Ground-state phases of $S = 1/2$ Heisenberg models on the body-centered cubic lattice

TL;DR

The work addresses the difficulty of simulating 3D frustrated quantum magnets by applying neural quantum states to $S=1/2$ Heisenberg models on the body-centered cubic lattice. Using a ViT-inspired variational ansatz with symmetry projections, the authors map the ground-state phase diagrams of cubic and tetragonal $J_1$-$J_2$ models up to hundreds of spins. They report a first-order Néel–collinear transition at $(J_2/J_1)_c = 0.705 \pm 0.005$ for the cubic model and a first-order Néel–chain transition at $(J_{2ab}/J_1)_c = 1.0375 \pm 0.0125$ for the tetragonal model, with no evidence for a quantum paramagnetic phase in NCCVO-relevant regimes. These findings imply that the tetragonal minimal model alone cannot explain NCCVO’s low-temperature behavior and highlight the need to include additional interactions or effects; the results also demonstrate the viability of neural quantum states for exploring ground states of 3D frustrated quantum magnets.

Abstract

Simulating low-temperature properties of three-dimensional frustrated quantum magnets is challenging due to the sign problem and the system sizes required to mitigate substantial finite-size effects. However, there are many experimental examples of three-dimensional crystals that could host exotic low-temperature states of matter, such as quantum spin liquids. We calculate the ground-state phase diagrams of frustrated quantum spin models on the body-centered cubic lattice using neural quantum states. First, we study the antiferromagnetic $J_1-J_2$ model where we find a direct first-order phase transition between Néel and collinear long-range-ordered phases at $(J_2/J_1)_c = 0.705$, consistent with previous studies. Then, in a tetragonally-distorted variant, proposed as a minimal model of NaCa$_2$Cu$_2$(VO$_4$)$_3$, we find no evidence of a quantum paramagnetic ground state, with a first-order phase transition between Néel and chain phases at $(J_{2ab}/J_1)_c = 1.0375$. Therefore, the ground state of the tetragonally-distorted model does not reproduce the low-temperature magnetic properties of NaCa$_2$Cu$_2$(VO$_4$)$_3$, and the inclusion of other effects is necessary to rationalize experimental observations.

Figures (10)

• Figure 1: Ground states of the classical cubic and tetragonal $J_1-J_2$ models on the body-centered cubic lattice. The exchange interactions satisfied by the state are highlighted. (a) Antiferromagnetic $J_1$ interactions are satisfied in the Néel state, (b) antiferromagnetic $J_2$ interactions by the collinear state and (c) antiferromagnetic $J_{2ab}$ (blue), ferromagnetic $J_{2c}$ (yellow) by the chain state. The sites shown make up the two-site cubic unit-cell of the body-centered cubic lattice.
• Figure 2: (a) (Top) Variational energy per site for the cubic-symmetric model. (Bottom) Relative error of variational energies compared to exact diagonalization for the $(4,2,2)$ lattice. Error bars (smaller than point size in the top panel) are the Monte Carlo errors when computing expectation values from the optimized wavefunction. (b) Correlation ratios for Néel and collinear orders, showing a first-order transition between the two with a critical point, $J_2^c$, that has a strong system size dependence for lattices with minimum side length $2$. The correlation ratios increase with system size in the respective phases, indicating a long-range-ordered phase in the thermodynamic limit. Exact values computed from exact diagonalization are shown. (c) The change in $J_2^c$ with system size where $L = N^{1/3}$. Error bars are given by the spacing of $J_2$ points for which we obtain converged variational solutions. Our values of $J_2^c$ for system sizes with a minimum side length of $4$ are consistent with values in the literature obtained via other methods schmidt2002oitmaa2004majumdar2009farnell2016ghosh2019(gray band), which are well-approximated by the RPA value pantic2014.
• Figure 3: (a) (Top) Variational energy per site for the tetragonal model. (Bottom) Relative error of variational energies compared to exact diagonalization for the $(4,2,2)$ lattice. Error bars are from Monte Carlo sampling of the optimized wavefunction. (b) Correlation ratios for the respective ordered phases, generally showing an increase with system size, indicative of long-range-ordered phases in the thermodynamic limit, and no significant shift in the critical point, $J_{2ab}^c$. Exact values from exact diagonalization are shown. (c) The change in $J_{2ab}^c$ with system size where $L = N^{1/3}$. Error bars are given by the spacing of $J_{2ab}$ points for which we obtain converged variational solutions. The values are consistent with $J_{2ab}^c = 1.0375 \pm 0.0125$, higher than $J_{2ab}^c = 1$ expected for the classical Ising model.
• Figure 4: Finding the optimal symmetry sector for the cubic model. (a-c) Variational energies obtained in different symmetry sectors for $(4,4,2), J_2 = 0.2$. The mean is the final energy of 5 different runs with error bars given by the standard deviation. We also plot the energy obtained by projecting the wavefunction from the previous symmetry stage without any optimization. First, (a), optimizations are carried out for all momenta in the irreducible Brillouin zone. Then, (b), starting from the optimized states, further optimizations/projections are carried out for all irreps of the little group of $\mathbf{k} = \Gamma = (0,0,0)$, giving us the optimal space-group sector. Finally, (c), further optimizations/projections are carried out in the spin-parity antisymmetric/symmetric sectors. Since projections find the same optimal symmetry sector as running further optimizations, for $N \geq 128$ and all system sizes on the tetragonal model we only run optimizations for different momenta then project the optimized states to find the optimal symmetry sectors.
• Figure 5: Comparison of the relative difference of variational energies obtained using different patchings for the (4,2,2) lattice, computed relative to a (1,1,1) patch. The shaded regions correspond to the error, due to fluctuations during the final iterations of the optimization. There is no systematic difference between the patchings.