A two-dimensional realization of the parity anomaly

Abstract

Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.

Figures (10)

• Figure 1: (\ref{['fig:coupled_wire_model:dimensions']}) Synthetic two-dimensional quantum system composed of a continuous spatial dimension $x$ and a discrete synthetic dimension $m$. The kinetic energy and cosine lattice potential govern dynamics along $x$, while position-dependent hoppings along $m$ acquire phases from an effective vector potential. (\ref{['fig:coupled_wire_model:phase_diagram']}) Phase diagram showing tunable Chern numbers $\mathcal{C}=0$ and $1$ in gapped phases, separated by a critical point where the bulk gap closes at a single Dirac point. (\ref{['fig:coupled_wire_model:bandstructure']}) Band structure of the coupled-wire model for various values of $|\lambda|$ at coupling strength $U=4.8\,\hbar^2k^2/(2M)$. Color encodes quasi-momentum $q_m$ along the synthetic dimension. At the critical point $\lambda=0$, the bulk gap closes at a single Dirac point at the $M$ point $(q_x,q_m)=(k,\pi)$. (\ref{['fig:coupled_wire_model:Wannier_function_drift']}) Quantized Wannier function center drift across the phase transition. For each $q_m$, the effective potential is a cosine function whose minimum defines the Wannier function center. During an adiabatic evolution of $q_m$ from $0$ to $2\pi$, this minimum drifts by a quantized distance $-\mathcal{C}d$. At the critical point $\lambda=0$, the potential vanishes at $q_m=\pi$, leading to a discontinuity in the Wannier function center (dashed lines), reflecting a breaking of adiabaticity. However, the line $x=- q_md/\pi$ (solid line) remains a mirror symmetry axis, constraining the Wannier center to follow this line and yielding half-quantized drift despite non-adiabatic effects.
• Figure 2: Berry curvature in the critical regime and at criticality. In the critical regime, the Berry curvature profiles at $\lambda=-0.2$ (a) and $0.2$ (b) are combined to form the even and odd contributions $F_{\rm even}(q_x,q_m)$ and $F_{\rm odd}(q_x,q_m)$ (c and d, respectively). The even contribution extends over the entire Brillouin zone, while the odd one is localized close to the $M$ point. (e) Berry curvature at the critical point, similar to the even contribution in (c). Its integral over the Brillouin zone yields a half-quantized Chern number $\mathcal{C}=0.5$. The $M$ Dirac point is shown as a red cross.
• Figure 3: Experimental implementation of the coupled-wire model using ultracold dysprosium atoms. (\ref{['fig:implementation:geometry']}) Schematic of the synthetic geometry in the $xm$ plane, with sharp edges at $m=\pm J$. (\ref{['fig:implementation:image']}) Example of an absorption image of the atomic cloud after a Stern-Gerlach magnetic pulse and a time-of-flight expansion, showing the momentum distribution along $x$ and the spin distribution along $m$ (image averaged over 40 repetitions with parameters $\lambda=0$, $q_x=0$ and $U=4.8(2)\,E_{\text{rec}}$). (\ref{['fig:implementation:lasers']}) Two Raman beams (red arrows) counterpropagating along $x$ induce position-dependent couplings between neighboring spin states $m\rightarrow m+1$, while an optical lattice along $x$ creates a periodic potential (green arrows). An additional laser beam (orange arrow) generates a quadratic Zeeman field. (\ref{['fig:implementation:transitions']}) Illustration of two-photon transitions: Raman transitions modify the spin state, while the optical lattice leaves it unchanged. (\ref{['fig:implementation:couplings']}) Spin-dependent coupling strengths for lattice and Raman couplings. Laser polarizations are engineered to ensure optimal matching between lattice and Raman coupling algebras.
• Figure 4: Band structure of our implementation of the coupled-wire model for various values of the tuning parameter $\lambda$ at coupling strength $U=4.8\,E_{\text{rec}}$. The finite extent of the synthetic dimension causes the ground band to split into $2J+1$ subbands (blue lines). For $\lambda<0$ (topological phase), gapless chiral edge modes connect the ground and first excited bands, reflecting open boundary conditions and bulk-boundary correspondence. At criticality ($\lambda=0$), the bulk gap closes, up to finite-size effects, at a single point located at $q_x=k$.
• Figure 5: (\ref{['fig:Hall_measurement:Pim']}) Hall response measurement for $\lambda=-0.6$ and $U=4.8(2)\,E_{\text{rec}}$. The spin distributions $P(m)$ before ($q_x=0$) and after ($q_x=\pm2k$) one Bloch oscillation reveal the transfer of population across the synthetic dimension. (\ref{['fig:Hall_measurement:Cm']}) Local Chern marker $C(m^*)$ from the variation of cumulative probability $P(m>m^*)$ [Eq. \ref{['eq:Cmstar']}]. Bulk values are consistent with unit Chern number $\mathcal{C}=1$. Error bars denote standard error over $\approx 40$ measurements.