Riemann surface dynamics of periodic non-Hermitian Hamiltonians

Abstract

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.

Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

“…It should be thus identified with the classical action for the spectral branch terminating at u = e iπ/3 (e −iπ/3 ). In the same manner the analogous symmetry relations for the genus-2 cases in Reference [26] allow us to identify the classical actions for all the spectral branches in Figure 4. Quantizing these classical actions according to the Bohr-Sommerfeld rule,…”

“…These expressions fully define the second-order corrections in terms of the classical action and its derivatives with respect to u. These are easily obtained from the previous results, Equations (20)- (22), (36) and (37) (see Reference [26] for the genus-2 cases). Note that in the genus-1 cases the second derivative S 0 (u) can be replaced with S 0 (u) by using the Picard-Fuchs Equations (19) and (35).…”

“…Then we review how to map the partition function onto a Feynman propagator and derive a Hamilton operator from there. This mapping was pioneered by Edwards and Lenard [ 10 ] and subsequently used in several works as starting point [ 11 , 25 , 26 , 27 ]. If the system consists of cations and anions with the same valency and concentration, then the resulting Hamilton operator is Hermitian.…”

“…In equilibrium the system minimizes its free energy by choosing an appropriate boundary charge q . In [ 25 , 26 ] this minimum was found to generally be the non-polarized state of the channel, i.e., . Adiabatic charge transfer through the channel is associated with the boundary charge q sweeping through its full period.…”

“…Top left: (n 1 , n 2 ) = (2, 1), α = 200; top right:(3,1), α = 300; bottom left: (4, 1), α = 400; bottom right:(3,2), α = 400. Reproduced with permission from References[25,26].…”