Pyrochlore $S=\frac{1}{2}$ Heisenberg antiferromagnet at finite temperature

We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore $S=\frac{1}{2}$ quantum antiferromagnet, at finite temperature $T$: canonical typicality for a finite cluster of $2\times 2\times 2$ unit cells (i.e., 32 sites), a finite-$T$ matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T=0.57J$, which is stable across finite size clusters and converged in the series expansion. At $T\approx 0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47k_{B}ln2$, which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on $T$ of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material ${\mathrm{NaCaNi}}_2{\mathrm{F}}_7$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 1

Heat capacity per spin in different pyrochlore magnets, for a detailed account see Sec. 4. The red curve represents the converged part of our results from the NLCE for the $S=\frac{1}{2}$ Heisenberg model. Results for a single tetrahedron for $S=\frac{1}{2}$ (dashed) for $S=\infty$ (classical case, dots) have $T$ scaled to match in the high-$T$ limit. Symbols are for experiments on ${\mathrm{NaCaNi}}_2{\mathrm{F}}_7$ [7], a $S=1$ (approximate) Heisenberg magnet, and the Ising spin ice ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ [8], both scaled in $T$ such that their maxima coincide with that of the $S=\frac{1}{2}$ model (with a factor $ln2/ln3$ to account for the larger $S=1$ entropy). Inset similarly shows entropy per spin. We use natural units in the paper $k_B=1, \hslash =1, J=1$.

Figure 2

Left: Example interaction graph $G$ of an hourglass composed by two corner sharing tetrahedra. Right: Resulting automorphism graph $C$, where each node represents one automorphism, and commuting automorphisms are connected by an edge. The red colored nodes represent the largest clique in the graph, in which all automorphisms commute (red edges). The identity (0,1,2,3,4,5,6) commutes with all automorphisms and has therefore been omitted from the graph $C$. It is part of the largest clique. The largest clique consists of the following eight nontrivial permutations: (2, 0, 1, 3, 5, 6, 4), (1, 2, 0, 3, 6, 4, 5), (2, 0, 1, 3, 6, 4, 5), (0, 1, 2, 3, 6, 4, 5), (1, 2, 0, 3, 4, 5, 6), (2, 0, 1, 3, 4, 5, 6), (1, 2, 0, 3, 5, 6, 4), (0, 1, 2, 3, 5, 6, 4), which are marked in red, the blue edges indicate a smaller clique with five nontrivial permutations.

Figure 3

Comparison and convergence of heat capacity of the spin-$\frac{1}{2}$ pyrochlore Heisenberg model determined via different methods. Top: Numerical linked cluster expansion (NLCE) with clusters of $n=2,\cdots ,8$ complete tetrahedra, as well as their Euler transform for NLCE orders $n=6$, 7, and 8 (red curves). These results appear to be converged down to temperatures of about $T=0.25$. The blue curve shows the result for a 32 site (8 unit cells) cluster with periodic boundaries obtained from canonical typicality. The error bars (shaded blue area) reflect variations in results obtained from sampling over different random vectors. Bottom: DMRG results obtained from the numerical (spline) derivative of the energy ${\langle H\rangle }_{\beta }$ for different bond dimensions $\chi$ obtained by finite temperature DMRG (purification) and the blue crosses represent their extrapolation to $\chi \to \infty$ using a quadratic polynomial in $1/\chi$. The error bars indicate the distance of the extrapolated from the $\chi =10000$ results. Typicality and Euler transform of NLCE results as in top panel.

Figure 4

NLCE results for the thermodynamic entropy per lattice site $S/N$ as a function of temperature $T$. The yellow through dark blue curves show raw NLCE data for different expansion order $n$ up to $n=8$ tetrahedra. The brown, black, and red curves are the corresponding Euler transform, showing converged entropies down to $T\approx 0.25$ for $n=8$. These results agree with the temperature integral (light blue curve) of $C_V/T$ of the Euler $n=8$ data for the specific heat from Fig. 3. We also show the entropy obtained from the finite size 32 site cluster obtained from canonical typicality in the black curve.

