Order, disorder, and monopole confinement in the spin-$\frac{1}{2}$ XXZ model on a pyrochlore tube

We study the ground state and thermodynamic properties of the spin-half XXZ model, with an Ising interaction $J_z$ and a transverse exchange interaction $J_x$, on a pyrochlore tube obtained by joining together elementary cubes in a one-dimensional array. Periodic boundary conditions in the transverse directions ensure that the bulk of the system consists of corner-sharing tetrahedra, with the same local geometry as the pyrochlore lattice. We use exact diagonalization, the density matrix renormalization group (DMRG), and minimally entangled typical thermal states (METTS) methods to study the system. When $J_z$ is antiferromagnetic ($J_z>0$) and $J_x$ is ferromagnetic ($J_x<0$), we find a transition from a spin liquid to an XY ferromagnet, which has power-law correlations at $T=0$. For $J_z<0$ and $J_x>0$, spin-two excitations are found to have lower energy than spin-one at the transition away from the fully polarized state, showing evidence for incipient spin-nematic order. When both interactions are antiferromagnetic, we find a nondegenerate ground state with no broken symmetries and a robust energy gap. The low-energy spectra evolve smoothly from predominantly Ising to predominantly XY interactions. In the spin-liquid regime of small $|J_x|$, we study the confinement of monopole-antimonopole pairs and find that the confinement length scale is larger for $J_x<0$ than for $J_x>0$, although both length scales are very short. These results are consistent with a local spin-liquid phase for the Heisenberg antiferromagnet with no broken symmetries.

Figure 1

A pyrochlore tube lattice with length $L=4$ (cubic unit cells) in the long direction. There are four tetrahedra in each cube and the total number of sites $N=4\times 4\times L=64$. The side length of the cube is set to be 1 and hence the distance between the nearest neighbor sites of the pyrochlore lattice is $\sqrt{2}/4$. Primitive translational vectors for the 3D pyrochlore lattice (0, 1/2, 1/2), (1/2, 0, 1/2), and (1/2, 1/2, 0) are represented by red arrows in the figure.

Figure 2

The upper panel shows the three primary ground-state regimes of the model as a function of $\varphi$ defined by the relations $J_z=Jcos\varphi , J_x=Jsin\varphi$: (i) Ising ferromagnet (FM), also called all-in-all-out (AIAO) phase (see text), (ii) XY ferromagnet. and (iii) spin liquid. The nature of the spin-liquid phase and whether there are multiple phases within the spin-liquid region is a primary focus of our study. The lower panels show the phases with $|J_z|=1$. The phase transition points are inferred from the spectra of the 32-site cluster with periodic boundary conditions obtained by exact diagonalization (ED) as well as from a DMRG or ED calculation on a 96-site or 128-site cluster discussed in Sec. 4.

Figure 3

Low-lying spectrum of an $L=2$ periodic cluster with 32 spins obtained by exact diagonalization (ED). The states are characterized by their $S_z$ quantum numbers. Three classes of ground-state regimes are evident from the figure: a high-$S_z$ ground-state phase in the middle, which corresponds to an Ising ferromagnet ($0.63\pi \lesssim \varphi \lesssim 1.25\pi$), and two $S_z=0$ ground-state regimes on the sides ending at the highly degenerate spin ice states at $\varphi =0$ or $2\pi$. Energy levels at $S_z=\pm k$ are degenerate due to spin flip symmetry.

Figure 4

Upper: Ferromagnetic structure factor in the $x$ component as a function of coupling strength $J_x$ for $N=64$ and $N=96$ systems for $J_z=1$ and $J_x<0$. Lower: The logarithm of spin-spin correlations as a function of the logarithm of site distance. A linear relationship for $J_x=-0.2$ and $J_x=-0.5$ indicates a power lower decay of spin-spin correlations. $N=96$ and $N=64$ data are represented by circle and diamond symbols respectively.

Figure 5

Energy gap $\mathrm{\Delta }$ and entanglement entropy $S_{A|B}$ vs $J_x$ for $J_z=1$ and $J_x<0$. In the power-law XY FM phase $|J_x|>0.2$, the extrapolated energy gap closes and the entanglement entropy grows logarithmically with size. The energy gap is finite and nearly size independent in the spin-liquid phase.

