Higher-order Topological Parity Anomaly and Half-integer Hall Effect in High-dimensional Synthetic Lattices

Recent advances in constructing synthetic dimension provide a powerful tool for exploring exotic topological states of matter in high dimensions. Here we report that the parity anomaly and associated \textit{half-integer} quantized Hall conductance, arising in 2$j$+1 (space-time) dimensions with a single or odd number of Dirac cones, can be realized by the boundary states of $n$-th order topological insulators in (2$j$+$n$)-dimensional synthetic lattices. We establish a general bulk-boundary correspondence by integrating the ``nested" Wilson loop theory with the time-reversal polarization at highly-symmetric momenta, a set of $Z_2$ topological invariants are extracted which determines the number of higher-order-boundary Dirac cones and their locations. We develop a general construction procedure for Hamiltonians supporting such higher-order topological parity anomaly. Moreover, we propose an experimental implementation scheme based on photonic synthetic dimensions and provide a method for probing the associated half-integer Hall conductance by the transmission spectra. Our work offers the realization and characterization of parity anomaly in general high-dimensional higher-order topological insulators and opens an avenue for exploring fundamental physics and possible device applications enabled by manipulating Dirac cones.