Microscopic identification of dissipative modes in relativistic field theories

Yohei Saito, Hirotsugu Fujii, Kazunori Itakura A, Osamu Morimatsu

Published Jan 1, 2015• 1 citation

Abstract

We present an argument to support the existence of dissipative modes in relativistic field theories. For an O(N) ϕ4 theory in spatial dimension 1 ≤ d ≤ 3, the two-point function of ϕ is shown to develop a pole of the form p0 ∼ i−1(p2 + m2) at small energy p0 and momentum p, with a nonzero finite coefficient or relaxation constant,, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. In contrast, an NLO calcu-lation in the one-particle irreducible (1PI) framework fails to give a nonzero . The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent z for the two-point function as z = 2 − η. We also discuss possible improvement of the result. Subject Index A40, A52

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Microscopic identification of dissipative modes in relativistic field theories

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Published Jan 1, 2015• 1 citation

OpenAlex

Abstract

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Microscopic identification of dissipative modes in relativistic field theories

Yohei Saito, Hirotsugu Fujii, Kazunori Itakura A +1 more

We present an argument to support the existence of dissipative modes in relativistic field theories. For an | $O(N)$ || $\varphi ^4$ | theory in spatial dimension | $1\leq d\le 3$ |⁠, the two-point function of | $\varphi$ | is shown to develop a pole of the form | $p_0\sim i\Gamma ^{-1} ({\boldsymbol{{ {p}}}}^2+m^2)$ | at small energy | $p_0$ | and momentum | ${\boldsymbol{{ {p}}}}$ |⁠, with a nonzero finite coefficient or relaxation constant, | $\Gamma$ |⁠, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of | $1/N$ | expansion. In contrast, an NLO calculation in the one-particle irreducible (1PI) framework fails to give a nonzero | $\Gamma$ |⁠. The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent | $z$ | for the two-point function as | $z=2-\eta$ |⁠. We also discuss possible improvement of the result.We present an argument to support the existence of dissipative modes in relativistic field theories. For an |$O(N)$||$\varphi ^4$| theory in spatial dimension |$1\leq d\le 3$|⁠, the two-point function of |$\varphi$| is shown to develop a pole of the form |$p_0\sim i\Gamma ^{-1} ({\boldsymbol{{ {p}}}}^2+m^2)$| at small energy |$p_0$| and momentum |${\boldsymbol{{ {p}}}}$|⁠, with a nonzero finite coefficient or relaxation constant, |$\Gamma$|⁠, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of |$1/N$| expansion. In contrast, an NLO calculation in the one-particle irreducible (1PI) framework fails to give a nonzero |$\Gamma$|⁠. The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent |$z$| for the two-point function as |$z=2-\eta$|⁠. We also discuss possible improvement of the result.

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Mar 24, 20100 citations

We present an argument to support the existence of dissipative modes in relativistic field theories. For an |

O(N)

\varphi ^4

| theory in spatial dimension |

1\leq d\le 3

|⁠, the two-point function of |

\varphi

| is shown to develop a pole of the form |

p_0\sim i\Gamma ^{-1} ({\boldsymbol{{ {p}}}}^2+m^2)

| at small energy |

p_0

| and momentum |

{\boldsymbol{{ {p}}}}

|⁠, with a nonzero finite coefficient or relaxation constant, |

\Gamma

|⁠, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of |

1/N

| expansion. In contrast, an NLO calculation in the one-particle irreducible (1PI) framework fails to give a nonzero |

\Gamma

|⁠. The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent |

z

| for the two-point function as |

z=2-\eta

|⁠. We also discuss possible improvement of the result.

From quantum to classical dynamics: Dynamic crossover in the relativistic

O(N)

model

We investigate the transition from unitary to dissipative dynamics in the relativistic

O(N)

vector model with the

\lambda (\varphi^{2})^{2}

interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent

z

in the limit of large temperatures and in

2 \leq d \leq 4

spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.

O(N)

-symmetric field theory based on an expansion of the two-particle irreducible (2PI) effective action to next-to-leading order in

1/N

and highlight the analogies and differences to the corresponding theory for closed systems. A central result of this approach is the systematic derivation of an open-system Boltzmann equation, which takes a very different form from its closed-system counterpart due to the absence of well-defined quasiparticles. As a consequence of the general structure of its derivation, it applies to open, gapless field theories that satisfy certain testable conditions, which we identify here. Specifically for the

O(N)

model, we use scaling analysis and numerical simulations to show that interactions are screened efficiently at small momenta and, therefore, the late-time evolution is effectively collisionless. This implies that fluctuations induced by a quench dissipate into the environment before they thermalize. Goldstone's theorem also constrains the dynamics far from equilibrium, which is used to show that the order parameter equilibrates more quickly for quenches preserving the

O(N)

symmetry than those breaking it explicitly.

O(N)

symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.