Integrability and PT-Symmetry of N-Body Systems with Spin-Coupling  -Interactions

Abstract

We study the PT-symmetric boundary conditions for "spin"-related δ-interactions and the corresponding integrability for both bosonic and fermionic many-body systems. The spectra and bound states are discussed in detail for spin-1 2 particle systems. Integrable models play significant roles in statistical and condensed matter physics. Many of these models can be exactly solved in terms of an algebraic or coordinate "Bethe Ansatz method"[1], see e.g., [2,3] for spin chain and ladder models and [4,5] for (continuous variable) quantum mechanical models. In particular, the quantum mechanical solvable models describing a particle moving in a local singular potential concentrated at one or a discrete number of points have been extensively discussed, see e.g. [6,7,8] and references therein. One dimensional problems with point interactions at, say, the origin (x = 0) can be characterized by the boundary conditions imposed on the wave function at x = 0, which is equivalent to two particles with contact interactions. The integrability of one dimensional quantum mechanical many-body problems with general contact interactions between two particles, has been studied in [9] and the bound states and scattering matrices are calculated for both bosonic and fermionic statistics. The results are generalized to the case of quantum mechanical systems with "spin"-related contact interactions, namely, the boundary conditions describing the contact interactions are dependent on the spin states of the particles [10].Recently, the complex generalization of conventional quantum mechanics has been investigated [11]. In stead of the standard formulation of quantum mechanics in terms of HermitianHamiltonians, quantum mechanical models with space-time reflection symmetry (PT symmetry) have been constructed and studied for continuous interaction potentials [12]. For point interaction potentials a systematic description of the boundary conditions and the spectra properties for self-adjoint, PT-symmetric systems and systems with real spectra have been presented, and the corresponding integrability of one dimensional many body systems with these kinds of point 1

Non-Hermitian dynamics and $\mathcal{PT}$-symmetry breaking in interacting mesoscopic Rydberg platforms

“…in optics [11? -15], photonics [16][17][18][19], quantum many-body systems [7,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], systems with topological models [40][41][42][43][44][45][46][47][48][49][50] and in curved space [51]. The PT transition was verified experimentally for a cold atomic dissipative Floquet system, in which the PT symmetry transitions can occur by tuning either the dissipation strength or the coupling strength.…”

Nonlocal -symmetric potentials

“…Eigenvalues of PT symmetrical Hamiltonians were calculated in [13]. Integrability and PT symmetry of many-body systems with pseudo-Hermitian point interactions were studied in [14]. The application of the ideas of supersymmetric quantum mechanics [16] in constructing non-Hermitian PT -symmetric Hamiltonians has been considered in [17] and a formulation of PT -symmetric supersymmetry has been outlined in [18].…”