Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents

Phantom energy in the nonlinear response of a quantum many-body scar state

Quantum many-body scars are notable as nonthermal, low-entanglement states that exist at high energies. Here, we use attractively interacting dysprosium gases to create scar states that are stable enough to be driven into a strongly nonlinear regime while retaining their character. We measure how the kinetic and total energies evolve after quenching the confining potential. Although the bare interactions are attractive, the atoms behave as if they repel each other: Their kinetic energy paradoxically decreases as the gas is compressed. The missing "phantom" energy is quantified by benchmarking our experimental results against generalized hydrodynamics calculations. We present evidence that the missing kinetic energy is carried by undetected, very high-momentum atoms.

New Classical Integrable Systems from Generalized TT[over ¯]-Deformations

We introduce and study a novel class of classical integrable many-body systems obtained by generalized TT[over ¯] deformations of free particles. Deformation terms are bilinears in densities and currents for the continuum of charges counting asymptotic particles of different momenta. In these models, which we dub "semiclassical Bethe systems" for their link with the dynamics of Bethe ansatz wave packets, many-body scattering processes are factorized, and two-body scattering shifts can be set to an almost arbitrary function of momenta. The dynamics is local but inherently different from that of known classical integrable systems. At short scales, the geometry of the deformation is dynamically resolved: either particles are slowed down (more space available), or accelerated via a novel classical particle-pair creation and annihilation process (less space available). The thermodynamics both at finite and infinite volumes is described by the equations of (or akin to) the thermodynamic Bethe ansatz, and at large scales generalized hydrodynamics emerge.

Navier-Stokes Equations for Low-Temperature One-Dimensional Quantum Fluids

We consider one-dimensional interacting quantum fluids, such as the Lieb-Liniger gas. By computing the low-temperature limit of its (generalized) hydrodynamics we show how in this limit the gas is well described by a conventional viscous (Navier-Stokes) hydrodynamics for density, fluid velocity, and the local temperature, and the other generalized temperatures in the case of integrable gases. The dynamic viscosity is proportional to temperature and can be expressed in a universal form only in terms of the emergent Luttinger liquid parameter K and its density. We show that the heating factor is finite even in the zero temperature limit, which implies that viscous contribution remains relevant also at zero temperatures. Moreover, we find that in the semiclassical limit of small couplings, kinematic viscosity diverges, reconciling with previous observations of Kardar-Parisi-Zhang fluctuations in mean-field quantum fluids.

Iterative Construction of Conserved Quantities in Dissipative Nearly Integrable Systems

Integrable systems offer rare examples of solvable many-body problems in the quantum world. Because of the fine-tuned structure, their realization in nature and experiment is never completely accurate, and therefore effects of integrability are observed only transiently. One way to overcome this limitation is to weakly couple nearly integrable systems to baths and driving: this will stabilize integrable effects up to arbitrary time and encode them in the stationary state approximated by a generalized Gibbs ensemble. However, the description of such driven dissipative nearly integrable models is challenging and no exact analytical methods have been proposed so far. Here, we develop an iterative scheme in which integrability breaking perturbations (baths) determine the conserved quantities that play the leading role in a highly efficient truncated generalized Gibbs ensemble description. Our scheme paves the way for easier calculations in thermodynamically large systems and can be used to construct unknown conserved quantities.

Soliton gas: Theory, numerics, and experiments

The concept of soliton gas was introduced in 1971 by Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted (rarefied) soliton gas, solitons with random amplitude and phase parameters are almost nonoverlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schrödinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials, and generalized Gibbs ensembles are introduced and various open questions and future challenges are discussed.

Tunable Superdiffusion in Integrable Spin Chains Using Correlated Initial States

Although integrable spin chains host only ballistically propagating particles, they can still feature diffusive charge transfer. This diffusive charge transfer originates from quasiparticle charge fluctuations inherited from the initial state's magnetization Gaussian fluctuations. We show that ensembles of initial states with quasi-long-range correlations lead to superdiffusive charge transfer with a tunable dynamical exponent. We substantiate our prediction with numerical simulations and discuss how finite time and finite size effects might cause deviations.

Dynamics of magnetization at infinite temperature in a Heisenberg spin chain

Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the one-dimensional Heisenberg model were conjectured as to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we studied the probability distribution of the magnetization transferred across the chain's center, [Formula: see text]. The first two moments of [Formula: see text] show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments ruled out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide insights into universal behavior in quantum systems.

