Spin Liquids on the Tetratrillium Lattice

TL;DR

This work analyzes spin liquids on the tetratrillium lattice, relevant to the Langbeinite compound K$_2$Ni$_2$(SO$_4$)$_3$, using large-$N$ theory, classical Monte Carlo, and pseudo-Majorana FRG to probe classical and quantum regimes. It finds a gapped, flat-band bottom in the large-$N$ spectrum, classifying the ground state as a fragile classical spin liquid with exponentially decaying correlations and no finite-temperature transitions in the Ising or Heisenberg limits; spin structure factors show quantitative agreement with large-$N$ predictions. For the quantum $S= frac{1}{2}$ case, PMFRG reveals no ordering down to accessible temperatures, leaving the true quantum ground state unresolved and suggesting the potential for emergent gauge structures under perturbations. Overall, the study clarifies a three-dimensional fragile CSL on a non-bipartite tetrahedral lattice and highlights how quantum fluctuations could drive novel disordered phases with nontrivial emergent phenomena in real materials.

Abstract

The tetratrillium lattice has recently been proposed as responsible for the dynamical properties observed in the $S=1$ langbeinite compound K$_2$Ni$_2$(SO$_4$)$_3$. Here, we study in detail the classical spin liquid properties of this lattice of tri-coordinated tetrahedra using classical Monte Carlo and large-$N$ theory calculations. In the large-$N$ limit, we find that the system presents a gapped spectrum with flat bottom bands, giving rise to a fragile spin liquid with exponentially decaying correlations according to the classification of classical spin liquids. We confirm that this scenario also holds in the more realistic Ising and Heisenberg cases, for which the system does not exhibit any finite-temperature phase transition, and the low-temperature spin structure factors exhibit excellent quantitative agreement with the large-$N$ theory. We also provide insight into the quantum $S=1/2$ limit by performing pseudo-Majorana functional renormalization group calculations at finite temperatures, and discuss the possible phases that can arise in the ground state due to quantum fluctuations.

Figures (9)

• Figure 1: (a) Trillium lattice unit cell for $u=0.33554$ and (b) $u=0.59454$, where the 4 triangles in the unit cell are shown. Each site in the unit cell is shown in a different color. (c) Tetratrillium lattice unit cell using the same $u$ values [blue for (a) sites and yellow for (b) sites]. Only the 4 tetrahedra in the unit cell are displayed.
• Figure 2: (a) Spectrum $\omega(\mathbf{q})$ of the coupling matrix $\mathbf{J}(\mathbf{q})$ in the first Brillouin zone at $T=0$, composed of 8 bands with a gap denoted by the dashed black lines. There are 4 degenerate flat bands at the bottom (blue). (b) Values of $\lambda_i$ as a function of the temperature for the two symmetry-inequivalent sublattices and different system sizes.
• Figure 3: Spin structure factor on the tetratrillium lattice for three different planes in the Brillouin zone, displayed one in each column. Each row represents a different model, where the first three correspond to the classical Ising, Heisenberg, and Large-$N$ models at low temperatures, $T=0.01$. The last row corresponds to the PMFRG results for the quantum $S=1/2$ Heisenberg model at $T=0.1$. All panels are normalized individually to their respective maximums.
• Figure 4: (a) Constant energy updates on the trillium lattice and (b) how they translate to star updates on the tetratrillium lattice. (c) Two monomers in the tetratrillium lattice (light red tetrahedra) arise from flipping a bond connecting the two trillium lattices. Blue and yellow sites indicate up and down spins, and the red circles indicate the spins that have been flipped. (d) A loop move of monomers in the top panel is created by flipping the red bonds in the depicted state in the order of increasing thickness (from left to right). The arrows indicate identical sites due to periodic boundaries. This process is identical to a sequence of star updates in the bottom panel (from left to right).
• Figure 5: Top panel: acceptance ratio as a function of the temperature for the tetratrillium lattice, using single-spin Metropolis and star updates (orange and green, respectively), compared with the single-spin update on the trillium lattice (blue). Bottom panel: specific heat $c_v$ and entropy $s$ per site as a function of the temperature for the tetratrillium lattice for $L=2$, 3, and 12. The Pauling estimate is shown by a dashed black line. The blue dashed curve corresponds to the entropy in the trillium lattice.