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arXiv:2112.14475

A finite temperature version of the Nagaoka--Thouless theorem in the $\mathrm{SU}(n)$ Hubbard model

Tadahiro Miyao

Published Dec 29, 2021• 0 citations
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Abstract

The Aizenman--Lieb theorem for the $\mathrm{SU}(2)$ Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization $m(β, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(β, b)=\tanh(βb)$. In this manuscript, we present an extension of the Aizenman--Lieb theorem to the $\mathrm{SU}(n)$ 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.

DOI
10.48550/arxiv.2112.14475

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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 Lieb-Mattis theorem on the absence of one-dimensional ferromagnetism is extended here from ground states to T>0 by proving, inter alia, that M(\ensuremath{\beta},h), the magnetization of a quantum system in a field h>0, is always less than the pure paramagnetic value M0{\mathit{M}}_{0}M0​(\ensuremath{\beta},h)=tanh(\ensuremath{\beta}h), with \ensuremath{\beta}==1/kT. Our proof rests on a new formulation in terms of path integrals that holds in any dimension; another of its applications is that the Nagaoka-Thouless theorem on the Hubbard model also extends to T>0 in the sense that M(\ensuremath{\beta},h) exceeds M0{\mathit{M}}_{0}M0​(\ensuremath{\beta},h).
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.
We present an extension of Nagaoka's theorem in the SU(nnn) generalization of the infinite-UUU Hubbard model. It is shown that, when there is exactly one hole, the fully polarized states analogous to the ferromagnetic states in the SU(2) Hubbard model are ground states. For a restricted class of models satisfying the connectivity condition, these fully polarized states are the unique ground states up to the trivial degeneracy due to the SU(nnn) symmetry. We also give examples of lattices in which the connectivity condition can be verified explicitly. The examples include the triangular, Kagom\'e, and hypercubic lattices in $d\phantom{\rule{0.28em}{0ex}}(\ensuremath{\ge}\phantom{\rule{-0.16em}{0ex}}2)$ dimensions, among which the cases of d=2d=2d=2 and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.
In a recent paper by Aizenman and Lieb [Phys. Rev. Lett. 65, 1470 (1990)], it was reported that the Nagaoka ferromagnet preserves its ferromagnetic character at temperatures T>0 in any finite volume. It is shown here that at any T>0 the Nagaoka ferromagnet becomes a perfect paramagnet in the thermodynamic limit. By using an inequality for the free energy, this result is extended to the general case when the number of electrons per site goes to 1. At any T>0 the spontaneous magnetization m(T,\ensuremath{\rho}) tends to zero as the electron density \ensuremath{\rho} tends to 1. Thus, a rigorous upper bound on m(T,\ensuremath{\rho}) is obtained. The results are independent of the statistics.
A new viewpoint of universality is introduced in strongly correlated electron systems; Our description relies on the operator theoretic correlation inequalities. We explain the Marshall-Lieb-Mattis theorem and Lieb\rq{}s theorem from a viewpoint of universality; in addition, from the new perspective, we prove that Lieb\rq{}s theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. We also study Nagaoka-Thouless\rq{} theorem and its stabilities in terms of universality.
The Hubbard model is a "highly oversimplified model" for electrons in a solid which interact with each other through extremely short ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two pieces; H_hop which describes quantum mechanical hopping of electrons, and Hint which describes nonlinear repulsive interaction. Either H_hop or H_int alone is easy to analyze, and does not favor any specific order. But their sum H=H_hop+H_int is believed to exhibit various nontrivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting "universality classes" of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results on the Hubbard model which shed light on "physics" of this fascinating model. We mainly concentrate on magnetic properties of the model at its ground states. We discuss Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on decay of correlations at finite temperatures in two-dimensions, Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one-dimension, Lieb's important theorem for half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to nonexperts by describing basic definitions and elementary materials in detail.