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Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states

Michael P. Kaicher, Davide Vodola, and Simon B. Jäger
Phys. Rev. B 107, 165144 – Published 25 April 2023
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  • INTRODUCTION
  • THEORY
  • RESULTS
  • SUMMARY AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as 1/rα, where r denotes the distance between two spins and α is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of α. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of α>1. For α<1 we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime α≥1, and qualitative agreement with DMRG in the strong-long range regime α≤1. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.

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    • Received 7 January 2023
    • Revised 11 April 2023
    • Accepted 11 April 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.165144

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    FermionsPhase diagramsQuantum phase transitions
    1. Techniques
    Mean field theorySpin chains
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Michael P. Kaicher1,*,†, Davide Vodola2,†, and Simon B. Jäger3

    • 1Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain
    • 2Dipartimento di Fisica e Astronomia, Università di Bologna, I-40129, Bologna, Italy
    • 3Department of Physics and Research Center OPTIMAS, University of Kaiserslautern-Landau, D-67663 Kaiserslautern, Germany

    • *Corresponding author: michael.p.kaicher(at)gmail.com
    • †Now at: BASF Digital Solutions, Next Generation Computing, Pfalzgrafenstr. 1, D-67056, Ludwigshafen, Germany.

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    Vol. 107, Iss. 16 — 15 April 2023

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    Images

    • Figure 1
      Figure 1

      We plot the entanglement entropy SN/2 from the covariance matrix obtained through the ZT algorithm for a system size N=100, α∈{0.30,0.50,0.75,1.00,⋯,3.00}, and θ∈(0,π/2). For the color coding we use the color map “plasma” in the python package “matplotlib”. Black squares represent the quantum critical points θc∞/π in the thermodynamic limit, which are listed in Table 1, while the dashed line serves as a guide to the eye.

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    • Figure 2
      Figure 2

      For a system of size N=100 and exponents α∈[1,3], we plot (a) the energy E and (b) the entanglement entropy SN/2 (bottom), as defined in Eqs. (6) and (11), obtained from the covariance matrix of the ZT algorithm (solid lines) and compare it to DMRG (hollow markers).

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    • Figure 3
      Figure 3

      Example for the fit of Eq. (13) to the value of θmax obtained by maximization of the entanglement entropy S with FGS. The thresholds θc∞/π are shown for the respective cases α∈{1,2,3}, see Table 1 for more details.

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    • Figure 4
      Figure 4

      (a) Extracting the central charge. Using the ZT algorithm for various α, here exemplified by α∈{1,2,3}, we plot the entanglement entropy SN/2 against log(N). For each α we perform a linear regression fit, neglecting the system sizes N∈{20,30,40} to mitigate finite-size effects. (b) Central charge c obtained from finite-size scaling up to system size N=100 of FGS evolutions through the ZT algorithm (blue squares) for the AFM long-range TFIM. For comparison, DMRG results from finite-size scaling of system sizes of up to N=100 from Ref. [6] (“DMRG”, orange square) and [8] (“DMRG*”, green triangles) are included. The red horizontal line represents the value c=1/2, which describes the Ising universality class. Error bars represent the standard deviation from the linear regression fit.

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    • Figure 5
      Figure 5

      (a) Energy and (b) entanglement entropy obtained from the covariance matrix of the ITE algorithm (solid lines) and DMRG (empty markers) simulations for N=100 and α∈{0.3,0.5,0.75}.

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    • Figure 6
      Figure 6

      Violations to the area law. The effective central charge c [Eq. (12)] calculated from finite scaling of system sizes N∈{40,50,⋯,100} for 50 different values deep in the gapped region θ∈(0,π/4) for the GHF ITE algorithm. Error bars for the standard deviation are also included, but too small to be visible. Solid lines connecting the markers serve as a guide to the eye.

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    • Figure 7
      Figure 7

      Energy expectation EFGS (a) and entanglement entropy SFGS (b) obtained from GHF ZT simulations in relation to EDMRG and SDMRG calculated from DMRG simulation as function of θ. The results are obtained for the AFM-TFIM of size N=100 and various values of α in the weak-long-range regime. [(c),(d)] Same as (a) and (b), but for values of α in the strong-long-range regime, including GHF simulation results for both, the GHF ITE and ZT algorithms.

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