## Abstract

Statistical mechanics of a 1D multivalent Coulomb gas can be mapped onto non-Hermitian quantum mechanics. We use this example to develop the instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of the Seiberg-Witten duality, we treat momentum and coordinate as complex variables. Constant-energy manifolds are given by Riemann surfaces of genus g ≥ 1. The actions along principal cycles on these surfaces obey the ordinary differential equation in the moduli space of the Riemann surface known as the Picard-Fuchs equation. We derive and solve the Picard-Fuchs equations for Coulomb gases of various charge content. Analysis of monodromies of these solutions around their singular points yields semiclassical spectra as well as instanton effects such as the Bloch bandwidth. Both are shown to be in perfect agreement with numerical simulations.

Original language | English (US) |
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Pages (from-to) | 517-537 |

Number of pages | 21 |

Journal | Journal of Experimental and Theoretical Physics |

Volume | 117 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2013 |

### Bibliographical note

Funding Information:ACKNOWLEDGMENTS We are indebted to A. Gorsky for introducing us to the algebraic geometry methods and sharing his unpublished notes. The work was supported by U.S. – Israel Binational Science Foundation Grant 2008075 and by NSF grant DMR13066734. Research of P. K. at the Perimeter Institute is supported by the Govern ment of Canada through Industry Canada and by the Province of Ontario through the Ministry of Eco nomic Development and Innovation.