Exact bounds on the energy gap of transverse-field Ising chains by mapping to random walks

TL;DR

This paper derives exact bounds on the energy gap of transverse-field Ising chains using a mapping to random walks, applicable to both homogeneous and disordered systems, and explores implications for quantum computing.

Contribution

It introduces a method to obtain explicit bounds on the energy gap of Ising chains, valid for arbitrary couplings and fields, extending previous results to correlated disorder.

Findings

Bounds accurately predict finite-size scaling of the gap.

Finite-size scaling relates to sums of independent random variables.

Identifies logarithmic corrections at special points.

Abstract

Based on a relationship with continuous-time random walks discovered by Igl\'oi, Turban, and Rieger [Phys. Rev. E {\bf 59}, 1465 (1999)], we derive exact lower and upper bounds on the lowest energy gap of open transverse-field Ising chains, which are explicit in the parameters and are generally valid for arbitrary sets of possibly random couplings and fields. In the homogeneous chain and in the random chain with uncorrelated parameters, both the lower and upper bounds are found to show the same finite-size scaling in the ferromagnetic phase and at the critical point, demonstrating the ability of these bounds to infer the correct finite-size scaling of the critical gap. Applying the bounds to random transverse-field Ising chains with coupling-field correlations, a model which is relevant for adiabatic quantum computing, the finite-size scaling of the gap is shown to be related to that of sums of independent random variables. We determine the critical dynamical exponent of the model and reveal the existence of logarithmic corrections at special points.