STRONGLY CORRELATED
ELECTRONS

I did research on strongly correlated electron systems while I was a physics senior student and a junior research associate in the Department of Theoretical Physics of Comenius University in Bratislava. Professor Milan Noga, who was then my advisor, taught me the subject and suggested some of the problems.

I first looked at the high-temperature expansion of the Hubbard model with the goal to calculate magnetic susceptibility. A formula for magnetic susceptibility which was then published in the literature was flawed. This motivated Milan Noga to explore the topic more closely. Milan invented a nice representation of the creation and annihilation operators in terms of what he called gamma matrices (of Clifford algebra), and I used the representation to develop an algebraic diagrammatic technique for evaluating the partition function. We found a formula for magnetic susceptibility at high temperatures which was free of anomalies present in previous treatments. At high temperatures (and at strong correlation) the electronic motion that contributes to the equilibrium density matrix is confined to the nearest neighbors; at most, electrons travel around elementary plaquettes (two dimensional unit cells). In a magnetic field electrons carry the Peierls phase and so any two- dimensional motion perpendicular to the field exhibits the Aharonov-Bohm effect (only closed loops contribute to the partition function). This leads to terms in the partition function (and magnetization and susceptibility) that are periodic with magnetic field. The period is large for atomic lattices (thousands of Tesla), but should be observable in lithographically manufactured periodic structures with larger unit cells. The calculations and the Aharonov-Bohm effect on a lattice was reported in J. Fabian, Int. J. Mod. Phys. B 8, 1065 (1994).

The Hubbard model is still thought by some to be the key to high-temperature superconductivity. This belief was much stronger at the time I worked in this field and so I too tried to explore the superconductivity connection. First, there is a question whether the Hubbard-model-like superconductivity is similar to the BCS one. This question is not trivial, since the eigenstates of the Hubbard Hamiltonian cannot be directly solved by the BCS ansatz--the electron-electron interaction is repulsive while the phonon-mediated interaction is attractive. Well, I found a way around: the BCS solution with the repulsive interaction (positive U) now maximizes the energy, rather than minimizes as in the case of attractive forces. BCS thus works well for the maximum energy. However, because the maximum energy and the minimum (ground-state) energy are connected by a simple algebraic formula, the ground-state energy can be approximated as well. The calculation and the comparison with the exact ground-state energy in the one-dimensional case can be found in J. Fabian, Czech. J. Phys. 43, 1136 (1993). I just note that no implications for the ground state eigenstate (and, thus, for superconductivity) follow from my treatment, which is only good for the energy.

I also calculated the optical conductivity of the Hubbard model in the strong coupling regime and at high temperatures. The calculation was motivated by the experiments that showed that resistivity of the cuprates increases linearly with temperature over quite a large temperature scale: from 50 to 1000 K. This is surprising since in the usual metals phonon-limited resistivity rises linearly with temperature only at temperatures above Debye temperature (or slightly below).  To calculate the frequency- and temperature-dependent conductivity I used the Kubo formula (conductivity proportional to the current-current correlation function) and a terribly complicated, but the only systematic, diagrammatic perturbation technique developed by Zaitsev and Izyumov. I managed to go only through the first nonvanishing order. In this order conductivity is inversely proportional to temperature, but the temperature dependence is not in a relaxation time as in the Drude formula. My formula has no relaxation time: it has a delta function divided by temperature. I believe that this behavior is quite general and a simple Drude-like treatment of the optical conductivity of strongly correlated systems is not correct. This can be also seen from the fact that for a strongly correlated system the (frequency-) integrated conductivity is temperature dependent, while the integrated Drude formula is temperature independent. My calculation was published in J. Fabian, Phys. Lett. A 186, 269 (1994).