Dynamics of Symmetry Breaking during Quantum Real-Time Evolution in a Minimal Model System

One necessary criterion for the thermalization of a nonequilibrium quantum many-particle system is ergodicity. It is, however, not sufficient in cases where the asymptotic long-time state lies in a symmetry-broken phase but the initial state of nonequilibrium time evolution is fully symmetric with respect to this symmetry. In equilibrium, one particular symmetry-broken state is chosen as a result of an infinitesimal symmetry-breaking perturbation. From a dynamical point of view the question is: Can such an infinitesimal perturbation be sufficient for the system to establish a nonvanishing order during quantum real-time evolution? We study this question analytically for a minimal model system that can be associated with symmetry breaking, the ferromagnetic Kondo model. We show that after a quantum quench from a completely symmetric state the system is able to break its symmetry dynamically and discuss how these features can be observed experimentally.

Figure 1

Dynamics of symmetry breaking in the ferromagnetic Kondo model. The proposed setup is composed out of three elementary components. The system spin and the fermionic reservoir, coupled via a ferromagnetic exchange $J$, are supposed to realize the ferromagnetic Kondo model. During the dynamics of symmetry breaking in the presence of an infinitesimal local magnetic field $h$ the system spin develops a local magnetization. The antiferromagnetic coupling $J_{\mathrm{af}}$ to the initializer spin generates an initially rotationally symmetric spin singlet.

Figure 2

(a) Dynamics of the local magnetization in the ferromagnetic Kondo model for different magnetic field strengths obtained from the numerically exact solution of the one-loop flow equations for $g={10}^{-2}$. For times $t<t^{*}$ the local magnetization only acquires perturbative corrections in the presence of a small magnetic field. Beyond the time scale $t^{*}$, the magnetic field induces a local magnetic moment that saturates to a value independent of the magnetic field in the asymptotic long-time limit up to corrections that vanish in the zero field limit. (b) Asymptotic long-time value of the local magnetization $⟨S_z(t\to \infty )⟩=(1+f_{*})/2$. Comparison of the full numerically exact result for $f_{*}$ based on the one-loop flow equations (points) to the analytical estimate $f_{*}/g^2=3/2+\log (h/D)/[1+g\log (h/D)]$ (line) demonstrating the accuracy of both the numerical as well as the analytical result.

Figure 3

Influence of temperature onto dynamical symmetry breaking in the ferromagnetic Kondo model. Dynamics of the local magnetization for different temperatures with $g={10}^{-2}$ and $h/D={10}^{-4}$. For small times $t<t_T$, see Eq. (10); the real-time evolution of the magnetization is equivalent to the zero-temperature limit demonstrating the observability of dynamical symmetry breaking in the ferromagnetic Kondo model at nonzero temperatures. Increasing temperature such that $t_T\lesssim t_{*}$ destroys the signatures of symmetry breaking eventually leading to a complete suppression in the limit $T\gg h$.