# Pieter W. Claeys

pwclaeys@gmail.com# News

## Adiabatic landscape

30 April 2020 Closely related to the previous post: our paper on **Adiabatic landscape and optimal paths in ergodic systems** is now online at **arXiv:2004.13735**! A long time coming, but a lot of interesting results in this one.
A collaboration with Sho Sugiura from Harvard University, Anatoly Dymarsky from University of Kentucky and the Skolkovo Innovation Center, and Anatoli Polkovnikov from Boston University.

If we slightly deform a quantum Hamiltonian by manipulating some control parameter, its eigenstates will change according to some unitary transformation. In chaotic (ergodic) systems these unitary transformations are generally not expected to posses any underlying structure: their generator is highly nonlocal and divergent. Here, we show that the opposite is true: even if the system is ergodic, local approximations to the generator of these deformations are well-defined near points where the eigenspectrum of the Hamiltonian exhibits macroscopic degeneracies (corresponding to non-ergodic points), and these local approximations remain well-defined deep in the ergodic regime. This allows us to (i) find **optimal directions for quantum control**, where there's a remarkable anisotropy in the control landscape, (ii) identify which **dark states** acquire local dressing when moving from the non-ergodic to the ergodic regime, and can hence be efficiently prepared and exhibit anomalous entanglement properaties (similar to quantum scars), and (iii) show how the existence of such optimal directions and states goes hand in hand with the existence of **nearly-conserved operators**.

*Whether one is interested in quantum state preparation or in the design of efficient heat engines, adiabatic (reversible) transformations play a pivotal role in minimizing computational complexity
and energy losses. Understanding the structure of these transformations and identifying the systems for which such transformations can be performed efficiently and quickly is therefore of primary importance. In this paper we focus on finding optimal adiabatic paths in the space of couplings controlling the system’s Hamiltonian. More specifically, starting from a local Hamiltonian we analyze directions in the space of couplings along which adiabatic transformations can be accurately generated by local operators, which are both realizable in experiments and easy to simulate numerically. We consider a fairly generic non-integrable 1D Ising model parametrized by two independent couplings, corresponding to longitudinal and transverse magnetic fields. We find regions in the space of couplings characterized by a very strong anisotropy of the variational adiabatic gauge potential (AGP), generating the adiabatic transformations, which allows us to define optimal adiabatic flows. We find that these flows generally terminate at singular points characterized by extensive degeneracies in the spectrum, splitting the parameter space into adiabatically disconnected regions. The anisotropy can be related to singularities in the AGP, and we identify special robust weakly-thermalizing and non-absorbing many-body “dark” states, which are annihilated by the singular part of the AGP and show that their existence extends deep into the ergodic regime.*

## 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

4 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.*

## Editors' Suggestion in Physical Review B

10 January 2020 More good news from APS: our paper on **Variational Schrieffer-Wolff transformations for quantum many-body dynamics** has been published in **Physical Review B** and chosen as **Editors' Suggestion**! It is currently highlighted on the Physical Review B homepage, with an illustration of our method: a physical model is translated to a Hamiltonian matrix, that matrix is subsequently simplified (returning an effective Hamiltonian), and we can efficiently calculate dynamics using the effective model.
In other words: *The description of nonequilibrium dynamics in interacting quantum systems is a long-standing challenge within condensed matter physics. In this theoretical work, the authors introduce a method to block-diagonalize low-energy sectors of strongly interacting Hamiltonians by variationally computing generators of rotations, equivalent to a nonperturbative dressing of quasiparticle excitations. This procedure allows for the calculation of effective dynamics including quenches and response functions, as is demonstrated in a Fermi-Hubbard model and an integrability-broken XY model, both of which demonstrate excellent convergence.*

## Cavendish Laboratory

15 November 2019 Part two of the postdoc experience: I've started a new postdoc as research associate at the **Cavendish Laboratory** at the **University of Cambridge**! I've joined the Theory of Condensed Matter group, where I will be working with Austen Lamacraft, Claudio Castelnovo, and Nigel Cooper.

## Variational Schrieffer-Wolff transformations

31 October 2019New joint work with Jonathan Wurtz and Anatoli Polkovnikov! Our paper on **Variational Schrieffer-Wolff Transformations for Quantum Many-Body Dynamics** is now available on **arXiv**.

In this paper, we propose a new method for **simulating the dynamics of quantum systems**. This is generally a hard problem because of the massive number of equations needing to be solved, and we show how this can be simplified by finding **effective models for parts of the system**. This leads to a large reduction in the number of equations, at the cost of leading to more involved models, and allowed us to get surprisingly accurate results for quantum systems with 18 (Fermi-Hubbard model) and 144 sites (integrability-broken XY spin model). We used this method to describe effects such as the (dis-)appearance of localization and correlation spreading out in a one-dimensional lattice, as in the figure below, leading to some interesting results!

*Building on recent results for adiabatic gauge potentials, we propose a variational approach for computing the generator of Schrieffer-Wolff transformations. These transformations consist of block diagonalizing a Hamiltonian through a unitary rotation, which leads to effective dynamics in a computationally tractable reduced Hilbert space. The generators of these rotations are computed variationally and thus go beyond standard perturbative methods; the error is controlled by the locality of the variational ansatz. The method is demonstrated on two models. First, in the attractive Fermi-Hubbard model with on-site disorder, we find indications of a lack of observable many-body localization in the thermodynamic limit due to the inevitable mixture of different spinon sectors. Second, in the low-energy sector of the XY spin model with a broken U(1)-symmetry, we analyze ground state response functions by combining the variational SW transformation with the truncated spectrum approach.*

## Editors' Suggestion in Physical Review Letters

30 August 2019 Our paper on **counterdiabatic driving in quantum many-body systems** has been published in **Physical Review Letters** and chosen as **Editors' Suggestion**!
Our work is currently highlighted on the Physical Review Letters homepage and according to the website: *Our Suggestions will be based on the potential interest in the results presented and, importantly, on the success of the paper in communicating its message, in particular to readers from other fields.*
Thanks to everyone involved!

## Visiting UNB

26 August 2019This week I'll be visiting the **University of New Brunswick** in Canada, visiting the De Baerdemacker group and talking to a crowd of unexpecting chemists about transitionless driving in quantum many-body systems.

## Conferences: DYNQOS19 & Lindau

16 June 2019Catch me in Germany this month - next week, I am attending the **Engineering Nonequilibrium Dynamics of Open Quantum Systems** (DYNQOS19) workshop at the **Max Planck Institute** in Dresden. On Tuesday, I will be talking about our recent results on **Floquet-engineering counterdiabatic protocols in quantum many-body systems**, and on Thursday I will be chairing the late afternoon session. More information about the program here.

After this conference, on to the next, since I will also be presenting this topic as a **poster** at the **69th Lindau Nobel Laureate Meeting** (see above)!

## Counterdiabatic driving, adiabatic gauge potentials, and Floquet-engineering

9 April 2019Publication news - our preprint about **counterdiabatic driving in quantum many-body systems** has appeared on **arXiv**! A new topic for me, albeit with some interesting connections with my Ph.D. research. Joint work with Mohit Pandey, Dries Sels, and Anatoli Polkovnikov.

When changing a system from state A to state B, this is usually done extremely slow (*adiabatically*), carefully attempting not to disturb the system. However, in practice we can't take arbitrarily long due to system constraints, external noise,... One interesting way of speeding up such processes is through *counterdiabatic driving*, where we explicitly counteract the forces arising when changing a system a finite rate. However, it is generally extremely hard to (a) know what forces to apply and (b) know how to realize these forces. Mathematically, everything is encoded in the *adiabatic gauge potential*, and in this preprint we show how this potential can be approximated in a surprisingly accurate way and how the resulting forces and counterdiabatic driving can be realized through a 'simple' shaking of the system.

**Floquet-engineering counterdiabatic protocols in quantum many-body systems**: Counterdiabatic (CD) driving presents a way of generating adiabatic dynamics at arbitrary pace, where excitations due to non-adiabaticity are exactly compensated by adding an auxiliary driving term to the Hamiltonian. While this CD term is theoretically known and given by the adiabatic gauge potential, obtaining and implementing this potential in many-body systems is a formidable task, requiring knowledge of the spectral properties of the instantaneous Hamiltonians and control of highly nonlocal multibody interactions. We show how an approximate gauge potential can be systematically built up as a series of nested commutators, remaining well-defined in the thermodynamic limit. Furthermore, the resulting CD driving protocols can be realized up to arbitrary order without leaving the available control space using tools from periodically-driven (Floquet) systems. This is illustrated on few- and many-body quantum systems, where the resulting Floquet protocols significantly suppress dissipation and provide a drastic increase in fidelity.

