Proposal for realizing unpaired Weyl points in a three-dimensional periodically driven optical Raman lattice

TL;DR

This work addresses realizing unpaired Weyl points and the chiral magnetic effect in a controllable quantum platform. It proposes a three-dimensional periodically driven optical Raman lattice (ORL) using four-level ultracold atoms, where adiabatic modulation of Raman and Zeeman terms yields a Floquet low-energy sector hosting eight Weyl points with total chirality $\chi_{\mathrm{tot}}=2\nu_3$ (here tunable to $\nu_3=3$, so $\chi_{\mathrm{tot}}=6$). A synthetic magnetic field implemented via laser-assisted tunneling induces CME-like charge transport, with a quantized pumped charge per cycle in the weak-field regime given by $\Delta Q=\frac{\chi_{\mathrm{tot}} B}{8\pi}$. The authors verify adiabaticity through spin-pumping diagnostics and discuss realistic experimental parameters for alkali atoms (e.g., $^{40}$K), indicating feasibility with current ultracold-atom technology. Overall, the proposal provides a practical platform to study chiral anomaly and nonequilibrium topological phenomena in Floquet-engineered systems.

Abstract

In static lattice systems, the Nielsen-Ninomiya theorem enforces the pairing of Weyl points with opposite chiralities, which precludes the chiral magnetic effect (CME) in equilibrium. Periodic driving provides a viable route to circumvent this no-go constraint. Here, we propose a scheme to realize and control unpaired Weyl points using ultracold atoms in a three-dimensional (3D) optical Raman lattice under continuous periodic driving. By engineering distinct relative symmetries between the lattice and multiple Raman potentials, the configuration generates an effective 3D spin-orbit coupling and yields a tunable topological-insulator phase. Through adiabatic periodic modulation of this system, we show that eight Weyl points emerge in the quasienergy spectrum of the low-energy sector, whose net chirality can be precisely tuned. A nonzero total chirality directly corresponds to the formation of unpaired Weyl points. Furthermore, by implementing a synthetic magnetic field via laser-assisted tunneling in this setup, we demonstrate that the chirality imbalance drives a quantized charge current in the weak-field regime, providing a direct signature of the CME. We verify that the adiabatic condition of the driving protocol, as well as the proposed experimental preparation and detection techniques, are within reach of current ultracold-atom experiments. This work establishes a realistic and controllable platform for exploring chiral-anomaly physics and nonequilibrium topological phenomena linked to Weyl fermions.

Figures (5)

• Figure 1: Schematic of the ORL scheme. (a) Experimental setup. A bias magnetic field is applied along the $y$ direction. In the $x$-$y$ plane, four incident laser beams are retroreflected by mirrors to form standing waves. Two beams at frequency $\omega_1$ are incident along $+x$ ($y$‑polarized, amplitude $E_{xy}$) and $+y$ ($z$‑polarized, amplitude $E_{yz}^{(1)}$), respectively, while the other two at frequency $\omega_2$ are both $z$‑polarized and incident along $+x$ (amplitude $E_{xz}$) and $+y$ (amplitude $E_{yz}^{(2)}$), respectively. Along the $z$ direction, three standing waves are created by counter‑propagating beam pairs: one at frequency $\omega_3$ containing both $x$ and $y$ polarization components (amplitudes $E_{zx}^{(1)}$ and $E_{zy}^{(1)}$), and two $x$‑polarized waves of equal amplitude $E_{zx}^{(2)}$ at frequencies $\omega_4$ and $\omega_4+2\Delta_{\text{ZM}}$, respectively. (b) Optical transitions in a four‑level system generating Raman potentials for alkali atoms. Top: States belonging to different orbital manifolds ($g$ and $e$) are coupled via two‑photon Raman transitions. Bottom: The spin-flip transitions $\ket{g,\sigma}\leftrightarrow\ket{e,\bar{\sigma}}$ are realized through double‑$\Lambda$ coupling configurations, which generate the potentials $\mathcal{M}_{x,y}$, whereas the spin-conserved transitions $\ket{g,\sigma}\leftrightarrow\ket{e,\sigma}$ are realized via single‑$\Lambda$ configurations, giving rise to the potentials $\mathcal{M}_{z,0}$. For alkali atoms, these potentials receive contributions from both the $D_1$ ($S_{1/2}\to P_{1/2}$) and $D_2$ ($S_{1/2}\to P_{3/2}$) lines. Here, $\sigma=\{\uparrow,\downarrow\}$ labels the spin state and $\bar{\sigma}$ denotes the opposite spin.
• Figure 2: Unpaired Weyl points in the quasienergy spectrum of the low‑energy sector. (a) Phase diagram as a function of the mass parameter $m^{(1)}_z$. The red dot marks the value $m_z^{(1)}=0.13E_r$ used in (b)-(c). (b) Left: Locations and chiralities of the Weyl points in the 3D BZ. Right: Quasienergy spectrum of the low‑energy Floquet operator $\tilde{U}_{\mathbf{k}}$ near the Weyl point located at $(0,0,\pi)$. (c) Spin texture of the low‑energy Floquet eigenstates around the Weyl point at $\mathbf{k}=(0,0,\pi)$, yielding $(\mu_x,\mu_y,\mu_z)=(1,-1,-1)$ and hence the chirality $\chi=\mu_x\mu_y\mu_z=+1$. All other parameters are fixed as $(V_0,V_z,M_1,M_2,m_z^{(2)})=(2,4,1,1,0.13)E_r$ and $\eta_0=0.15$.
• Figure 3: Spin pumping and adiabatic condition. (a) Quasienergy spectrum of the low-energy sector along the momentum line $(k_x,k_y)=(0,\pi)$ exhibit opposite winding numbers in the two spin sectors, corresponding to pumped charges $Q_{\uparrow}=1$ (red) and $Q_{\downarrow}=-1$ (blue). (b) Pumped spin $\Delta S$ calculated from the full ORL Hamiltonian versus the driving period $T$. The dashed line marks the quantized value $\Delta S= 1$. The deviation from quantization drops below $1\%$ for $T\geq500 E^{-1}_r$. All parameters are the same as in Fig. \ref{['fig2']}.
• Figure 4: Quantized charge current as a manifestation of the CME. (a) Laser-assisted tunneling scheme. A linear tilt potential $\Delta$ along $y$ suppresses the bare hopping, which is resonantly restored by two running-wave beams with wavevectors $\mathbf{k}_1, \mathbf{k}_2$ and frequencies $\omega_1,\omega_2$. (b) Phase diagram as a function of the mass parameter $m^{(1)}_z$ for $\theta=0$. The red dot marks the value $m_z^{(1)}=0.2E_r$ used in (c). (c) Left: Normalized pumped charge $\Delta Q$ per driving cycle versus magnetic‑field strength $B$. Right: Instantaneous energy gap between the low‑ and high‑energy bands as a function of $B$. All other parameters are taken as $(t_0,t_{z},t_{\rm so},t_{\rm soz}, \eta_0t_{\text{on}}, m_z^{(2)}) = (0.065,0.097,0.050,0.050,0.10,0.13)E_r$, with the driving period $T=500 E_r^{-1}$
• Figure 5: Quasienergy spectra of the Floquet operator computed from the full ORL Hamiltonian (solid line) and the corresponding tight-binding model (dashed line).