Out-of-time-ordered correlators for Wigner matrices

Abstract

We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables $A$ and $B$ in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Elect. J. Prob. 27, 1-38, 2022]) or the Hilbert-Schmidt norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Forum Math., Sigma 10, E96, 2022]).

Figures (2)

• Figure 1: The two curves show the behaviour of $\mathcal{C}_{A,B}(t)$ in two different scenarios for two commuting traceless observables $A,B$ normalized to $\langle A^2 \rangle = \langle B^2 \rangle =1$. The black curve represents the case $A = B$ with $\mathrm{rank}(A) = N^{a}$, $a \in [0,1]$ where the OTOC exhibits a large peak of size $N^{1-a}$ around the scrambling time $t_* \sim 1$. Afterwards, it decays to its thermal limiting value (normalized to one) around the relaxation time $t_{**} \sim N^{\frac{1-a}{3}}$. The red curve represents the case where $AB = 0$. Here, both $t_*$ and $t_{**}$ are of order one, independent of the ranks of $A$ and $B$. For more details see Section \ref{['rmk:interpret']}.
• Figure 2: Depicted are four curves illustrating the influence of $\beta = 1/T$ on the OTOC $\mathcal{C}_{A,A}^{(\beta)}(t)$ up to intermediate times for Example 1 from Section \ref{['rmk:interpret']} (i.e. normalized to $\langle A^2 \rangle = 1$ with $\mathrm{rank} \, A = N^{\frac{1-a}{2}}$ and $a \in [0,1]$). As $\beta$ increases, the characteristic rank-dependent peak of size $\sim N^{1-a}$ around the scrambling time $t_* \sim 1$ becomes more pronounced and very slightly shifted to the left.

Theorems & Definitions (34)

• Theorem 2.2: OTOC for Wigner matrices
• Remark 2.3
• Definition 3.1: $\ell$-weighted Schatten norms
• Lemma 3.2: $M$-bound
• Theorem 3.3: Averaged multi-resolvent local law with Schatten norms
• Remark 3.4: Optimality
• Remark 3.5: Extensions
• Proposition 4.1: Step 1: Global law
• Lemma 4.2: see Lemma 3.2 in edgeETH
• Proposition 4.3: Step 2: Characteristic flow