SU(4) topological resonating valence bond spin liquid on the square lattice

We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate $\mathbf{6}$ representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to ${\mathbb{Z}}_2$, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.

Figure 1

(a) RVB state defined as a resonance state between SU(4) singlet bond configurations on a 2D square lattice. Singlets (shown as ellipses) are made from two antisymmetric 6 representations (represented as two-box Young tableaus). (b) SU(4) RVB state represented as a 2D PEPS tensor network.

Figure 2

(a) NN RVB tensor $T_0$ with only one virtual $\mathbf{6}$ state. (b) $T_1,T_2$, and $T_3$ tensors with three virtual $\mathbf{6}$ states. (c) In the iPEPS method one constructs first a bilayer $\mathbb{E}=A\otimes A^{*}$ tensor by contracting the physical index. (d) Corner transfer matrix renormalization group (CTMRG) algorithm for a rotation-invariant tensor $\mathbb{E}$. A unique corner matrix $C$ and a unique side tensor $T$ are renormalized by adding $\mathbb{E}$ iteratively. (e) Transfer matrix $\mathcal{T}=T\otimes T$. (f) $|{\mathrm{\Psi }}_{\mathrm{Env}}\rangle$ (infinite) quantum chain obtained by contracting the $\mathbb{E}$ TN on half of the 2D plane (here from top to bottom). (g) ${\sigma }_b$ boundary operator on a $N_v=3$ ring. The $D^2$ degrees of freedom on each site have been reshaped as $D\times D$.

Figure 3

Environment EE $S_{\text{Env}}$ plotted as a function of the logarithm of the correlation length $\xi$. Linear fits enable one to extract the central charge $c$ of the wave functions defined by $T_0, T_2$, and $T_3$; a higher order correction $a{\xi }^{-b}$ is included for $T_1$. We find $c\simeq 1$ in all cases.

Figure 4

(a) Correlation length versus environment dimension for $\varphi =0$, and various values of $\theta$ interpolating $A(\theta ,0)$ between $T_2$ ($\theta =0$) and $T_0$ ($\theta =\pi /2$). The $\chi \to \infty$ correlation length ${\xi }_{\infty }$ (for $\theta \ne 0,\pi /2$) is obtained from an exponential fit $\xi \left(\chi \right)={\xi }_{\infty }+aexp(-\chi /\ell )$. (b) Correlation length ${\xi }_{\infty }$ versus $\theta$ (for $\varphi =0$). The correlation length seems to diverge for $\theta <\pi /2$ (in the vicinity of the NN RVB) in the hatched region. In contrast, no critical region was found in the vicinity of the critical point $\theta =\varphi =0$ (blue line). Note that the correlation length has a minimum around $\theta =\pi /8$.

Figure 5

Renyi entanglement entropies $S_q$ at $q=1/3$ versus the half-cylinder circumference $N_v$ for PEPS parametrized by (a) $\varphi =0$ and $\theta =\pi /8$, (b) $\varphi =0$ and $\theta =3\pi /8$, and (c) $\varphi =\pi /4$ and $\theta =\pi /2$, obtained at the $\chi$ values indicated on the plots. Squares (circles) label the odd (even) sector. The stars stand for the average of the two sectors and the dashed lines are linear fits. The arrow points to the value $-ln2$.

Figure 6

(a) Modular matrix $S$ of a critical or a trivial gapped phase. (b) Modular matrix $S$ of a topological phase. (c) Trace of the modular matrix $S$ obtained by TRG for $A(\theta ,0)$ versus $\theta$. The value 2 corresponds to a topological phase while 4 implies a trivial phase. We show two sets of data obtained for two ranges of the cut parameter $\chi$ and we note that larger $\chi$ are needed when the correlation length grows.