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Physical review. B, Solid state

Polaron effects in the thermodynamics of the narrow-band Hubbard chain

U. Bernstein, G. Beni

Published Apr 15, 1976• 4 citations
Publisher

Abstract

The thermodynamic effects of polaronic interactions in the Hubbard model are studied by high-temperature perturbation theory. Two electron-lattice interaction terms are added to the Hubbard Hamiltonian. The transfer term is treated as the perturbation, so the results are valid for narrow bandwidths and high temperatures. A one-particle Green's-function method is used to find an explicit expression for the secondorder change in the partition function, and a simple analytic expression is given for the weak-binding limit. The magnetic susceptibility and specific heat are computed numerically and given for several cases of polaron binding energies and molecular energy levels. The results are compared with the limit of no electron-lattice coupling and it is shown that they cannot be explained as a simple renormalization of the Hubbard parameters $t$ and $U$.

DOI
10.1103/physrevb.13.3431

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Soviet physics. Doklady•Oct 1, 1984•0 citations
The thermodynamic effects of polaronic interactions in the Hubbard model are studied by high-temperature perturbation theory. Two electron-lattice interaction terms are added to the Hubbard Hamiltonian. The transfer term is treated as the perturbation, so the results are valid for narrow bandwidths and high temperatures. A one-particle Green's-function method is used to find an explicit expression for the secondorder change in the partition function, and a simple analytic expression is given for the weak-binding limit. The magnetic susceptibility and specific heat are computed numerically and given for several cases of polaron binding energies and molecular energy levels. The results are compared with the limit of no electron-lattice coupling and it is shown that they cannot be explained as a simple renormalization of the Hubbard parameters ttt and UUU.
The dimerized-chain Hubbard model is characterized by a Coulomb energy UUU and two alternating transfer constants, an intradimer constant ttt, and a smaller interdimer constant ${t}^{\ensuremath{'}}$. We have used thermodynamic perturbation theory to treat the half-filled-band case, taking the interdimer transfer as the perturbation term. The unperturbed problem is a lattice of isolated dimers. We give the energy, specific heat, and magnetic susceptibility as functions of temperature for the unperturbed case and for the first nonvanishing order in the perturbation. The results are consistent with the U=0U=0U=0 case, which is solved exactly. The interdimer transfer ${t}^{\ensuremath{'}}$ has the effect of distorting the peaks in the specific heat and reducing the susceptibility; this is shown graphically.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc{U}_{c}Uc​ and exhibits strong indications of a lower critical occupation number nc{n}_{c}nc​. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.
We consider a narrow one-dimensional tight-binding chain with intra-atomic Coulomb repulsions described in the Hubbard model. The electrons at each site are furthermore assumed to be coupled to a high-frequency local oscillator. In the limit where the oscillator frequency is large compared to the bandwidth, and in the extreme-band-narrowing regime, for various electron densities, we find that: the half-filled-band ground state is an antiferromagnetic insulator; the one-third-filled-band ground state is a diamagnetic insulator; and the one-quarter-filled-band ground state is a weakly coupled system of bound singlet pairs somewhat reminiscent of superconductivity.
For an infinite linear chain we consider the strong-coupling ($U\ensuremath{\rightarrow}\ensuremath{\infty}$) limit of an extended Hubbard Hamiltonian which takes into account the nearest-neighbor interatomic Coulomb repulsion. It is shown that the magnetic susceptibility obeys a Curie law, and, to this effect, a theorem is proved pertaining to the partition function. The specific heat is computed for the quarter-filled band to lowest order in the transfer integral. The density-density correlation function and the specific heat are also obtained in the atomic limit for an arbitrary density.
We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}$(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is briefly discussed.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.