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arXiv:1202.3158

Properties of Hubbard models with degenerate localized single particle eigenstates

Andreas Mielke

Published Feb 14, 2012• 0 citations
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Abstract

We consider the repulsive Hubbard model on a class of lattices or graphs for which there is a large degeneracy of the single particle ground states and where the projector onto the space of single particle ground states is highly reducible. This means that one can find a basis in the space of the single particle ground states such that the support of each single particle ground state belongs to some small cluster and these clusters do not overlap. We show how such lattices can be constructed in arbitrary dimensions. We construct all multi-particle ground states of these models for electron numbers not larger than the number of localized single particle eigenstates. We derive some of the ground state properties, esp. the residual entropy, i.e. the finite entropy density at zero temperature.

DOI
10.48550/arxiv.1202.3158

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We consider the repulsive Hubbard model on a class of lattices or graphs for which there is a large degeneracy of the single particle ground states and where the projector onto the space of single particle ground states is highly reducible. This means that one can find a basis in the space of the single particle ground states such that the support of each single particle ground state belongs to some small cluster and these clusters do not overlap. We show how such lattices can be constructed in arbitrary dimensions. We construct all multi-particle ground states of these models for electron numbers not larger than the number of localized single particle eigenstates. We derive some of the ground state properties, esp. the residual entropy, i.e. the finite entropy density at zero temperature.
The lowest eigenstates of the hopping matrix on the line graph of a cubic lattice with periodic boundary conditions are highly degenerate, they form a lowest flat band. Further, these states are localized. If one considers a repulsive bosonic Hubbard model on this lattice, it is possible to construct exact multi-particle ground states simply by putting particles in the localized single particle ground states such that they avoid each other. This can be done up to a certain critical particle number [Formula: see text]. We prove that at this particle number the ground state entropy is subextensive [Formula: see text]. For lower densities, the entropy is extensive. We further show that the problem is related to the number of 4-cycle decompositions of the cubic lattice with periodic boundary conditions.
We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices---sawtooth chain and two kagom\'e chains---with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential ${\ensuremath{\mu}}_{0}$. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.
We study the repulsive Hubbard model both analytically and numerically on a family of highly frustrated lattices which have one-electron states localized on isolated trapping cells. We construct and count exact many-electron ground states for a wide range of electron densities and obtain closed-form expressions for the low-temperature thermodynamic quantities which are universal for all lattices of the family. Furthermore, we find that saturated ferromagnetism is obtained only for sufficiently high electron densities and large Hubbard repulsion UUU while there is no finite average moment in the ground states at lower densities.
An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.
We consider the antiferromagnetic Heisenberg model and the repulsive Hubbard model for a class of frustrated lattices with a completely dispersionless (flat) lowest one-particle (either one-magnon or one-electron) band. We construct exact many-particle ground states for a wide range of particle densities, calculate their degeneracy, and, as a result, obtain closed-form expressions for the low-temperature thermodynamic quantities around a particular value of the magnetic field h sat or the chemical potential μ 0 . We confirm our analytic findings by numerical data for finite lattices.
Spinless fermions on highly frustrated lattices are characterized by the lowest single-particle band which is completely flat.Concrete realizations are provided by the sawtooth chain and the kagom é lattice.For these models a real-space picture is given in terms of localized states.Furthermore, we find a finite zero-temperature entropy for a suitable choice of the chemical potential.The entropy is computed numerically at finite temperature and one observes a strong cooling effect during adiabatic changes of the chemical potential.We argue that the localized states, the associated zerotemperature entropy as well as the large temperature variations carry over to the repulsive Hubbard model.The relation to flat-band ferromagnetism is also discussed briefly.