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arXiv:2207.05502 · Physical review. E

Typical perturbation theory: Conditions, accuracy, and comparison with a mesoscopic case

Mats H. Lamann, Jochen Gemmer B

Published Nov 22, 2022• 2 citations

Abstract

The perturbation theory based on typicality introduced by Dabelow and Reimann [Phys. Rev. Lett. 124, 120602 (2020)0031-900710.1103/PhysRevLett.124.120602] and further refined by Dabelow et al. [Phys. Rev. Res. 2, 033210 (2020)2643-156410.1103/PhysRevResearch.2.033210; J. Stat. Mech. (2021) 0131061742-546810.1088/1742-5468/abd026] provides a powerful tool since it is intended to be applicable to a wide range of scenarios while relying on only a few parameters. Even though the authors present various examples to demonstrate the effectiveness of the theory, the conditions used in its derivation are often not thoroughly checked. It is argued that this is justified (without analytical reasoning) by the robustness of the theory. In the paper at hand, said perturbation theory is tested on three spin-based models. The following criteria are taken into focus: the fulfillment of the conditions, the accuracy of the predicted dynamics, and the relevance of the results with respect to a mesoscopic case.

DOI: 10.1103/physreve.106.054148

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The perturbation theory based on typicality introduced by Dabelow and Reimann [Phys. Rev. Lett. 124, 120602 (2020)0031-900710.1103/PhysRevLett.124.120602] and further refined by Dabelow et al. [Phys. Rev. Res. 2, 033210 (2020)2643-156410.1103/PhysRevResearch.2.033210; J. Stat. Mech. (2021) 0131061742-546810.1088/1742-5468/abd026] provides a powerful tool since it is intended to be applicable to a wide range of scenarios while relying on only a few parameters. Even though the authors present various examples to demonstrate the effectiveness of the theory, the conditions used in its derivation are often not thoroughly checked. It is argued that this is justified (without analytical reasoning) by the robustness of the theory. In the paper at hand, said perturbation theory is tested on three spin-based models. The following criteria are taken into focus: the fulfillment of the conditions, the accuracy of the predicted dynamics, and the relevance of the results with respect to a mesoscopic case.

Abstract We substantially extend our relaxation theory for perturbed many-body quantum systems from ((2020) Phys. Rev. Lett. 124 120602) by establishing an analytical prediction for the time-dependent observable expectation values which depends on only two characteristic parameters of the perturbation operator: its overall strength and its range or band width. Compared to the previous theory, a significantly larger range of perturbation strengths is covered. The results are obtained within a typicality framework by solving the pertinent random matrix problem exactly for a certain class of banded perturbations and by demonstrating the (approximative) universality of these solutions, which allows us to adopt them to considerably more general classes of perturbations. We also verify the prediction by comparison with several numerical examples.

In a recent Letter [Dabelow and Reimann, Phys. Rev. Lett. 124, 120602 (2020)], a perturbative relaxation theory is applied to the Bose-Hubbard model and compared to data from a quantum simulator experiment [Trotzky et al. Nature Phys 8, 325 (2012)]. In this Comment, I argue that the discrepancy found is due to the quasi-local observable measured in the experiment being affected by a harmonic trapping potential that is unaccounted for in the analytic calculation. I support my claim with quasi-exact numerics computed using a parallel time-evolving block decimation algorithm (TEBD), and show that the relaxation theory gives accurate results in the correct limit when compared to an appropriate local observable. Finally, I note that my numerics disagree with the t-DMRG calculations of Trotzky et al. but give better qualitative agreement with their experimental results.

A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first

n

terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.