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Extremely correlated limit of the Hubbard model

Edward Perepelitsky

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

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.

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•Jan 1, 2018•0 citations
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.
A strong-coupling series expansion for the Green's function and the extremely correlated Fermi liquid (ECFL) theory are used to calculate the moments of the electronic spectral functions of the infinite-UUU Hubbard model. Results from these two complementary methods agree very well at both low densities, where the ECFL solution is the most accurate, and at high to intermediate temperatures, where the series converge. We find that a modified first moment, which underestimates the contributions from the occupied states and is accessible in the series through the time-dependent Green's function, best describes the peak location of the spectral function in the strongly correlated regime. This is examined by the ECFL results at low temperatures, where it is shown that the spectral function is largely skewed towards the occupied states.
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 consider a doped Mott insulator in the large dimensionality limit within both the recently developed extremely correlated Fermi liquid (ECFL) theory and the dynamical mean-field theory (DMFT). We show that the general structure of the ECFL sheds light on the rich frequency dependence of the DMFT self-energy. Using the leading Fermi liquid form of the two key auxiliary functions introduced in the ECFL theory, we obtain an analytical ansatz, which provides a good quantitative description of the DMFT self-energy down to hole doping level $\ensuremath{\delta}\ensuremath{\simeq}0.2$. In particular, the deviation from Fermi liquid behavior and the corresponding particle-hole asymmetry developing at a low-energy scale are well reproduced by this ansatz. The DMFT being exact at large dimensionality, our study also provides a benchmark of the ECFL in this limit. We find that the main features of the self-energy and spectral line shape are well reproduced by the ECFL calculations in the $O({\ensuremath{\lambda}}^{2})$ minimal scheme, for not too low doping level $\ensuremath{\delta}\ensuremath{\gtrsim}0.3$. The DMFT calculations reported here are performed using a state-of-the-art numerical renormalization-group impurity solver, which yields accurate results down to an unprecedentedly small doping level $\ensuremath{\delta}\ensuremath{\lesssim}0.001$.
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.
We develop a diagrammatic approach for calculating the high temperature expansion of dynamic correlation functions, such as the electron Green's function and the time-dependent density-density and spin-spin correlation functions, for the infinite-U Hubbard Model with any number of spin species. The formalism relies on the use of restricted lattice sums, in which distinct vertices of the diagram represent distinct sites on the lattice. We derive a new formula for the restricted lattice sum of a disconnected diagram consisting of several connected components, and use it to prove the linked cluster theorem with respect to "generalized connected diagrams", formed by overlapping the original connected components on the lattice. This enables us to express all quantities as a sum over these generalized connected diagrams. We compute the Green's function to 4th order in βt for the case of m spin species on a d-dimensional hypercube by hand. We take the m\to\infty limit, enabling us to obtain expressions for the Dyson-Mori self-energy to 4th order in βt for the case of an infinite number of spin species. This may have connections to slave boson techniques used for the study of this model. Our approach is computationally more efficient than any used previously for the calculation of the high temperature expansion of dynamic correlation functions, and high order results for both the Green's function and the time-dependent density-density and spin-spin correlation functions shall be presented in a separate paper [25].
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.