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Closely competing valence bond crystal orders in the ground state of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study

Rong Cheng, Tao Li

TL;DR

The paper tackles the long-standing problem of the ground state of the spin-1/2 pyrochlore antiferromagnetic Heisenberg model by performing large-scale unrestricted variational optimization of RVB wave functions. Using NN-RVB and a generalized RVB framework with up to $N_v=4N^2$ parameters and a finite-depth BFGS optimizer, the authors uncover a maximally resonating valence bond crystal (VBC) with $2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}$ periodicity and a four-level symmetry-breaking hierarchy, lowering the energy beyond previous RVB benchmarks. The NN-RVB state captures the qualitative features and yields energies very close to the generalized RVB results, with extrapolated $E/N$ around $-0.4846$ to $-0.4848$ in the thermodynamic limit, on clusters up to $N=2048$ sites. The work reveals near-degeneracy with a dressed hard hexagon VBC state and shows that a tiny next-nearest-neighbor coupling can favor the maximally resonating VBC, while also demonstrating that other competing states, such as the monopole flux state, are energetically less favorable. Together, these findings provide a new, highly scalable benchmark for frustrated three-dimensional magnets and offer insights into the possible ground-state orders in pyrochlore materials. Key methodological advances include unrestricted optimization of a highly parameterized RVB ansatz within the $S=1/2$ PAFH framework and the demonstration that a maximally resonating VBC can emerge as a robust ground state in a three-dimensional, strongly frustrated quantum magnet. The results have potential implications for experimental pyrochlore systems and breathing pyrochlores, where hierarchical VBC patterns and symmetry breaking may play a role in the observed behaviors and could inform future material design and interpretation of spectroscopic data.

Abstract

The spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-$\frac{1}{2}$ PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with $2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}$ periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in $\langle \hat{\mathbf{S}}^{2}_{u} \rangle$ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.

Closely competing valence bond crystal orders in the ground state of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study

TL;DR

The paper tackles the long-standing problem of the ground state of the spin-1/2 pyrochlore antiferromagnetic Heisenberg model by performing large-scale unrestricted variational optimization of RVB wave functions. Using NN-RVB and a generalized RVB framework with up to parameters and a finite-depth BFGS optimizer, the authors uncover a maximally resonating valence bond crystal (VBC) with periodicity and a four-level symmetry-breaking hierarchy, lowering the energy beyond previous RVB benchmarks. The NN-RVB state captures the qualitative features and yields energies very close to the generalized RVB results, with extrapolated around to in the thermodynamic limit, on clusters up to sites. The work reveals near-degeneracy with a dressed hard hexagon VBC state and shows that a tiny next-nearest-neighbor coupling can favor the maximally resonating VBC, while also demonstrating that other competing states, such as the monopole flux state, are energetically less favorable. Together, these findings provide a new, highly scalable benchmark for frustrated three-dimensional magnets and offer insights into the possible ground-state orders in pyrochlore materials. Key methodological advances include unrestricted optimization of a highly parameterized RVB ansatz within the PAFH framework and the demonstration that a maximally resonating VBC can emerge as a robust ground state in a three-dimensional, strongly frustrated quantum magnet. The results have potential implications for experimental pyrochlore systems and breathing pyrochlores, where hierarchical VBC patterns and symmetry breaking may play a role in the observed behaviors and could inform future material design and interpretation of spectroscopic data.

Abstract

The spin- antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin- PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.

Paper Structure

This paper contains 13 sections, 44 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The RVB wave functions for the ground state of the spin-$\frac{1}{2}$ PAFH and their optimization
  3. The maximally resonating VBC state as a candidate ground state of the spin-1/2 PAFH
  4. The ground state energy as obtained from unrestricted optimization of the generalized RVB ansatz
  5. Symmetry breaking pattern in the ground state of the spin-$\frac{1}{2}$ PAFH
  6. $\langle \hat{\mathbf{S}}_{u}^{2}\rangle$ as a diagnosis of the symmetry breaking pattern in the ground state of the spin-$\frac{1}{2}$ PAFH
  7. The local spin correlation pattern in the ground state of the $L=2$ and the $L=4$ pyrochlore cluster
  8. An isolated spin chain state or a maximally resonating VBC state?
  9. NN-RVB ansatz for the maximally resonating VBC state and its extension on larger pyrochlore clusters
  10. A comparison with the dressed hard hexagon VBC state
  11. A comparison with the monopole flux state
  12. A summary of numerical results
  13. Conclusions and Discussions

Figures (10)

  • Figure 1: Illustration of the pyrochlore lattice. The pyrochlore lattice is a three dimensional network made of conner-sharing tetrahedrons. $a_{1},a_{2},a_{3}$ denote the three basis vectors of the pyrochlore lattice. The pyrochlore lattice can also be viewed as the stacking of kagome planes oriented in the direction of the four facets of an elementary tetrahedron, namely in the $[1,0,0]$, $[0,1,0]$, $[0,0,1]$ and the $[1,1,1]$ direction.
  • Figure 2: Convergence of the variational energy of the spin-$\frac{1}{2}$ PAFH on a $L=4$ pyrochlore cluster with the optimization steps. Shown here is the result obtained from the unrestricted optimization of a generalized RVB ansatz which contains as many as $N_{v}=262144$ variational parameters. The purple horizontal line marks the energy obtained from the mVMC package with additional symmetry projection. Here we only show the evolution of the variational energy in the last 50000 optimization steps for brevity.
  • Figure 3: The local spin correlation pattern in the ground state of the spin-$\frac{1}{2}$ PAFH on the $L=2$ pyrochlore cluster. Since the spin correlation pattern is translational symmetric, we have only illustrated the result in a single pyrochlore unit cell. Here we use the thickness of the bond to denote the strength of the NN spin correlation on it. The number beside the legend marks the expectation value of the spin correlation on the given bond. We note that the blue bonds actually form closed loops of length $4$ under periodic boundary condition.
  • Figure 4: The local spin correlation pattern in the maximally resonating VBC state. The unit cell is doubled in all the three basis vector directions of the pyrochlore lattice and thus contains in total $32$ lattice sites. For sake of clearance, we have illustrated the two layers of the enlarged unit cell in the $\vec{a}_{3}$ direction separately. Here we use both color and thickness of the bond to distinguish valence bonds of different strength. The number beside the legend denotes the expectation value of the NN spin correlation on the given bond. The blue shaded hexagonal areas mark the location of the $6$-spin resonance process. (c)Illustration of the arrangement of the resonating rings on a kagome plane of pyrochlore lattice in the maximally resonating VBC state. We note that there are four set of kagome planes in the pyrochlore lattice, directed respectively in the $[1,0,0]$, $[0,1,0]$, $[0,0,1]$ and the $[1,1,1]$ direction.
  • Figure 5: The convergence of the variational energy of the spin-$\frac{1}{2}$ PAFH with optimization steps on the $L=4$, $L=6$ and the $L=8$ pyrochlore cluster. The results are obtained by optimizing the NN-RVB ansatz. Here we use the optimized NN-RVB ansatz on the $L=4$ cluster as the initial guess for the $L=6$ cluster. The optimized ansatz for the $L=6$ cluster is again used as the initial guess for the $L=8$ cluster. For both the $L=4$ and the $L=6$ cluster, we have shown here the evolution of the variational energy during the last 10000 optimization steps. For the $L=8$ cluster, we have only conducted 600 optimization steps as a result of the rapid convergence of the variational energy itself and the heavy computational cost on such a large cluster.
  • ...and 5 more figures