Exact Results on Itinerant Ferromagnetism and the 15-puzzle Problem

Abstract

We apply a result from graph theory to prove exact results about itinerant ferromagnetism. Nagaoka's theorem of ferromagnetism is extended to all non-separable graphs except single polygons with more than four vertices by applying the solution to the generalized 15-puzzle problem, which studies whether the hole's motion can connect all possible tile configurations. This proves that the ground state of a $U\to \infty$ Hubbard model with one hole away from the half filling on a 2D honeycomb lattice or a 3D diamond lattice is fully spin-polarized. Furthermore, the condition of connectivity for $N$-component fermions is presented, and Nagaoka's theorem is also generalized to $SU(N)$-symmetric fermion systems on non-separable graphs.

References in corpus (18)

• Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms
• From Nagaoka's ferromagnetism to flat-band ferromagnetism and beyond: An introduction to ferromagnetism in the Hubbard model
• Exact SO(5) Symmetry in spin 3/2 fermionic system
• Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms
• Random walks on graphs: ideas, techniques and results
• Phases of the infinite U Hubbard model
• Ground-State Ferromagnetic Transition in Strongly Repulsive One-Dimensional Fermi Gases
• Ferromagnetism in itinerant two-dimensional t2g systems
• Exact Results on Itinerant Ferromagnetism in Multi-orbital Systems on Square and Cubic Lattices
• The inverse Mermin-Wagner theorem for classical spin models on graphs
• Analytic theory of Hund's metals: a renormalization group perspective
• Nagaoka states in the SU($n$) Hubbard model
• Classical Antiferromagnetism in Kinetically Frustrated Electronic Models
• Large N
• Mott made easy
• Sign problem free quantum Monte-Carlo study on thermodynamic properties and magnetic phase transitions in orbital-active itinerant ferromagnets
• Exact results for itinerant ferromagnetism in a $t_{2g}$ orbital system on cubic and square lattices
• Kinetic Magnetism at the Interface Between Mott and Band Insulators

Cited by in corpus (22)

• Equation of State and Thermometry of the 2D SU($N$) Fermi-Hubbard Model
• Ferromagnetism in the SU($n$) Hubbard model with a nearly flat band
• Ferromagnetism in Quantum Dot Plaquettes
• Ferromagnetism in $d$-dimensional SU($n$) Hubbard models with nearly flat bands
• Flat-band Ferromagnetism of the SU$(N)$ Hubbard Model on the Tasaki Lattice
• Exact eigenstates of multicomponent Hubbard models: SU($N$) magnetic $\eta$ pairing, weak ergodicity breaking, and partial integrability
• Ferromagnetism in the SU(N) Kondo lattice model -- SU(N) double exchange and supersymmetry
• Magnetic phase diagram of the infinite-U Hubbard model with nearest- and next nearest-neighbor hoppings
• Phase transitions in the decomposition of $SU(N)$ representations
• On the Divergence of the Ferromagnetic Susceptibility in the SU(N) Nagaoka-Thouless Ferromagnet
• Exact hole-induced $SU(N)$ flavor-singlets in certain $U=\infty$ $SU(N)$ Hubbard models
• Frustration induced Itinerant Ferromagnetism of Fermions in Optical Lattice
• Flat-band ferromagnetism in the SU($N$) Hubbard and Kondo lattice models
• An algebraic approach to revealing magnetic structures of ground states in many-electron systems
• Nonabelian ferromagnets with three-body interactions
• Itinerant ferromagnetism in an SU(3) Fermi-Hubbard model at finite temperatures: A dynamical mean-field theory study
• Quantum permutation puzzles with indistinguishable particles
• Enhanced Itinerant Ferromagnetism in Hole-doped Transition Metal Oxides: Beyond the Canonical Double Exchange Mechanism
• Optimal array geometries for kinetic magnetism and Nagaoka polarons
• Shaping Magnetic Order by Local Frustration for Itinerant Fermions on a Graph
• Software Adaptation and Generalization of Physically-Constrained Games
• Beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion