Orthogonality Catastrophe in Dissipative Quantum Many-Body Systems

We present an analog of the phenomenon of orthogonality catastrophe in quantum many-body systems subject to a local dissipative impurity. We show that the fidelity $F(t)$, giving a measure for distance of the time-evolved state from the initial one, displays a universal scaling form $F(t)\propto t^{\theta }{e}^{-\gamma t}$, when the system supports long-range correlations, in a fashion reminiscent of traditional instances of orthogonality catastrophe in condensed matter. An exponential falloff at rate $\gamma$ signals the onset of environmental decoherence, which is critically slowed down by the additional algebraic contribution to the fidelity. This picture is derived within a second-order cumulant expansion suited for Liouvillian dynamics, and substantiated for the one-dimensional transverse field quantum Ising model subject to a local dephasing jump operator, as well as for $XY$ and $XX$ quantum spin chains, and for the two-dimensional Bose gas deep in the superfluid phase with local particle heating. Our results hint that local sources of dissipation can be used to inspect real-time correlations and to induce a delay of decoherence in open quantum many-body systems.

Figure 1

Comparison between the fidelity $F(t)$ for a quantum Ising chain with a spin-dephasing impurity (red line) and for the same Ising chain with a local defect on the transverse field, at zero temperature [29] (green line) and at finite temperature [30] (blue line). The asymptotic behavior of the three curves is highlighted on the right. The local Lindbladian channel results in a slower decay of the fidelity compared to the other two cases. Close to $t\lesssim (\sqrt{\theta }+\theta )/\gamma$ the fidelity in the dissipative Ising model transits from a concave universal behavior to the usual convex character typical of isolated systems.

Figure 2

The rate of exponential decay $\gamma$ and its derivative as a function of the gap ($g-1$) in the paramagnetic phase of the one-dimensional quantum Ising chain subject to local spin dephasing. Close to the critical point, $g\to 1$, the latter (blue line) exhibits a logarithmic divergence, while the former (red line) is continuous.