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Resonant Valance Bond Ground States on Corner-sharing Lattices

Zhao Zhang, Cecilie Glittum

TL;DR

This work addresses how resonating valence bond (RVB) ground states emerge in corner-sharing lattices in the $U\to\infty$ limit by studying a quasi-1D pyrochlore stripe that interpolates between previously studied lattices. The authors develop an exact solution for a chain of tetrahedra where each tetrahedron hosts a spin-$\tfrac{1}{2}$ monomer and a spin-$0$ dimer, revealing an exponentially degenerate manifold of partial RVB ground states and solving the infinite-chain ground state via basis transformations; the results agree with extrapolated exact diagonalization. They extend the analysis to the pyrochlore stripe, showing that the ground state comprises dimer-monomer configurations with holon-induced domain walls, and that the local SU(2) structure organizes into spin-doublet representations with explicit energy spectra. The paper also discusses the implications for periodic boundary conditions and finite doping, suggests potential nested Bethe Ansatz approaches, and indicates that the observed partial RVB physics may generalize to other corner-sharing simplex networks. Overall, the work unifies two prior RVB constructions on corner-sharing lattices and provides a concrete, exactly solvable framework for partially RVB ground states in quasi-1D geometries, with a clear path toward higher dimensions and integrability analyses.

Abstract

The Hubbard model in the $U\to\infty$ limit has recently been shown to have resonant valence bond (RVB) ground states on the corner-sharing sawtooth and pyrochlore lattices in the dilute doping limit of a single vacancy. The two results were obtained by different approaches which do not apply to one another. We make the first step towards unifying them by studying the quasi-1D lattice of a pyrochlore stripe, where all corners are not shared between two tetrahedra, and the valence bond configurations are not fixed by the location of the vacancy. 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-$1/2$ monomer and one spin-$0$ 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.

Resonant Valance Bond Ground States on Corner-sharing Lattices

TL;DR

This work addresses how resonating valence bond (RVB) ground states emerge in corner-sharing lattices in the limit by studying a quasi-1D pyrochlore stripe that interpolates between previously studied lattices. The authors develop an exact solution for a chain of tetrahedra where each tetrahedron hosts a spin- monomer and a spin- dimer, revealing an exponentially degenerate manifold of partial RVB ground states and solving the infinite-chain ground state via basis transformations; the results agree with extrapolated exact diagonalization. They extend the analysis to the pyrochlore stripe, showing that the ground state comprises dimer-monomer configurations with holon-induced domain walls, and that the local SU(2) structure organizes into spin-doublet representations with explicit energy spectra. The paper also discusses the implications for periodic boundary conditions and finite doping, suggests potential nested Bethe Ansatz approaches, and indicates that the observed partial RVB physics may generalize to other corner-sharing simplex networks. Overall, the work unifies two prior RVB constructions on corner-sharing lattices and provides a concrete, exactly solvable framework for partially RVB ground states in quasi-1D geometries, with a clear path toward higher dimensions and integrability analyses.

Abstract

The Hubbard model in the limit has recently been shown to have resonant valence bond (RVB) ground states on the corner-sharing sawtooth and pyrochlore lattices in the dilute doping limit of a single vacancy. The two results were obtained by different approaches which do not apply to one another. We make the first step towards unifying them by studying the quasi-1D lattice of a pyrochlore stripe, where all corners are not shared between two tetrahedra, and the valence bond configurations are not fixed by the location of the vacancy. 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- monomer and one spin- 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.

Paper Structure

This paper contains 8 sections, 28 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The RVB Ground State on the Sawtooth Lattice
  3. The monomer-dimer ground states on the Pyrochlore Stripe
  4. Symmetries and Irreducible Representations
  5. Energy Level Ordering of Irreducible Representations
  6. Exact Ground States in the Thermodynamic Limit
  7. Conclusion
  8. Funding information

Figures (8)

  • Figure 1: Effective hopping strength in the presence of vacancies. Due to the fermionic statistics, particle-hole transformation introduces a $\pi$-flux through odd-length loops containing a holon.
  • Figure 2: The ground states in the single holon sector of (a) a single triangle as a superposition of dimer configurations, and (b) the sawtooth lattice as a superposition of domain wall configurations.
  • Figure 3: Ground state energy $E_{\rm GS}$ versus $1/L$ from exact diagonalization for the sawtooth chain (blue) and the pyrochlore stripe (red). Dashed lines give the upper bound from the largest system studied, solid lines give the lower bound through extrapolation from the two biggest system sizes studied. Stars mark the analytical results for an infinite chain.
  • Figure 4: (a) A pyrochlore stripe consisting of 4 tetrahedra, showing one possible dimer configuration in the ground states having a single vacancy. (b) The effective hopping constants inside the $k$th tetrahedron, where the hopping constant between two sites $j$ and $j^\prime$ is defined by the convention $(-1)^{j-j^\prime}t$. (c) The four degenerate ground states of a single tetrahedron. The large arrows represent a spin-singlet dimer and the small arrows represent a spin-doublet. Unlike in the sawtooth lattice which allows a fixed convention of the dimer direction, the direction of the dimers here are specified by arrows.
  • Figure 5: An orthonormal basis for the subspace with two spin-ups and one spin-down in (a) the 8-dimensional spin-doublet sector, and (b) the 4-dimensional spin-quadruplet sector. The grey triangle represents that the spins at the three corners are symmetrized.
  • ...and 3 more figures