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 nonperturbative relations among the operator product expansion coefficients. Finally, the large N limit is considered.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 nonperturbative relations among the operator product expansion coefficients. Finally, the large N limit is considered.

Physical review. EFeb 13, 20252 citations

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

Gianluca Costagliola

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.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.

arXiv:1511.02921Physical review. D/Physical review. D.Mar 22, 201613 citations

Operator product expansion coefficients in the exact renormalization group formalism

Carlo Pagani, Hidenori Sonoda

We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $\ensuremath{\epsilon}$ expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the Lee-Yang model in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $\ensuremath{\epsilon}$ expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the Lee-Yang model in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.

arXiv:2001.07015Physical review. D/Physical review. D.May 11, 202017 citations

Conformal perturbation of off-critical correlators in the 3D Ising universality class

Michele Caselle, Gianluca Costagliola, Nicodemo Magnoli

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.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.

arXiv:1605.05133Physical review. D/Physical review. D.Jul 11, 201612 citations

Dynamic perturbation theory

Uwe Claus Täuber

In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.

Cambridge University Press eBooksMar 6, 20141 citation

Critical dynamics and renormalization-group methods for the kinetic Ising model

Gene F. Mazenko, Oriol T. Valls

The theory of the critical dynamics of the single spin-flip kinetic Ising model is examined. Various unproven assumptions and misconceptions are uncovered and discussed. In particular, we point out that the condition of detailed balance places important constraints on the renormalization-group analysis of this problem. The relationship among different fixed-point dynamical operators and dynamic universality classes is clarified. The role and convergence of perturbation-theory expansions which treat the bare coupling between cells as a small parameter in the real-space renormalization-group treatment are analyzed. We give reasons for believing that this approach cannot compete with methods using direct high-temperature expansions. We also comment on the usefulness of other methods in treating this problem.The theory of the critical dynamics of the single spin-flip kinetic Ising model is examined. Various unproven assumptions and misconceptions are uncovered and discussed. In particular, we point out that the condition of detailed balance places important constraints on the renormalization-group analysis of this problem. The relationship among different fixed-point dynamical operators and dynamic universality classes is clarified. The role and convergence of perturbation-theory expansions which treat the bare coupling between cells as a small parameter in the real-space renormalization-group treatment are analyzed. We give reasons for believing that this approach cannot compete with methods using direct high-temperature expansions. We also comment on the usefulness of other methods in treating this problem.

Physical review. B, Condensed matterFeb 1, 198514 citations

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.