Abstract
It is well known that, using the conventional non-Hermitian but PTâsymmetric BoseâHubbard Hamiltonian with real spectrum, one can realize the BoseâEinstein condensation (BEC) process in an exceptional-point limit of order N. Such an exactly solvable simulation of the BEC-type phase transition is, unfortunately, incomplete because the standard version of the model only offers an extreme form of the limit, characterized by a minimal geometric multiplicity K = 1. In our paper, we describe a rescaled and partitioned direct-sum modification of the linear version of the BoseâHubbard model, which remains exactly solvable while admitting any value of Kâ„1. It offers a complete menu of benchmark models numbered by a specific combinatorial scheme. In this manner, an exhaustive classification of the general BEC patterns with any geometric multiplicity is obtained and realized in terms of an exactly solvable generalized BoseâHubbard model.
Highlights
The conceptual appeal of P T âsymmetry currently influences several different areas of theoretical and/or experimental physics [1,2] as well as of the related mathematical physics [3,4,5]
The birth of the concept dates back to the mathematics of perturbation theory [6,7,8,9,10], but the real start of popularity was only inspired by Bender and Boettcherâs 1998 conjecture [11] that P T âsymmetry of a Hamiltonian H could play a key role in a ânon-Hermitianâ formulation of quantum mechanics of bound states
Let us follow the guidance provided by Graefe et al [17], who performed a detailed perturbationâapproximation analysis of the P T âsymmetric BoseâHubbard (BH) system living in a small vicinity of its BoseâEinstein condensation (BEC) dynamical singularity
Summary
The conceptual appeal of P T âsymmetry (i.e., of the parity times time reversal symmetry) currently influences several different areas of theoretical and/or experimental physics [1,2] as well as of the related mathematical physics [3,4,5]. Bender with Boettcher [11] were probably the first who managed to simulate this process (leading to a quantum phase transition, i.e., to an abrupt loss of the observability of the energy), using various elementary one-dimensional single-particle local potentials. This proved inspiring and influenced the model building efforts in multiple areas of realistic phenomenological considerations. In both of these model-building arrangements, the process of the condensation is attributed, in Katoâs language [18], to the presence of higher-order exceptional points. We intend to study the former mechanism of the quantum phase transition in more detail, extending the scope of the approach to certain more general dynamical scenarios characterized by nontrivial, optional geometric multiplicities of the generic exceptional-point degeneracies
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