Random Matrix Statistics in Propagating Correlation Fronts of Fermions

We theoretically study propagating correlation fronts in noninteracting fermions on a one-dimensional lattice starting from an alternating state, where the fermions occupy every other site. We find that, in the long-time asymptotic regime, all the moments of dynamical fluctuations around the correlation fronts are described by the universal correlation functions of Gaussian orthogonal and symplectic random matrices at the soft edge. Our finding here sheds light on a hitherto unknown connection between random matrix theory and correlation propagation in quantum dynamics.

Domain Wall Dynamics in Classical Spin Chains: Free Propagation, Subdiffusive Spreading, and Soliton Emission

The nonequilibrium dynamics of domain wall initial states in a classical anisotropic Heisenberg chain exhibits a striking coexistence of apparently linear and nonlinear behaviors: the propagation and spreading of the domain wall can be captured quantitatively by linear, i.e., noninteracting, spin wave theory absent its usual justifications; while, simultaneously, for a wide range of easy-plane anisotropies, emission can take the place of stable solitons-a process and objects intrinsically associated with interactions and nonlinearities. The easy-axis domain wall only has transient dynamics, the isotropic one broadens diffusively, while the easy-plane one yields a pair of ballistically counterpropagating domain walls which, unusually, broaden subdiffusively, their width scaling as t^{1/3}.

Exact Large-Scale Fluctuations of the Phase Field in the Sine-Gordon Model

We present the first exact theory and analytical formulas for the large-scale phase fluctuations in the sine-Gordon model, valid in all regimes of the field theory, for arbitrary temperatures and interaction strengths. Our result is based on the ballistic fluctuation theory combined with generalized hydrodynamics, and can be seen as an exact "dressing" of the phenomenological soliton-gas picture first introduced by Sachdev and Young [Phys. Rev. Lett. 78, 2220 (1997)PRLTAO0031-900710.1103/PhysRevLett.78.2220], to the modes of generalized hydrodynamics. The resulting physics of phase fluctuations in the sine-Gordon model is qualitatively different, as the stable quasiparticles of integrability give coherent ballistic propagation instead of diffusive spreading. We provide extensive numerical checks of our analytical predictions within the classical regime of the field theory by using Monte Carlo methods. We discuss how our results are of ready applicability to experiments on tunnel-coupled quasicondensates.

All Local Conserved Quantities of the One-Dimensional Hubbard Model

We present the exact expression for all local conserved quantities of the one-dimensional Hubbard model. We identify the operator basis constructing the local charges and find that nontrivial coefficients appear in the higher-order charges. We derive the recursion equation for these coefficients, and some of them are explicitly given. There are no other local charges independent of those we obtained.

Unusual ergodic and chaotic properties of trapped hard rods

We investigate ergodicity, chaos, and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly nonergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.

Nonlinear Fluctuating Hydrodynamics for Kardar-Parisi-Zhang Scaling in Isotropic Spin Chains

Finite temperature spin transport in integrable isotropic spin chains is known to be superdiffusive, with dynamical spin correlations that are conjectured to fall into the Kardar-Parisi-Zhang (KPZ) universality class. However, integrable spin chains have time-reversal and parity symmetries that are absent from the KPZ (Kardar-Parisi-Zhang) or stochastic Burgers equation, which force higher-order spin fluctuations to deviate from standard KPZ predictions. We put forward a nonlinear fluctuating hydrodynamic theory consisting of two coupled stochastic modes: the local spin magnetization and its effective velocity. Our theory fully explains the emergence of anomalous spin dynamics in isotropic chains: it predicts KPZ scaling for the spin structure factor but with a symmetric, quasi-Gaussian, distribution of spin fluctuations. We substantiate our results using matrix-product states calculations.

Transport and Entanglement across Integrable Impurities from Generalized Hydrodynamics

Quantum impurity models (QIMs) are ubiquitous throughout physics. As simplified toy models they provide crucial insights for understanding more complicated strongly correlated systems, while in their own right are accurate descriptions of many experimental platforms. In equilibrium, their physics is well understood and have proven a testing ground for many powerful theoretical tools, both numerical and analytical, in use today. Their nonequilibrium physics is much less studied and understood. However, the recent advancements in nonequilibrium integrable quantum systems through the development of generalized hydrodynamics (GHD) coupled with the fact that many archetypal QIMs are in fact integrable presents an enticing opportunity to enhance our understanding of these systems. We take a step towards this by expanding the framework of GHD to incorporate integrable interacting QIMs. We present a set of Bethe-Boltzmann type equations which incorporate the effects of impurity scattering and discuss the new aspects which include entropy production. These impurity GHD equations are then used to study a bipartioning quench wherein a relevant backscattering impurity is included at the location of the bipartition. The density and current profiles are studied as a function of the impurity strength and expressions for the entanglement entropy and full counting statistics are derived.

Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials

We identify a class of trapping potentials in cubic nonlinear Schrödinger equations (NLSEs) that make them nonintegrable, but prevent the emergence of power spectra associated with ergodicity. The potentials are characterized by equidistant energy spectra (e.g., the harmonic-oscillator trap), which give rise to a large number of resonances enhancing the nonlinearity. In a broad range of dynamical solutions, spanning the regimes in which the nonlinearity may be either weak or strong in comparison with the linear part of the NLSE, the power spectra are shaped as narrow (quasidiscrete), evenly spaced spikes, unlike generic truly continuous (ergodic) spectra. We develop an analytical explanation for the emergence of these spectral features in the case of weak nonlinearity. In the strongly nonlinear regime, the presence of such structures is tracked numerically by performing simulations with random initial conditions. Some potentials that prevent ergodicity in this manner are of direct relevance to Bose-Einstein condensates: they naturally appear in 1D, 2D, and 3D Gross-Pitaevskii equations (GPEs), the quintic version of these equations, and a two-component GPE system.

Generalized Thermalization in Quantum-Chaotic Quadratic Hamiltonians

Thermalization (generalized thermalization) in nonintegrable (integrable) quantum systems requires two ingredients: equilibration and agreement with the predictions of the Gibbs (generalized Gibbs) ensemble. We prove that observables that exhibit eigenstate thermalization in single-particle sector equilibrate in many-body sectors of quantum-chaotic quadratic models. Remarkably, the same observables do not exhibit eigenstate thermalization in many-body sectors (we establish that there are exponentially many outliers). Hence, the generalized Gibbs ensemble is generally needed to describe their expectation values after equilibration, and it is characterized by Lagrange multipliers that are smooth functions of single-particle energies.

Real-time broadening of bath-induced density profiles from closed-system correlation functions

The Lindblad master equation is one of the main approaches to open quantum systems. While it has been widely applied in the context of condensed matter systems to study properties of steady states in the limit of long times, the actual route to such steady states has attracted less attention yet. Here, we investigate the nonequilibrium dynamics of spin chains with a local coupling to a single Lindblad bath and analyze the transport properties of the induced magnetization. Combining typicality and equilibration arguments with stochastic unraveling, we unveil for the case of weak driving that the dynamics in the open system can be constructed on the basis of correlation functions in the closed system, which establishes a connection between the Lindblad approach and linear response theory at finite times. In this way, we provide a particular example where closed and open approaches to quantum transport agree strictly. We demonstrate this fact numerically for the spin-1/2 XXZ chain at the isotropic point and in the easy-axis regime, where superdiffusive and diffusive scaling is observed, respectively.

Matrix Product Symmetries and Breakdown of Thermalization from Hard Rod Deformations

We construct families of exotic spin-1/2 chains using a procedure called "hard rod deformation." We treat both integrable and nonintegrable examples. The models possess a large noncommutative symmetry algebra, which is generated by matrix product operators with a fixed small bond dimension. The symmetries lead to Hilbert space fragmentation and to the breakdown of thermalization. As an effect, the models support persistent oscillations in nonequilibrium situations. Similar symmetries have been reported earlier in integrable models, but here we show that they also occur in nonintegrable cases.

Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.

Integrable Digital Quantum Simulation: Generalized Gibbs Ensembles and Trotter Transitions

The Trotter-Suzuki decomposition is a promising avenue for digital quantum simulation (DQS), approximating continuous-time dynamics by discrete Trotter steps of duration τ. Recent work suggested that DQS is typically characterized by a sharp Trotter transition: when τ is increased beyond a threshold value, approximation errors become uncontrolled at large times due to the onset of quantum chaos. Here, we contrast this picture with the case of integrable DQS. We focus on a simple quench from a spin-wave state in the prototypical XXZ Heisenberg spin chain, and study its integrable Trotterized evolution as a function of τ. Because of its exact local conservation laws, the system does not heat up to infinite temperature and the late-time properties of the dynamics are captured by a discrete generalized Gibbs ensemble (dGGE). By means of exact calculations we find that, for small τ, the dGGE depends analytically on the Trotter step, implying that discretization errors remain bounded even at infinite times. Conversely, the dGGE changes abruptly at a threshold value τ_{th}, signaling a novel type of Trotter transition. We show that the latter can be detected locally, as it is associated with the appearance of a nonzero staggered magnetization with a subtle dependence on τ. We highlight the differences between continuous and discrete GGEs, suggesting the latter as novel interesting nonequilibrium states exclusive to digital platforms.

