Chaos and Dynamics in Correlated Quantum Matter

The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as $A$ and $B$, is computed analytically using a Coulomb gas method. It is shown that this probability, for large $N$, goes as $\exp[-\beta N^2\Phi(\zeta)]$, where the parameter $\beta$ is the Dyson index of the ensemble, $\zeta$ is the large deviation parameter, while the rate function $\Phi(\zeta)$ is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems $1$ and $2$, obtained by further partitioning the subsystem $A$, using the properties of the density matrix’s partial transpose $\rho_{12}^\Gamma$. The density of states of $\rho_{12}^\Gamma$ is found to be close to the Wigner’s semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter $\rho_{12}^\Gamma$. Log negativity is used to quantify the entanglement between subsystems $1$ and $2$. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.

Many-body interference occurs as a fundamental process during the evolution of a quantum system consisting of two or more indistinguishable particles. The (measurable) consequences of this interference, as a function of the particles’ mutual indistinguishability, was studied for non-interacting photons transmitted through beam-splitter arrays. However, the role of many-body interference in the dynamics of interacting particles, e.g. cold atoms in optical lattices, had so far remained unclear. We identify a quantifier of the particles’ mutual indistinguishability attuned to time-continuously evolving systems of (interacting) particles, which predicts the complex dynamical behaviour of observables influenced by genuine few-body interference. Our measure allows a systematic exploration of the role of many-body interference in the non-, weakly, and strongly interacting regimes, as well as a study of the dependence of different spectral properties on the particles mutual indistinguishability.

We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple 1D lattice Floquet models without time-reversal symmetry. Computing the spectral form factor $K(t)$ analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, $t_\text{Th}$. The $t_\text{Th}$ diverges logarithmically with system size, and for a large system, two regimes clearly emerge: for $t \ll t_Th$, $K(t)$ shows a universal behaviour which reflects the many-body spatial structure of the system; for $t \gg t_Th$, $K(t)$ agrees with the form obtained using RMT.

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure the fluctuations of the dynamics in equilibrium of the system. We found that close to integrability the equipartitioned dynamics is characterized by a power-law distribution of excursion times far off equilibrium. Long excursions arise from sticky dynamics close to coherent localized time-periodic solutions present in the phase space. Measuring the exponent allows to predict the transition into non-ergodic dynamics. In this contribution, we discuss our findings in the framework of the Fermi-Pasta-Ulam (FPU) and the Klein-Gordon (KG) lattices. There, we found that in the limit of small and large energies respectively, the systems show signatures of ergodicity breaking due to $q$-breathers (FPU) and discrete breathers (KG).

We find that localised perturbations in a chaotic classical many-body system-- the classical Heisenberg spin chain -- spread ballistically with a finite speed and little change in form as a function of distance from the origin of the perturbation even when the local spin dynamics is diffusive. We study this phenomenon by shedding light on the two complementary aspects of this butterfly effect-- the rapid growth of perturbations and its simultaneous ballistic (light-cone) spread, as characterised by the Lyapunov exponents and the butterfly speed respectively. We connect this to recent studies of the out-of-time-ordered correlator (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a correlator in our system and demonstrate that most of the interesting qualitative features -- with the exception of the physics of entanglement -- are present in the classical system. We present mostly results for the infinite temperature case but also discuss finite temperatures, where our results remain valid.

We present a minimal model for quantum chaos in a spatially extended many-body system. It consists of a chain of sites with nearest-neighbour coupling under Floquet time evolution. Quantum states at each site span a q-dimensional Hilbert space and time evolution for a pair of sites is generated by a $q^2 \times q^2$ random unitary matrix. The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbour on one side during the first half of the evolution period, and to its neighbour on the other side during the second half of the period. We show how dynamical behaviour averaged over realisations of the random matrices can be evaluated using diagrammatic techniques, and how this approach leads to exact expressions in the large-$q$ limit. We give exact results for the spectral form factor, relaxation of local observables, bipartite entanglement growth and operator spreading.

Using large-scale simulations based on matrix product state and quantum Monte Carlo techniques, we study the superfluid to Bose glass-transition for one-dimensional attractive hard-core bosons at zero temperature, across the full regime from weak to strong disorder. As a function of interaction and disorder strength, we identify a Berezinskii-Kosterlitz-Thouless critical line with two different regimes. At small attraction where critical disorder is weak compared to the bandwidth, the critical Luttinger parameter $K_c$ takes its universal Giamarchi-Schulz value $K_c=3/2$. Conversely, a non-universal $K_c>3/2$ emerges for stronger attraction where weak-link physics is relevant. In this strong disorder regime, the transition is characterized by self-similar power-law distributed weak links with a continuously varying characteristic exponent $\alpha$. Elmer V.H. Doggen, Gabriel Lemarié, Sylvain Capponi and Nicolas Laflorencie, Phys. Rev. B 96, 180202(R) (2017)

