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arXiv:2111.06108 · Physical review. D/Physical review. D.

Onset of quantum chaos in disordered CFTs

Micha Berkooz A, Adar Sharon, Navot Silberstein, Erez Y. Urbach

Published Aug 9, 2022• 9 citations
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Abstract

We study the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ in quantum field theories with spacetime-independent disorder interactions. Generically ${\ensuremath{\lambda}}_{L}$ can only be computed at isolated points in parameter space, and little is known about the way in which chaos grows as we deform the theory away from weak coupling. In this paper, we describe families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow ${\ensuremath{\lambda}}_{L}$ from weak to strong coupling. We find surprising behaviors in some cases, including a discontinuous transition into chaos. We also derive self-consistency equations for the two- and four-point functions for products of $N$ nontrivial conformal field theories deformed by disorder at leading order in $1/N$.

DOI
10.1103/physrevd.106.045007

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We study the Lyapunov exponent \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ in quantum field theories with spacetime-independent disorder interactions. Generically \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ can only be computed at isolated points in parameter space, and little is known about the way in which chaos grows as we deform the theory away from weak coupling. In this paper, we describe families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ from weak to strong coupling. We find surprising behaviors in some cases, including a discontinuous transition into chaos. We also derive self-consistency equations for the two- and four-point functions for products of NNN nontrivial conformal field theories deformed by disorder at leading order in 1/N1/N1/N.
We study the quantum Lyapunov exponent λ_{L} in theories with spacetime-independent disorder. We first derive self-consistency equations for the two- and four-point functions for products of N models coupled by disorder at large N, generalizing the equations appearing in SYK-like models. We then study families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow λ_{L} from weak to strong coupling. We find interesting behaviors, including a discontinuous transition into chaos, mimicking classical KAM theory.
A bstract We discuss aspects of the quantum Lyapunov exponent λ L in theories with an exactly marginal SYK-like random interaction, where λ L can be computed as a continuous function of the interaction strength J \mathcal{J} J J . In 1 d , we prove a conjecture from [1] which states that at small J \mathcal{J} J J , λ L can be found by considering a specific limit of the four-point function in the decoupled theory. We then provide additional evidence for the 2 d version of this conjecture by discussing new examples of Lyapunov exponents which can be computed at weak coupling.
A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.
A bstract Fast scrambling is a distinctive feature of quantum gravity, which by means of holography is closely tied to the behaviour of large- c conformal field theories. We study this phenomenon in the context of semiclassical Liouville theory, providing both insights into the mechanism of scrambling in CFTs and into the structure of Liouville theory, finding that it exhibits a maximal Lyapunov exponent despite not featuring the identity in its spectrum. However, as we show, the states contributing to the relevant correlation function can be thought of as dressed scramblons. At a technical level we first use the path integral picture in order to derive the Euclidean four-point function in an explicit compact form. Next, we demonstrate its equivalence to a conformal block expansion, revealing an explicit but non-local map between path integral saddles and conformal blocks. By analytically continuing both expressions to Lorentzian times, we obtain two equivalent formulations of the OTOC, which we use to study the onset of chaos in Liouville theory. We take advantage of the compact form in order to extract a Lyapunov exponent and a scrambling time. From the conformal block expansion formulation of the OTOC we learn that scrambling shifts the dominance of conformal blocks from heavy primaries at early times to the lightest primary at late times. Finally, we discuss our results in the context of holography.
Abstract We study aspects of chaos and thermodynamics at strong coupling in a scalar model using LCT numerical methods. We find that our eigenstate spectrum satisfies Wigner-Dyson statistics and that the coefficients describing eigenstates in our basis satisfy Random Matrix Theory (RMT) statistics. At weak coupling, though the bulk of states satisfy RMT statistics, we find several scar states as well. We then use these chaotic states to compute the equation of state of the model, obtaining results consistent with Conformal Field Theory (CFT) expectations at temperatures above the scale of relevant interactions. We also test the Eigenstate Thermalization Hypothesis by computing the expectation value of local operators in eigenstates, and check that their behavior is consistent with thermal CFT values at high temperatures. Finally, we compute the Spectral Form Factor (SFF), which has the expected behavior associated with the equation of state at short times and chaos at long times. We also propose a new technique for extracting the connected part of the SFF without the need of disorder averaging by using different symmetry sectors.
We study scrambling in a model consisting of a number NNN of MMM-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large NNN and large MMM. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to 1M\frac{1}{M}M1​. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a T4T^4T4 dependence on temperature TTT.