Riemann surface dynamics of periodic non-Hermitian Hamiltonians

A number of physical problems, including statistical mechanics of 1D multivalent Coulomb gases, may be formulated in terms of non-Hermitian quantum mechanics. We use this example to develop a non-perturbative method of instanton calculus for non-Hermitian Hamiltonians. This can be seen as an extension of semiclassical methods in conventional quantum mechanics. Treating momentum and coordinate as complex variables yields a Riemann surface of constant complex energy. The classical and instanton actions are given by periods of this surface; we show how to obtain these via methods from algebraic topology. We demonstrate the accuracy of this analytic procedure in comparison with numerical simulations for a class of periodic non-Hermitian Hamiltonians, as well as the validity of the Bohr-Sommerfeld quantization and Gamow's formula in these cases.