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First-order phase transitions in the square-lattice easy-plane J-Q model

Nisheeta Desai and Ribhu K. Kaul
Phys. Rev. B 102, 195135 – Published 19 November 2020
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  • INTRODUCTION
  • THE MODEL
  • NUMERICAL SIMULATIONS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study the quantum phase transition between the superfluid and valence bond solid in easy-plane J-Q models on the square lattice. The Hamiltonian we study is a linear combination of two model Hamiltonians: (1) an SU(2) symmetric model, which is the well known J-Q model that does not show any direct signs of a discontinuous transition on the largest lattices and is presumed continuous and (2) an easy-plane version of the J-Q model, which shows clear evidence for a first-order transition even on rather small lattices of size L≈16. A parameter 0≤λ≤1 [λ=0 being the easy-plane model and λ=1 being the SU(2) symmetric J-Q model] allows us to smoothly interpolate between these two limiting models. We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) to investigate the nature of this transition as λ is varied—here we present studies for λ=0,0.5,0.75,0.85,0.95, and 1. While we find that the first-order transition weakens as λ is increased from 0 to 1, we find no evidence that the transition becomes continuous until the SU(2) symmetric point, λ=1. We thus conclude that the square-lattice superfluid-VBS transition in the two-component easy-plane model is generically first order.

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    • Received 21 September 2020
    • Revised 29 October 2020
    • Accepted 30 October 2020

    DOI:https://doi.org/10.1103/PhysRevB.102.195135

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum phase transitions
    1. Physical Systems
    Mott superfluidsQuantum spin models
    1. Techniques
    Heisenberg modelQuantum Monte CarloXY model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Nisheeta Desai and Ribhu K. Kaul

    • Department of Physics & Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA

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    Vol. 102, Iss. 19 — 15 November 2020

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    • Figure 1
      Figure 1

      Phase diagram of HλJQ described by Eq. (1) as a function of λ and g≡Q/J. Using the model HλJQ we can access the phase boundary between the Néel and VBS phases. The transition at λ=0 was demonstrated to be first order previously [28]. We find that this transition is first order for all values of λ<1. The signals of first-order behavior that we detect in our QMC simulations vanish at the symmetric point λ=1 even on the largest lattices simulated here. We note that when λ=1 the model has a higher SU(2) symmetry; everywhere else in this phase diagram it only has the generic U(1)×Z2 easy-plane symmetry.

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    • Figure 2
      Figure 2

      Quantum Monte Carlo results for the transition from superfluid to the VBS phase for λ=0.5: (a),(b) show the quantities Rm⊥2 and Rϕx2, respectively, defined by Eq. (11) cross for different L at the transition point. The x̂ axis on these graphs is identical to the one shown in (c). (c) The same data as shown in (a),(b) but here together, suggesting an accurate estimate for the critical coupling can be obtained from the crossing of Rm⊥2 and Rϕx2 for a given value of L. (d) Values of coupling at the crossings of L and L/2 for Rm⊥2 and Rϕx2 are plotted vs 1/L. gc(L) from crossing analysis of Rm⊥2−Rϕx2 as suggested in (c) is shown to fit to a form gc(L)=gc*+CLe where gc*=12.111(3). This fitting has been done for L≤64 since larger sizes deviate from this fitting form. We demonstrate in Fig. 7 that this deviation arises due to the formation of double peaks in the histograms for the Monte Carlo estimators, a classic sign of a first-order transition.

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    • Figure 3
      Figure 3

      Finite-size scaling of the superfluid and VBS order parameters at the critical point with extrapolations to the thermodynamic limit for λ=0.5,0.75 and 1. Shown are 〈m⊥2〉 and 〈ϕx2〉 as a function of L evaluated at the finite size pseudocritical coupling gc(L). These couplings gc(L) are determined by estimating where Rm2 and Rϕ2 for the same value of L cross each other as described in Fig. 2. Dashed lines show extrapolations of the finite size data. Extrapolations have been carried out for two different fit forms, (a) power law: C0+C1Le1 (left) and (b) polynomial: C0+C1L+C2L2 (right). The value of χ2 per degree of freedom for these fits is shown in the table below the figure, indicating the reliability of the fits. The biggest system size used for the fits is L=128. We find that the numerical values to which 〈m⊥2〉c and 〈ϕx2〉c extrapolate depend on the fit form itself and are inconsistent with the stochastic errors (shown in the legend). In both fit forms the extrapolated order parameters go unambiguously to finite values for λ=0.5 and λ=0.75. For λ=1 on the other hand they are consistent with a zero extrapolated value. The 〈m⊥2〉 data shows this effect much more clearly than in the 〈ϕx2〉, where it is nonetheless also evident.

