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

Róbert Juhász
Phys. Rev. B 106, 064204 – Published 10 August 2022

Abstract

Based on a relationship with continuous-time random walks discovered by Iglói, Turban, and Rieger [Phys. Rev. E 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.

  • Received 30 June 2022
  • Revised 25 July 2022
  • Accepted 25 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.064204

©2022 American Physical Society

Physics Subject Headings (PhySH)

General PhysicsCondensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Róbert Juhász*

  • Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary

  • *juhasz.robert@wigner.hu

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Issue

Vol. 106, Iss. 6 — 1 August 2022

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