## Lindau Nobel Laureate Meeting

18 March 2019 Excellent news and something I'm already looking forward to: I have been selected to participate in the **69th Lindau Nobel Laureate Meeting**! From 31 June to 5 July I will be joining almost 600 other young scientists in Lindau, Germany, to discuss physics and attend lectures, discussion sessions and masterclasses with over 40 Nobel Laureates in Physics and related fields. This selection followed an initial selection and nomination by the Research Foundation Flanders (FWO Vlaanderen), whose funding is gratefully acknowledged.

*Once every year, more than 30 Nobel Laureates convene in Lindau to meet the next generation of leading scientists: 500-600 undergraduates, PhD students, and post-doc researchers from all over the world. The Lindau Nobel Laureate Meetings foster the exchange among scientists of different generations, cultures, and disciplines.*

## APS March Meeting

4 March 2019Tomorrow I will be presenting our work on Floquet resonances in the central spin model in the APS March Meeting currently taking place in Boston, during the session **Non-Equilibrium Physics in AMO Systems II**. According to the abstract:

*Adiabatically varying the driving frequency of a periodically driven many-body quantum system can induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian, corresponding to adiabatic transitions in the Floquet spectrum and presenting a general tool in quantum many-body control. Using the central spin model as an application, we show how such controlled driving processes can lead to a polarization-based decoupling of the central spin from its decoherence-inducing environment at resonance. While it is generally impossible to obtain the exact Floquet Hamiltonian in driven interacting systems, we exploit the integrability of the central spin model to show how techniques from quantum quenches can be used to explicitly construct the Floquet Hamiltonian in a restricted many-body basis and model Floquet resonances.*.

## Editors' Suggestion in Physical Review B

6 February 2019 More good news: our paper on **integrability and duality in spin chains** has also been published in Physical Review B. It has been chosen as an **Editors' Suggestion**, leading to a nice figure and teaser on the Physical Review B homepage.

Joint work with Eyzo Stouten, Jean-Sébastien Caux and Vladimir Gritsev.

## XYZ Richardson-Gaudin models

6 February 2019 Our recent paper on **general XYZ Richardson-Gaudin models** with Stijn De Baerdemacker, Alexandre Faribault and his Ph.D. student Claude Dimo has now been published in Journal of Physics A as a **Letter** and has been awarded the **IOPselect label**. According to the the IOP site, our paper belongs to a group of '*articles from the last 12 months that have been chosen by our editors for their novelty, significance and potential impact on future research.*'

## Publication news

16 October 2018 Some news about active collaborations: our paper about the most general **XYZ Richardson-Gaudin model in an arbitrary magnetic field** just appeared on arXiv. While XYZ models generally do not allow for an arbitrary magnetic field, we show how such a field can be introduced through the restriction to spin-1/2 models, which relaxes the usual integrability constraints. Joint work with Stijn De Baerdemacker, Alexandre Faribault and his Ph.D. student Claude Dimo from the University of Lorraine, following my one-week stay at Lorraine earlier this year. Hopefully, this should allow for a more realistic description of the decoherence of central spin models. All comments are welcome!

On to the next state of the publication process - the paper with Eyzo Stouten, Mikhail Zvonarev, Jean-Sébastien Caux and Vladimir Gritsev on generalized Lieb-Liniger models just got accepted for publication in Journal of Physics A: Mathematical and Theoretical and should appear online shortly.

## Postdoc and website

26 September 2018This week my FWO Ph.D. fellowship officially ended, and after some interesting years I’ve had to say bye to both Ghent University and Ghent itself. Moving on to postdoc life, I’ve just arrived in Boston to start a postdoc with Anatoli Polkovnikov at the Condensed Matter Theory group at Boston University, with support from the Belgian American Education Foundation. Hopefully more updates soon!

And in other (more obvious) news, I have a personal website!