Figure 5

Magnetic susceptibility $\chi /N$ as a function of temperature $T$. We show raw data from different NLCE orders (yellow, green) as well as their Euler transform (red). The blue curve shows the result in the finite size cluster with $N=32$ spins and the black crosses with error bars indicate the diagrammatic Monte Carlo results from Ref. [91]. We also show DMRG results for the $N=32$ site cluster (blue crosses) and for the $N=48$ site cluster (green crosses) extrapolated to infinite bond dimension, where the error bars indicate the distance of the extrapolated value to the largest bond dimension $\chi =10000$.

Figure 6

Static spin structure factor $S\left(\vec{Q}\right)$ for different temperatures calculated with finite-$T$ DMRG in the $N=32$, upper rows ($N=48$, lower rows) site cluster using bond dimension of $\chi =8000$ ($\chi =10000$). Even rows show the $(h,h,l)$ plane in momentum space, odd rows the $(h,l,0)$ plane. The DMRG ansatz for the two different clusters has a different symmetry.

Figure 7

Total spin ${\langle {\sum }_{\text{tet}}{\vec{S}}_i\rangle }_{\beta }$ of a tetrahedron in the 32 (blue) site cluster as a function of temperature $T$. The red curves show the Euler transform of NLCE results and the black curve is the total spin of a single tetrahedron (four site cluster) for comparison.

Figure 8

Magnetothermodynamics of the $S=\frac{1}{2}$ pyrochlore Heisenberg magnet. Top: heat capacity $C_V/N$ per lattice site at varying fields as a function of temperature. We show data obtained from the Euler transform of the $n=8$ tetrahedra from NLCE. The shown temperature range corresponds to the part of the curve where the $n=7$ Euler transform agrees with the $n=8$ result to ensure convergence. Red crosses indicate the position of the maximum of the specific heat and the inset shows the temperature of the maximum as a function of the applied field $h$. Bottom: Entropy per site $S/N$ at varying fields as a function of temperature. We show only the converged part of the $n=8$ Euler transform of our NLCE results. Legends for magnetic field $h$ apply for both panels.

Figure 9

Magnetization $m_z/N$ as a function of $T$. We show results obtained from the $n=8$ Euler transform of the NLCE including data from clusters of up to $n=8$ tetrahedra, in the range where they agree with the $n=7$ Euler transform. At low fields $h=0.4, h=0.8$, we observe a local maximum of the finite temperature magnetization (inset). The dashed lines correspond to the magnetization per site for a single tetrahedron.

Figure 10

Convergence comparison of different NLCE expansions. Left: Expansion using clusters with complete tetrahedra up to eighth order. Center: Expansion using complete unit cells up to sixth order (100 clusters with $n=6$ unit cells: included 48 clusters completely with ED and 8 clusters in combination with canonical typicality). Right: Expansion using single sites, up to 14th order. For all expansions the highest order Euler series acceleration is shown. The expansion based on complete tetrahedra consistently yields superior convergence and is used in this work. Dashed lines show the height and position of the Schottky anomaly as extracted from the left panel for comparison. The red dashed lines correspond to the eighth order Euler transform of the tetrahedra expansion, showing that all expansions agree with this result in the converged regime.

Figure 11

Convergence comparison of different NLCE expansions for the susceptibility. Left: Expansion using clusters with complete tetrahedra up to eighth order. Center: Expansion using complete unit cells up to sixth order (100 clusters with $n=6$ unit cells: included 48 clusters completely with ED). Right: Expansion using single sites, up to 14th order. For all expansions the highest order Euler series acceleration is shown. The expansion based on complete tetrahedra consistently yields superior convergence and is used in the remainder of this work. Dashed lines show the height and position of the Schottky anomaly as extracted from the left panel for comparison.

Figure 12

Number of connected $|{\mathcal{C}}_n|$ and topologically distinct clusters $|{\mathcal{T}}_n|$ for the three different expansions.

Figure 13

All six topologically distinct clusters with six tetrahedra.

Figure 14

All 24 topologically distinct clusters with eight tetrahedra.

Figure 15

48 site cluster used in our finite temperature DMRG simulations. We use periodic boundary conditions (periodic bonds not shown for clarity).