Topological phases protected by projective space-time inversion symmetry in alkaline-earth-metal-like atoms

TL;DR

The paper investigates topological phases protected by projective space-time inversion symmetry (P T) in spinless alkaline-earth-like atom systems, demonstrating how interactions map to spinful DIII-class physics in a synthetic lattice. It proposes a quantum simulation scheme using four coupled SSH chains realized as a spinless optical-clock lattice with a uniform plaquette flux, where a Z2 gauge field and projective symmetry yield $(\\mathcal{P}T)^2=-1$. Using DMRG, the authors analyze the interacting regime, showing that interorbital spin-exchange interactions split the fourfold edge degeneracy into twofold and, at stronger coupling, drive transitions to symmetry-broken CDW/ODW/ORS/SRS phases, with a detailed phase diagram and critical properties (central charges on the order of $c\\approx 1.0$ to $3.94$ along phase boundaries). The work provides a framework for realizing and probing interacting topological phases beyond conventional classifications in ultracold atoms, with observable signatures in edge occupations and entanglement spectra, and highlights the practical tunability via orbital Feshbach resonances and clock-laser parameters.

Abstract

An important aspect in categorizing topological phases is whether the system is spinless or spinful, given that these classes exhibit distinct symmetry algebras, leading to disparate topological classifications. By utilizing the projective presentation strategy, the topological phases of spinless (or spinful) systems can be emulated using spinful (or spinless) systems augmented with gauge fields. In this study, we propose to implement the topological phases safeguarded by the unique projective space-time inversion symmetry inherent to spinful models, using synthetic spinless alkaline-earth-metal-like atoms. Employing the separation of orbital and nuclear-spin degrees of freedom, the model is configured as a rectangular tube penetrated by a uniform magnetic flux through each plaquette, which simulates a spinless ladder endowed with projective space-time inversion symmetry satisfying the algebraic properties of a spinful model. For interacting topological phases with interorbital spin-exchange interactions, which also adhere to space-time inversion symmetry, the four-fold degeneracy of edge modes is split into two pairs of edge modes with two-fold degeneracy.We map the complete phase diagram in the end and discover that these interacting topological phases ultimately evolve into distinct charge-density-wave phases via spontaneous symmetry breaking.

Figures (6)

• Figure 1: Illustration of the proposed model. (a) Ultracold alkaline-earth-like atoms are loaded onto one-dimensional (1D) spin-dependent optical superlattices created by a series of interfering laser beams denoted by $R_L$. A narrow $\pi$-polarized clock laser, $R_C^{\pi}$, induces a single-photon transition between the clock-state manifolds within the same hyperfine spin states. By adjusting the angle $\eta$ between the clock laser and the laser generating the optical lattice, the photon recoil momentum can be tuned as $k_{\mathrm{C}}=2 \pi /\lambda_{\mathrm{C}}\cos \eta$ and thus the phases carried by the coupling terms are tunable. Microwaves (MW) are employed to couple the two hyperfine spin states. (b) Energy level diagram and relevant coupling. The two hyperfine spin states are coupled by microwaves. States within the $| ^1S_0 \rangle$ and $| ^3P_0 \rangle$ manifolds are coherently coupled by the spin-conserving clock laser ($R_C^{\pi}$, indicated by the red curves). Additionally, interactions within the clock-state manifolds can couple different spin states across different orbitals (depicted by the green curves). (c) Configuration for the tight-binding model of the 1D optical clock. It consists of a square tube with four legs (namely, $| g\uparrow\rangle$, $|e\uparrow \rangle$ and $| g\downarrow \rangle$, and $|e\downarrow \rangle$), where each on-page plaquette is penetrated by the same synthetic magnetic flux $\phi =\pi \lambda _{\mathrm{L}}/\lambda _{\mathrm{C}}\cos \eta$. Gray cubes represent the unit cells of the configuration, while colored spheres represent the sites.
• Figure 2: Variations of entanglement spectrum and von Neumann entropy. (a) The lowest four levels in the entanglement spectrum $\xi_i$ ($i=1,2,3,4$), and (b) the von Neumann entropy $S_{\mathrm{vN}}$ as functions of $V_{\rm ex}$ with $U=0.0$, $L=56$ and $\rho=0.5$ under open boundary conditions.
• Figure 3: Chemical potential and edge modes. (a) Chemical potential $\mu(N)$ for a chain with $L=56$ lattice sites under open boundary conditions. (b)-(e) The edge-mode density distributions $\Delta \hat{n}_{j\alpha \sigma}$ of the four mid-gap modes for the TI state. Here, $V_{\mathrm{ex}}=0.2$ and $U=0.0$.
• Figure 4: Order parameters induced by interaction. (a) The density distribution $\langle\hat{n}_{j\alpha}\rangle$ in ODW phase with $V_{\text{ex}}=0.0$ and $U=1.8$. (b) $\langle\hat{n}_{j}\rangle$ in CDW phase with $V_{\text{ex}}=-0.2$ and $U=-0.6$. (c) The local order parameter $\langle \hat{\rho}_{j\pm}\rangle$ in ORS phase with $V_{\text{ex}}=-1.9$ and $U=1.0$. (d) $\langle \hat{\rho}_{j\pm}\rangle$ in the SRS phase with $V_{\text{ex}}=1.5$ and $U=0.15$. Here, $L=56$ and $\rho=0.5$.
• Figure 5: The phase diagram of Hamiltonian (\ref{['H1']}) on $V_{\mathrm{ex}}-U$ plane with half filling $\rho=0.5$ in the thermodynamic limit, which contains TI (topological insulator), SRS (spin rung-singlet), ORS (orbital rung-singlet), ODW (orbital density wave) and CDW (charge density wave) phases.