Weak Integrability Breaking: Chaos with Integrability Signature in Coherent Diffusion.

We study how perturbations affect dynamics of integrable many-body quantum systems, causing transition from integrability to chaos. Looking at spin transport in the Heisenberg chain with impurities we find that in the thermodynamic limit transport gets diffusive already at an infinitesimal perturbation. Small extensive perturbations therefore cause an immediate transition from integrability to chaos. Nevertheless, there is a remnant of integrability encoded in the dependence of the diffusion constant on the impurity density, namely, at small densities it is proportional to the square root of the inverse density, instead of to the inverse density as would follow from Matthiessen's rule. We show that Matthiessen's rule has to be modified in nonballistic systems. Results also highlight a nontrivial role of interacting scattering on a single impurity, and that there is a regime where adding more impurities can actually increase transport.

Exact Localized and Ballistic Eigenstates in Disordered Chaotic Spin Ladders and the Fermi-Hubbard Model.

We demonstrate the existence of exact atypical many-body eigenstates in a class of disordered, interacting one-dimensional quantum systems that includes the Fermi-Hubbard model as a special case. These atypical eigenstates, which generically have finite energy density and are exponentially many in number, are populated by noninteracting excitations. They can exhibit Anderson localization with area-law eigenstate entanglement or, surprisingly, ballistic transport at any disorder strength. These properties differ strikingly from those of typical eigenstates nearby in energy, which we show give rise to diffusive transport as expected in a chaotic quantum system. We discuss how to observe these atypical eigenstates in cold-atom experiments realizing the Fermi-Hubbard model, and comment on the robustness of their properties.

Kardar-Parisi-Zhang Physics in the Quantum Heisenberg Magnet.

Equilibrium spatiotemporal correlation functions are central to understanding weak nonequilibrium physics. In certain local one-dimensional classical systems with three conservation laws they show universal features. Namely, fluctuations around ballistically propagating sound modes can be described by the celebrated Kardar-Parisi-Zhang (KPZ) universality class. Can such a universality class be found also in quantum systems? By unambiguously demonstrating that the KPZ scaling function describes magnetization dynamics in the SU(2) symmetric Heisenberg spin chain we show, for the first time, that this is so. We achieve that by introducing new theoretical and numerical tools, and make a puzzling observation that the conservation of energy does not seem to matter for the KPZ physics.

Interaction instability of localization in quasiperiodic systems.

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.

Spin diffusion from an inhomogeneous quench in an integrable system.

Generalized hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg XXZ spin 1/2 chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetization and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to 2/3, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.

Fractality in nonequilibrium steady states of quasiperiodic systems.

We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.

Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System.

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects, we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.

Dissipative Remote-State Preparation in an Interacting Medium.

Standard quantum state preparation methods work by preparing a required state locally and then distributing it to a distant location by a free-space propagation. We instead study procedures of preparing a target state at a remote location in the presence of an interacting background medium on which no control is required, manipulating only local dissipation. In mathematical terms, we characterize a set of reduced steady states stabilizable by local dissipation. An explicit local method is proposed by which one can construct a wanted one-site reduced steady state at an arbitrary remote site in a lattice of any size and geometry. In the chain geometry we also prove uniqueness of such a steady state. We demonstrate that the convergence time to fixed precision is smaller than the inverse gap, and we study robustness of the scheme in different medium interactions.

Relaxation times of dissipative many-body quantum systems.

We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than ∼1/L. In integrable systems with boundary dissipation one typically observes scaling of ∼1/L(3), while in chaotic ones one can have faster relaxation with the gap scaling as ∼1/L and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.

Large-deviation statistics of a diffusive quantum spin chain and the additivity principle.

Using the large-deviation formalism, we study the statistics of current fluctuations in a diffusive nonequilibrium quantum spin chain. The boundary-driven XX chain with dephasing consists of a coherent bulk hopping and a local dissipative dephasing. We analytically calculate the exact expression for the second current moment in a system of any length and then numerically demonstrate that in the thermodynamic limit, higher-order cumulants and the large-deviation function can be calculated using the additivity principle or macroscopic hydrodynamic theory. This shows that the additivity principle can also hold in systems that are not purely stochastic, and can in particular be valid in quantum systems. We also show that in large systems, the current fluctuations are the same as in the classical symmetric simple exclusion process.

Exact large-deviation statistics for a nonequilibrium quantum spin chain.

We consider a one-dimensional XX spin chain in a nonequilibrium setting with a Lindblad-type boundary driving. By calculating large-deviation rate function in the thermodynamic limit, a generalization of free energy to a nonequilibrium setting, we obtain a complete distribution of current, including closed expressions for lower-order cumulants. We also identify two phase-transition-like behaviors in either the thermodynamic limit, at which the current probability distribution becomes discontinuous, or at maximal driving, when the range of possible current values changes discontinuously. In the thermodynamic limit the current has a finite upper and lower bound. We also explicitly confirm nonequilibrium fluctuation relation and show that the current distribution is the same under mapping of the coupling strength Γ→1/Γ.

Eigenvalue statistics as an indicator of integrability of nonequilibrium density operators.

