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Journal of Physics C Solid State Physics

Renormalisation group method for the polaron effect in the Hubbard Hamiltonian

D. Schmeltzer, Werner R. Hanke

Published Nov 20, 1982• 5 citations
Publisher

Abstract

The one-dimensional Hubbard Hamiltonian coupled to a boson field of Einstein oscillators is investigated using the real-space renormalisation group (RG) technique. Choosing N=3 sites per RG bloc, the lowest energy states retained are constructed from direct products of electronic (half-filled band case) and the Einstein oscillator states. The authors scale always to a local picture which is solved exactly. In the limit of 'positive' U going to zero, the effective gap (obtained by elimination of the electron-phonon interaction) also tends to zero. The general behaviour of the effective gap (positive or negative) is separated by a line in the plane (U, lambda / omega 0 1 2/), where lambda is the electron-phonon coupling and omega 0 is the frequency.

DOI
10.1088/0022-3719/15/32/004

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arXiv:cond-mat/9703118•Physical review. B, Condensed matter•Aug 15, 1997•188 citations
The one-dimensional Hubbard Hamiltonian coupled to a boson field of Einstein oscillators is investigated using the real-space renormalisation group (RG) technique. Choosing N=3 sites per RG bloc, the lowest energy states retained are constructed from direct products of electronic (half-filled band case) and the Einstein oscillator states. The authors scale always to a local picture which is solved exactly. In the limit of 'positive' U going to zero, the effective gap (obtained by elimination of the electron-phonon interaction) also tends to zero. The general behaviour of the effective gap (positive or negative) is separated by a line in the plane (U, lambda / omega 0 1 2/), where lambda is the electron-phonon coupling and omega 0 is the frequency.
The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.
The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (ty{\mathit{t}}_{\mathit{y}}ty​=tx{\mathit{t}}_{\mathit{x}}tx​); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=ty{\mathit{t}}_{\mathit{y}}ty​/tx{\mathit{t}}_{\mathit{x}}tx​, is different from the unity and zero the ground state is a conductor up to a critical value (U/t)c1{)}_{\mathit{c}1})c1​, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t)c2{)}_{\mathit{c}2})c2​\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.
A new real-space renormalisation group method is developed and tested for the case of the Anderson-Hubbard model. The development is concentrated on obtaining and subsequent analysis of one-dimensional recurrence relations which take into account the effects of interaction and temperature. The results are presented in a form that naturally permits the extension of the given treatment to the case of higher dimensions and off-diagonal bond percolation type of disorder. Although the method itself is not limited to low temperatures, actual computations were made for the case of low temperatures and weak-electron-electron coupling. For this case, the scaling expressions obtained for the ground-state energy and the energy gap are in satisfactory agreement with available exact results.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term VVV is added to the Hubbard interaction.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.