[Paper Review] Finite-Temperature Scrambling of a Random Hamiltonian

Sagar Vijay, Ashvin Vishwanath|arXiv (Cornell University)|Mar 22, 2018

Stochastic processes and statistical mechanics1 references3 citations

TL;DR

This paper presents an exact calculation of finite-temperature out-of-time-ordered correlation functions (OTOCs) for a quantum system with a Hamiltonian drawn from the Gaussian unitary ensemble (GUE), in the large-N limit. It reveals that the OTOC exhibits early-time quadratic growth and saturates to an asymptotic value with a power-law decay in time, where the exponent depends on temperature and operator class. Crucially, it identifies a coherent scrambling regime where the OTOC satisfies a unitarity-breaking condition, indicating effective information scrambling on a timescale $ t_c \sim J^{-1} $, which matches the scrambling time $ t_s $ at low temperatures.

ABSTRACT

We study the finite-temperature scrambling behavior of a quantum system described by a Hamiltonian chosen from a random matrix ensemble. This effectively (0+1)-dimensional model admits an exact calculation of various ensemble-averaged out-of-time-ordered correlation functions in the large-$N$ limit, where $N$ is the Hilbert space dimension. For a Hamiltonian drawn from the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all temperatures. In addition to an early time quadratic growth of the averaged out-of-time-ordered commutator (OTOC), we determine that the OTOC saturates to its asymptotic value as a power-law in time, with an exponent that depends both on temperature, and on one of four classes of operators appearing in the correlation function, that naturally emerge from this calculation. Out-of-time-ordered correlation functions of operators that are distributed around the thermal circle take a time $t_{s}\sim β$ to decay at low temperatures. We generalize these exact results, by demonstrating that out-of-time-ordered correlation functions averaged over any ensemble of Hamiltonians that are invariant under unitary conjugation $H ightarrow {U} H {U}^{\dagger}$, exhibit power-law decay to an asymptotic value. We argue that this late-time behavior is a generic feature of unitary dynamics with energy conservation. Finally, by comparing the OTOC with a commutator-anticommutator correlation function, we examine whether there is a time window over which a typical Hamiltonian behaves as a "coherent scrambler" in the language of Ref. \cite{Kitaev_IAS, Kitaev_Suh}.

Motivation & Objective

• To understand the universal finite-temperature scrambling dynamics of quantum systems with random Hamiltonians.
• To compute ensemble-averaged out-of-time-ordered correlation functions (OTOCs) exactly in the large-N limit for a GUE-distributed Hamiltonian.
• To determine whether typical random Hamiltonians exhibit coherent scrambling behavior, defined by a breakdown of unitarity in the scattering matrix at early times.
• To generalize the late-time power-law decay of OTOCs to any Hamiltonian ensemble invariant under unitary conjugation.
• To compare the OTOC dynamics with a commutator-anticommutator correlation function to assess the existence of a coherent scrambling window.

Proposed method

• The study employs a (0+1)-dimensional model with a Hamiltonian sampled from the Gaussian unitary ensemble (GUE), enabling exact calculations in the large-N limit.
• It computes the ensemble-averaged OTOC $ \overline{C_\beta(t)} = \frac{1}{2} \left\langle \left| [V(t), W(0)] \right|^2 \right\rangle_\beta $ using random matrix theory techniques.
• The analysis includes the calculation of the quantity $ D_\beta(t) $, defined as $ \left\langle W(0)V(t)^2W(0) \right\rangle_\beta $, to assess the coherent scrambling condition $ |D_\beta(t)| \gg \overline{C_\beta(t)} $.
• The authors derive exact expressions for $ \overline{D_\beta(t)} $ and $ \overline{C_\beta(t)} $ in the large-N limit, involving modified Bessel functions $ I_1 $ and $ I_2 $, and analyze their asymptotic behavior.
• The paper classifies operators into four distinct classes based on their trace and product structure, which determine the power-law decay exponent of the OTOC.
• It establishes that power-law decay to an asymptotic value is a generic feature of unitary dynamics with energy conservation, valid for any unitarily invariant Hamiltonian ensemble.

Experimental results

Research questions

• RQ1 Does the OTOC of a random Hamiltonian exhibit universal late-time power-law decay, and if so, what determines the decay exponent?
• RQ2 Can a typical random Hamiltonian act as a 'coherent scrambler' in the sense of Kitaev and Suh, satisfying the condition $ |D_\beta(t)| \gg \overline{C_\beta(t)} $ at early times?
• RQ3 How does the coherent scrambling time $ t_c $ compare to the scrambling time $ t_s $, and does it scale as $ J^{-1} $ at low temperatures?
• RQ4 What role do the four distinct classes of operators—defined by their trace and product structure—play in determining the late-time OTOC behavior?
• RQ5 Is the power-law decay of OTOCs to an asymptotic value a generic feature of unitary dynamics with energy conservation, or specific to random matrix ensembles?

Key findings

• The OTOC exhibits early-time quadratic growth, $ \overline{C_\beta(t)} \sim J^2 t^2 $, at all temperatures, indicating rapid operator spreading.
• The OTOC saturates to its asymptotic value with a power-law decay in time, $ \sim t^{-\alpha} $, where the exponent $ \alpha $ depends on both temperature and the operator class.
• At low temperatures, the coherent scrambling condition $ |D_\beta(t)| \gg \overline{C_\beta(t)} $ is satisfied for disjoint, traceless operators over a window $ 0 \ll t \ll J^{-1} $, indicating effective information scrambling.
• The coherent scrambling time $ t_c $, defined by the onset of this condition, is found to be of the same order as the scrambling time $ t_s \sim J^{-1} $, implying that scrambling is coherent at low temperatures.
• Out-of-time-ordered correlation functions for operators distributed on the thermal circle decay over a time $ t_s \sim \beta $ at low temperatures, indicating a thermal-scale timescale for decay.
• The power-law decay of the OTOC to an asymptotic value is shown to be a generic feature of unitary dynamics with energy conservation, extending beyond random matrix ensembles to any unitarily invariant Hamiltonian ensemble.