Degenerate subspace localization and local symmetries (2024)

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Degenerate subspace localization and local symmetries

Peter Schmelcher
Phys. Rev. Research 6, 023188 – Published 20 May 2024
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Degenerate subspace localization and local symmetries (1)

Abstract
Authors
Article Text
  • INTRODUCTION
  • WEAK COUPLING EXPANSION
  • DEGENERATE SUBSPACE LOCALIZATION
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • References

    Degenerate subspace localization and local symmetries (2)

    Abstract

    Domain specific localization of eigenstates has been a persistent observation for systems with local symmetries. The underlying mechanism for this localization behavior has, however, remained elusive. We provide here an analysis of a local reflection symmetric tight-binding Hamiltonian which attempts at identifying the key features that lead to the localized eigenstates. A weak coupling expansion of closed-form expressions for the eigenvectors demonstrates that the degeneracy of on-site energies occurring at the center of the locally symmetric domains represents the nucleus for eigenstates spreading across the domain. Since the symmetry-related subdomains constituting a locally symmetric domain are isospectral, we encounter pairwise degenerate eigenvalues that split linearly with an increasing coupling strength of the subdomains. The coupling to the (nonsymmetric) environment in an extended setup then leads to the survival of a certain system specific fraction of linearly splitting eigenvalues. The latter go hand in hand with the eigenstate localization on the locally symmetric domain. We provide a brief outlook addressing possible generalizations of local symmetry transformations while maintaining isospectrality.

    • Degenerate subspace localization and local symmetries (3)
    • Degenerate subspace localization and local symmetries (4)
    • Degenerate subspace localization and local symmetries (5)
    • Received 19 January 2024
    • Revised 28 March 2024
    • Accepted 1 April 2024

    DOI:https://doi.org/10.1103/PhysRevResearch.6.023188

    Degenerate subspace localization and local symmetries (6)

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas

    Localization

    1. Techniques

    Perturbation theoryTight-binding model

    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Authors & Affiliations

    Peter Schmelcher*

    • Zentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany and The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
    • *Peter.Schmelcher@physnet.uni-hamburg.de

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    Vol. 6, Iss. 2 — May - July 2024

    Subject Areas
    • Quantum Physics
    Degenerate subspace localization and local symmetries (7)
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    • Degenerate subspace localization and local symmetries (11)

      Figure 1

      Eigenstate maps showing the absolute values of the eigenvector components with varying site index (horizontal axis) for an increasing degree of excitation (vertical axis), i.e., an increasing energy, on gray scale for different 12-dimensional TB Hamiltonians. (a)Absence of local symmetries. Diagonal values of the TB Hamiltonian are 0.8,2.4,2.9,5.0,1.9,3.0,2.5,4.0,1.8,0.9,3.1,4.9 and the off-diagonal coupling value is ε=0.15. (b)Three consecutive four-dimensional domains of local reflection symmetry constituting a 12-dimensional TB Hamiltonian for ε=0.15. The diagonal values are 0.8,2.4,2.4,0.8,1.9,3.0,3.0,1.9,3.2,0.9,0.9,3.2, i.e., the LS domains reside on the sites 03,47, and 811, respectively. (c)Same diagonal values as in (b)but for ε=0.45 for all intradomain couplings and ε=0.1 for all interdomain couplings. (d)shows a sketch of the TB setup indicating the numerically labeled sites within a domain of local symmetry by the same symbols (×,,). This setup belongs to subfigures (b)and (c).

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      Figure 2

      Energy eigenvalue spectra (a,b) for a varying center coupling value εc=00.5. (a)A single reflection LS domain consisting of eight sites and a subdomain coupling strength ε=0.4. Diagonal values of the TB-Hamiltonian are 1.9,1.4,1.1,1.5,1.5,1.1,1.4,1.9. We observe the pairwise degeneracy of the eigenvalues for εc=0 and their linear splitting with increasing value of εc (note that this splitting is very small for the pair with the largest eigenvalues and barely visible on the scale of the figure). (b)A stack of subfigures showing the eigenvalue spectrum for a single reflection LS domain embedded into an asymmetric environment. Diagonal values of the ten-dimensional TB-Hamiltonian are 6.0,13.0,10.0,5.0,8.0,8.0,5.0,10.0,18.0,9.0, containing a six-dimensional reflection symmetric domain. As for (a)the center coupling obeys εc=00.5 and the remaining couplings are ε=0.5. (c)The eigenstate map of the setup (b)for εc=ε=0.5. Eigenvector components are shown with varying site index on the horizontal axis, and sorted w.r.t. increasing energy eigenvalues from bottom to top along the vertical axis. ( d) shows a sketch of the TB setup indicating the numerically labeled sites and the domain with local symmetry by the symbol and the nonsymmetric sites/domains with the symbol , all over belonging to subfigures (b)and (c).

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      Figure 3

      A 24-dimensional TB Hamiltonian consisting of four different six-dimensional neighboring reflection-based LS domains. The four domains involve the sites [05],[611],[1217],[1823]. (a)Energy eigenvalue spectra with varying (all four) center coupling strengths εc=00.5. (b)The corresponding eigenstate map for ε=εc=0.5. (c)shows a sketch of the TB setup indicating the numerically labeled sites and drawing sites within domains with a local symmetry by the same symbols (×,,,).

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