Classification of trivial spin-1 tensor network states on a square lattice

Classification of possible quantum spin liquid (QSL) states of interacting spin-1/2's in two dimensions has been a fascinating topic of condensed matter for decades, resulting in enormous progress in our understanding of low-dimensional quantum matter. By contrast, relatively little work exists on the identification, let alone classification, of QSL phases for spin-1 systems in dimensions higher than one. Employing the powerful ideas of tensor network theory and its classification, we develop general methods for writing QSL wave functions of spin-1 respecting all the lattice symmetries, spin rotation, and time reversal with trivial gauge structure on the square lattice. We find $2^5$ distinct classes characterized by five binary quantum numbers. Several explicit constructions of such wave functions are given for bond dimensions $D$ ranging from two to four, along with thorough numerical analyses to identify their physical characters. Both gapless and gapped states are found. The topological entanglement entropy of the gapped states is close to zero, indicative of topologically trivial states. In $D=4$, several different tensors can be linearly combined to produce a family of states within the same symmetry class. A rich “phase diagram” can be worked out among the phases of these tensors, as well as the phase transitions among them. Among the states we identified in this putative phase diagram is the plaquette-ordered phase, gapped resonating valence bond phase, and a critical phase. A continuous transition separates the plaquette-ordered phase from the resonating valence bond phase.

Figure 1

(a) Schematic figure of lattice symmetry operations on a square lattice. $T_{1,2}$: translation along the $x,y$ directions, $C_4$: $\pi /2$ spatial rotation about the origin, $\sigma$: reflection about the $y=x$ axis. (b) Site tensor transformation under the $C_4$ operation. (c) Site tensor under the $\sigma$ operation. Virtual legs are labeled by $l,r,u,d$, and the physical leg by $p$.

Figure 2

(a) Spin, (b) dimer, (c) vector chirality correlators, and (d) entanglement entropy (EE) for the PEPS made of ${\mathbf{T}}_3^{\left(11\right)}$. All correlations are measured with the finite-size PEPS of dimension $L_x\times L_y$. Measurement distance $R_{ij}$ is chosen as $L_x=L_y=L=3R_{ij}$ for successively increasing linear size $L$. Numerical data (blue circles) are fitted very well to exponential functions (red solid line). Best-fit exponential functions are given for each figure. EE is measured on a cylinder geometry (periodic boundary condition along the $y$ direction) reaching the limit of $L_x\to \infty$, and the topological entanglement entropy is extracted from the fitting, $S_{vN}(L_y=0)=0.06$.

Figure 3

(a) Spin, (b) dimer, and (c) vector chirality correlators for PEPS made of ${\mathbf{T}}_3^{\left(22\right)}$. The same system size is employed as in Fig. 2. Numerical results (blue circle) are fitted to algebraic (spin, dimer) or exponential (spin chirality) functions (red solid line). Best-fit power-law or exponential functions are shown for each figure.

Figure 4

Snapshots of (a) a loop configuration and (b) a dimer configuration that forms the tensor network state formed by tensors ${\mathbf{T}}_4^{\left(2\right)}$ and ${\mathbf{T}}_4^{\left(3\right)}$, respectively. Each red loop in (a) indicates a matrix product state defined in Eq. (4.5), and each red ellipsis in (b) denotes the singlet made out of two spin-1's.

Figure 5

Quantum phase diagram of the class characterized by $({\theta }_{C_4},{\theta }_{\sigma },{\chi }_{\mathcal{T}},{\chi }_{\sigma \mathcal{T}},{\chi }_{\theta =2\pi })=(-1,-1,+1,+1,+1)$ as a function of two tunable parameters $\theta$ and $\varphi$ [Eq. (4.4)]. On the $\theta =0$ line, the PEPS is made of solely ${\mathbf{T}}_4^{\left(1\right)}$ while ${\mathbf{T}}_4^{\left(2\right)}$ and ${\mathbf{T}}_4^{\left(3\right)}$ at the point $(\theta ,\varphi )=(\pi /2,0)$ and $(\pi /2,\pi /2)$, respectively. Here, the blue dot at $(\theta ,\varphi )=(0.26\pi ,0.15\pi )$ denotes the energetically most favorable state for $J_1\text{−}{J}_2$ Hamiltonian with $J_2=0.54J_1$ and its energy density is $E=-1.44J_1$. Along the dotted purple lines specified by (1)–(3), various correlations and physical quantities are measured to identify the quantum phases, and results are shown in Fig. 6.

Figure 6

Dimer correlation functions for several (a) $\varphi$'s at $\theta =0.5\pi$ [(1)-line in Fig. 5] and (b) $\theta$'s at $\varphi =0.5\pi$ [(2)-line in Fig. 5]. Right panels of (a) and (b) are the plaquette order parameter [see Eq. (4.8)] and the energy density for Heisenberg exchange interaction, respectively. Here, $\langle {\mathrm{PO}}_0\rangle =-5/4$ denotes the expectation of the plaquette order parameter for the state made of ${\mathbf{T}}_4^{\left(2\right)}$, i.e., $(\theta ,\varphi )=(0.5\pi ,0)$. As for the correlations, the system size is chosen to be $L_x=L_y=3R_{ij}$ for each $R_{ij}$, while $L_x=L_y=40$ is fixed for measuring the plaquette order parameter and energy density.

Figure 7

Dimer correlation functions along the line (3) in Fig. 5, where the legend specifies the parameters $(\theta ,\varphi )$. The linear system size was chosen to be $L_x=L_y=3R_{ij}$.