Higher-order topological parity anomaly and half-integer quantum 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 quantized Hall conductance, arising in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>2</a:mn> <a:mi>j</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> (space-time) dimensions with a single or odd number of Dirac cones, can be realized by the boundary states of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>n</b:mi> <b:mi>th</b:mi> </b:mrow> </b:math> -order topological insulators in ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>2</c:mn> <c:mi>j</c:mi> <c:mo>+</c:mo> </c:mrow> <c:mi>n</c:mi> </c:math> )-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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>Z</d:mi> <d:mn>2</d:mn> </d:msub> </d:math> topological invariants are extracted, which determines the number and location of higher-order-boundary Dirac cones that are protected by time-reversal and reflection symmetries. 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 photonic device applications enabled by manipulating Dirac cones.