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arXiv:1511.02921 · Physical review. D/Physical review. D.

Operator product expansion coefficients of the 3D Ising model with a trapping potential

Gianluca Costagliola

Published Mar 22, 2016• 13 citations

Abstract

Recently the operator product expansion coefficients of the 3D Ising model universality class have been calculated by studying via Monte Carlo simulation the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a nonuniform potential acting as a trap. This setting is described by the trap size scaling ansatz, which can be combined with the general framework of the conformal perturbation in order to write down the correlators $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\epsilon}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$, from which the operator product expansion coefficients can be estimated. We find ${C}_{\ensuremath{\sigma}\ensuremath{\epsilon}}^{\ensuremath{\sigma}}=1.051(3)$, in agreement with the results already known in the literature, and ${C}_{\ensuremath{\epsilon}\ensuremath{\epsilon}}^{\ensuremath{\epsilon}}=1.32(15)$, confirming and improving the previous estimate obtained in the uniform perturbation case.

DOI: 10.1103/physrevd.93.066008

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Recently the operator product expansion coefficients of the 3D Ising model universality class have been calculated by studying via Monte Carlo simulation the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a nonuniform potential acting as a trap. This setting is described by the trap size scaling ansatz, which can be combined with the general framework of the conformal perturbation in order to write down the correlators

⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\sigma}(0)⟩

,

⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\epsilon}(0)⟩

and

⟨\ensuremath{\epsilon}(\mathbf{r})\ensuremath{\epsilon}(0)⟩

, from which the operator product expansion coefficients can be estimated. We find

{C}_{\ensuremath{\sigma}\ensuremath{\epsilon}}^{\ensuremath{\sigma}}=1.051(3)

, in agreement with the results already known in the literature, and

{C}_{\ensuremath{\epsilon}\ensuremath{\epsilon}}^{\ensuremath{\epsilon}}=1.32(15)

, confirming and improving the previous estimate obtained in the uniform perturbation case.

Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where

t

is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.

Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to

L=1600

. As a check, also simulations of the spin-

\frac{1}{2}

Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of

N

-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where

L

is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading

{Z}_{2}

-even scalar $\ensuremath{\epsilon}$.

We consider the critical relaxation of the Ising model, the so-called model A, and study its operator product expansion. Within perturbation theory, we focus on the operator product expansions of the two-point function and the response function. At the fixed point, we normalize the coefficients and the scaling variables so that the result displays universality. The role of the fluctuation-dissipation theorem is also discussed, and it is shown that it provides non-perturbative relations among the operator product expansion coefficients. Finally, the large N limit is considered.

We simulate the improved

(q+1)

-state clock model on the simple-cubic lattice at the critical point on lattices of a linear size up to

L=960

. We compute operator product expansion coefficients for the three-dimensional

XY

universality class. These are compared with highly accurate estimates obtained by using the conformal bootstrap method. We find that the results are consistent.

Using the nonperturbative functional renormalization group (FRG) within the Blaizot-M\'endez-Galain-Wschebor approximation, we compute the operator product expansion (OPE) coefficient

c_{112}

associated with the operators

\mathcal{O}_1\sim\varphi

and

\mathcal{O}_2\sim\varphi^2

in the three-dimensional

\mathrm{O}(N)

universality class and in the Ising universality class (

N=1

) in dimensions

c_{112}

. When available, exact results and estimates from the conformal bootstrap and Monte-Carlo simulations compare extremely well to our results, while FRG is able to provide values across the whole range of

d

and

N

considered.