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arXiv:2202.01611 · Physical review. B./Physical review. B

Divergence of magnetic susceptibility in the SU( N ) Nagaoka-Thouless ferromagnet

Rajiv R. P. Singh, J. Oitmaa

Published Jul 29, 2022• 4 citations
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Abstract

Using finite-temperature strong-coupling expansions for the SU($N$) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-$U$ limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all $N$. 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($N$) parameter $N$ as $1/N$ implying that ferromagnetism is gradually weakened with increasing $N$ as the characteristic temperature scale for ferromagnetic order goes down.

DOI
10.1103/physrevb.106.014424

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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≤\ensuremathρ\ensuremath≤10\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}10\ensuremath≤\ensuremathρ\ensuremath≤1 and all NNN. We express the ferromagnetic susceptibility of the model as a Curie term plus \ensuremathΔ\ensuremathχ\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}\ensuremathΔ\ensuremathχ, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to \ensuremathΔ\ensuremathχ\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}\ensuremathΔ\ensuremathχ are non-negative in the limit that the hole density \ensuremathδ=1\ensuremath−\ensuremathρ\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}\ensuremathδ=1\ensuremath−\ensuremathρ 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\ensuremathΔ/Texp\mathrm{\ensuremath{\Delta}}/Texp\ensuremathΔ/T as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ 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.
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.
We develop finite-temperature strong-coupling expansions for the SU(N)\mathrm{SU}(N)SU(N) Hubbard model in powers of $\ensuremath{\beta}t, w=exp(\ensuremath{-}\ensuremath{\beta}U)$ and $\frac{1}{\ensuremath{\beta}U}$ for arbitrary filling. The expansions are done in the grand canonical ensemble and are most useful at a density of one particle per site, where for UUU larger than or of order of the bandwidth, the expansions converge over a wide temperature range ${t}^{2}/U\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}10U$. By taking the limit $w\ensuremath{\rightarrow}0$, valid at temperatures much less than UUU, the expansions turn into a high-temperature expansion for a dressed SU(N)\mathrm{SU}(N)SU(N) Heisenberg model that includes nearest-neighbor exchange, further neighbor exchanges, and ring exchanges known from the T=0T=0T=0 perturbation theory of the SU(2) Hubbard model. Below a filling of one particle per site, the $w\ensuremath{\rightarrow}0$ limit corresponds to an effective $t\text{\ensuremath{-}}J$ model. The onset of strong correlations can be identified by a plateaulike behavior in the entropy as a function of temperature. At small deviations from one particle per site, the expansions can be arranged in powers of a small parameter $\ensuremath{\delta}=1\ensuremath{-}n$, the deviation from one particle per site, where the leading $\ensuremath{\beta}t$-dependent terms correspond to holes sloshing around in a disordered SU(N)\mathrm{SU}(N)SU(N) background. We use these expansions to calculate the thermodynamic properties of the model at moderate and high temperatures over a wide parameter range.
The occurrance of ferromagnetism for the t-J model on a square lattice is investigated using high-temperature series for the Helmholtz free energy F and the uniform magnetic susceptibility ${\mathrm{\ensuremath{\chi}}}_{0}$. For J=0 fourteenth-order series have been calculated for F and ${\mathrm{\ensuremath{\chi}}}_{0}$ and we estimate F(T=0) to find the ground-state energy. A region of fully polarized ferromagnetism is found for J0. A ferrimagnetic region is observed for larger J/t where ${\mathrm{\ensuremath{\chi}}}_{0}$ is divergent, but F(T=0) is below that of spinless fermions. Our results do not support Nagaoka's theorem at a thermodynamic density of holes.
A finite temperature version of the Nagaoka--Thouless theorem in the SU(n)\mathrm{SU}(n)SU(n) Hubbard model
The Aizenman--Lieb theorem for the SU(2)\mathrm{SU}(2)SU(2) Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization m(β,b)m(β, b)m(β,b) of the system in the presence of a field bbb surpasses the pure paramagnetic value m0(β,b)=tanh⁡(βb)m_0(β, b)=\tanh(βb)m0​(β,b)=tanh(βb). In this manuscript, we present an extension of the Aizenman--Lieb theorem to the SU(n)\mathrm{SU}(n)SU(n) Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.
The Nagaoka $(U=\ensuremath{\infty})$ limit of the Hubbard model on a square lattice is mapped onto the itinerant-localized Kondo model at infinitely strong coupling. Such a model is well suited to perform quantum Monte Carlo (QMC) simulations to compute spin correlation functions. For periodic boundary conditions, this model is shown to exhibit no short-range ferromagnetic (FM) spin correlations at any doping $\ensuremath{\delta}\ensuremath{\ge}0.01$ and at finite temperature T=0.1t.T=0.1t.T=0.1t. Our simulations give no indication that there is a tendency towards ferromagnetic ordering in the ground state, with more than one hole. Employing on the other hand the open boundary conditions (or mixed boundary conditions) may result in the qualitatively different results for the thermodynamic limit depending on the way one chooses to approach this limit. These observations imply that the relevant thermodynamic limit remains unclear.
We study the ferromagnetic instability in an SU(NNN) Fermi-Hubbard model on the hypercubic lattice. Combining dynamical mean-field theory with continuous-time quantum Monte Carlo simulations, we find that, in the strong-coupling regime at low temperatures, ferromagnetically ordered (FM) states develop away from the commensurate fillings. In the particle-doped SU(333) system near one-third filling, the FM state is accompanied by a spontaneous flavor-selective Mott state, where two of the three flavors are Mott insulating while the remaining flavor is metallic. Since particles in the metallic flavor can almost freely move on the lattice without correlation effects, the ordered state is stabilized by the kinetic-energy gain of the doped particles. This is similar to the generalized Nagaoka ferromagnetism discussed in the one-hole-doped system at one-third filling. In the SU(444) case, we find that six distinct types of FM states appear as the particle density varies. The results uncover the nature of the FM state in the SU(NNN) Fermi-Hubbard systems and highlight the rich magnetic behavior enabled by enlarged internal symmetries.