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arXiv:2207.09498 · Physical review. B./Physical review. B

Exact hole-induced resonating-valence-bond ground state in certain U = ∞ Hubbard models

Kyung-Su Kim

Published Apr 3, 2023• 10 citations
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Abstract

We prove that the motion of a single hole induces the nearest-neighbor resonating-valence-bond ground state in the $U=\ensuremath{\infty}$ Hubbard model on a triangular cactus---a treelike variant of a kagome lattice. The result can be easily generalized to $t\ensuremath{-}J$ models with antiferromagnetic interactions $J\ensuremath{\ge}0$ on the same graphs. This is a weak converse of Nagaoka's theorem of ferromagnetism on a bipartite lattice.

DOI
10.1103/physrevb.107.l140401

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We prove that the motion of a single hole induces the nearest-neighbor resonating-valence-bond ground state in the U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞ Hubbard model on a triangular cactus---a treelike variant of a kagome lattice. The result can be easily generalized to t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J models with antiferromagnetic interactions J\ensuremath≥0J\ensuremath{\ge}0J\ensuremath≥0 on the same graphs. This is a weak converse of Nagaoka's theorem of ferromagnetism on a bipartite lattice.
We prove that the motion of a single hole induces S U ( N ) flavor singlets in the U = ∞ S U ( N ) (Fermi) Hubbard model on a Husimi-like tree graph. The result is generalized to certain t − J models with singlet hopping terms typically neglected in the literature. This is an S U ( N ) generalization of the “counter-Nagaoka theorem” introduced in []. Our results suggest the existence of resonating flavor singlet (RFS)-like polarons in the t − J models on a more realistic nonbipartite lattice. Such RFS polarons may be relevant for a novel strong-coupling mechanism of superconductivity or other exotic fractionalized phases of matter. Published by the American Physical Society 2024
An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond (RVB) state. We start with the t-J Hamiltonian for this model describing a single hole in the system and derive exact ground and excited states of the Hamiltonian when the hole is static. The local spin distortion around a hole is termed a spin polaron. The dynamics of the hole is studied through construction of exact eigenstates describing spin polarons in extended states. Coherent states for all momentum wave vectors in the chain direction exist for certain spin polarons. In the Nagaoka limit, the RVB polaronic states have energy greater than that of the ferromagnetic state. For exchange interaction J\ensuremath{\ne}0, the RVB states have lower energy. Other exact states obtained include states in which the hole motion is localized. Structural similarity of the present model to the arrangement of Cu spins in La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​ is also highlighted.
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.
Using exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.
The Hubbard model in the U\to∞ U → ∞ limit has been known to have resonating valence bond (RVB) ground states on certain corner-sharing simplex lattices. Examples include both the quasi-1D sawtooth lattice with open boundary and a larger class of higher dimensional lattices without boundaries. The two types of results were obtained by different approaches which do not apply to one another. In the second class of lattices, the simplest simplex is a tetrahedron. We hereby generalize both results by studying the singly hole-doped system on the quasi-1D lattice of a tetrahedron chain, which can be considered a stripe of the pyrochlore or checkerboard lattices. The energy level ordering of irreducible representations of each tetrahedron shows that a chain of them has exponentially degenerate partial RVB or dimer-monomer ground states where each tetrahedron hosts one spin- 1/2 1 / 2 monomer and one spin- 0 0 dimer. The exact ground states in the infinitely long chain limit are analytically solved by introducing basis transformations between local Hilbert spaces of neighboring tetrahedra, and its energy agrees with the extrapolation of numerical exact diagonalization results of finite sized systems.