# Pieter W. Claeys

pwclaeys@gmail.com# News

## Adiabatic eigenstate deformations

10 April 2020 Our paper on **Adiabatic eigenstate deformations as a sensitive probe for quantum chaos** is now online at **arXiv:2004.05043**!
A collaboration with Mohit Pandey, David Campbell, and Anatoli Polkovnikov from Boston University, and Dries Sels from Harvard University and Universiteit Antwerpen.

Whereas **classical chaos** is expressed through an exponential **sensitivity of particle trajectories** to initial conditions, **quantum chaos** is usually encoded in the eigenstates of the Hamiltonian - the effects of which subsequently appear in quantum dynamics. Chaos leads to ergodicity, one of the fundamental concepts in various fields of physics, which can be contrasted with the non-ergodic behaviour of integrable systems. Here, we propose an extremely sensitive probe for quantum chaos, in line with both quantum and classical definitions: the **sensitivity of eigenstates** to small perturbations on the underlying Hamiltonian. Quantum chaos manifests itself in a vast range of different phenomena, each relevant up to a particular time scale, and the sensitivity of our proposed measure is argued to follow from the fact that this measure is sensitive to dynamics at exponentially large time scales.

*In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis. Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor. In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos. We are able to detect transitions from non-ergodic to ergodic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures. Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point. In some cases, small integrability-breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.*

## Maximum velocity quantum circuits

04 March 2020 New work on **maximum velocity quantum circuits**, in collaboration with Austen Lamacraft! Our paper just appeared on **arXiv**.

If we perturb a physical system at a given point in space and time, how does this perturbation influence this system at a different point in space and time? If the points are far away, it generally doesn't. Perturbations grow with a finite "**butterfly velocity**" (think chaos and the butterfly effect), and it takes a finite time for any local effect to become noticeable at a different location. This is usually not a very 'sharp' effect: the butterfly velocity is smeared out by a diffusively-widening front. Here, we calculate **out-of-time-order correlators**, a measure for chaos and the scrambling of quantum information, in systems where this butterfly velocity is maximal and dynamics are governed by so-called quantum circuits. Due to geometric constraints, no such widening front is possible, and these effects become much more clearly outlined: all effects are focused on the light cone x=v_{B}t, so called since the butterfly velocity behaves as an effective light speed, and decay exponentially away from this light cone. We explicitly relate the rate at wich excitations in such a system relax towards thermal equilibrium with the rate at which OTOCs decay away from this light-cone, connecting **scrambling** with **thermalization**.

*We consider the long-time limit of out-of-time-order correlators (OTOCs) in two classes of quantum lattice models with time evolution governed by local unitary quantum circuits and maximal butterfly
velocity v _{B} = 1 . Using a transfer matrix approach, we present analytic results for the long-time value of the OTOC on and inside the light cone. First, we consider ‘dual-unitary’ circuits with various levels of ergodicity, including the integrable and non-integrable kicked Ising model, where we show exponential decay away from the light cone and relate both the decay rate and the long-time value to those of the correlation functions. Second, we consider a class of kicked XY models similar to the integrable kicked Ising model, again satisfying v_{B} = 1, highlighting that maximal butterfly velocity is not exclusive to dual-unitary circuits.*

## Integrability and dark states in an anisotropic central spin model

29 January 2020 Some more work on a familiar topic: our paper on **Integrability and dark states in an anisotropic central spin model** just appeared on the **arXiv**! Together with Tamiro Villazon and Anushya Chandran from Boston University, we found that **the central spin model with XX Heisenberg interactions is integrable**. The central spin model with fully isotropic (XXX) interactions has long been known to be integrable and was one of the models I kept returning to during my Ph.D. research on Richardson-Gaudin models, but it came as a complete surprise to me that the XX model also is. Even more remarkable, all eigenstates have a special structure and can be classified as either **bright** or **dark** states. The bright states are experimentally accessible by tuning the central magnetic field, while in the dark states the central spin effectively decouples from its environment, leading to all kinds of interesting physical phenomena.

Also: my first last author paper!

*Central spin models describe a variety of quantum systems in which a spin-1/2 qubit interacts with a bath of surrounding spins, as realized in quantum dots and defect centers in diamond. We show that the fully anisotropic central spin Hamiltonian with (XX) Heisenberg interactions is integrable. Building on the class of integrable Richardson-Gaudin models, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. These states divide into two exponentially large classes: bright states, where the qubit is entangled with the bath, and dark states, where it is not. We discuss how dark states limit qubit-assisted spin bath polarization and provide a robust long-lived quantum memory for qubit states.*