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arXiv:2610.26838

Field theory guides Monte Carlo in the one-dimensional long-range Ising model

António Antunes, José Matos

Published Oct 19, 2026• 0 citations
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Abstract

The statistical long-range Ising (LRI) model in one dimension provides a family of some of the simplest non-exactly solvable critical points. These critical points are described by long-range conformal field theories (CFTs) that are well understood analytically with perturbative expansions and some exact results. Guided by this, we revisit the Monte Carlo (MC) approach to the corresponding lattice model which has many favorable features compared to other systems studied with MC methods. In particular, with very modest computational cost, we are able to estimate scaling dimensions of leading and subleading operators as well as the associated three-point functions, and even certain four-point functions. These results help highlight the fact that 1d LRI CFTs are a phenomenal playground to more generally test non-perturbative methods and ideas for strongly coupled CFTs.

DOI
10.48550/arxiv.2610.26838

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The statistical long-range Ising (LRI) model in one dimension provides a family of some of the simplest non-exactly solvable critical points. These critical points are described by long-range conformal field theories (CFTs) that are well understood analytically with perturbative expansions and some exact results. Guided by this, we revisit the Monte Carlo (MC) approach to the corresponding lattice model which has many favorable features compared to other systems studied with MC methods. In particular, with very modest computational cost, we are able to estimate scaling dimensions of leading and subleading operators as well as the associated three-point functions, and even certain four-point functions. These results help highlight the fact that 1d LRI CFTs are a phenomenal playground to more generally test non-perturbative methods and ideas for strongly coupled CFTs.
The long-range Ising model is the prototype of criticality with power-law interactions. In each dimension its interacting critical points form a one-parameter family of nonlocal conformal field theories, weakly coupled at both ends but not in between. We study this family in one, two and three dimensions from several independent directions. We extend the expansion about the generalized free field from three to eight loops and expand about the short-range end. We show that Hamiltonian truncation applies to this nonlocal theory once a truncation artifact is corrected. Where they can be compared, the resummed expansions, Hamiltonian truncation, the numerical renormalization group and Monte Carlo simulations agree on the dimension of the leading Z2\mathbb{Z}_2Z2​-even scalar at the sub-percent level, as universality requires.
We investigate the one-dimensional Ising model with long-range interactions decaying as 1/r^{1+s} 1 / r 1 + s . In the critical regime, for 1/2 ≤ s ≤ 1 1 / 2 ≤ s ≤ 1 , this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with s s . For s;gt1 s > 1 the model has instead no phase transition at finite temperature, as in the short-range case. In the standard field-theoretic description, involving a generalized free field with quartic interactions, the critical model is weakly coupled near s=1/2 s = 1 / 2 but strongly coupled in the vicinity of the short-range crossover at s=1 s = 1 . We introduce a dual formulation that becomes weakly coupled as s \to 1 s → 1 . Precisely at s=1 s = 1 , the dual description becomes an exactly solvable conformal boundary condition of the two-dimensional free scalar. We present a detailed study of the dual model and demonstrate its effectiveness by computing perturbatively the CFT data near s=1 s = 1 , up to next-to-next-to-leading order in 1-s 1 − s , by two independent approaches: (i) standard renormalization of our dual field-theoretic description and (ii) the analytic conformal bootstrap. The two methods yield complete agreement.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
Abstract Analytic predictions have been derived recently by Dohm and Wessel (2021 Phys. Rev. Lett. 126 060601) from anisotropic φ 4 theory and conformal field theory for the amplitude F c of the critical free energy of finite anisotropic systems in the two-dimensional Ising universality class. These predictions employ the hypothesis of multiparameter universality. We test these predictions by means of high-precision Monte Carlo (MC) simulations for F c of the Ising model on a square lattice with isotropic ferromagnetic couplings between nearest neighbors and with an anisotropic coupling between next-nearest neighbors along one diagonal. We find remarkable agreement between the MC data and the analytical prediction. This agreement supports the validity of multiparameter universality and invalidates two-scale-factor universality as F c is found to exhibit a nonuniversal dependence on the microscopic couplings of the scalar φ 4 model and the Ising model. Our results are compared with the exact result for F c in the three-dimensional φ 4 model with a planar anisotropy in the spherical limit. The critical Casimir amplitude is briefly discussed.
We present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on 643{64}^{3}643 and 1283{128}^{3}1283 simple-cubic lattices. The eigenvalues of a linearized transformation matrix, constructed with the use of 53 even operators and 46 odd operators, are shown to be free of truncation errors. Our estimate of the critical coupling is 0.221 652\ifmmode\pm\else\textpm\fi{}0.000 003\ifmmode\pm\else\textpm\fi{}0.000 001 where the first error is statistical and the second due to the finite number of blocking steps. The results for the relevant exponents are \ensuremath{\nu}=0.624(2) and \ensuremath{\eta}=0.026(3). This estimate for \ensuremath{\nu} lies (2--3)\ensuremath{\sigma} below that obtained from other methods. The correction-to-scaling exponent is found to lie in the range \ensuremath{\omega}=0.8--0.85. We also find that the subleading magnetic exponent is relevant and present evidence that it corresponds to a redundant eigenvector of the majority-rule blocking transformation.