Chiral Spin Liquid in Rydberg Atom Arrays

Abstract

Despite long-standing theoretical interest, the chiral spin liquid, a topologically ordered phase, has yet to be observed experimentally. Here we surprisingly find its emergence in an experimentally realized dipolar $\text{XY}$ model when Rydberg atoms are arranged in a breathing kagome lattice. Using the infinite density matrix renormalization group, we numerically calculate the ground state's chiral order parameter, spin-spin correlations, Chern number, and entanglement spectrum. Our numerical results provide strong evidence for the chiral spin liquid phase. Furthermore, we identify a quantum phase transition from a Dirac spin liquid to a chiral spin liquid as the lattice geometry is tuned from the isotropic kagome to the breathing kagome lattice.

Figures (7)

• Figure 1: (a) Rydberg atoms are arranged in a breathing kagome lattice with a zoomed-in view of a unit cell. Within each unit cell, site 2 and site 3 are displaced from their isotropic kagome lattice positions by $h a$, while site 1 remains fixed. Atomic displacements modulate the long-range dipolar interactions, e.g., splitting the nearest-neighbor coupling into two distinct strengths $J_1 \sim 1/[(1-h)a]^3$ and $J_1^\prime \sim 1/[(1+h)a]^3$. An external magnetic field $\boldsymbol{B}$ is applied perpendicular to the atomic plane. (b) Dipolar interactions between atom $i$ and atom $j$, giving rise to an effective spin-$1/2$ XY model. (c) Cylindrical geometry employed in the iDMRG calculations. Different cylindrical geometries, YC$2m$ and YC$2m$-2, correspond to periodic boundary conditions along $\boldsymbol{a}_2$, obtained by identifying site $x$ with site $\alpha$ or site $\beta$ [see (a)], respectively. A flux $\theta$ can be inserted by modifying the dipolar interaction terms to probe the properties of distinct spin liquid phases. (d) Schematic phase diagram of the model in Eq. (\ref{['Hxy']}) with respect to the parameter $h$.
• Figure 2: (a) The spin-spin correlations $\left\langle\sigma^\nu_i \sigma^\nu_j\right\rangle(\nu=x,z)$ plotted as colored dashed or solid bonds between two neighbors with mean values (blue solid, blue dashed, red solid, red dashed): $\left\langle\sigma_i^x \sigma_j^x\right\rangle = (-0.336,-0.324,-0.204,-0.217)$, $\left\langle\sigma_i^z \sigma_j^z\right\rangle = (-0.332,-0.321,-0.177,-0.189)$. For separations beyond these neighbors, the magnitude of all correlations is less than $0.06$. In addition, the chiral order parameters $\chi^\triangleright$ and $\chi^\triangleleft$ are provided. Note that both states $\psi_1$ and $\psi_s$ yield qualitatively similar results. The spin structure factors (b) $S^{xx}(\boldsymbol{k})$ and (c) $S^{zz}(\boldsymbol{k})$ exhibit peaks at the $M_E$ point of the extended Brillouin zone (white dashed line). The unit of $k_x$ and $k_y$ is $1/a$. (d) Spin pumping under adiabatically inserted flux $\theta$. Starting from one ground state, the flux $\theta$ is increased in steps of $\pi/3$. The adiabaticity is verified by the fidelity $\left|\left\langle\psi(\theta)|\psi(\theta+\pi/3)\right|\right\rangle|^2\approx 1$ between adjacent flux steps. Entanglement spectra of the YC12 cylinder for (e) $\psi_1$ and (f) $\psi_s$, resolved by $\sigma^z$ and $p_y = 0, 2\pi/6, \ldots, 5\times 2\pi/6$. The unit of $p_y$ is $1/(2a)$. The spectra display the chiral spin liquid characteristic counting $\{1,1,2,3,5,7,\ldots\}$, marked in black. The spin-pumping results are obtained on the YC10 cylinder with a bond dimension of $3000$, whereas all other results are extracted from the YC12 cylinder with a bond dimension of $10^{4}$. Here, we set $h=0.3$.
• Figure 3: (a) The chiral order parameter, (b) the correlation length, and (c) the fidelity between neighboring points, with respect to the parameter $h$, revealing the continuous phase transition from the Dirac spin liquid to the chiral spin liquid. (d) The transfer matrix spectrum exhibits a gapped excitation energy for the model at $h=0.3$ (gray and red circular markers). The red markers correspond to internode scattering. For comparison, we also plot the spectrum of internode scattering represented by the blue square markers for the lattice at $h=0.1$ in the Dirac spin liquid phase. Prior to the loss of adiabaticity at $\theta = \pi/2$, a linear dispersion indicative of a gapless Dirac point is visible. The cylinder employed here is the YC8-2 lattice, with a bond dimension of $5000$ for calculations of the phase diagram and the spectrum at $h=0.3$, and with a bond dimension of $7000$ for calculating the spectrum at $h=0.1$.
• Figure 4: (a) Phase diagram of ${H}_\text{tot}$ on the YC8-2 cylinder at $h=0.3$. The inset shows the light-shift pattern assigned to each atom in the unit cell. (b) TDVP simulation of the quasi-adiabatic protocol with $\delta_0 = 10$ and $\tau=0.6$ for a finite lattice of $N=42$ sites. The inset shows the light-shift pattern applied to the lattice, with blue (red) circles denoting $\eta_i=1 (-1)$. As the addressing light is ramped down, $P_z$ approaches zero and a nonzero chiral order parameter in the bulk $\chi^\triangleright_\text{bulk}$ emerges, where $\chi^\triangleright_\text{bulk}$ is the average over the three central $\triangleright$ triangles marked gray in the inset. (c) Bulk chiral order parameter of the prepared state at $t=6$ as a function of $h$. We see that $|\chi^\triangleright_\text{bulk}|$ exhibits a significant rise as $h$ increases across the transition point. The left and right insets show the spatially resolved chiral order parameter at each triangle and the correlations $\left\langle\sigma_i^x \sigma_j^x\right\rangle$ at $h=0$ and $h=0.3$, respectively. Site coordinates are fixed to those of the isotropic kagome lattice ($h = 0$) to facilitate comparison across geometries. Each bond represents the correlation between connected sites; circles indicate correlations relative to a fixed reference site highlighted by the dark red circle. The time-evolved state is computed with a bond dimension of 1024.
• Figure S 1: The strengths of the dipolar interactions up to the eighth-neighbor are shown as a function of the parameter $h$. The $n$th largest interaction of the isotropic kagome lattice may split into two distinct values in the breathing kagome lattice: a stronger interaction $J_n$ (solid line) and a weaker interaction $J_n^\prime$ (dashed line). The dot-dashed line indicates the truncation threshold employed in the iDMRG calculations.