In the version provided by Sriednicki [1], the celebrated Eigenstate Thermalization Hypothesis (ETH) relies in another, more fundamental property of eigenstates in first-quantized quantum systems describing particle systems with chaotic classical limit: Berry's conjecture. The later states that in the bulk and far from boundaries, chaotic (ergodic) eigenstates are well described in the semiclassical limit $\hbar \to 0$ by a superposition of plane waves with locally defined wavenumber and random phases, the (even more) celebrated Random Wave Model [2]. In its modern formulation where it is extended to deal with systems with arbitrary confinement and including subtle correlations induced by the normalization condition [3], the RWM is based on two separated and essential ingredients. First, it is assumed (and still unproven) that when probed by local observables, chaotic eigenstates appear as Gaussian Random Fields. Second, all the microscopic and system-specific features of the system are encoded in the two-point correlation function of the field, that in turn is constructed from the exact microscopic Green (or Wigner) function, which in the semiclassical limit takes the well-known Bessel form in the bulk. In this work, we use recently developed methods for second-quantized systems with classical (mean-field) chaotic limit [4], and attempt to check the Gaussian conjecture and the universality of the two-point correlation in the large particle number limit $N \to \infty$, thus taking the first steps towards the construction of a RWM in Fock space. [1] M. Srednicki, Phys. Rev. E 50, 888 (1994) [2] M. Berry, J. Phys. A 10, 2083 (1977) [3] J.-D. Urbina, K. Richter, Adv. Phys. 62, 363 (2013) [4] T. Engl, J.-D. Urbina, K. Richter, Phil. Trans. R. Soc. A 374 , 20150159 (2016)

We study the non-equilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterised by a non-Abelian $S_3$ symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localised (MBL) phase. Using exact diagonalisation and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the $ℤ_3$ clock symmetry or a $ℤ_2$ chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic edge zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.

The investigation of the scrambling of information in interacting quantum systems has recently attracted a lot of attention as a manifestation of quantum chaos [1]. However, the unitarity of the time evolution renders time ordered observables inadequate for the quantification of quantum chaos. To capture the effect of the latter, one can make use of the so-called out-of-time-ordered correlators (OTOCs) whose short-time behavior can be directly related to the instability of a corresponding classical chaotic system with characteristic sensitivity to initial conditions given by the Lyapunov exponent. In order to investigate the connection between the classically unstable motion and the OTOCs we have used a momentum-truncated model describing attractively interacting bosons on a ring. This model, though integrable, has an instability that renders the quantum system unstable when the interactions are switched on [2]. We show that this local instability of the classical system is sufficient to reproduce the short-time behavior of the OTOCs as predicted for chaotic systems, where the classical stability exponent takes the role of the Lyapunov exponent. [1] Maldacena et al. High Energ. Phys. (2016) 2016: 106 [2] Dvali et al. Phys. Rev. D 88, 124041 (2013)

A key challenge in the search for new non-equilibrium phases of matter is the tendency of closed many-body systems to indefinitely absorb energy from a driving field. Generically this leads to an infinite temperature state where any interesting quantum, and in particular topological, effects are washed out. Here we show that in fact heating can be used as a resource for establishing universal prethermal behavior which exhibits topological phenomena. The prethermalization regime which we consider occurs for low driving frequencies, and persists throughout a long time window. Recently such prethermal states were found in one dimensional topological pumps [Lindner, Berg, Rudner, PRX 2017]. We provide bounds on the lifetimes of states, study different manifestations of universal prethermal behavior in a variety of systems, and discuss probes for observing topological properties.

Out of time order correlators (OTOCs), which measure the spreading of information in quantum systems, have drawn a lot of attention recently. However, the numerical calculation of OTOCs is extremely challenging, which renders the verification of theoretical predictions difficult. We tackle this problem within the Schroedinger picture in combination with a matrix-product state formulation of the time dependent variational principle (TDVP). First, we benchmark this technique by comparing the results with exact Krylov space time evolution results for small chains. Second, we calculate the OTOCs for system sizes which are unreachable by exact methods and analyze the hydrodynamic spreading of the light cone front.

We analyze the effect of a central spin equally coupled to all spins of the random field Heisenberg chain. We discover that if the coupling strength to the central spin is sufficiently small compared to the disorder strength, localization persists. Within this many-body localized phase of the central spin model, the central spin affects the magnitude of logarithmic entanglement growth. We show how its polarization can be employed to detect whether the surrounding system is localized. Sufficiently strong coupling eventually delocalizes the model accompanied by a mobility edge.

Motivated by quantum quench experiments in cold-atom systems, we study the case in which a cloud of fermions is suddenly subjected to a steep linear potential (effectively a strong electric field). We argue that this system, even in the absence of disorder, should exhibit an entanglement entropy that grows logarithmically with time, and we back up this statement with numerical results obtained via time-evolving block decimation (TEBD). We discuss the relationship between this phenomenon and conventional disorder-induced many-body localisation, with especial emphasis on the nature of the energy-level statistics. Collaborators: Maxi Schulz, Roderich Moessner, and Frank Pollmann.