We propose to quantify the complexity of nonequilibrium steady state density operators, as well as of long-lived Liouvillian decay modes, in terms of the level spacing distribution of their spectra. Based on extensive numerical studies in a variety of models, some solvable and some unsolved, we conjecture that the integrability of density operators (e.g., the existence of an algebraic procedure for their construction in finitely many steps) is signaled by a Poissonian level statistics, whereas in the generic nonintegrable cases one finds level statistics of a Gaussian unitary ensemble of random matrices. Eigenvalue statistics can therefore be used as an efficient tool to identify integrable quantum nonequilibrium systems.

Excitation energy transfer efficiency: equivalence of transient and stationary setting and the absence of non-Markovian effects.

We analyze efficiency of excitation energy transfer in photosynthetic complexes in transient and stationary setting. In the transient setting, the absorption process is modeled as an individual event resulting in a subsequent relaxation dynamics. In the stationary setting the absorption is a continuous stationary process, leading to the nonequilibrium steady state. We show that, as far as the efficiency is concerned, both settings can be considered to be the same, as they result in almost identical efficiency. We also show that non-Markovianity has no effect on the resulting efficiency, i.e., corresponding Markovian dynamics results in identical efficiency. Even more, if one maps dynamics to appropriate classical rate equations, the same efficiency as in quantum case is obtained.

Coexistence of diffusive and ballistic transport in a simple spin ladder.

We show that in a nonintegrable spin ladder system with the XX type of coupling along the legs and the XXZ type along the rungs there are invariant subspaces that support ballistic magnetization transport. In the complementary subspace the transport is found to be diffusive. This shows that (i) quantum chaotic systems can possess ballistic subspaces, and (ii) diffusive and ballistic transport modes can coexist in a rather simple nonintegrable model. In the limit of an infinite anisotropy in rungs the system studied is equivalent to the one-dimensional Hubbard model.

Transport properties of a boundary-driven one-dimensional gas of spinless fermions.

We analytically study a system of spinless fermions driven at the boundary with an oscillating chemical potential. Various transport regimes can be observed: At zero driving frequency the particle current through the system is independent of the system's length; at the phase-transition frequency, being equal to the bandwidth, the current decays as ~n(-α) with the chain length n, α being either 2 or 3; below the transition the scaling of the current is ~n(-1/2), indicating anomalous transport, while it is exponentially small ~exp(-n/2ξ) above the transition. Therefore, by a simple change of frequency of the a.c. driving of the system one can vary transport from ballistic, anomalous to insulating.

Non-Markovian behavior of small and large complex quantum systems.

The channel induced by a complex system interacting strongly with a qubit is calculated exactly under the assumption of randomness of its eigenvectors. The resulting channel is represented as an isotropic time-dependent oscillation of the Bloch ball, leading to non-Markovian behavior, even in the limit of infinite environments. Two contributions are identified: one due to the density of states and the other due to correlations in the spectrum. Prototype examples, one for chaotic and the other for regular dynamics are explored.

Spin transport in a one-dimensional anisotropic Heisenberg model.

We analytically and numerically study spin transport in a one-dimensional Heisenberg model in linear-response regime at infinite temperature. It is shown that as the anisotropy parameter Δ is varied spin transport changes from ballistic for Δ<1 to anomalous at the isotropic point Δ=1, to diffusive for finite Δ>1, ending up as a perfect isolator in the Ising limit of infinite Δ. Using perturbation theory for large Δ a quantitative prediction is made for the dependence of diffusion constant on Δ.

Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation.

We study a one-dimensional XX chain under nonequilibrium driving and local dephasing described by the Lindblad master equation. The analytical solution for the nonequilibrium steady state found for particular parameters in a previous study [M. Znidaric J. Stat. Mech. (2010) L05002] is extended to arbitrary coupling constants, driving, and a homogeneous magnetic field. All one-, two-, and three-point correlation functions are explicitly evaluated. It is shown that the nonequilibrium stationary state is not Gaussian. Nevertheless, in the thermodynamic and weak-driving limit it is only weakly correlated and can be described by a matrix product operator ansatz with matrices of fixed dimension 4. A nonequilibrium phase transition at zero dephasing is also discussed. It is suggested that the scaling of the relaxation time with the system size can serve as a signature of a nonequilibrium phase transition.

Thermalization and ergodicity in one-dimensional many-body open quantum systems.

Using an approach based on the time-dependent density-matrix renormalization-group method, we study the thermalization in spin chains locally coupled to an external bath. Our results provide evidence that quantum chaotic systems do thermalize, that is, they exhibit relaxation to an invariant ergodic state which, in the bulk, is well approximated by the grand canonical state. Moreover, the resulting ergodic state in the bulk does not depend on the details of the baths. On the other hand, for integrable systems we found that the invariant state in general depends on the bath and is different from the grand canonical state.

Long-range order in nonequilibrium interacting quantum spin chains.

We conjecture that nonequilibrium boundary conditions generically trigger long-range order in nonequilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin-1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long-range spin-spin correlations in an integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long-range order also typically emerges after breaking the integrability of the model.