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It is well known that using the conventional non-Hermitian but ${\calPT}-$symmetric Bose-Hubbard Hamiltonian with real spectrum one can realize theBose-Einstein 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 offersan extreme form of the limit characterized by a minimal geometric multiplicity$K=1$. In our paper we describe a rescaled and partitioned direct-summodification of the linear version of the Bose-Hubbard model which remainsexactly solvable while admitting any value of $K\geq 1$. It offers a completemenu of benchmark models numbered by a specific combinatorial scheme. In thismanner, an exhaustive classification of the general BEC patterns with anygeometric multiplicity is obtained and realized in terms of an exactly solvablegeneralized Bose-Hubbard model.

quantum reports

Bose-Einstein condensation processes with nontrivial geometric  multiplicites realized via ${\cal PT}-$symmetric and exactly solvable  linear-Bose-Hubbard building blocks