Exact study of charge-spin separation, pairing fluctuations, and pseudogaps in four-site Hubbard clusters

Abstract

An exact study of charge-spin separation, pairing fluctuations and pseudogaps is carried out by combining the analytical eigenvalues of the four-site Hubbard clusters with the grand canonical and canonical ensemble approaches in a multidimensional parameter space of temperature (T ), magnetic field (h), on-site interaction (U ) and chemical potential (µ). Our results, near the average number of electrons N ≈ 3, strongly suggest the existence of a critical parameter Uc(T ) for the localization of electrons and a particle-hole binding (positive) gap ∆ e−h (T ) > 0 at U > Uc(T ), with a zero temperature quantum critical point, Uc(0) = 4.584. For U < Uc(T ), particle-particle pair binding is found with a (positive) pairing gap ∆ P (T ) > 0. The ground state degeneracy is lifted at U > Uc(T ) and the cluster becomes a Mott-Hubbard like insulator due to the presence of energy gaps at all (allowed) integer numbers (1 ≤ N ≤ 8) of electrons. In contrast, for U ≤ Uc(T ), we find an electron pair binding instability at finite temperature near N ≈ 3, which manifests a possible pairing mechanism, a precursor to superconductivity in small clusters. Rigorous criteria for the existence of many-body Mott-Hubbard like particle-hole and particle-particle pairings, spin-spin pairing, (spin) pseudogap and (spin) antiferromagnetic critical crossover temperatures, at which the corresponding pseudogaps disappear, are also formulated. In particular, the resulting phase diagram consisting of charge and spin pseudogaps, antiferromagnetic correlations, hole pairing with competing hole-rich ( N = 2), hole-poor ( N = 4) and magnetic ( N = 3) regions in the ensemble of clusters near 1/8 filling closely resembles the phase diagrams and inhomogeneous phase separation recently found in the family of doped high Tc cuprates.

An exact study of pairing fluctuations and phase diagrams in four-site Hubbard Nanoclusters

“…We show only the hole-doped region (N 4) away from half-filling since there is particle-hole symmetry. The true gaps [11] for N = 1, 2 begin to develop at infinitesimal U > 0. The energy gap at < N > ≈ 3 opens at a finite, critical U c = 4.584 and stays open for relatively large U.…”

“…In the former, near < N > ≈ 3 within an interval μ + < μ < μ − in the chemical potential, we observe charge pairing followed by spin pairing at very low-temperature. In the latter we only have a MH like charge gap [11] in a paramagnetic region where< N > ≈ 3.…”

“…Note that onsite parallel spin occupations are forbidden by the Pauli exclusion principle. Our calculations are based on the exact analytical expressions for the eigenvalue E n of the nth many-body eigenstate of the 4-site Hubbard cluster [11] with energies measured in units of t, the hopping parameter. (t has been set to 1 in most of the results that follow.)…”

“…(t has been set to 1 in most of the results that follow.) Thermodynamic properties are calculated by using the grand partition function Z (where the number of particles N and the projection of spin s z are allowed to fluctuate) and the grand canonical potential U [10,11] is, …”

“…Our study has uncovered important answers to these questions [10,11]. Also, the 4-site (square) cluster is the basic building block of the CuO 2 planes in the HTSCs and it can be used as a block reference to build up larger super blocks of desirable sizes in two dimensions (2D).…”