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arXiv:2204.08599 · Physical review. B./Physical review. B

Extremely correlated Fermi liquid theory for the U=∞, d=∞ Hubbard model to O(λ3)

S. Shears, Edward Perepelitsky, Michael Arciniaga, B. Sriram Shastry

Published Jul 6, 2022• 2 citations
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Abstract

We present the $O({\ensuremath{\lambda}}^{3})$ results from the $\ensuremath{\lambda}$ expansion in the extremely correlated Fermi-liquid theory applied to the infinite-dimensional $t\ensuremath{-}J$ model (with $J=0$) and compare the results with the earlier $O({\ensuremath{\lambda}}^{2})$ results as well as the results from the dynamical mean-field theory. We focus attention on the $T$ dependence of the resistivity $\ensuremath{\rho}(T)$, the Dyson self-energy, and the quasiparticle weight $Z$ at various densities. The comparison shows that all the methods display quadratic-in-$T$ resistivity followed by a quasilinear-in-$T$ resistivity characterizing a strange metal and gives an estimate of the different scales of these variables relative to the exact results.

DOI
10.1103/physrevb.106.035108

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We present the O(\ensuremathλ3)O({\ensuremath{\lambda}}^{3})O(\ensuremathλ3) results from the \ensuremathλ\ensuremath{\lambda}\ensuremathλ expansion in the extremely correlated Fermi-liquid theory applied to the infinite-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model (with J=0J=0J=0) and compare the results with the earlier O(\ensuremathλ2)O({\ensuremath{\lambda}}^{2})O(\ensuremathλ2) results as well as the results from the dynamical mean-field theory. We focus attention on the TTT dependence of the resistivity \ensuremathρ(T)\ensuremath{\rho}(T)\ensuremathρ(T), the Dyson self-energy, and the quasiparticle weight ZZZ at various densities. The comparison shows that all the methods display quadratic-in-TTT resistivity followed by a quasilinear-in-TTT resistivity characterizing a strange metal and gives an estimate of the different scales of these variables relative to the exact results.
We present the results for the low-energy properties of the infinite-dimensional $t\text{\ensuremath{-}}J$ model with J=0J=0J=0, using $O({\ensuremath{\lambda}}^{2})$ equations of the extremely correlated Fermi liquid formalism. The parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$ is analogous to the inverse spin parameter 1/(2S)1/(2S)1/(2S) in quantum magnets. The present analytical scheme allows us to approach the physically most interesting regime near the Mott insulating state $n\ensuremath{\lesssim}1$. It overcomes the limitation to low densities $n\ensuremath{\lesssim}0.7$ of earlier calculations, by employing a variant of the skeleton graph expansion, and a high-frequency cutoff that is essential for maintaining the known high-TTT entropy. The resulting quasiparticle weight ZZZ, the low $\ensuremath{\omega},T$ self-energy, and the resistivity are reported. These are quite close at all densities to the exact numerical results of the $U=\ensuremath{\infty}$ Hubbard model, obtained using the dynamical mean field theory. The present calculation offers the advantage of generalizing to finite TTT rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising TTT. The present scheme is generalizable to finite dimensions and a nonvanishing JJJ.
Recent progress in extremely correlated Fermi liquid theory (ECFL) and the dynamical mean field theory (DMFT) enables us to accurately compute in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit the resistivity of the $t\ensuremath{-}J$ model after setting $J\ensuremath{\rightarrow}0$. This is also the $U=\ensuremath{\infty}$ Hubbard model. Since JJJ is set to zero, our study isolates the dynamical effects of the single occupation constraint enforced by the projection operator originally introduced by Gutzwiller. We study three densities n=.75,.8,.85n=.75,.8,.85n=.75,.8,.85 that correspond to a range between the overdoped and optimally doped Mott insulating state. We delineate four distinct regimes separated by three crossovers, which are characterized by different behaviors of the resistivity $\ensuremath{\rho}$. We find at the lowest temperature TTT a Gutzwiller correlated Fermi liquid regime with $\ensuremath{\rho}\ensuremath{\propto}{T}^{2}$ extending up to an effective Fermi temperature that is dramatically suppressed from the noninteracting value by the proximity to half filling, $n\ensuremath{\sim}1$. This is followed by a Gutzwiller correlated strange metal regime with $\ensuremath{\rho}\ensuremath{\propto}(T\ensuremath{-}{T}_{0})$, i.e., a linear resistivity extrapolating back to $\ensuremath{\rho}=0$ at a positive T0{T}_{0}T0​. At a higher temperature scale this crosses over into the bad metal regime with $\ensuremath{\rho}\ensuremath{\propto}(T+{T}_{1})$, i.e., a linear resistivity extrapolating back to a finite resistivity at T=0T=0T=0 and passing through the Ioffe-Regel-Mott value where the mean free path is a few lattice constants. This regime finally gives way to the high TTT metal regime, where we find $\ensuremath{\rho}\ensuremath{\propto}T$, i.e., a linear resistivity extrapolating back to zero at T=0T=0T=0. The present work emphasizes the first two, i.e., the two lowest temperature regimes, where the availability of an analytical ECFL theory is of help in identifying the changes in related variables entering the resistivity formula that accompanies the onset of linear resistivity, and the numerically exact DMFT helps to validate the results. We also examine thermodynamical variables such as the magnetic susceptibility, compressibility, heat capacity, and entropy and correlate changes in these with the change in resistivity. This exercise casts valuable light on the nature of charge and spin correlations in the Gutzwiller correlated strange metal regime, which has features in common with the physically relevant strange metal phase seen in strongly correlated matter.
In this work, we describe the simplifications to the Extremely Correlated Fermi Liquid Theory (ECFL) \\cite{ECFL, Monster} which occur in the limit of infinite spatial dimensions. In particular, we show that the single-particle electron Green's function G(k) can be written in terms of two momentum-independent self-energies. Moreover, we elucidate the nature of the ECFL \\lambda expansion in the limit of infinite dimensions and carry out this expansion explicitly to O(\\lambda^2). Additionally, we demonstrate the vanishing of vertex corrections to the optical conductivity in general and to each order in \\lambda in the limit of infinite dimensions. We generalize the ECFL formalism to the infinite-U Anderson impurity model (AIM) , and demonstrate a Dynamical Mean-Field Theory (DMFT) like mapping between the ECFL objects of the infinite-dimensional t-J model and the infinite-U AIM, and show that this mapping holds to each order in \\lambda. We compute the spectral function for the AIM to O (\\lambda^2) and make comparisons with results obtained through Numerical Renormalization Group (NRG) computations. Finally, we develop a novel formalism for the high-temperature expansion of dynamical correlation functions in the infinite-U Hubbard model which is more efficient than any used previously and gives results for an arbitrary number of spin species. We use it to calculate the single-particle Green's function G(k) to fourth order in (\\beta t) for m spin-species on a d-dimensional hypercube.
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.
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​.
The Extremely Correlated Fermi Liquids (ECFL) theory is reviewed as a framework for understanding the ttt-JJJ model in metallic systems close to the Mott insulating limit. This overview presents the underlying ideas and the resulting equations in a form accessible to nonexperts. We compare theoretical results with all available resistivity data for single-layer High-Tc_{c}c​ systems, and with some spectral data. The highlighted results include a density dependent quasilinear T-dependence in resistivity, an unusually small quasiparticle weight, and distinct low-temperature emergent scales that dominate transport, thermodynamics and spectral properties of single-layer High Tc_cc​ systems. Suggestions are made for further experiments to probe the physics of these challenging quantum many-body systems.