Quantum quench in interacting field theory: a self-consistent approximation

TL;DR

The paper investigates quantum quenches in an interacting φ^4 field theory using a self-consistent Hartree–Fock approximation. It shows that a composite quench of mass and coupling effectively shifts the mass to a self-consistent value $m^*$, preserving stationary, thermal-like correlators and enabling relaxation even in low dimensions where free theories fail. The authors develop both perturbative (Dyson) and nonperturbative time-evolution approaches, including a slab-Quench mapping and Keldysh formalism, to demonstrate that local observables thermalize with a momentum-dependent effective temperature determined by the mass shift. They also analyze divergences and renormalization, provide asymptotic results across dimensions, and compare quasi-adiabatic and exact methods, finding strong agreement with the proposed ansatz. Overall, the work advances understanding of non-equilibrium dynamics and thermalization in interacting quantum field theories, highlighting the central role of interaction-induced mass shifts.

Abstract

We study a composite quantum quench of the energy gap and the interactions in the interacting φ^4 model using a self-consistent approximation. Firstly we review the results for free theories where a quantum quench of the energy gap or mass leads for long times to stationary behaviour with thermal characteristics. An exception to this rule is the 2d case with zero mass after the quench. In the composite quench however we find that the effect of the interactions in our approximation is simply to effectively change the value of the mass. This means on the one hand that the interacting model also exhibits the same stationary behaviour and on the other hand that this is now true even for the massless 2d case.

Figures (13)

• Figure 1: Top: Spacetime plot of the deep quench propagator $C_{dq}(r,t)$ in $1d$ and for $m=1$, as obtained by numerical integration of (\ref{['integral1']}). The horizon effect is clearly demonstrated. Outside the horizon the value is exactly zero. Bottom: Time dependence of $C_{dq}(r,t)$ (blue line) at fixed distance $r=r_0=2$, denoted by the vertical red line in the above figure. The dashed lines give the large time asymptotic expressions. Notice the decaying oscillations $\sim t^{-1/2} \cos{2 m t}$ (purple line) around the stationary value $\sim e^{-mr}$ (red line).
• Figure 2: Images required for the slab with Dirichlet (a) or periodic (b) boundary conditions.
• Figure 3: Effective temperature as a function of the final mass $\bar{\beta} m_0 = F_d (m/m_0)$ in units of the initial mass $m_0=1$. Inset: Asymptotic behaviour for small $m$. Notice the logarithmic corrections in $2d$.
• Figure 4: The Schwinger-Keldysh contour for a quantum quench.
• Figure 5: First order Feynman diagram.