A Dipolar Chiral Spin Liquid on the Breathed Kagome Lattice

Abstract

Continuous control over lattice geometry, when combined with long-range interactions, offers a powerful yet underexplored tool to generate highly frustrated quantum spin systems. By considering long-range dipolar antiferromagnetic interactions on a breathed Kagome lattice, we demonstrate how these tools can be leveraged to stabilize a chiral spin liquid. We support this prediction with large-scale density-matrix renormalization group calculations and explore the surrounding phase diagram, identifying a route to adiabatic preparation via a locally varying magnetic field. At the same time, we identify the relevant low-energy degrees of freedom in each unit cell, providing a complementary language to study the chiral spin liquid. Finally, we carefully analyze its stability and signatures in finite-sized clusters, proposing direct, experimentally viable measurements of the chiral edge mode in both Rydberg atom and ultracold polar molecule arrays.

Figures (12)

• Figure 1: (a) Schematic of a tweezer array platform where the position of the dipolar spins can be arbitrarily chosen. (b) Top view of the the two-dimensional breathed Kagome lattice, where the distance between the Kagome unit cells is scaled by the breathing parameter $\beta$. We label the small and large triangles by $\blacktriangle$ and $\bigtriangledown$, respectively. (c) A cluster composed of $N=75$ spins that exhibits signatures of the CSL: a lack of magnetic ordering (in single and two-site spin operators), time reversal symmetry breaking (i.e. non-zero chirality), and additional static and dynamic responses (see Figs. \ref{['fig:Cluster_Stability']} and \ref{['fig:Cluster_Dynamics']}). (d) Phase diagram with respect to breathing $\beta$ and the strength $h_z$ of a local magnetic field pattern. There is a direct, second order transition between the paramagnet and the CSL at $h_z \sim 0.01$ for $\beta = 1.5$ (yellow dashed line, details in Fig. \ref{['fig:LocalField']}.
• Figure 2: Full characterization of the CSL state at breathing $\beta = 1.5$ on a YC12=0 cylinder with R2 interaction cutoff. (a) Local spin and chiral correlations for the iMPS ground state. Black rhombus corresponds to the iMPS unit cell, with dashed lines indicating the periodic boundary condition of the cylinder. Each circle encodes the single-site magnetization $\langle \sigma^z \rangle$, while the lines encode the spin-spin correlations $\langle \sigma_i^z \sigma_j^z \rangle$. The system does not display any magnetic ordering, up to small finite-size and convergence effects ($\sim 1\%$ of correlation.) . The color of the small ($\blacktriangle$) and large ($\bigtriangledown$) triangles corresponds to their chiralities---the homogeneous value indicates time reversal symmetry breaking. (b) Two-point correlation functions for $\sigma_z, \sigma_x, \text{ and } \chi_\blacktriangle$. All spin correlations decay to zero at long distances, but the chiral-chiral correlations approach a non-zero value, indicating TRS-breaking. (c) Spin pumping along the cylinder and state overlap per cylinder ring upon the adiabatic threading of $\theta$ flux through the cylinder (top and bottom, respectively). At $\theta = 2 \pi$, a total of $1/2$ spin magnetization has been pumped, and the resulting ground state is orthogonal to the original state, indicating that the ground state is in a different topological sector (top schematic). At $\theta=4\pi$, the final and the initial states are the same, and an integer total magnetization has been pumped. Adiabaticity of our protocol is ensured by the large overlap between $\ket{\psi(\theta)}$ with the previous state $\ket{\psi(\theta - \Delta \theta)}$. (d)[(e)] The entanglement spectrum of the vacuum [semion] as a function of momentum $k$ around the cylinder and the total magnetization quantum number $S_z$ (color). Both sectors display a low-energy chiral edge mode. The number of modes of lowest $|S_z|$ states (boxes) agrees with the expected SU$_1$(2) WZW edge theory.
• Figure 3: (a) Phase diagram with respect to the breathing parameter $\beta$ as measured by the (1) chirality $\langle \chi_\blacktriangle \rangle$, (2) translation symmetry breaking order parameter $O^{\circ / \diamondsuit_{SSB}}$, and (3) correlation length $\xi$. The nature of the transition is determined by the behavior of the order parameter $\chi_\blacktriangle$ and $\xi$. The transition from the KSL into the CSL at $\beta=1.3$ appears second order owing to the continuous behavior of the chirality and the associated diverging correlation length across the transition. These considerations also apply to the Glider-VBS to VBS transition at $\beta=2.5$. The transition from the CSL into the Glider-VBS pattern appears first order due to the discontinuous change of the chirality and the lack of a diverging correlation length. (b) Correlations for the different observed ground states at the $\beta$ values indicated by the colored vertical lines: (b1) the Kagome Spin Liquid (KSL), (b2) Glider-VBS, and (b3) VBS phases. The KSL exhibits homogeneous spin-spin correlations, without breaking any spin rotation or lattice translation. In the Glider-VBS phase, the correlations no longer obey translation symmetry, but instead obey a glider symmetry composed of a translation and a mirror transformation (dashed line). In the VBS state, this glider symmetry is also broken.
• Figure 4: (a) Schematic of the basis we consider for our $\beta \to \infty$ perturbative effective model. For each triangle composed of 3 spins, we project into the 4 lowest-lying, degenerate states, which can be described by 2 spin-$1/2$ d.o.f.s, $\hat{s}$ and $\hat{\eta}$. $\hat{s}$ encodes the spin rotation properties of this manifold, while $\hat{\eta}$ encodes the chirality and the planar distribution of correlations in the triangle. In our effective model, each pair of spins lives on the vertex of a triangular lattice. (b) Projection of the full wavefunction into this low-energy subspace. Even away from the $\beta \rightarrow \infty$ limit, the overlap of the wavefunction with this subspace remains high ($\gtrsim 92 \%$ at the Kagome point $\beta=1$). (c) Using our second-order perturbative effective Hamiltonian $H_{\text{eff}}$, the phase diagram (with respect to $\beta$) of our model closely matches that of the full DMRG calculation for YC8 cylinders with R1 cutoff. This underscores that the $\hat{s}$ and $\eta$ spins are the relevant degrees of freedom that inform the physics across all values of $\beta$.
• Figure 5: Phase diagram with respect to breathing $\beta$ and Ising coupling strength $\Delta$. All numerics were performed on a YC8 cylinder geometry with R1 interaction cutoff. We observe a robust region of the CSL up to large values of $\Delta$, and for large $\Delta$ and $\beta$, we see the emergence of a new ordered phase characterized by Ising AFM order which breaks translation symmetry. (Inset) The two-point $\langle S^z_0S^z_i\rangle$ correlation function, with respect to a fixed site (chosen to be 0, bottom left white site), highlights the presence of long-range Ising AFM order.