Theoretical studies of PT symmetric topological gapless photonic crystals

Abstract

Topological gapless materials such as topological semimetals and topological metals have attracted lots of attention in recent years due to their unique topological transport properties and topologically protected surface states. As a kind of the representatives, the Weyl semimetals, which have gapless points in the bulk energy band structures, allow us to realize Weyl fermions which have not been observed in high energy physics. Such gapless modes especially gapless points in solid-state materials broaden the scope of the type of elementary particles in high energy physics. Although it is not difficult to realize band degeneracies in electronic materials, to ensure that the Fermi levels are exactly located at the degenerate points is not easy. This is one of the main obstacles to realize the theoretically predicted topological gapless models in electronic materials. Fortunately, photonic crystals(PhCs) which are periodically arranged materials can be artificially manufactured. Actually, they can be used to simulate and study the topological gapless and gapped band structures. Due to the existence of translational symmetry, PhCs possess photonic band structures which can be used to control the propagation of electromagnetic fields in these materials. Traditional photonic band structures with a photonic band gap in PhCs have been theoretically and experimentally studied for decades. The difficulty in realizing certain types of topological band structures in electronic materials has disappeared when we simulate them in PhCs since the frequencies of electromagnetic fields propagating in PhCs can be controlled. Previous simulations on topological gapped systems result in topological protected one-way edge states in PhCs which can be used to achieve uni-directional back-scatter free waveguides. Classification of topological gapped and gapless systems with respect to time reversal symmetry, particle-hole symmetry, chiral symmetry and several spatial symmetries has been fully conducted in theory and partially achieved in experiments. Discrete symmetries always lead to degenerate bands in the band structures of periodic materials. Therefore the gapless modes in both electronic materials and PhCs are always protected by discrete symmetries. Previously people have studied PhCs with time reversal symmetry and inversion symmetry(sometimes called parity symmetry in high energy physics). Here we consider a new type of symmetry, which is the combination of time reversal symmetry(T) and parity symmetry(P), the PT symmetry. A Hamiltonian, or more generally a linear operator, which has PT symmetry is not necessarily to be Hermitian. Although the operator itself is not Hermitian, its eigenvalues are kept to be real as long as the operator is invariant under PT symmetry. Thus it represents a real physical system. We find that a two-dimensional PhC with hexagonal lattice structure which has PT symmetry will have gapless points in bulk band structure even both T symmetry and P symmetry are broken down. Furthermore, these gapless points are topologically protected by PT symmetry and characterized by a Z2 topological charge. As a matter of fact, they are stable against local perturbations as long as the perturbations do not break PT symmetry.