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    • Figure 4
      Figure 4

      This is the same kind of analysis as Fig. 3 but now for λ=0.85,0.95. We have shown these two values of λ separately to avoid overcrowding. The largest value of L used for this analysis is L=96. It can be inferred that both the order parameters extrapolate to a finite value for λ=0.85 in the thermodynamic limit using both the fitting forms. For λ=0.95 they clearly extrapolate to a finite value in the power law fit. However, they extrapolate to a very small positive value in the polynomial fit which is within four times the error bar, therefore we are unable to draw a very reliable conclusion here. Figure 5 shows the evolution of the extrapolated quantities from Figs. 3 and 4 as a function of λ for λ=0.5,0.75,0.85,0.95,1.0.

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    • Figure 5
      Figure 5

      〈m2〉c and 〈ϕ2〉c as found from extrapolations in Figs. 3 and 4 using the power law and polynomial fitting forms plotted as a function of λ. The error bars shown are stochastic errors; there are in addition systematic errors associated with the extrapolating function used. To estimate the systematic error we note that power law form overestimates and the polynomial underestimates the extrapolated order parameters; they thus provide a window for the order parameters in the thermodynamic limit. For all λ<1 both Néel and VBS order parameters extracted from both fit forms are positive indicating that both order parameters are finite at the transition: The transition is hence of first order. The extrapolated order parameters can be seen to approach zero as λ→1 (this is smoother for 〈m2〉c given the larger values compared to 〈ϕ2〉c). This indicates that the first-order transition continuously evolves to a second order transition as λ approaches 1. For λ=1 we find that polynomial extrapolation gives a negative value and the power law gives a positive value consistent with the most extensive studies that find a continuous transition with SU(2) symmetry [15].

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    • Figure 6
      Figure 6

      (a),(b) Crossings of Lρs for λ=0.5 and λ=0.75 indicating a transition from a magnetic to nonmagnetic phase. The black stars denote points where the curves of L and L/2 cross. (c) ρs extracted at these crossing points is fit to a power law and is shown to extrapolate to a finite value in the thermodynamic limit for both λ=0.5 and λ=0.75 (the same analysis for λ=0.85,0.95 has been shown in Fig. 10). (d) The value of the coupling g at these crossing points, gc(L), is shown to extrapolate to gc*=12.11(2) and gc*=15.49(1).

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    • Figure 7
      Figure 7

      Histograms (first row) and time series data for L=96 (second row) of observables close to the critical point (g≈12.1) for λ=0.5. Here m̃⊥2 and ϕ̃x2 are, respectively, m⊥2 and ϕx2 normalized so that the maximum value is 1.0. This data has been collected for less than 5000 MC steps per bin. The histograms show double peaked behavior and time series data shows switching between two orders.

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    • Figure 8
      Figure 8

      Histograms (left) and Monte Carlo histories (right) of ρs for λ=0.75 near the critical point g≈15.4. The histogram data has been collected for 1000 MC steps per bin. The double peak in the histograms that is barely visible for L=96 just starts to appear for L=128. Switching between the two values of ρs depicted in the time series data also indicates first-order behavior.

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    • Figure 9
      Figure 9

      Observables ρs, m⊥2, and ϕx2 plotted vs β can be seen to saturate on increasing β. This data has been taken at g=12 for λ=0.5 and g=16 for λ=0.75. These observables can be seen to saturate before β=L.

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    • Figure 10
      Figure 10

      This analysis is the same one shown in Fig, 6 for λ=0.85,0.95. We have put these in a separate figure to avoid overcrowding the former. Here too, the stiffness extracted at the transition point goes to a finite value as L→∞.

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