Prepared for submission to American Journal of Modern Physics Theoretical Physics Fermionic and bosonic partition functions at imaginary chemical potential as Bloch functions Evangelos G. Filothodoros 1 1 Institute of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece. Correspondence should be addressed to Evangelos G. Filothodoros; efilotho@physics.auth.gr ORCID:https://orcid.org/0000-0002-5898-7288 Abstract We point out that the phase transitions of the d +1 Gross-Neveu and CP N−1 models at finite temperature and imaginary chemical potential can be mapped to transformations of Hubbard-like regular hexagonal to square lattice with the intermediate steps to be specific surfaces (irregular hexagonal kind) with an ordered construction based on the even indexed Bloch-Wigner-Ramakrishnan polylogarithm function. The zeros and extrema of the Clausen Cl d (θ) function play an important role to the analysis since they allow us not only to study the fermionic and bosonic theories and their phase transitions but also the possibility to explore the existence of conductors arising from the correspondence between the partition functions of the two models and the Bloch and Wannier functions that play a crucial role in the tight-binding approximation in solid state physics. Keywords: Fermion-boson map; Hubbard lattice; conductor; Bloch function; Wannier function 1. Introduction In the realm of condensed matter physics the Hubbard model stands as a cornerstone in the study of electronic correlations within quantum materials [1, 2]. It describes electrons in a solid that interact with each other through short-range repulsive interactions. Specifically, the Hubbard model introduces a contact interaction between par- ticles of opposite spin on each site of a lattice. When applied to electron systems, these interactions are typically expected to be repulsive, arising from the screened Coulomb interaction. In this work, we believe that if we exam- ine in detail the setup of the higher dimensional thermal windows for fermions and bosons at imaginary chemical potential, we may connect them to the transformation of Hubbard-like lattices to square lattice and the appearance of insulator’s identities. An Appendix contains some technical details and useful formulae for the Bloch-Wigner function. 1 arXiv:2405.09686v1 [hep-th] 15 May 2024