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Higher-order Topological Parity Anomaly and Half-integer Hall Effect in High-dimensional Synthetic Lattices
Xian-Hao Wei, Xi-Wang Luo, Guang-Can Guo, Zheng-Wei Zhou
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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 Z₂ 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.
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