Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study

Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the fractional quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by exact diagonalization of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral projected entangled pair states (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with density matrix renormalization group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range “gossamer tails”. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)${}_2$ Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge $c=3/2$, as expected for a Moore-Read chiral spin liquid. Using the PEPS bulk-edge correspondence, we argue the “weak” criticality of the bulk is in fact a finite-$D$ artifact of the chiral PEPS, which quickly becomes (practically) irrelevant as the PEPS bond dimension $D$ is increased. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.

Figure 1

CTMRG for the chiral PEPS with one-site unit cell. (a) $A_1+i{A}_2$ PEPS tensor $\mathcal{A}$. (b) Double tensor $E$ obtained by tracing out physical indices $E={\sum }_s{\overline{\mathcal{A}}}^s{\mathcal{A}}^s$ (or $E={\sum }_s{\overline{\mathcal{B}}}^s{\mathcal{B}}^s)$, where ${\overline{\mathcal{A}}}^s\left({\overline{\mathcal{B}}}^s\right)$ is the complex conjugate of ${\mathcal{A}}^s\left({\mathcal{B}}^s\right)$. (c) CTMRG environment for $2\times 2$ cluster, constructed from the (real) corner matrix $C$ and the (hermitian) boundary tensor $T$, depicted separately in (d) and (e), respectively. The environment bond dimension is chosen to be $\chi =k{D}^2(k\in {\mathbb{N}}_{+})$. The energy density is calculated by inserting either the identity operator or the local hamiltonian operator in the red shaded $2\times 2$ cluster. (f) and (g) are the transfer matrices constructed from the $T$ tensor only, and from the $T$ and $E$ tensors, respectively. The maximal correlation length can be obtained from the largest two eigenvalues of both transfer matrices (see text).

Figure 2

Lanczos ED of the spin-1 chiral HAFM on a $4\times 4$ (16-site) (a) and $\sqrt{20}\times \sqrt{20}$ (20-site) (b) tori. The various columns correspond to different IRREPs of the space group and different symbols are used to distinguish eigenstates with different (total) spin quantum numbers. Momenta corresponding to the respective Brillouin zones are shown on the right. The GS energy has been subtracted for clarity.

Figure 3

iPEPS variational energies per site vs $D^2/\chi$. Results for $D=5$ and $D=6$ have been extrapolated to the $\chi \to \infty$ limit. A comparison to DMRG data obtained on cylinders of width $W$ (see text) and plotted vs $1/W\left(\times 4\right)$ is shown. The DMRG data are extrapolated using linear and exponential fits, providing approximate bounds for the thermodynamic energy.

Figure 4

Inverse of iPEPS (maximal) correlation length $1/{\xi }_{\mathrm{MPS}}^{\left(1\right)}$ (see text) vs $D^2/\chi$. The data for the four best optimized PEPS of bond dimension $D=5$ and 6 are shown with different symbols according to legend. Dashed lines are simple power-law fits, $1/\xi \simeq {(D^2/\chi )}^{\alpha },\alpha <1$.

Figure 5

Maximal and next four subleading correlation lengths ${\xi }_{\mathrm{MPS}}^{\left(n\right)},n=1,\cdots ,5$, in the $D=6B_1+i{B}_2$ PEPS (optimized up to $\chi =108)$ vs $\chi$ in log-log scale, and sorted according to the degeneracy $g$ of the corresponding TM eigenvalue $|t_n|$. The dashed (straight) line corresponds to a simple power-law divergence, $\xi =A{\chi }^{\alpha }$ and $\alpha =0.46$. For comparison, spin and dimer correlation lengths are also shown.

Figure 6

Spin-spin, dimer-dimer, and chiral-chiral correlations (in absolute value) vs distance in the $D=6,B_1+i{B}_2$, chiral PEPS. Dashed lines are simple exponential fits of the short-range decay of the spin-spin and dimer-dimer correlations. The correlation lengths are extracted from the exponential decay at large distances.

Figure 7

Weight associated to the exponential decay with the maximal correlation length, as a function of the (maximal) correlation length, in the $B_1+i{B}_2$ chiral PEPS of bond dimension $D=5$ and 6. (a) Semilogarithmic plot. The straight lines are exponential fits $w\left(\xi \right)\sim exp(-\xi /\lambda )$. (b) Log-log plot. The straight lines are power-law fits $w\left(\xi \right)\sim 1/{\xi }^{\alpha }$.

