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Spectral crossovers and universality in quantum spin chains coupled to random fields

Debojyoti Kundu, Santosh Kumar, and Subhra Sen Gupta
Phys. Rev. B 105, 014205 – Published 21 January 2022
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  • INTRODUCTION
  • METHODOLOGY: THE SPIN HAMILTONIAN, ITS…
  • RANDOM MATRIX THEORY (RMT)
  • CALCULATIONS AND RESULTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter λ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.

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    • Received 23 July 2021
    • Accepted 9 December 2021

    DOI:https://doi.org/10.1103/PhysRevB.105.014205

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Physical Systems
    1-dimensional spin chainsQuantum spin modelsStrongly correlated systems
    1. Techniques
    Exact diagonalizationRandom field spin modelsRandom matrix theorySpin lattice models
    Interdisciplinary Physics

    Authors & Affiliations

    Debojyoti Kundu*, Santosh Kumar†, and Subhra Sen Gupta‡

    • Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh 201314, India

    • *debojyoti.kundu.physics@gmail.com
    • †Corresponding author: skumar.physics@gmail.com
    • ‡Corresponding author: subhro.sengupta@gmail.com

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    Vol. 105, Iss. 1 — 1 January 2022

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    Images

    • Figure 1
      Figure 1

      NNSD for N=16. (a) Shows Poissonian distribution when magnetic field is zero; (b) NNSD follows GOE for h=0.2; and (c) shows how Poisson distribution is recovered due to eigenvector localization for a typical large magnetic field (h=2.5).

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    • Figure 2
      Figure 2

      Variation of DKL(PGOE||P) [KL divergence of NNSD P(s) with respect to PGOE(s)] with increasing value of the symmetry-breaking crossover parameter h for the (a) N=14 and (b) N=16 systems.

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    • Figure 3
      Figure 3

      NNSD for N=14. Poissonian-to-GOE crossover with increasing h. (a), (c) Show the two limiting cases Poissonian and GOE, respectively and (b) shows one of the intermediate cases. We have used fP→O(λ,s) to fit this crossover and the values of the RMT crossover parameter λ are determined.

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    • Figure 4
      Figure 4

      Plots of NNSD for N=16, similar to the N=14 cases in Fig. 3.

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    • Figure 5
      Figure 5

      Variation of DKL(PGUE||P) [KL divergence of NNSD P(s) with respect to PGUE(s)] with the increasing value of symmetry-breaking crossover parameter Jt for (a) N=14 and (b) N=16 systems.

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    • Figure 6
      Figure 6

      NNSD for N=14; GOE-to-GUE crossover with increasing scalar spin-chirality amplitude (Jt) by keeping the system at constant random magnetic field (h=0.2). (a), (c) Show the two limiting cases GOE and GUE, respectively, and (b) shows one of the intermediate states. We have used fO→U(λ,s) to fit this crossover and the value of the RMT crossover parameter λ is determined.

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    • Figure 7
      Figure 7

      Plots of NNSD for N=16, similar to the N=14 cases in Fig. 6.

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    • Figure 8
      Figure 8

      Inverted basis GOE distribution for the (a) N=14, (b) N=16, and (c) N=18 systems, at the crossover points.

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    • Figure 9
      Figure 9

      Inverted basis GUE distribution for the (a) N=14, (b) N=16, and (c) N=18 systems, at the crossover points.

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    • Figure 10
      Figure 10

      Variation of RMT crossover parameter λ [of the crossover function fP→O(λ,s)] with (a) physical crossover parameter h, for the N=14, 16, and 18 (inverted basis) systems. Poissonian-to-GOE crossover is observed when λ gets saturated, for N=14 the crossover value is h=0.17, for N=16 it is h=0.13, and for N=18 it is h=0.085. (b) The coinciding linear region of the plots is fitted with the scaling function nah. λ is plotted against nah. The universal scaling exponent a=0.23. The dashed green line is a fit based on the data points occurring in the linear regime.

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    • Figure 11
      Figure 11

      Variation of RMT crossover parameter λ [of the crossover function fO→U(λ,s)] with (a) physical crossover parameter Jt for N=14, 16, and 18 (inverted basis) systems. GOE-to-GUE crossover is observed when λ gets saturated, for N=14 the crossover value is Jt=0.62, for N=16 it is Jt=0.42, and for N=18 it is Jt=0.28. (b) The coinciding linear region of the plots is fitted with the scaling function naJt. λ is plotted against naJt. The universal scaling exponent a=0.25. The dashed green line is a fit based on the data points occurring in the linear regime.

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    • Figure 12
      Figure 12

      (a) Poissonian → (b) intermediate → (c) GOE crossover for probability distribution of ratios with the increasing value of h for the N=14 system.

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    • Figure 13
      Figure 13

      (a) Poissonian → (b) intermediate → (c) GOE crossover for probability distribution of ratios with the increasing value of h for the N=16 system.

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    • Figure 14
      Figure 14

      (a) Poissonian → (b) intermediate → (c) GOE crossover for probability distribution of ratios with the increasing value of h for the N=18 system.

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    • Figure 15
      Figure 15

      Ratio distribution for N=14; (a)–(c) Show the GOE-to-GUE crossover with the increasing value of physical crossover parameter Jt, while the value of the range of the random magnetic field (h) is kept at 0.2. FO→U(λ,r) interpolates this crossover properly.

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    • Figure 16
      Figure 16

      Ratio distribution for the N=16 system, similar to the N=14 cases in Fig. 15.

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    • Figure 17
      Figure 17

      Ratio distribution for the N=18 system, similar to the N=14 cases in Fig. 15.

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    • Figure 18
      Figure 18

      Scaling in ratio distribution for the Poissonian-to-GOE crossover. λ of the interpolating matrix model's level-spacing ratio distribution EP→O(λ,r) is plotted against the scaled physical crossover parameter h. Here the universal scaling exponent a=0.25. The dashed green line is a fit based on the data points occurring in the linear regime.

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    • Figure 19
      Figure 19

      Scaling in ratio distribution for the GOE-to-GUE crossover. λ of the interpolating function FO→U(λ,r) is plotted against the scaled physical crossover parameter Jt. Here the universal scaling exponent a=0.26. The dashed green line is a fit based on the data points occurring in the linear regime.

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