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Quantum criticality of two-dimensional quantum magnets with long-range interactions

S. Fey, Sebastian C. Kapfer, K. P. Schmidt

arXiv:1802.06684v1cond-mat.str-el

TL;DR

Long-range interaction calculations require evaluating high-dimensional nested sums to obtain gap-series coefficients. The paper uses Monte Carlo integration and Dlog-Padé extrapolations; converging Padé families provide estimates of critical couplings and exponents, with poor integration convergence when cancellations are strong.

Abstract

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We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents, while the system remains in the universality class of the nearest-neighbor model in the frustrated cases independent of the long-range nature of the interaction.

EVALUATION OF NESTED INFINITE SUMS USING MONTE CARLO INTEGRATION

Monte Carlo integration evaluates the high-dimensional nested sums needed for the long-range Ising-model gap series. The method uses importance sampling and tailored moves, while convergence is poor when cancellations make the integral much smaller than its absolute integral.

EVALUATION OF NESTED INFINITE SUMS USING MONTE CARLO INTEGRATION

  • At order 9 on two-dimensional lattices, the nested sums reach 18 dimensions, motivating Monte Carlo evaluation of the infinite embedding sums.The calculation samples graph embeddings to evaluate coefficients of the gap series.Source: EVALUATION OF NESTED INFINITE SUMS… · p. 6
  • Importance sampling estimates the embedding sum using a probability weight, with an analytically tractable reference integrand used to eliminate the uncomputable partition function.The sampled configurations encode concrete lattice coordinates for graph sites, with contributions set to zero when sites overlap.Source: EVALUATION OF NESTED INFINITE SUMS… · p. 6
  • The sampling probabilities are calibrated by balancing the magnitudes of the target and reference integrands before independent random streams are used for the calculation.A brief preliminary run determines the balancing constant.Source: EVALUATION OF NESTED INFINITE SUMS… · p. 7
  • The sampling scheme combines single-site shifts with two distance-changing moves to help the system escape low-weight configurations.The distance-changing moves have probabilities 0.2 and 0.1, respectively; low-weight states require these moves for rapid relaxation while preserving detailed balance.Source: EVALUATION OF NESTED INFINITE SUMS… · p. 7, Figure S1 · p. 7
  • For α = 4, the order-8 five-vertex-graph calculation shows Monte Carlo error decaying asymptotically as the inverse square root of the number of steps.The example tracks running means across 50 seeds.Source: EVALUATION OF NESTED INFINITE SUMS… · p. 7, Figure S2 · p. 8
  • Convergence is poor when I[f] is much smaller than I[|f|], a problem especially for antiferromagnetic interactions on both lattices.On the square lattice, the gap momentum k = (π, π) makes hopping contributions depend on cos[(π, π)^T · Δiν].Source: EVALUATION OF NESTED INFINITE SUMS… · p. 7

LIST OF COEFFICIENTS

The section presents gap-series coefficients for both lattices and both interaction signs, organized by momentum and lattice.

LIST OF COEFFICIENTS

  • Gap-series coefficients are tabulated for both lattices and for ferromagnetic and antiferromagnetic interactions, sorted by momentum and lattice.The tables report coefficient errors in parentheses, defined by the standard deviation across random seeds.Source: LIST OF COEFFICIENTS · p. 8
  • For ferromagnetic interactions, the first-order gap coefficient has analytic expressions for the square and triangular lattices at k = (0, 0)^T.The corresponding lattice-specific expressions are introduced in the text.Source: LIST OF COEFFICIENTS · p. 9
  • The first-order coefficients are also separately given for the antiferromagnetic cases.Source: LIST OF COEFFICIENTS · p. 9

EXTRAPOLATION

The gap series is extrapolated to estimate critical points and exponents, using the expected power-law gap closure and DlogPadé approximants. At α = 10/3, the analysis also estimates a multiplicative logarithmic correction by fixing the exponent to its mean-field value.

EXTRAPOLATION

  • Near λc, power-law behavior makes the logarithmic derivative develop a pole at the critical value.The gap series supplies the input for this analysis.Source: EXTRAPOLATION · pp. 9–11
  • DlogPadé extrapolants use physical poles to estimate λc and the associated critical exponent.The method applies Padé extrapolation to the logarithmic derivative.Source: EXTRAPOLATION · p. 11
  • At α = 10/3, the logarithmic correction exponent p is estimated by fixing θ to its mean-field value 1/2.The paper notes that extracting p from a high-order series is demanding.Source: EXTRAPOLATION · pp. 11–12
  • The reported λc and zν estimates use the highest-order Padé approximants from converging polynomial-order families, summarized by their mean and standard deviation.The families compared are [M, M −2], [M, M + 2], [M, M −1], [M, M + 1], and [M, M].Source: EXTRAPOLATION · p. 12
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