# Pieter W. Claeys

pwclaeys@gmail.com# News

## Leeds-Loughborough-Nottingham seminar

20 October 2021I will be presenting our recent preprint on the **Absence of superdiffusion in certain random spin models** in the Leeds-Loughborough-Nottingham seminar series on Wednesday 3rd of November, 3pm UK time. This online seminar series is run by the Universities of Leeds (Dr. Papic), Loughborough (Dr. Lazarides) and Nottingham (Prof. Garrahan), with weekly seminars focusing on non-equilibrium physics. More info, including a Zoom link and recordings of previous talks, can be found on the seminar series website.

**Abstract.**The dynamics of spin at finite temperature in the spin-1/2 Heisenberg chain was found to be superdiffusive in numerous recent numerical and experimental studies. Theoretical approaches to this problem have emphasized the role of nonabelian SU(2) symmetry as well as integrability, but the associated methods cannot be readily applied when integrability is broken. After an introduction to superdiffusion I will examine spin transport in such a spin-1/2 chain in which the exchange couplings fluctuate in space and time, breaking integrability but not spin symmetry, showing that operator dynamics in the strong noise limit can be analyzed using conventional perturbation theory. I will argue that the spin dynamics undergo enhanced diffusion with some interesting transient behavior rather than superdiffusion, comparing the dynamics with both a hydrodynamic approach and tensor network simulations.

## Absence of superdiffusion in certain random spin models

15 October 2021New preprint on **Absence of superdiffusion in certain random spin models** on arXiv today! With Austen Lamacraft and Jonah Herzog-Arbeitman we investigate the fate of superdiffusion in a Heisenberg model with noisy exchange coupling. If you would like the paper explained to you, Austen recently gave a talk at KITP that remains available to watch here.

Even though quantum dynamics is a complicated problem, the large-scale dynamics can usually be understood in terms of a few simple types of motion, including diffusion and sound waves, with the number and nature of these modes determined by conservation laws. In the simplest case of a single conservation law, diffusion of the conserved quantity is the norm — the complicated quantum system essentially undergoes a random walk. This simple picture applies even in the case of a multi-component conserved quantity such as the spin density in an isotropic paramagnet: within linear response each component diffuses separately. An exception to this phenomenology has recently attracted a lot of attention: in the one-dimensional spin-1/2 Heisenberg model, spin dynamics were found to behave superdiffusively. Spin dynamics occur faster than diffusion, in a way characteristic of the Kardar–Parisi–Zhang universality class. It is currently believed that superdiffusion arises through the combination of nonabelian symmetry and integrability. However, a fully microscopic calculation of the KPZ scaling function for any model is still lacking.

In this work we propose such a microscopic model for a Heisenberg model with a noisy exchange coupling — breaking integrability but preserving total spin symmetry. In this model, different approaches have predicted either (a weaker form of) superdiffusion or regular diffusion with small corrections. In this work we introduce a graphical picture for spin dynamics that holds near the strong noise limit. Noise causes the spin components to diffuse independently, and the Heisenberg terms introduce an interaction where spin components can either split or merge with some probability set by the interaction strength. Such a model can then be tackled using conventional tools. We find that the diffusion constant does grow in time, but that this is only a transient phenomenon: regular diffusion persists at long times, albeit with an enhanced diffusion constant. We were able to get analytic predictions for both the diffusion constant and the correlation profile, and these are in excellent agreement with numerical simulations using tensor networks. Our findings are consistent with the hydrodynamic prediction of regular diffusion with subleading corrections, and indicate the absence of superdiffusion in this class of random spin models.

*The dynamics of spin at finite temperature in the spin-1/2 Heisenberg chain was found to be superdiffusive in numerous recent numerical and experimental studies. Theoretical approaches to this problem have emphasized the role of nonabelian SU(2) symmetry as well as integrability but the associated methods cannot be readily applied when integrability is broken. We examine spin transport in a spin-1/2 chain in which the exchange couplings fluctuate in space and time around a nonzero mean J, a model introduced by De Nardis et al. [Phys. Rev. Lett. 127, 057201 (2021)]. We show that operator dynamics in the strong noise limit at infinite temperature can be analyzed using conventional perturbation theory as an expansion in J. We find that regular diffusion persists at long times, albeit with an enhanced diffusion constant. The finite time spin dynamics is analyzed and compared with matrix product operator simulations.*

## Dissipative dynamics in open XXZ Richardson-Gaudin models

05 August 2021New work with Austen Lamacraft on **Dissipative dynamics in open XXZ Richardson-Gaudin models** now on **arXiv**! Extending on previous work by Austen and Daniel Rowlands, we consider open systems with collective dissipation that can be mapped to Richardson-Gaudin models. While this mapping was previously known, the exact solution remained surprisingly unexplored, and we here investigate the physics of such collective models.

Moving from closed systems to open systems, the usual description in terms of a Hermitian Hamiltonian no longer applies and we rather need to consider a non-Hermitian Liouvillian. While we lose most of the salient properties of Hermiticity, the Liouvillian that we obtain turns out to be *pseudo-Hermitian*. This can be interpreted as an additional symmetry of the system, and as we change the coupling to the environment the dynamics of the system qualitatively changes if this symmetry is broken. At weak coupling the system essentially behaves as a closed system with weak dissipation, but as the coupling to the environment is increased at some point the system becomes overdamped, and all correlations exhibit purely exponential decay. Mathematically, the Liouvillian passes through exceptional points and PT-symmetry/pseudo-Hermiticity is spontaneously broken. We also uncover some additional structure in the spectrum, showing how at large coupling to the environment the spectrum exhibits features of the homogeneous model — where we can exactly solve the system using collective spins.

*In specific open systems with collective dissipation the Liouvillian can be mapped to a non-Hermitian Hamiltonian. We here consider such a system where the Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is pseudo-Hermitian/PT-symmetric, and as the strength of the coupling to the environment is increased the spectrum in a fixed symmetry sector changes from a broken pseudo-Hermitian phase with complex conjugate eigenvalues to a pseudo-Hermitian phase with real eigenvalues, passing through a series of exceptional points and associated dissipative quantum phase transitions. The homogeneous limit supports a nontrivial steady state, and away from this limit this state gives rise to a slow logarithmic growth of the decay rate (spectral gap) with system size. Using the exact solution, it is furthermore shown how at large coupling strengths the ratio of the imaginary to the real part of the eigenvalues becomes approximately quantized in the remaining symmetry sectors.*