Parametric correlations versus fidelity decay: The symmetry breaking case

We provide formulas for fidelity decay and parametric energy correlations for random matrix ensembles where time-reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities exists. Fidelity freeze is observed for systems with even and odd spin.

Figure 1

Ensemble average of the fidelity amplitude $f(\lambda ,\lambda ,\tau )$ (solid line) with $H_0$ taken from the GOE (case I) and the perturbation taken from the GUE for different perturbation strengths $\lambda$. The results $f(\lambda ,0,\tau )$ (a pure GOE perturbation, dashed lines) and $f(0,\lambda ,\tau )$ (a purely perpendicular perturbation, dashed-dotted lines) are shown as well for the same parameter.

Figure 2

Fidelity amplitude $f({\lambda }_{\parallel },{\lambda }_{\perp },\tau )$ for case I (upper picture) and for case II (bottom picture) for different values of ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ and for fixed overall perturbation $\lambda$ $\equiv$ $\sqrt{{\lambda }_{\parallel }^2+{\lambda }_{\perp }^2}$ $=0.1$. The values of ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ are given by the five ratios ${\lambda }_{\parallel }^2/{\lambda }_{\perp }^2$ $=$ $\infty$ (solid line), 2 (dotted line), 1 (dashed line), $1/2$ (dashed-dotted line), and 0 (dashed-dotted-dotted-dotted line).

Figure 3

Same as Fig. 2 but for a strong overall perturbation $\lambda =1.5$. Fidelity amplitude $f({\lambda }_{\parallel },{\lambda }_{\perp },\tau )$ is shown for case I (upper panel) and case II (lower panel) and for the ratios ${\lambda }_{\parallel }^2/{\lambda }_{\perp }^2$ $=$ $\infty$ (solid line), 2 (dotted line), 1 (dashed line), $1/2$ (dashed-dotted line), and 0 (dashed-dotted-dotted-dotted line). In the lower panel for comparison fidelity amplitude $f(\sqrt{2}\lambda ,\tau /2)$ for a GUE with a GUE perturbation is shown as well (thinner solid line).

Figure 4

Cross form-factor $\tilde{K}({\lambda }_{\parallel },{\lambda }_{\perp },\tau )$ for case I (upper panel) and case II (lower panel) for different values of ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ and for small overall perturbation $\lambda =0.1$. The values of ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ are given by the the ratios ${\lambda }_{\parallel }^2/{\lambda }_{\perp }^2$ $=\infty$ (solid line), 2 (dotted line), 1 (dashed line), $1/2$ (dashed-dotted line), and 0 (dashed-dotted-dotted-dotted line).