Magnetization of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the triangular lattice

After decades of debate, now there is a rough consensus that at zero temperature the spin-$1/2$ Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\circ }$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond $E_b$ and magnetization per spin $M_0$ in the thermodynamic limit are obtained with high precision. The former is estimated to be $E_b=-0.18334\left(10\right)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as $M_0=0.161\left(5\right)$. It is about $32\%$ of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about $20\%$ smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.

Figure 1

Schematic diagram of the PESS wave function ansatz on the infinite triangular lattice. The blue lines are the bonds of the lattice, which are marked by $x,y$, and $z$, respectively. The green lines represent virtual bonds of the wave function. The tensors sitting at the center of the triangles are the simplex tensors $S$ and the tensors covered by red circle are the projection tensors $A$. The rhombus with dashed lines marks a $3\times 3$ unit cell of the trial wave function. The physical indices are perpendicular to the plane and not shown here.

Figure 2

(a) Converting the tensor network skeleton from the honeycomb lattice to a square lattice by one-step contraction, e.g., in the direction of the bonds surrounded by dashed ellipses. (b) The $3\times 3$ unit cell obtained after deformation in the reduced network $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$. Here the environment tensors of the unit cell are shown explicitly, e.g., $L^{(3,1)}$ are the edge tensors associated with the left of $T^{(3,1)}$.

Figure 3

Ground-state energy $E_b$ for $D=10,11,12$, and 13 is plotted as a function of $\chi$. The numerical error is on the fifth digit and they are not shown for a clear vision. The data obtained by GFMC [15] and SE [21] are also shown for comparison. Our data are obviously below that from GFMC but agree well with that by SE (within the error bar).

Figure 4

$M$, marked as $\square$, is plotted as a function of $1/D$. All the data points have already been extrapolated to the infinite-$\chi$ limit. The lines are different numerical fittings: solid line is obtained from all $D$, while dot-dashed line and dashed line are obtained from even and odd $D$ only, respectively.

Figure 5

$E_b$ and $M$ are shown as a function of $\chi$ for $6\times 6$ and $3\times 3$ unit cells. Our results show that they agree well, suggesting that the $3\times 3$ unit cell is large enough for TAHM.

Figure 6

Correlation measured by mutual information in one triangle of the ground-state wave function. Here, $I^{\left(2\right)}$ denotes the total pair correlation, namely $I^{\left(2\right)}=I_{ab}+I_{bc}+I_{ca}$. See Eqs. (8) and (9).