Effective size of a parity-time symmetric dimer

Abstract: Parity-time (PT) symmetric dimers were introduced to highlight the unusual properties of non-Hermitian systems that are invariant after a combined parity and time reversal operation. They are also the building blocks of a variety of symmetry and topologically protected structures, especially on integrated photonic platforms. As the name suggests, it consists of two coupled oscillators, which can be optical, mechanical, electronic, and so on in nature. In this article, we show that its effective size, defined by the number of lattice sites inversely proportional to the lattice momentum, is surprisingly three instead of two from the perspective of energy quantization. More specifically, we show analytically that the complex energy levels of a one-dimensional concatenated chain with $N$ PT-dimers are determined by a system size of $1+2N$, which reduces to three in the case of a single PT-dimer. We note that while energy quantization conditions were established in various non-Hermitian systems, exact and explicitly quantized complex energies as reported here are still scarce. In connection, we also discuss the other symmetries of a PT-dimer and concatenated PT-dimer chain, including non-Hermitian particle-hole symmetry and chiral symmetry.