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arXiv:1605.08213 · Physical review. B./Physical review. B

Low-energy physics of the t−J model in d=∞ using extremely correlated Fermi liquid theory: Cutoff second-order equations

B. Sriram Shastry, Edward Perepelitsky

Published Jul 28, 2016• 18 citations
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Abstract

We present the results for the low-energy properties of the infinite-dimensional $t\text{\ensuremath{-}}J$ model with $J=0$, 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 $1/(2S)$ 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-$T$ entropy. The resulting quasiparticle weight $Z$, 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 $T$ rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising $T$. The present scheme is generalizable to finite dimensions and a nonvanishing $J$.

DOI
10.1103/physrevb.94.045138

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We present the results for the low-energy properties of the infinite-dimensional t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model with J=0J=0J=0, using O(\ensuremathλ2)O({\ensuremath{\lambda}}^{2})O(\ensuremathλ2) equations of the extremely correlated Fermi liquid formalism. The parameter \ensuremathλ\ensuremath∈[0,1]\ensuremath{\lambda}\ensuremath{\in}[0,1]\ensuremathλ\ensuremath∈[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≲1n\ensuremath{\lesssim}1n\ensuremath≲1. It overcomes the limitation to low densities n\ensuremath≲0.7n\ensuremath{\lesssim}0.7n\ensuremath≲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 \ensuremathω,T\ensuremath{\omega},T\ensuremathω,T self-energy, and the resistivity are reported. These are quite close at all densities to the exact numerical results of the U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞ 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.
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.
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=0J=0J=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 TTT dependence of the resistivity $\ensuremath{\rho}(T)$, 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.
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​.
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.
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.
In this work, we apply the extremely correlated Fermi liquid (ECFL) theory into studying the ttt-t′t't′-JJJ model in one and two dimensions. In 1-d, we combine ECFL and time-dependent density matrix renormalization group (tDMRG) method. The two methods provide a unique insight into the strong momentum dependence of the self-energy of this prototypical non-Fermi liquid, described at low energies as aTomonaga-Luttinger liquid. We also demonstrate its intimate relationship to spin-charge separation,i.e. the splitting of Landau quasiparticles of higher dimensions into two constituents, driven bystrong quantum fluctuations inherent in one dimension. In 2-d, low energy properties of the metallic state of the two-dimensional ttt-JJJ model are presented for second neighbor hopping with hole-doping (t′leq0t'\\leq0t′leq0) and electron-doping (t′>0t'>0t′>0), with various superexchange energy JJJ. The density and temperature dependent spectral properties, resistivity and Hall response are calculated. The spectral features display high thermal sensitivity at modest T for density ngeq0.8n\\geq 0.8ngeq0.8, implying a suppression of the effective Fermi-liquid temperature by two orders of magnitude relative to the bare bandwidth. Flipping the sign of t′t't′ or varying doping delta\\deltadelta changes the curvature of the resistivity versus T curves between convex and concave, consistent to experimental findings.