Two coupled particles of identical mass but opposite charge are studied, with a constant transverse external magnetic field and an external potential, interacting with a bath of harmonic oscillators. We show that the problem cannot be mapped to a one-dimensional problem like the one in Ao (Phys. Rev. Lett. 72, 1898 (1994)), it strictly remains two-dimensional. We calculate the effective action both for the case of linear coupling to the bath and without a linear coupling using imaginary time path integral at finite temperature. At zero temperature we use Leggett’s prescription to derive the effective action. In the limit of zero magnetic field we recover a two-dimensional version of the result derived in Chudnovsky (Phys. Rev. B, 54, 5777 (1996)) for the case of two identical particles. We find that in the limit of strong dissipation, the effective action reduces to a two-dimensional version of the Caldeira–Leggett form in terms of the reduced mass and the magnetic field. The case of ohmic dissipation with the motion of the two particles damped by the ohmic frictional constant η is studied in detail.
Partition function of the two-dimensional nearest neighbour Ising models for finite lattices in a non-zero magnetic field#
