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Physical review. B, Condensed matter

Spin-disordered ground state for a nearly half-filled Hubbard model

S. E. Barnes

Published May 1, 1990• 1 citation
Publisher

Abstract

Following Nagaoka, the ground state of the large-U nearly half-filled Hubbard model is ``ferromagnetic.'' However, because of the anisotropic exchange energy ${\mathit{J}}_{\mathit{A}}$, the physical Nagaoka ground state is nondegenerate and comprises a totally disordered spin state, i.e., a linear combination of all possible spin arrangements. For this ground state the susceptibility is Pauli-like but of a magnitude associated with a narrow band of width \ensuremath{\sim}${\mathit{J}}_{\mathit{A}}$ while the holons, i.e., the charge excitations, correspond to spinless fermions with a band of width 2zt. Associated with the spinon sector are very poorly defined spin waves, i.e., bosons, with a quadratic dispersion relation. These spin waves do not become soft at any finite temperature, rather, they soften as they approach the metal-insulator transition, i.e., as the number of holes ${\mathit{n}}_{0}$ becomes small. Examined is the problem of constructing a mean-field slave-boson approach for the present problem.

DOI
10.1103/physrevb.41.8991

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Following Nagaoka, the ground state of the large-U nearly half-filled Hubbard model is ``ferromagnetic.'' However, because of the anisotropic exchange energy JA{\mathit{J}}_{\mathit{A}}JA​, the physical Nagaoka ground state is nondegenerate and comprises a totally disordered spin state, i.e., a linear combination of all possible spin arrangements. For this ground state the susceptibility is Pauli-like but of a magnitude associated with a narrow band of width \ensuremath{\sim}JA{\mathit{J}}_{\mathit{A}}JA​ while the holons, i.e., the charge excitations, correspond to spinless fermions with a band of width 2zt. Associated with the spinon sector are very poorly defined spin waves, i.e., bosons, with a quadratic dispersion relation. These spin waves do not become soft at any finite temperature, rather, they soften as they approach the metal-insulator transition, i.e., as the number of holes n0{\mathit{n}}_{0}n0​ becomes small. Examined is the problem of constructing a mean-field slave-boson approach for the present problem.
We study the metal-insulator transition and magnetic ordering in the Hubbard model using the particle-hole mapping. The analysis simplifies near the ferromagnetic limit. We find that the two-dimensional Hubbard model has a charge excitation gap at half filling for any finite U in this region on both the bipartite square lattice and the nonbipartite triangular lattice. In some cases, the system goes through a first-order phase transition to become a paramagnetic metal as Sz{\mathit{S}}_{\mathit{z}}Sz​ is lowered. We also discuss the extension to the doped case. We find that in the large U limit, a single doped hole has a bandwidth of order of J rather than t at Sz{\mathit{S}}_{\mathit{z}}Sz​=0.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
Nagoaka's result that the ground state of the near-neighbor-hopping Hubbard model, for a nearly half-filled band and for interaction strength much greater than hopping energy, is ferromagnetic, is shown to be valid for bosons with spin, as well as fermions, in a finite system. In the thermodynamic limit, the boson system is shown to be nonmagnetic. Ferromagnetism is argued to exist at nonzero temperatures for fermions in an infinite system, and upper and approximate lower bounds are obtained for the Curie temperature.
We study the magnetic properties of itinerant quantum magnetic particles, described by a generalized Hubbard model with large spin (S>1/2)(S>1/2)(S>1/2), which may be realized in optical lattices of laser-cooled atom systems. In fermion systems (half-integer spins), an extended form of Nagaoka ferromagnetism may be realized. However, as novel aspects of the large-spin cases, we found that the condition on the lattice connectivity is more stringent than in the case of S=1/2S=1/2S=1/2 particles and that the system shows a peculiar degenerate structure of the ground state in which the ferromagnetic state is included. In contrast, it turns out that the ground state of itinerant bosonic systems (integer spins) has a degenerate structure similar to that of fermion system with S>1/2S>1/2S>1/2 regardless of the shape, connectivity, or filling of the lattice, and that the state with the maximum total spin is always one of the ground states. Because the system consists of 2S+12S+12S+1 types of particles and we study a SU(2S+1)\text{SU}(2S+1)SU(2S+1) invariant model, the degeneracy of the ground state is given by the multiplets of the fully symmetric Young tableau of SU(2S+1)\text{SU}(2S+1)SU(2S+1) if the state with maximum total spin belongs to the ground state.
The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​. This change is probably associated with a Mott metal-insulator transition (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​ but vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.