Figure 6

Upper: Nematic structure factor $S_{\text{nematic}}$ as a function of coupling strength $J_x$ for $N=64$ and $N=96$ systems. Overlap of different lattice sizes curves imply that it is a short-ranged order. Lower: The logarithm of bond-bond nematic correlations as a function of the logarithm of bond distance. The slopes are $-13,-7.9,-6.8$ for $J_x=0.2,1.0,10.0$, indicating correlation lengths are $0.08,0.13,0.15$ in units of a cubic unit cell respectively for these cases. Correlation lengths in terms of nearest neighbor distance are 0.21, 0.36, and 0.42. $N=96$ and $N=64$ data are represented by circle and diamond symbols, respectively.

Figure 7

Energy gap $\mathrm{\Delta }$ and entanglement entropy $S_{A|B}$ vs $J_x$ for $J_x>0$ and $J_z=1$. A robust energy gap is observed for all $J_x$ values and entanglement entropy saturates ultimately when lattice size increases.

Figure 8

Excitation energy $\mathrm{\Delta }$ for $S_z=63$ and $S_z=62$ excitations from the ferromagnetic state $S_z=64$ for $J_z=-1$ and $J_x>0$. The results are essentially size independent and were checked up to 128-site periodic cluster.

Figure 9

Lowest energy states, relative to the ferromagnetic state, in different $S_z$ sectors for the 32-site periodic cluster. Close to the transition at $\varphi /\pi \approx 0.61$ we observe an even-odd effect, i.e., a differing energy dependence whenever $S_z$ is even or odd, indicative of nematic order.

Figure 10

[(a), (b)] Probability of monopole-antimonopole pair separations $P$ vs pair separations $r$ and [(c), (d)] logarithm of the probability $ln\left(P\right)$ vs separation $r.$

Figure 11

Energy $E$, heat capacity $C$, and entropy $S$ per site as a function of temperature $T$ for fixed $J_z=1$ and several different spin X and Y coupling strengths $J_x=0,-1/11,-0.2$. Different lattice sizes $N=64$ (square point) and $N=96$ (solid lines) are shown in all the panels. The green triangular QMC data points for the 3D pyrochlore lattice are obtained from Ref. [32].

Figure 12

Energy $E$, heat capacity $C$, and entropy $S$ per site as a function of temperature $T$ for fixed $J_z=1$ and several different spin X and Y coupling strengths $J_x=0,0.1,0.3,1.0$. Different lattice sizes $N=64$ (square point) and $N=96$ (solid lines) are shown in all the panels. NLC data [19] for the Heisenberg antiferromagnet ($J_x=1.0$) for the 3D pyrochlore lattice are shown by green triangles.

Figure 13

Bipartite entanglement entropy $S_{A|B}$ as a function of the logarithm of lattice size $ln\left(N\right)$ for XY ferromagnetic phase $J_x=-0.5,J_z=1$. The slope of the fitting curve indicates central charge $c\approx 1$.

Figure 14

We use sites (0,0,0) and (2,0,0) in a pyrochlore tube lattice with length $L=4$ (cubic unit cells) in the long direction as examples and list the six nearest neighbors in the 3D pyrochlore lattice and their equivalent sites in the tube geometry studied. The last column shows whether the nearest neighbor is present in the tube geometry with periodic boundary conditions in the transverse direction but open boundary conditions in the long direction.

Figure 15

The logarithm of spin-spin correlation $\text{ln}(S_x{S}_x\left(r\right))$ vs the logarithm of sites distance $\text{ln}\left(r\right)$ on a 96-site pyrochlore lattice. $J_z=1$ and the value of $J_x$ is shown in the legend. Different truncation error cutoff $\epsilon$ curves are on top of each other, indicating that $\epsilon ={10}^{-6}$ as we use in the main text is small enough to capture the physics.

Figure 16

Exponential decay of spin-spin and bond-bond correlations on a 96-site pyrochlore lattice with $J_z=1,J_x>0$. $\alpha$ is the spin index. Spin $x$ and $z$ are symbolized by circles and triangles respectively. $J_x=0.2,1.0,10.0$ data are represented by different colors.