Figure 8

Structure factor of the $D=6B_1+i{B}_2$ chiral PEPS along the path in the Brillouin zone shown in the inset, for different values of the channel environment bond dimension $\chi =87$ (blue), 131 (red), 158 (yellow), and 185 (purple). Due to finite-$\chi$ effects we find slightly negative values (of the order of the truncation error) at the $\mathrm{\Gamma }$ point; in order to better compare the different $\chi$ we have shifted the curves by this small offset.

Figure 9

(a) Cylinder transfer matrix. (b) Boundary (right) vector obtained by contracting $N_v(=6$ here) environment $T$ tensors on a ring is reshaped into the RDM ${\rho }_R$.

Figure 10

Entanglement spectrum of the $B_1+i{B}_{2}D=6$ chiral PEPS for $N_v=6$ (see text) vs edge momentum $K$, computed for $\chi =216$. Different symbols correspond to different values of $|S_z|$, showing that the spectrum is composed of exact $\text{SU}\left(2\right)$ multiplets with integer (a) and half-integer spins (b). Dashed lines correspond to the low-energy chiral CFT modes.

Figure 11

Close-up of the low (quasi-)energy entanglement spectrum of the $D=6$ chiral PEPS for $N_v=6$ vs $K\left[\mathrm{mod}\pi \right]$ (see text). Comparison of the spectra for $\chi =144$ [(a) and (b)] and $\chi =216$ [(c) and (d)] is shown showing an almost convergence with $\chi$. The expected $j=0$ and $j=1/2$ chiral modes of the WZW $\text{SU}{\left(2\right)}_2$ theory appear at the bottom of the ${\mathbb{Z}}_2$-even [(a) and (c)] and ${\mathbb{Z}}_2$-odd [(b) and (d)] sectors of the ES, respectively. The SU(2)-multiplet content of each group of levels is indicated, in agreement with the CFT prediction (except for the blue boxes where a few levels are missing).

Figure 12

(a) The bipartite entanglement entropy as a function of the correlation length for MPS approximations (with increasing bond dimension ${\chi }^{\prime }=50\to 100)$ of the fixed point of $\rho \approx exp(-H_b)$; the three data sets are for different values of the bond dimension $\chi$ of the MPO approximation for $\rho$: $\chi =87$ (blue), 131 (red), and 158 (yellow). The lines are fits to the data points with $S\propto \frac{c}{6}ln\left(\xi \right)$, where we find $c_{\chi =87}=1.187,c_{\chi =131}=1.429$, and $c_{\chi =158}=1.462$. (b) The spectrum of $H_b$ as computed with the quasiparticle Ansatz on the MPO approximation $\rho \approx exp(-H_b)$ with bond dimension $\chi =158$ in the thermodynamic limit.

Figure 13

Comparison between the inverse iPEPS (maximal) correlation length ${\xi }_{\mathrm{E}}^{\left(1\right)}$ (see text) of the spin-1 chiral HAFM (a) and the inverse iPEPS (dimer) correlation length ${\xi }_{\mathrm{dimer}}$ of the spin-1/2 chiral HAFM (b). The data for the four best optimized spin-1 PEPS of bond dimension $D=5$ and 6 are shown in (a). In (b), open and closed symbols correspond to two different sets of the Hamiltonian parameters given by $J_1=K_c=1,J_2=0$ (open symbols) and $J_1=0.89,J_2=0.42$, and $K_c=0.375$ (closed symbols). Dashed lines are simple power-law fits, $1/\xi \simeq {(D^2/\chi )}^{\alpha },\alpha <1$.

Figure 14

The fixed point environments for strips with a single line (a) or with two lines (ladder) (b) of $E$ tensors are obtained as leading eigenvectors of the transfer matrices marked by dotted boxes. Strips to compute spin-spin (c), (longitudinal) dimer-dimer (d), and chiral-chiral (e) correlations, inserting two spin, dimer or chiral operators within the $E$ tensor(s) at distance $d$.

Figure 15

Complete entanglement spectrum of the $B_1+i{B}_{2}D=6$ chiral PEPS for $N_v=6$ vs edge momentum $K$. Different symbols correspond to different values of $|S_z|$, showing that the spectrum is composed of exact $\text{SU}\left(2\right)$ multiplets with integer (a) and half-integer spins (b). Dashed lines correspond to the low-energy chiral CFT modes.

Figure 16

Comparison between entanglement spectra of the $B_1+i{B}_{2}D=6$ chiral PEPS (for $N_v=6)$ computed using the iPEPS $T$ tensor obtained either with $\chi =D^2=36$ ((a) and (c)) or with $\chi =4D^2=144$ ((b) and (d)).