# Non-equilibrium Quantum Dynamics and Transport

The study of closed quantum systems out-of-equilibrium, with or without disorder, opens the door to new and exciting conceptual questions about dynamics, thermalization, universality and dynamical phase transitions beyond traditional condensed matter paradigms. In addition to being theoretically appealing, these questions are strongly motivated by the rapid progress in ultra-cold atomic, molecular, and trapped ion systems that offer promising experimental platforms to investigate the dynamics of closed quantum systems. More generally, recent experimental advances in ultrafast optics and pump-probe spectroscopy, solid-state qubits, ultracold gases or quantum nanostructures have opened up exciting avenues for non-equilibrium quantum physics, with many open questions such as quantum transport beyond linear-response or the existence of novel dynamical phases of matter with no equilibrium counterparts. A central theme in my research is studying the non-equilibrium dynamics of low-dimensional strongly-correlated quantum systems, with some emphasis on entanglement spreading, quantum quenches, hydrodynamics and far-from-equilibrium transport.

**Periodically driven (Floquet) quantum systems **

Recent years have witnessed substantial progress in understanding the dynamics of periodically driven (Floquet) systems. Low dimensional quantum critical systems are a natural setting in which to study Floquet dynamics, as many insights into the non-equilibrium dynamics of many-body systems have come from the study of Conformal Field Theories (CFTs) in 1+1d. Despite the naive expectation that such gapless systems should absorb energy and simply heat up when subjected to a boundary drive, we have identified distinct regimes in which the system shows universal features that can be understood using tools of field theory and scaling theory. Recently, I also studied the dynamical phase transitions separating distinct Floquet phases of matter by developing novel renormalization group techniques. We proposed a general picture of “Floquet quantum criticality” in terms of a distinct type of domain wall associated with time translational symmetry-breaking. Even more recently, we also uncovered a new class of “Thouless pumps” in interacting, translation-invariant Floquet systems, that are intrinsically-Floquet integrable systems with chiral (topologically non-trivial) quasiparticles.

*Floquet Dynamics of Boundary-Driven Systems at Criticality*, W. Berdanier, M. Kolodrubetz, R. Vasseur & J.E. Moore, Phys. Rev. Lett.**118**, 260602 (2017).- Floquet Quantum Criticality, W. Berdanier, M. Kolodrubetz, S.A. Parameswaran & R. Vasseur,
**.** - Strong-Disorder Renormalization Group for Periodically Driven Systems, William Berdanier, Michael Kolodrubetz, S. A. Parameswaran & Romain Vasseur, Phys. Rev B
**98**, 174203 (2018). - Integrable Many-Body Quantum Floquet-Thouless Pumps, A.J. Friedman, S. Gopalakrishnan and R. Vasseur, Phys. Rev. Lett.
**123**, 170603 (2019).

**Anomalous transport in one-dimensional quantum magnets **

The XXZ model is a canonical model of quantum magnetism. In one dimension, this model is integrable and has ballistically moving quasiparticles. Thus, energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. In collaboration with Sarang Gopalakrishnan, my group has worked on the theory of this anomalous transport using a kinetic picture based on generalized hydrodynamics and fluctuations. We computed the diffusion constant of the XXZ chain in the gapped regime at infinite temperature, and provided the first theoretical explanation for the anomalous dynamical exponent z=3/2 at the isotropic point — reminiscent of the KPZ (Kardar–Parisi–Zhang) universality class. As we showed recently, superdiffusive transport can be explained by the emergence of “soft” classical solitons. We also studied spin transport in the presence of a small magnetic field, and uncovered a dynamical phase transition in this model, with a “phase” characterized by anomalous local spin relaxation. In easy-plane XXZ spin chains, we found that the spin conductivity diverges at low frequencies, but for special “commensurate” parameters it has a finite limit. These results explain longstanding puzzles regarding the discontinuous parameter dependence of the dc response. Superdiffusive transport can also occur in the Hubbard model, suggesting KPZ scaling may be accessed in near-term experiments with optical lattice Hubbard emulators.

- Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains, S. Gopalakrishnan and R. Vasseur, Phys. Rev. Lett.
**122**, 127202 (2019). - Generalized hydrodynamics, quasiparticle diffusion, and anomalous local relaxation in random integrable spin chains, U. Agrawal, S. Gopalakrishnan and R. Vasseur, Phys. Rev. B
**99**, 174203 (2019). - Anomalous relaxation and the high-temperature structure factor of XXZ spin chains, S. Gopalakrishnan, R. Vasseur, and B. Ware, PNAS
**116**(33) 16250-16255 (2019). - Anomalous low-frequency conductivity in easy-plane XXZ spin chains, U. Agrawal, S. Gopalakrishnan, R. Vasseur and B. Ware, Phys. Rev. B
**101**, 224415 (2020). - Superdiffusion from emergent classical solitons in quantum spin chains, J. De Nardis, S. Gopalakrishnan, E. Ilievski and R. Vasseur, Phys. Rev. Lett.
**125**, 070601 (2020). - Spin crossovers and superdiffusion in the one-dimensional Hubbard model, M. Fava, B. Ware, S. Gopalakrishnan, R. Vasseur and S. Parameswaran, Phys. Rev. B
**102**, 115121 (2020).

**Hydrodynamics of weak integrability breaking**

Integrable systems have infinitely many conserved quantities, which spread ballistically in general. When integrability is broken, only a few of these conserved quantities survive. The remaining conserved quantities are generically transported diffusively. My group, together with S. Gopalakrishnan, has recently proposed a hydrodynamical framework to describe this crossover between generalized to conventional hydrodynamics in nearly integrable systems.

**Non-equilibrium steady-states and hydrodynamics **

Quantum integrable systems, such as the interacting Bose gas in one dimension and the XXZ quantum spin chain, have an extensive number of local conserved quantities that endow them with exotic thermalization and transport properties. We studied numerically and analytically non-equilibrium steady states in such systems involving ballistic heat and spin currents, and showed that in some cases, they can be described using an emergent thermodynamic description. We also provided a general solution (both numerically and analytically) of the “Bethe-Boltzmann” hydrodynamic equations that describe the evolution in time of integrable quantum systems from local to global equilibrium. The solution of these kinetic equations allowed us to model experiments on ultracold one-dimensional Bose gases.

*Expansion potentials for exact far-from-equilibrium spreading of particles and energy*, R. Vasseur, C. Karrasch and J.E. Moore, Phys. Rev. Lett.**115**, 267201 (2015).*Nonequilibrium quantum dynamics and transport: from integrability to**many-body localization*, R. Vasseur and J.E. Moore, J. Stat. Mech. (2016) 064010.*Bethe-Boltzmann Hydrodynamics and Spin Transport in the XXZ spin chain*, V.B. Bulchandani, R. Vasseur, C. Karrasch and J.E. Moore, Phys. Rev. B**97**, 045407 (2018).*Solvable Hydrodynamics of Quantum Integrable Systems*, V.B. Bulchandani, R. Vasseur, C. Karrasch and J.E. Moore, Phys. Rev. Lett.**119**, 220604 (2017).- Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Sarang Gopalakrishnan, David A. Huse, Vedika Khemani, and Romain Vasseur, Phys. Rev B
**98**, 220303(R) (2018).

**Quenching topological edge modes**

Quantum quenches – the abrupt change of a control parameter in a quantum system – provide a useful way to study the non-equilibrium physics of quantum systems and reveal the existence of new phenomena (dynamical transitions, non-thermal steady-states, prethermalization *etc*.). We are interested in finding new ways to probe the protected edge modes of topological insulators and superconductors using local quantum quenches, as an alternative to usual transport-based methods. The peculiar nature of topological edges leads to a remarkably robust universal post-quench dynamics, with clear non-equilibrium signatures that could be accessed in optical absorption experiments. This applies in particular to Majorana zero modes that emerge in certain types of topological superconducting materials. We found that the peculiar nature of Majorana zero modes leads to signatures in the dynamics following a quantum quench that are completely universal and robust against details of the setup.

**Local quantum quenches**

The study of two one-dimensional systems connected by some sort of interaction has become paradigmatic in modern quantum physics. It plays a particularly important role in the context of local quenches, transport through quantum dots, and the dynamics of magnetic impurities. An essential feature of these systems is the existence of crossover scales, which play a role similar to the Kondo temperature in the Kondo problem, and qualitatively separate weak and strong coupling regimes. Using the underlying integrability of such quantum impurity systems, we proposed new analytical techniques to study this dynamical crossover in non-equilibrium setups.

*Crossover physics in the non-equilibrium dynamics of quenched quantum impurity systems*, Romain Vasseur, Kien Trinh, Stephan Haas and Hubert Saleur. Phys. Rev. Lett.**110**, 240601 (2013).*Universal quench dynamics of interacting quantum impurity systems*, D.M. Kennes, V. Meden and R. Vasseur, Phys. Rev. B**90**, 115101 (2014).*Universal Entanglement Dynamics following a Local Quench*, R. Vasseur and H. Saleur, SciPost Phys. 3, 001 (2017).