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arXiv:1007.0400 · Physical Review B

Effect of spatial inhomogeneity on the mapping between strongly interacting fermions and weakly interacting spins

V. V. França, Klaus Capelle

Published Oct 5, 2010• 6 citations
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Abstract

A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J, and Heisenberg models. While for spatially homogeneous models in the thermodynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices, and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.

DOI
10.1103/physrevb.82.134405

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A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J, and Heisenberg models. While for spatially homogeneous models in the thermodynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices, and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.
We investigate entanglement of strongly interacting fermions in spatially inhomogeneous environments. To quantify entanglement in the presence of spatial inhomogeneity, we propose a local-density approximation (LDA) to the entanglement entropy, and a nested LDA scheme to evaluate the entanglement entropy on inhomogeneous density profiles. These ideas are applied to models of electrons in superlattice structures with different modulation patterns, electrons in a metallic wire in the presence of impurities, and phase-separated states in harmonically confined many-fermion systems, such as electrons in quantum dots and atoms in optical traps. We find that the entanglement entropy of inhomogeneous systems is strikingly different from that of homogeneous systems.
We review the dynamical mean-field theory of strongly correlated fermion systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact in the limit of large lattice coordination (or infinite spatial dimensions). This method can be used for the determination of phase diagrams and the calculation of thermodynamic properties, one-particle Green's functions, and response functions, using analytic and numerical techniques which are described. We review the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. Some applications of this method to other models are also reviewed. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are provided with this article.
The phase diagram of the attractive Hubbard model with spatially inhomogeneous interactions is obtained using a single-site dynamical mean-field theory-like approach. The model is characterized by three parameters: the interaction strength, the active fraction (fraction of sites with the attractive interaction), and electron filling. The calculations indicate that in a parameter regime with intermediate values of interaction strength (compared to the bare bandwidth of the electrons), and intermediate values of the active fraction, unconventional superconductivity is obtained. The results of this work are likely to be relevant to many systems with spatially inhomogeneous superconductivity such as strongly correlated oxides, systems with negative UUU centers, and, in future, cold atom optical lattices.
This dissertation summarizes my recent work regarding systems of strongly interacting fermionic atoms in optical lattices. This work addresses the combination of two experimental techniques that have been the subject of much recent research in ultracold atom physics. One is the use of optical lattices, which provide a means to realize diverse interaction configurations within a clean, controllable system. The other is the use of magnetically tunable Feshbach resonances to control the strength of the interatomic interaction. Together, these techniques offer the possibility of an experimental realization of many important model Hamiltonians of condensed matter physics, and may also lead to the discovery of new physics. Recent study of this system has shown that strong interactions near Feshbach resonance will lead to the population of multiple lattice bands, and that collisions between atoms on neighboring sites cannot be neglected. These effects lead to a complicated Hamiltonian, but one which can be simplified to an effective single-band model equivalent to the generalized Hubbard model (GHM), which is an extension of the Hubbard model that includes correlated hopping terms. My main results concern the study of this model. The strong correlations between the particles make it difficult to definitively determine the many body physics of the GHM. As a first approach to understanding the GHM in optical lattices, I focus mainly on cases where the problem is greatly simplified by allowing interactions among only small groups of lattice sites. This restriction can be implemented in experiments using an optical superlattice potential. Our results include a proposed scheme (based on double-well superlattices) to empirically verify that the GHM describes this system and to directly measure the various parameters of this model. Other results include exact solutions on four-site square plaquettes, which demonstrate that d-wave excitations can occur in the low-energy states. By using a superlattice to give an array of weakly coupled plaquettes, one can thus produce a d-wave superfluid state. This is of relevance to the study of high-Tc superconductors, although I note certain key differences between the sort of d-wave superfluid described here and that of the superconductors.
In this work, we study two models of strongly correlated fermions: Hubbard and FalicovKimball. We present a proof of the existence of real and ordered solutions to the nested Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown, where U is the strength of the interaction between fermions at the same site. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. For the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model. We also obtain a lower bound for the ground state energy of the Falicov-Kimball model. This bound is given by a bulk term, plus a term proportional to the boundary. A numerical value for the coefficient of the boundary term is determined. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions.
Two aspects of strongly correlated electron systems are studied in this thesis. Hubbard-type models are thought to be at the basis of high-critical-temperature superconductivity. One interaction which has not had much study is the nearest-neighbour hopping of on-site singlet pairs. We refine earlier renormalization group arguments and, using the density-matrix renormalization group method, numerically confirm their prediction that a spin gap opens at infinitesimal pair-hopping amplitude V > 0 in the one-dimensional tight-binding model. We also find a phase separation transition at a finite V > 0 as well as a spin-gap transition at a finite V < 0. The exotic magnetic behaviour of heavy-fermion materials involves an interplay between the screening of local moments from each other and the formation of a magnetic state of long range order. While the single-impurity Kondo Hamiltonian is thought to model some aspects of this behaviour, the properties of the Kondo lattice model away from half-filling are largely unknown. We determine the presence of a spin-gap region in the phase diagram of the one-dimensional Heisenberg-Kondo lattice model and make predictions about certain concealed dimerization order parameters.