ABSTRACT
PT-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside the conventional equilibrium statistical mechanics of Hermitian systems. PT-symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The correlation functions of these models display a much richer set of behaviours than Hermitian systems, displaying sinusoidally modulated exponential decay, as in a dense fluid, or even sinusoidal modulation without decay. Classical spin models with PT-symmetry include Z(N) models with a complex magnetic field, the chiral Potts model and the anisotropic next-nearest-neighbour Ising model. Quantum many-body problems with a non-zero chemical potential have a natural PT-symmetric representation related to the sign problem. Two-dimensional quantum chromodynamics with heavy quarks at non-zero chemical potential can be solved by diagonalizing an appropriate PT-symmetric Hamiltonian.
ABSTRACT
Quantum mechanics and classical mechanics are distinctly different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum and classical mechanics into the complex domain. These complex extensions continue to exhibit a correspondence, and this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship requires the use of asymptotics beyond all orders.
