnitude relative to the bare bandwidth. The cotangent Hall angle exhibits a T 2 behavior at low T , followed by an interesting kink at higher T . The Hall number exhibits strong renormalization due to correlations. Flipping the sign of changes the curvature of the resistivity versus T curves between convex and concave. Our results provide a natural route for understanding the observed difference in the temperature dependent resistivity of strongly correlated electron-doped and hole-doped matter.
Superconductivity in the t-J model is studied by extending the recently introduced extremely correlated fermi liquid theory. Exact equations for the Greens functions are obtained by generalizing Gor’kov’s equations to include extremely strong local repulsion between electrons of opposite spin. These equations are expanded in a parameter λ representing the fraction of double occupancy, and the lowest order equations are further simplified near Tc, resulting in an approximate integral equation for the superconducting gap. The condition for Tc is studied using a model spectral function embodying a reduced quasiparticle weight Z near half-filling, yielding an approximate analytical formula for Tc. This formula is evaluated using parameters representative of single layer High-Tc systems. In a narrow range of electron densities that is necessarily separated from the Mott–Hubbard insulator at half filling, we find a typical Tc ∼ 102 K.