Integrability Breaking from Backscattering

We analyze the onset of diffusive hydrodynamics in the one-dimensional hard-rod gas subject to stochastic backscattering. While this perturbation breaks integrability and leads to a crossover from ballistic to diffusive transport, it preserves infinitely many conserved quantities corresponding to even moments of the velocity distribution of the gas. In the limit of small noise, we derive the exact expressions for the diffusion and structure factor matrices, and show that they generically have off diagonal components. We find that the particle density structure factor is non-Gaussian and singular near the origin, with a return probability showing logarithmic deviations from diffusion.

Observation of hydrodynamization and local prethermalization in 1D Bose gases

Hydrodynamics accurately describe relativistic heavy-ion collision experiments well before local thermal equilibrium is established¹. This unexpectedly rapid onset of hydrodynamics-which takes place on the fastest available timescale-is called hydrodynamization^2-4. It occurs when an interacting quantum system is quenched with an energy density that is much greater than its ground-state energy density^5,6. During hydrodynamization, energy gets redistributed across very different energy scales. Hydrodynamization precedes local equilibration among momentum modes⁵, which is local prethermalization to a generalized Gibbs ensemble^7,8 in nearly integrable systems or local thermalization in non-integrable systems⁹. Although many theories of quantum dynamics postulate local prethermalization^10,11, the associated timescale has not been studied experimentally. Here we use an array of one-dimensional Bose gases to directly observe both hydrodynamization and local prethermalization. After we apply a Bragg scattering pulse, hydrodynamization is evident in the fast redistribution of energy among distant momentum modes, which occurs on timescales associated with the Bragg peak energies. Local prethermalization can be seen in the slower redistribution of occupation among nearby momentum modes. We find that the timescale for local prethermalization in our system is inversely proportional to the momenta involved. During hydrodynamization and local prethermalization, existing theories cannot quantitatively model our experiment. Exact theoretical calculations in the Tonks-Girardeau limit¹² show qualitatively similar features.

Dynamical Fermionization in One-Dimensional Spinor Gases at Finite Temperature

Following the removal of axial confinement the momentum distribution of a Tonks-Girardeau gas approaches that of a system of noninteracting spinless fermions in the initial harmonic trap. This phenomenon, called dynamical fermionization, has been experimentally confirmed in the case of the Lieb-Liniger model and theoretically predicted in the case of multicomponent systems at zero temperature. We prove analytically that for all spinor gases with strong repulsive contact interactions at finite temperature the momentum distribution after release from the trap asymptotically approaches that of a system of spinless fermions at the same temperature but with a renormalized chemical potential which depends on the number of components of the spinor system. In the case of the Gaudin-Yang model we check numerically our analytical predictions using the results obtained from a nonequilibrium generalization of Lenard's formula describing the time evolution of the field-field correlators.

Relaxation of Phonons in the Lieb-Liniger Gas by Dynamical Refermionization

Motivated by recent experiments, we investigate the Lieb-Liniger gas initially prepared in an out-of-equilibrium state that is Gaussian in terms of the phonons, namely whose density matrix is the exponential of an operator quadratic in terms of phonon creation and annihilation operators. Because the phonons are not exact eigenstates of the Hamiltonian, the gas relaxes to a stationary state at very long times whose phonon population is a priori different from the initial one. Thanks to integrability, that stationary state needs not be a thermal state. Using the Bethe-ansatz mapping between the exact eigenstates of the Lieb-Liniger Hamiltonian and those of a noninteracting Fermi gas and bosonization techniques we completely characterize the stationary state of the gas after relaxation and compute its phonon population distribution. We apply our results to the case where the initial state is an excited coherent state for a single phonon mode, and we compare them to exact results obtained in the hard-core limit.

