Real-time correlators in chaotic quantum many-body systems

Abstract

We study real-time local correlators $⟨\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)⟩$ in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbf{x},t)$. The relevant trajectories involve the operator contracting to a point at both the initial and final time and so are structurally different from those dominating the out-of-time-order correlator. In the absence of conservation laws, correlations decay exponentially: $⟨\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)⟩\sim \exp (-s_{\mathrm{eq}}r(\mathbf{v})t)$, where $\mathbf{v}=\mathbf{x}/t$ defines a spacetime ray, and $r(\mathbf{v})$ is an associated decay rate. We express $r(\mathbf{v})$ in terms of cost functions for various spacetime structures. In 1+1D, operator histories can show a phase transition at a critical ray velocity $v_c$, where $r(\mathbf{v})$ is nonanalytic. At low $v$, the dominant Feynman histories are "fat": the operator grows to a size of order $t^{\alpha }\gg 1$ before contracting to a point again. At high $v$ the trajectories are "thin": the operator always remains of order-one size. In a Haar-random unitary circuit, this transition maps to a simple binding transition for a pair of random walks (the two spatial boundaries of the operator). In higher dimensions, thin trajectories always dominate. We discuss ways to extract the butterfly velocity $v_B$ from the time-ordered correlator, rather than the OTOC. Correlators in the random circuit may alternatively be computed with an effective Ising-like model: a special feature of the Ising weights for the Haar brickwork circuit gives $v_c=v_B$. This work addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in quantum field theories.

29 pages, 15 figures

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• Emergent random matrix universality in quantum operator dynamics
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