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arXiv:1008.1272 · Physical Review B

Approximate solution of variational wave functions for strongly correlated systems: Description of bound excitons in metals and insulators

Balázs Hetényi

Published Sep 3, 2010• 21 citations
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Abstract

An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase $(U<{U}_{c})$ corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the $U>{U}_{c}$ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.

DOI
10.1103/physrevb.82.115104

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An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase (U<Uc)(U<{U}_{c})(U<Uc​) corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the U>UcU>{U}_{c}U>Uc​ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.
Two very different methods — exact diagonalization on finite chains and a variational method — are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodic Anderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (gn d ) as a function of various parameters. In the absence of on-site Coulomb interaction (U f ) between f electrons, the two methods yield similar results. The double occupancy of d levels remains always finite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains gives the same result for finite U f , while the Gutzwiller method leads to a Brinkman-Rice transition at a critical value (U ), which depends on U f and V.
Two very different methods – exact diagonalization on finite chains and a variational method – are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodic Anderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (νd) as a function of various parameters. In the absence of on-site Coulomb interaction (Uf ) between f electrons, the two methods yield similar results. The double occupancy of d levels remains always finite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains gives the same result for finite Uf , while the Gutzwiller method leads to a Brinkman-Rice transition at a critical value (U c d ), which depends on Uf and V .
We explore the phase diagram of the Hubbard model by generalizing the Gutzwiller variational wave function to take into account nearest-neighbor spin correlations. The energies of various variational states are calculated and compared. At half filling the lowest energy state has long-range antiferromagnetic correlations, in agreement with previous results. Away from half filling, when the antiferromagnet no longer has the lowest energy, the Gutzwiller approximation yields a lower energy than that found for small spin polarons.
This work covers three lectures on Gutzwiller wave functions for correlated electron systems. In the first lecture, some basic ideas on model Hamiltonians and their motivation are outlined, Gutzwiller's variational approach to Landau Fermi-liquid theory is sketched, and, as a first application, the theory is applied to the one-band Hubbard model where the Brinkman–Rice metal-to-insulator transition is found. The second lecture is devoted to the perturbative calculation of expectation values. In this rather technical part the variational analog of Wick's theorem, Feynman diagrams, and the linked-cluster theorem are introduced. The important differences between the variational approach and the standard Green function technique are the absence of a time-dependence and the flexibility to select the expansion parameter of the variational perturbation theory. When this flexibility is used appropriately, not a single self-energy diagram must be calculated in the limit of large dimensions, i.e. in the limit of large lattice coordination number the expectation values can be evaluated exactly for general Hubbard-type models for all interaction strengths and band fillings. The third lecture covers the application of the theory to ferromagnetism of nickel. After a discussion of the nickel problem, multi-band Hubbard models and Gutzwiller wave functions are introduced and parameterized. The favourable comparison between the results from Gutzwiller theory and a large number of experiments shows that correlations play an important role in itinerant ferromagnets.
The Gutzwiller wave function for a strongly correlated model can, if supplemented with a long-range Jastrow factor, provide a proper variational description of Mott insulators, so far unavailable. We demonstrate this concept in the prototypical one-dimensional t-t(') Hubbard model, where at half-filling we reproduce all known phases, namely, the ordinary Mott undimerized insulator with power-law spin correlations at small t(')/t, the spin-gapped metal above a critical t(')/t and small U, and the dimerized Mott insulator at large repulsion.
Recently a new phenomenological Hamiltonian was proposed to describe the superconducting cuprates in which correlations and on-site Coulomb repulsion are introduced by partial Gutzwiller projection. This Gossamer Hamiltonian has an exact ground state and differs from the t-J and Hubbard Hamiltonians in possessing a powerful attractive interaction among electrons responsible for Cooper pairing in the d-wave channel. It is a faithful description for a superconductor with strong on-site electronic repulsion. The superconducting tunneling gap remains intact and despite on-site repulsion. Near half-filling the Gossamer superconductor with strong repulsion has suppressed photoemission intensities and superfluid density, is unstable toward an antiferromagnetic insulator and possesses an incipient Mott-Hubbard gap. The Gossamer technique can be applied to metallic ground states thus possibly serving as an apt description of strongly correlated metals. Such a Gossamer metallic phase, just as the Gossamer superconducting one, becomes arbitrarily hard to differentiate from an insulator as one turns the Coulomb correlations up near half-filling. Both the metallic and superconducting states undergo a quantum phase transition to an antiferromagnetic insulator as one increases the on-site Coulomb repulsion. In the Gossamer model we reach the critical point at half-filling by fully projecting out the double occupancy. Such a critical point might be the Anderson Resonating Valence bond state.