Anomalous relaxation of density waves in a ring-exchange system

We present the analysis of the slowing down exhibited by stochastic dynamics of a ring-exchange model on a square lattice, by means of numerical simulations. We find the preservation of coarse-grained memory of initial state of density-wave types for unexpectedly long times. This behavior is inconsistent with the prediction from a low frequency continuum theory developed by assuming a mean-field solution. Through a detailed analysis of correlation functions of the dynamically active regions, we exhibit an unconventional transient long ranged structure formation in a direction which is featureless for the initial condition, and argue that its slow melting plays a crucial role in the slowing-down mechanism. We expect our results to be relevant also for the dynamics of quantum ring-exchange dynamics of hard-core bosons and more generally for dipole moment conserving models.

Anomalous transport from hot quasiparticles in interacting spin chains

Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticles both in integrable and near-integrable limits, with an emphasis on the role of hydrodynamic fluctuations. We also comment on recent developments including non-linear response, full-counting statistics and far-from-equilibrium transport. We provide an overview of recent numerical and experimental results on transport in XXZ spin chains.

Thermalization of Interacting Quasi-One-Dimensional Systems

Many experimentally relevant systems are quasi-one-dimensional, consisting of nearly decoupled chains. In these systems, there is a natural separation of scales between the strong intrachain interactions and the weak interchain coupling. When the intrachain interactions are integrable, weak interchain couplings play a crucial part in thermalizing the system. Here, we develop a Boltzmann-equation formalism involving a collision integral that is asymptotically exact for any interacting integrable system, and apply it to develop a quantitative theory of relaxation in coupled Bose gases in the experimentally relevant Newton's cradle setup. We find that relaxation involves a broad spectrum of timescales. We provide evidence that the Markov process governing relaxation at late times is gapless; thus, the approach to equilibrium is generally nonexponential, even for spatially uniform perturbations.

Long-lived solitons and their signatures in the classical Heisenberg chain

Motivated by the Kardar-Parisi-Zhang (KPZ) scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well known to be nonintegrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with ln(1+S_{i}·S_{j}) interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons have a long-lived Heisenberg counterpart when they are not too narrow and not too fast moving. Finally, we demonstrate their presence in thermal states of the Heisenberg chain, even when the typical soliton width is larger than the spin correlation length, and argue that these excitations likely underlie the KPZ scaling.

New trends in quantum integrability: recent experiments with ultracold atoms

Over the past two decades quantum engineering has made significant advances in our ability to create genuine quantum many-body systems using ultracold atoms. In particular, some prototypical exactly solvable Yang-Baxter systems have been successfully realized allowing us to confront elegant and sophisticated exact solutions of these systems with their experimental counterparts. The new experimental developments show a variety of fundamental one-dimensional (1D) phenomena, ranging from the generalized hydrodynamics to dynamical fermionization, Tomonaga-Luttinger liquids, collective excitations, fractional exclusion statistics, quantum holonomy, spin-charge separation, competing orders with high spin symmetry and quantum impurity problems. This article briefly reviews these developments and provides rigorous understanding of those observed phenomena based on the exact solutions while highlighting the uniqueness of 1D quantum physics. The precision of atomic physics realizations of integrable many-body problems continues to inspire significant developments in mathematics and physics while at the same time offering the prospect to contribute to future quantum technology.

Nonlinear Transport by Bethe Bound States

We consider nonlinear ballistic spin transport in the XXZ spin chain and derive an analytical result for the nonlinear Drude weight D^{(3)} at infinite temperatures. In contrast to the linear Drude weight D^{(1)}, we find that the result not only depends on anisotropy but also on the string length of the quasiparticles transporting the spin current. Our result provides further insights into transport by quasiparticles and raises questions about Luttinger liquid universality.

Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion

The Kardar-Parisi-Zhang (KPZ) universality class describes the coarse-grained behavior of a wealth of classical stochastic models. Surprisingly, KPZ universality was recently conjectured to also describe spin transport in the one-dimensional quantum Heisenberg model. We tested this conjecture by experimentally probing transport in a cold-atom quantum simulator via the relaxation of domain walls in spin chains of up to 50 spins. We found that domain-wall relaxation is indeed governed by the KPZ dynamical exponent z = 3/2 and that the occurrence of KPZ scaling requires both integrability and a nonabelian SU(2) symmetry. Finally, we leveraged the single-spin-sensitive detection enabled by the quantum gas microscope to measure an observable based on spin-transport statistics. Our results yield a clear signature of the nonlinearity that is a hallmark of KPZ universality.

Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy and Generalized Hydrodynamics

We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally nonlocal and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how generalized hydrodynamics emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that generalized hydrodynamics in the weak interaction regime is robust under nonintegrable perturbations.

