The quantum Ising model: finite sums and hyperbolic functions

TL;DR

This paper derives exact formulas involving hyperbolic functions for finite-size quantum Ising chains, enabling precise calculations of critical properties and fast adiabatic control.

Contribution

It introduces new closed-form expressions for sums related to the quantum Ising model, extending classical mathematical tables and facilitating quantum critical and control analyses.

Findings

Exact sums for hyperbolic functions derived

Closed-form expressions for fidelity susceptibility obtained

Coefficients for counterdiabatic Hamiltonian calculated

Abstract

We derive exact closed-form expressions for several sums leading to hyperbolic functions and discuss their applicability for studies of finite-size Ising spin chains. We show how they immediately lead to closed-form expressions for both fidelity susceptibility characterizing the quantum critical point and the coefficients of the counterdiabatic Hamiltonian enabling arbitrarily quick adiabatic driving of the system. Our results generalize and extend the sums presented in the popular Gradshteyn and Ryzhik Table of Integrals, Series, and Products.