Spin-ordered ground state and thermodynamic behaviors of the spin-$\frac{3}{2}$ kagome Heisenberg antiferromagnet

Three different tensor network (TN) optimization algorithms are employed to accurately determine the ground state and thermodynamic properties of the spin-3/2 kagome Heisenberg antiferromagnet. We found that the $\sqrt{3}\times \sqrt{3}$ state (i.e., the state with ${120}^{\circ }$ spin configuration within a unit cell containing 9 sites) is the ground state of this system, and such an ordered state is melted at any finite temperature, thereby clarifying the existing experimental controversies. Three magnetization plateaus ($m/m_s=1/3,23/27$, and 25/27) were obtained, where the 1/3-magnetization plateau has been observed experimentally. The absence of a zero-magnetization plateau indicates a gapless spin excitation that is further supported by the thermodynamic asymptotic behaviors of the susceptibility and specific heat. At low temperatures, the specific heat is shown to exhibit a $T^2$ behavior, and the susceptibility approaches a finite constant as $T\to 0$. Our TN results of thermodynamic properties are compared with those from high-temperature series expansion. In addition, we disclose a quantum phase transition between $q=0$ state (i.e., the state with ${120}^{\circ }$ spin configuration within a unit cell containing three sites) and $\sqrt{3}\times \sqrt{3}$ state in a spin-3/2 kagome XXZ model at the critical point ${\mathrm{\Delta }}_c=0.54$. This study provides reliable and useful information for further explorations on high-spin kagome physics.

Figure 1

(a) Tensor network representation of the spin-3/2 quantum KHAF. There are six inequivalent tensors, where each of tensors X, Y, and Z has three physical indices and three geometrical indices, and each of tensors A, B, and C has only three geometrical indices. Each bond has a diagonal matrix ${\lambda }_i(i=1,2,...9)$, and the length unit $a=1$ is also indicated. The spin configurations of (b) $q=0$ and (c) $\sqrt{3}\times \sqrt{3}$ states are depicted. (d) $m_x$ and $m_z$ components of the local magnetization obtained by cluster update ($D=16$); the three orientations are ${120}^{\circ }$ to each other. We schematically plot them as pointing up, right-down, and left-down, respectively. (e) A phase diagram for the spin-3/2 quantum XXZ model with the anisotropic parameter $\mathrm{\Delta }$ on kagome lattice. When $0\le \mathrm{\Delta }<0.54$, the ground state should be in the $q=0$ state; when $0.54\le \mathrm{\Delta }\le 1$, the ground state is in the $\sqrt{3}\times \sqrt{3}$ state. A quantum phase transition occurs at the critical point ${\mathrm{\Delta }}_c=0.54$.

Figure 2

The energies of the $\sqrt{3}\times \sqrt{3}$ and $q=0$ states as function of bond dimension $D$ calculated by three different (simple, cluster, and full) update schemes for the spin-3/2 KHAF model.

Figure 3

The magnetization curve $m/m_s$ ($m_s$ the saturation magnetization) versus the magnetic field, where there exist three magnetization plateaus, i.e., $m/m_s=1/3$, 23/27, and 25/27, but no $m=0$ plateau.

Figure 4

Specific heat $C$ versus $T$ for the spin-3/2 KHAF model. The solid (red) line represents the results obtained by the cluster update (second-order TSD, with $D=16$). The dashed (black) line is calculated by the tenth-order high-temperature series expansion. Inset: the low-temperature part of $C$, which can be well fitted with a polynomial $C=7.80{(T/J)}^2-64.7{(T/J)}^{5/2}+188{(T/J)}^3-222{(T/J)}^{7/2}+92.4{(T/J)}^4$.

Figure 5

Zero-field magnetic susceptibility $\chi$ as a function of temperature $T$ for the spin-3/2 KHAF model. The result (red solid line) is obtained by the cluster update scheme (second-order TSD and $D=16$). The dashed and dot-dashed lines (black) are calculated by the tenth-order high-temperature series expansion under different parameters Pade[5,5] and Pade[4,6]. Inset: the low-temperature part of $\chi$ versus $T$, showing that at $T\to 0$, $\chi \to 0.164$.

Figure 6

The energy difference $\delta e=(e_{q=0}^0-e_{\sqrt{3}\times \sqrt{3}}^0)/s^2$ between $q=0$ and $\sqrt{3}\times \sqrt{3}$ states as a function of $\mathrm{\Delta }$. The phase transition point is ${\mathrm{\Delta }}_c=0.54$.

Figure 7

Graphic representation for the simple update scheme. (a) The imaginary time evolution for tensor X; the tensors Y and Z are treated similarly. (b) Operations on tensor A; tensors B and C are treated similarly.

Figure 8

Graphical representation for the cluster update scheme. (a) The tensor network contains three inequivalent kinds of hexagons, marked by $N_1$, $N_2$, $N_3$, respectively. Each hexagon contains nine physical indices and six geometrical indices. (b) Take unit cell $N_1$ as an example. We contract the double-layer tensor cluster to get tensor W, which will later be canonicalized for updating the effective environment ${\tilde{\lambda }}_9$, as well as the tensors $X$ and $A$. (c) After the “canonicalization,” we contract the hexagon tensor cluster with its conjugate layer, leaving only ${\tilde{\lambda }}_9$ and one physical bond open, and obtain the reduced density matrix $M^1$.

Figure 9

(a) Graphic illustration of a higher-order singular value decomposition. (b) Tensor network representation of the initial density operator. (c, d) After some transformations, one obtains the initial density operator:

Figure 10

Graphic representation for the full update scheme. (a) Tensors E1, E2, $\cdots$, E6 are environmental matrices obtained by the iTEBD algorithm. Tensors X and A are the “system” tensors that need to be transformed and optimized, where tensor X has three physical indices but tensor A does not. The right panel shows the QR decomposition of tensor X. (b) By cutting the bond between tensors Q and R, and contracting the whole tensor cluster, one get an environment tensor K, with which one optimizes the decimation matrix variationally.