Fragmentation and Emergent Integrable Transport in the Weakly Tilted Ising Chain

We investigate emergent quantum dynamics of the tilted Ising chain in the regime of a weak transverse field. Within the leading order perturbation theory, the Hilbert space is fragmented into exponentially many decoupled sectors. We find that the sector made of isolated magnons is integrable with dynamics being governed by a constrained version of the XXZ spin Hamiltonian. As a consequence, when initiated in this sector, the Ising chain exhibits ballistic transport on unexpectedly long timescales. We quantitatively describe its rich phenomenology employing exact integrable techniques such as generalized hydrodynamics. Finally, we initiate studies of integrability-breaking magnon clusters whose leading-order transport is activated by scattering with surrounding isolated magnons.

Macroscopic Effects of Localized Measurements in Jammed States of Quantum Spin Chains

A quantum jammed state can be seen as a state where the phase space available to particles shrinks to zero, an interpretation quite accurate in integrable systems, where stable quasiparticles scatter elastically. We consider the integrable dual folded XXZ model, which is equivalent to the XXZ model in the limit of large anisotropy. We perform a jamming-breaking localized measurement in a jammed state. We find that jamming is locally restored, but local observables exhibit nontrivial time evolution on macroscopic, ballistic scales, without ever relaxing back to their initial values.

Global Quenches after Localized Perturbations

We investigate the effect of a single spin flip preceding a global quench between translationally invariant local Hamiltonians in spin-1/2 chains. The effect of the localized perturbation does not fade away however large the distance from the perturbation is. In particular, translational invariance is not restored and the infinite-time limit depends on whether the spin was flipped or not. We argue that this phenomenon is more general than the particular example considered and we conjecture that it is triggered by topological properties, specifically, the existence of "semilocal charges."

Out-of-Time-Ordered Crystals and Fragmentation

Is a spontaneous perpetual reversal of the arrow of time possible? The out-of-time-ordered correlator (OTOC) is a standard measure of irreversibility, quantum scrambling, and the arrow of time. The question may be thus formulated more precisely and conveniently: can spatially ordered perpetual OTOC oscillations exist in many-body systems? Here we give a rigorous lower bound on the amplitude of OTOC oscillations in terms of a strictly local dynamical algebra allowing for identification of systems that are out-of-time-ordered (OTO) crystals. While OTOC oscillations are possible for few-body systems, due to the spatial order requirement OTO crystals cannot be achieved by effective single or few body dynamics, e.g., a pendulum or a condensate. Rather they signal perpetual motion of quantum scrambling. It is likewise shown that if a Hamiltonian satisfies this novel algebra, it has an exponentially large number of local invariant subspaces, i.e., Hilbert space fragmentation. Crucially, the algebra, and hence the OTO crystal, are stable to local unitary and dissipative perturbations. A Creutz ladder is shown to be an OTO crystal, which thus perpetually reverses its arrow of time.

Absence of Normal Fluctuations in an Integrable Magnet

We investigate dynamical fluctuations of transferred magnetization in the one-dimensional lattice Landau-Lifshitz magnet with uniaxial anisotropy, representing an emblematic model of interacting spins. We demonstrate that the structure of fluctuations in thermal equilibrium depends radically on the characteristic dynamical scale. In the ballistic regime, typical fluctuations are found to follow a normal distribution and scaled cumulants are finite. In stark contrast, on the diffusive and superdiffusive timescales, relevant, respectively, for the easy-axis and isotropic magnet at vanishing total magnetization, typical fluctuations are no longer Gaussian and, remarkably, scaled cumulants are divergent. The observed anomalous features disappear upon breaking integrability, suggesting that the absence of normal fluctuations is intimately tied to the presence of soliton modes. In a nonequilibrium setting of the isotropic magnet with weakly polarized step-profile initial state we find a slow drift of dynamical exponent from the superdiffusive towards the diffusive value.

Duality between Weak and Strong Interactions in Quantum Gases

In one-dimensional quantum gases there is a well known "duality" between hard core bosons and noninteracting fermions. However, at the field theory level, no exact duality connecting strongly interacting bosons to weakly interacting fermions is known. Here we propose a solution to this long-standing problem. Our derivation relies on regularizing the only pointlike interaction between fermions in one dimension that induces a discontinuity in the wave function proportional to its derivative. In contrast to all known regularizations our potential is weak for small interaction strengths. Crucially, this allows one to apply standard methods of diagrammatic perturbation theory to strongly interacting bosons. As a first application we compute the finite temperature spectral function of the Cheon-Shigehara model, the fermionic model dual to the celebrated Lieb-Liniger model.

