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Physical review. B, Condensed matter

Diamagnetic susceptibility of the ∞-UHubbard model

A. J. Schofield, J. M. Wheatley, Tao Xiang

Published Oct 15, 1991• 17 citations
Publisher

Abstract

The diamagnetic susceptibility of the \ensuremath{\infty}-U Hubbard model is investigated in one and two dimensions. In one dimension, it is shown that the change current flowing in the ground state of a ring threaded by a flux tube vanishes in the thermodynamic limit. Similar behavior is inferred for the single-hole problem on a two-dimensional square lattice, from exact-diagonalization results on small clusters. Effectively the applied field is cancelled by an induced fictitious field and the susceptibility vanishes. Our results suggest that the Nagaoka ferromagnet is unstable with respect to a low-spin state level crossing to a chiral spin state occurring at infinitesimally small diamagnetic coupling.

DOI
10.1103/physrevb.44.8349

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The diamagnetic susceptibility of the \ensuremath{\infty}-U Hubbard model is investigated in one and two dimensions. In one dimension, it is shown that the change current flowing in the ground state of a ring threaded by a flux tube vanishes in the thermodynamic limit. Similar behavior is inferred for the single-hole problem on a two-dimensional square lattice, from exact-diagonalization results on small clusters. Effectively the applied field is cancelled by an induced fictitious field and the susceptibility vanishes. Our results suggest that the Nagaoka ferromagnet is unstable with respect to a low-spin state level crossing to a chiral spin state occurring at infinitesimally small diamagnetic coupling.
The solution of a three-body Schr\"odinger equation (for two holes and a spin flip) shows that in the U=\ensuremath{\infty} Hubbard model the saturated ferromagnetic state with two holes is unstable (in a finite volume and for periodic boundary conditions). The ground state carries a finite momentum, and becomes degenerate with the Nagaoka state in the thermodynamic limit. This is in agreement with exact diagonalization results for small lattices where, in addition, we find that the true two-hole ground state is a singlet. As a result, our notions of ferromagnetism in the Hubbard model might need revision.
Using finite-temperature strong-coupling expansions for the SU(NNN) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-UUU limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all NNN. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU(NNN) parameter NNN as 1/N1/N1/N implying that ferromagnetism is gradually weakened with increasing NNN as the characteristic temperature scale for ferromagnetic order goes down.
The infinite-U one-dimensional Hubbard model is 'solvable' in the limit that the nearest-neighbour hopping is dominant and infinitesimal longer-range hopping lifts the one-dimensional spin degeneracy. In this limit the Hubbard model maps onto an effective spin interaction: cyclic or ring exchange with a variety of lengths and strengths. For bipartite geometries we find a region of stable ferromagnetism, which originates from the mechanism intrinsic to Nagaoka's ferromagnetism. We look at connectivities that limit exactly to the square-lattice connectivity as the range of the infinitesimal hopping diverges. When we extend this hopping range, we find that the stable region of ferromagnetism shrinks, eventually vanishing as the infinitesimal hopping range diverges and we limit to the square-lattice geometry. Since the calculation involves a non-trivial density of particles in the thermodynamic limit, it suggests how the proposed region of ferromagnetism may become lost for the case of the two-dimensional square lattice, as numerical simulations predict.
We investigate the instability of the saturated ferromagnetic ground state (Nagaoka state) in the Hubbard model on various lattices in dimensions d=2d=2d=2 and d=3d=3d=3. A variational resolvent approach is developed for the Nagaoka instability both for $U=\ensuremath{\infty}$ and for $U<\ensuremath{\infty}$ which can easily be evaluated in the thermodynamic limit on all common lattices. Our results significantly improve former variational bounds for a possible Nagaoka regime in the ground-state phase diagram of the Hubbard model. We show that a pronounced particle-hole asymmetry in the density of states and a diverging density of states at the lower band edge are the most important features in order to stabilize Nagaoka ferromagnetism, particularly in the low-density limit.