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arXiv:2107.06482 · SciPost Physics

Real-time dynamics of the $O(4)$ scalar theory within the fRG approach

Yang-yang Tan, Yongrui Chen, Wei‐jie Fu

Published Jan 18, 2022• 22 citations
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Abstract

In this paper, the real-time dynamics of the O(4) O ( 4 ) scalar theory is studied within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The flow equations for the effective action and its n n -point correlation functions are derived in terms of the "classical'' and "quantum’’ fields, and a concise diagrammatic representation is presented. An analytic expression for the flow of the four-point vertex is obtained. Spectral functions with different values of temperature and momentum are obtained. Moreover, we calculate the dynamical critical exponent for the phase transition near the critical temperature in the O(4) O ( 4 ) scalar theory in 3+1 3 + 1 dimensions, and the value is found to be z\simeq 2.023 z ≃ 2.023 .

DOI
10.21468/scipostphys.12.1.026

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Real-time dynamics of the O(4)O(4)O(4) scalar theory within the fRG approach
In this paper, the real-time dynamics of the O(4) O ( 4 ) scalar theory is studied within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The flow equations for the effective action and its n n -point correlation functions are derived in terms of the "classical'' and "quantum’’ fields, and a concise diagrammatic representation is presented. An analytic expression for the flow of the four-point vertex is obtained. Spectral functions with different values of temperature and momentum are obtained. Moreover, we calculate the dynamical critical exponent for the phase transition near the critical temperature in the O(4) O ( 4 ) scalar theory in 3+1 3 + 1 dimensions, and the value is found to be z\simeq 2.023 z ≃ 2.023 .
The Schwinger-Keldysh functional renormalization group developed by Y.-y. Tan [Real-time dynamics of the O ( 4 ) scalar theory within the fRG approach, ] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of O ( ∂ 2 ) in the derivative expansion for the O ( N ) symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension d and the field component number N . It is found that one has z ≥ 2 in the whole range of 2 ≤ d ≤ 4 for the case of N = 1 , while in the case of N = 4 , the dynamic critical exponent turns to z < 2 when the dimension approach towards d = 2 . Published by the American Physical Society 2024
We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
A bstract We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
We study perturbative Wilsonian renormalization group (RG) for the scalar ϕ 4 theory at finite temperature to one loop order in the Schwinger-Keldysh closed-time-path (CTP) formalism. By explicitly integrating out the UV modes, we show how effective interactions coupling the two branches of the CTP contour arise. This provides a microscopic derivation of the type of effective CTP actions used in the literature. While the full effective theory has no critical points, we instead proceed to study critical properties of the RG flow in a reduced space of complex couplings that characterize certain time-ordered correlators and show the existence of two novel fixed points in d = 4 spacetime dimensions. We relate one to the Wilson-Fisher fixed point known to exist only in 4 − ε dimensions, and the other to the standard Gaussian fixed point.
We employ the functional renormalization group approach formulated on the Schwinger-Keldysh contour to calculate real-time correlation functions in scalar field theories. We provide a detailed description of the formalism, discuss suitable truncation schemes for real-time calculations, as well as the numerical procedure to self-consistently solve the flow equations for the spectral function. Subsequently, we discuss the relations to other perturbative and nonperturbative approaches to calculate spectral functions and present a detailed comparison and benchmark in d=0+1d=0+1d=0+1 dimensions.
We apply Renormalization Group techniques to the Real Time formulation of thermal field theory. Due to the separation between the T = 0 and the T = 0 parts of the propagator in this formalism, one can derive exact evolution equations for the Green functions describing the effect of integrating out thermal fluctuations of increasing wavelengths, the initial conditions being the renormalized Green functions of the T = 0 theory. As a first application, we study the phase transition for the real scalar theory, computing the order of the transition, the critical temperature, and critical exponents, in different approximations to the evolution equations for the scalar potential. CERN-TH/96-23 The dynamics of a second order, or weakly first order, phase transition is governed by long wavelength fluctuations of the order parameter. These fluctuations, whose length scale is much larger than the inverse temperature of the system, are essentially classical, since the