Integrable spin chain with Hilbert space fragmentation and solvable real-time dynamics

We revisit the so-called folded XXZ model, which was treated earlier by two independent research groups. We argue that this spin-1/2 chain is one of the simplest quantum integrable models, yet it has quite remarkable physical properties. The particles have constant scattering lengths, which leads to a simple treatment of the exact spectrum and the dynamics of the system. The Hilbert space of the model is fragmented, leading to exponentially large degeneracies in the spectrum, such that the exponent depends on the particle content of a given state. We provide an alternative derivation of the Hamiltonian and the conserved charges of the model, including an alternative interpretation of the so-called "dual model" considered earlier. We also construct a nonlocal map that connects the model with the Maassarani-Mathieu spin chain, also known as the SU(3) XX model. We consider the exact solution of the model with periodic and open boundary conditions, and also derive multiple descriptions of the exact thermodynamics of the model. We consider quantum quenches of different types. In one class of problems the dynamics can be treated relatively easily: we compute an example for the real-time dependence of a local observable. In another class of quenches the degeneracies of the model lead to the breakdown of equilibration, and we argue that they can lead to persistent oscillations. We also discuss connections with the TT[over ¯] and hard rod deformations known from quantum field theories.

Nonequilibrium Dynamics and Weakly Broken Integrability

Motivated by dynamical experiments on cold atomic gases, we develop a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. Using the exact matrix elements of the underlying integrable model, we establish an analytical approach to real-time dynamics. The method addresses a broad range of timescales, from the intermediate regime of prethermalization to late-time thermalization. Predictions are given for the time evolution of physical quantities, including effective temperatures and thermalization rates. The approach provides conceptual links between perturbed quantum many-body dynamics and classical Kolmogorov-Arnold-Moser theory. In particular, we identify a family of perturbations which do not cause thermalization in the weakly perturbed regime.

Hydrodynamic nonlinear response of interacting integrable systems

We develop a formalism for computing the nonlinear response of interacting integrable systems. Our results are asymptotically exact in the hydrodynamic limit where perturbing fields vary sufficiently slowly in space and time. We show that spatially resolved nonlinear response distinguishes interacting integrable systems from noninteracting ones, exemplifying this for the Lieb-Liniger gas. We give a prescription for computing finite-temperature Drude weights of arbitrary order, which is in excellent agreement with numerical evaluation of the third-order response of the XXZ spin chain. We identify intrinsically nonperturbative regimes of the nonlinear response of integrable systems.

Spatiotemporal Crossover between Low- and High-Temperature Dynamical Regimes in the Quantum Heisenberg Magnet

The stranglehold of low temperatures on fascinating quantum phenomena in one-dimensional quantum magnets has been challenged recently by the discovery of anomalous spin transport at high temperatures. Whereas both regimes have been investigated separately, no study has attempted to reconcile them. For instance, the paradigmatic quantum Heisenberg spin-1/2 chain falls at low temperature within the Tomonaga-Luttinger liquid framework, while its high-temperature dynamics is superdiffusive and relates to the Kardar-Parisi-Zhang universality class in 1+1 dimensions. This Letter aims at reconciling the two regimes. Building on large-scale matrix product state simulations, we find that they are connected by a temperature-dependent spatiotemporal crossover. As the temperature T is reduced, we show that the onset of superdiffusion takes place at longer length and timescales ∝1/T. This prediction has direct consequences for experiments including nuclear magnetic resonance: it is consistent with earlier measurements on the nearly ideal Heisenberg S=1/2 chain compound Sr_{2}CuO_{3}, yet calls for new and dedicated experiments.

Emergent hydrodynamics in a strongly interacting dipolar spin ensemble

Conventional wisdom holds that macroscopic classical phenomena naturally emerge from microscopic quantum laws^1-7. However, despite this mantra, building direct connections between these two descriptions has remained an enduring scientific challenge. In particular, it is difficult to quantitatively predict the emergent 'classical' properties of a system (for example, diffusivity, viscosity and compressibility) from a generic microscopic quantum Hamiltonian^7-14. Here we introduce a hybrid solid-state spin platform, where the underlying disordered, dipolar quantum Hamiltonian gives rise to the emergence of unconventional spin diffusion at nanometre length scales. In particular, the combination of positional disorder and on-site random fields leads to diffusive dynamics that are Fickian yet non-Gaussian^15-20. Finally, by tuning the underlying parameters within the spin Hamiltonian via a combination of static and driven fields, we demonstrate direct control over the emergent spin diffusion coefficient. Our work enables the investigation of hydrodynamics in many-body quantum spin systems.

