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Ising tricriticality in the extended Hubbard model with bond dimerization

Satoshi Ejima, Fabian H. L. Essler, Florian Lange, and Holger Fehske
Phys. Rev. B 93, 235118 – Published 10 June 2016
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  • INTRODUCTION
  • MODEL
  • DMRG TREATMENT
  • FIELD THEORY ANALYSIS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We explore the quantum phase transition between Peierls and charge-density-wave insulating states in the one-dimensional, half-filled, extended Hubbard model with explicit bond dimerization. We show that the critical line of the continuous Ising transition terminates at a tricritical point, belonging to the universality class of the tricritical Ising model with central charge c=7/10. Above this point, the quantum phase transition becomes first order. Employing a numerical matrix-product-state based (infinite) density-matrix renormalization group method we determine the ground-state phase diagram, the spin and two-particle charge excitations gaps, and the entanglement properties of the model with high precision. Performing a bosonization analysis we can derive a field description of the transition region in terms of a triple sine-Gordon model. This allows us to derive field theory predictions for the power-law (exponential) decay of the density-density (spin-spin) and bond-order-wave correlation functions, which are found to be in excellent agreement with our numerical results.

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    • Received 5 April 2016

    DOI:https://doi.org/10.1103/PhysRevB.93.235118

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Charge density wavesCritical phenomenaQuantum phase transitions
    1. Techniques
    Density matrix renormalization groupExtended Hubbard modelMatrix product states
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Satoshi Ejima1, Fabian H. L. Essler2, Florian Lange1,3, and Holger Fehske1

    • 1Institut für Physik, Ernst-Moritz-Arndt-Universität Greifswald, 17489 Greifswald, Germany
    • 2The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom
    • 3Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

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    Vol. 93, Iss. 23 — 15 June 2016

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    Images

    • Figure 1
      Figure 1

      iDMRG ground-state phase diagram of the 1D EHM with bond dimerization (2). The red solid line gives the PI-CDW phase boundaries for δ/t=0.2. The quantum phase transition is continuous (first order) below (above) the tricritical Ising point [Ut,Vt] marked by the asterisk. For comparison results for the BOW-CDW (blue dashed line), SDW-BOW (green dotted line), and SDW-CDW (green dashed-dotted line) transitions of the pure EHM (δ=0) were included [22]. (Inset) PI-CDW transition for δ/t=0.1 and 0.2 in the weak-coupling regime. As expected, decreasing δ/t, the transition lines come closer to BOW-CDW transition line of the pure EHM.

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    • Figure 2
      Figure 2

      Correlation length ξχ (top) and entanglement spectrum εα (bottom) as a function of V/t for U/t=4 (left) and U/t=12 (right), where δ/t=0.2. Data are obtained by iDMRG. Dashed lines give the BOW-CDW (SDW-CDW) transition for U/t=4 (U/t=12) in the EHM [22].

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    • Figure 3
      Figure 3

      Charge (Δc), spin (Δs), and neutral (Δn) gaps in dependence on V/t for (a) U/t=4 and (b) U/t=12. Again, δ/t=0.2. The dimerized PI (CDW) phase is marked in gray (white). Note the jump of the spin gap, δs≡Δs(Vc+)−Δs(Vc−), at Vc/t.

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    • Figure 4
      Figure 4

      Central charge c*(L) along the PI-CDW transition line for δ/t=0.2. DMRG data (obtained with PBC) indicate the Ising universality class (c=1/2) for U<Ut and, most notably, a tricritical Ising point with c=7/10 at Ut (red dotted line). (Inset) Jump-value of the spin gap for U≳Ut. The infinite MPS data point to a first-order transition.

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    • Figure 5
      Figure 5

      Density-density correlation functions at the tricritical Ising point for δ/t=0.2. Data obtained by iDMRG with χ=1600. The correlation functions (symbols) show a power-law decay, in accordance with the field theory predictions, Eqs. (26) and (27).

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    • Figure 6
      Figure 6

      BOW correlation functions at the tricritical Ising point for δ/t=0.2 computed by iDMRG with χ=1600. (Top) The asymptotic values for the two-point functions of staggered and smooth combinations of the BOW density are estimated by fitting to Eqs. (33) and (34). (Bottom) log-log plots of the same correlation functions with the asymptotic values subtracted show power-law decay compatible with Ising tricriticality.

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    • Figure 7
      Figure 7

      Spin correlation function (symbols) at the tricritical Ising point for δ/t=0.2 using the iDMRG with χ=1600, showing exponential decay. The line is a fit to Eq. (35).

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    • Figure 8
      Figure 8

      Density-density correlation functions at the Ising transition point (Vc≃2.503) for U/t=4 and δ/t=0.2, using the iDMRG with χ=1600. (a) The correlator of the staggered combination is in excellent agreement with Eq. (A2) with 4Ã2≈1.535. (b) Correlations of the smooth combination njsm are plotted separately for odd and even ℓ with j=1 and 2. The data are in excellent agreement with the prediction Eq. (A3).

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    • Figure 9
      Figure 9

      BOW correlations at the Ising transition point for U/t=4 and δ/t=0.2. The correlators exhibit a power-law decay consistent with the field theory predictions, Eq. (A4).

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