Generalized hydrodynamics in strongly interacting 1D Bose gases

[Figure: see text].

Stability of Superdiffusion in Nearly Integrable Spin Chains

Superdiffusive finite-temperature transport has been recently observed in a variety of integrable systems with non-Abelian global symmetries. Superdiffusion is caused by giant Goldstone-like quasiparticles stabilized by integrability. Here, we argue that these giant quasiparticles remain long-lived and give divergent contributions to the low-frequency conductivity σ(ω), even in systems that are not perfectly integrable. We find, perturbatively, that σ(ω)∼ω^{-1/3} for translation-invariant static perturbations that conserve energy and σ(ω)∼|logω| for noisy perturbations. The (presumable) crossover to regular diffusion appears to lie beyond low-order perturbation theory. By contrast, integrability-breaking perturbations that break the non-Abelian symmetry yield conventional diffusion. Numerical evidence supports the distinction between these two classes of perturbations.

Dissipative Hot-Spot-Enabled Shock and Bounce Dynamics via Terahertz Quantum Quenches in Helical Edge States

We study quantum quenches of helical liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized "hot spots" that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to frequency doubling that could be detected experimentally from emitted terahertz radiation.

Anomalous Hydrodynamics in a One-Dimensional Electronic Fluid

We construct multimode viscous hydrodynamics for one-dimensional spinless electrons. Depending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broad regime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydrodynamic modes leads to anomalous scaling of physical observables and waves propagating in the fluid. In the four-mode regime, all modes are ballistic and acquire Kardar-Parisi-Zhang (KPZ)-like broadening with asymmetric power-law tails. "Heads" and "tails" of the waves contribute equally to thermal conductivity, leading to ω^{-1/3} scaling of its real part. In the three-mode regime, the system is in the universality class of a classical viscous fluid [O. Narayan and S. Ramaswamy, Anomalous Heat Conduction in One-Dimensional Momentum-Conserving Systems, Phys. Rev. Lett. 89, 200601 (2002).PRLTAO0031-900710.1103/PhysRevLett.89.200601, H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014).JSTPBS0022-471510.1007/s10955-014-0933-y]. Self-interaction of the sound modes results in a KPZ-like shape, while the interaction with the heat mode results in asymmetric tails. The heat mode is governed by Levy flight distribution, whose power-law tails give rise to ω^{-1/3} scaling of heat conductivity.

Constructing Integrable Lindblad Superoperators

We develop a new method for the construction of one-dimensional integrable Lindblad systems, which describe quantum many body models in contact with a Markovian environment. We find several new models with interesting features, such as annihilation-diffusion processes, a mixture of coherent and classical particle propagation, and a rectified steady state current. We also find new ways to represent known classical integrable stochastic equations by integrable Lindblad operators. Our method can be extended to various other situations and it establishes a structured approach to the study of solvable open quantum systems.

Exact Real-Time Longitudinal Correlation Functions of the Massive XXZ Chain

We apply a recently developed thermal form factor expansion method to evaluate the real-time longitudinal spin-spin correlation functions of the spin-1/2 XXZ chain in the antiferromagnetically ordered regime at zero temperature. An analytical result incorporating all types of excitations in the model is obtained, without any approximations. This allows for the accurate calculation of the real-time correlations in this strongly interacting quantum system for arbitrary distances and times.

Soliton gas in bidirectional dispersive hydrodynamics

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs. We construct kinetic equations for both types of bidirectional soliton gases and solve the respective shock-tube problems for the collision of two "monochromatic" soliton beams consisting of solitons of approximately the same amplitude and velocity. The corresponding weak solutions of the kinetic equations consisting of differing uniform states separated by contact discontinuities for the mean flow are constructed. Concrete examples of bidirectional Eulerian soliton gases for the defocusing nonlinear Schrödinger (NLS) equation and the resonant NLS equation are considered. The kinetic equation of the resonant NLS soliton gas is shown to be equivalent to that of the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq equations. The analytical results for shock-tube Riemann problems for bidirectional soliton gases are shown to be in excellent agreement with direct numerical simulations.