We determine the information scrambling rate $\lambda_L$ due to electron-electron Coulomb interaction in graphene. $\lambda_L$ characterizes the growth of chaos and gives information about the thermalization of a many body system. We demonstrate that $\lambda_L$ behaves for strong coupling similar to transport scattering rates. A weak coupling analysis however reveals that scrambling is related to dephasing or single particle relaxation, and that $\lambda_L$ is parametrically larger than the collision time relevant for hydrodynamic processes such as electrical conduction and viscous flow.

I report the development of a hybrid semiclassical method to study the dynamics of one dimensional systems in and out of equilibrium. Our method handles internal degrees of freedom completely quantum mechanically, and accounts efficiently for entanglement entropy generation by these. In non-equilibrium situations, we can follow time evolution up to timescales at which local thermalization occurs. I discuss two applications of the method. First we investigate the quench dynamics and phase fluctuations of a pair of tunnel coupled one dimensional Bose condensates described by the sine-Gordon model. As a second application, we study diffusive and ballistic spin transport in the O(3) non-linear sigma model generated by an inhomogeneous initial state. M. Kormos, C. P. Moca, G. Zarand, Semiclassical theory of front propagation and front equilibration following an inhomogeneous quantum quench arXiv:1712.09466 C. P. Moca, M. Kormos, G. Zarand, Semi-semiclassical theory of quantum quenches in one dimensional systems, Phys. Rev. Lett. 119, 100603 (2017)

A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry in 1985 within the so-called diagonal approximation, and the full RMT spectral form factor K(t) from semiclassics has been completed by Müller et al. in 2004. In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as ‘many-body localized phase’ and ‘ergodic phase’. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. In this poster, based on arXiv:1712.02665, show the first theoretical explanation for these observations. We compute K(t) explicitly in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.

Recently the Sachdev-Ye-Kitaev (SYK) model has attracted a lot of attention as a toy model for a non-fermi liquid. The SYK model remains solvable at strong coupling in a large-N limit and was found to be a maximally scrambling system which saturates the recently proposed universal bound on chaos in quantum systems. The simple structure of this model offers a unique opportunity to study heating dynamics and thermalization in systems without quasi particle excitations. In this work, we investigate a periodically driven SYK model, by employing a non-equilibrium field theoretic approach and solving the resulting Kadanoff Baym equations numerically on the real-time Keldysh contour. Using the Green's function, we determine the energy density of the system, as well as an effective temperature for weak driving. We find that the SYK model shows two dynamical regimes, including exponential heating at late times.

We have studied the non-equilibrium dynamics of a hole created in a fermionic or bosonic Mott insulator in the atomic limit, which corresponds to a degenerate spin system. We have shown that this system provides a striking example, when a single particle created in a featureless, infinite temperature spin bath not only exhibits non-classical dynamics, but also induces strong long-lived correlations between the surrounding spins. In the absence of interactions, the spin correlations arise purely from quantum interference, and the correlations are both ferromagnetic and antiferromagnetic, in contrast to the equilibrium Nagaoka effect. These results are relevant for a number of condensed matter spin systems and should be observable using state of the art bosonic or fermionic quantum gas microscopes.

Many-body quantum systems typically display diffusive dynamics and ballistic spreading of information. Here we address the open problem of how slow the dynamics can be after a generic breaking of integrability by local interactions. We use quasi-exact methods to show that global quench dynamics display very slow spreading of information, similar to that of disordered systems exhibiting a many-body localized phase. To address this behavior we employ degenerate perturbation theory to study the dynamics of quasiparticles in such systems. Our method reveals slow dynamical regimes along with accurate estimates of their delocalization time scales. As an example, we construct a large class of 1D models where, despite the absence of asymptotic localization, the transient dynamics is exceptionally slow, i.e. indistinguishable from many-body localized systems for the time scales accessible by numerical simulations and experiment.

We study the dynamics of microscopic quantum correlations, viz., bipartite entanglement and quantum discord, in Ising spin chain with periodically varying external magnetic field along the transverse direction. Depending upon system parameters, local quantum correlations in the evolved states of such systems may get saturated to non-zero values after sufficiently large number of driving cycles. Moreover, we investigate convergence of the local density matrices, from which the quantum correlations under study originate, towards the final steady-state density matrices as a function of driving cycles. We find that the geometric distance between the non-equilibrium and the steady-state reduced density matrices obey power-law scaling. The steady-state quantum correlations corresponding to various initial states in thermal equilibrium are studied as a function of drive time period of a square pulsed field. The steady-state quantum correlations are marked by presence of peaks in the frequency domain. The steady-state features can be further understood by probing band structures of Floquet Hamiltonian. Finally, we compare the steady state values of the local quantum correlations under study with that of the canonical Gibbs ensemble and infer about canonical ergodic properties. We find that depending upon the quantum phases of the initial state and the pathway of the driving Hamiltonian the final values of the long-time quantum correlations may correspond to a canonical Gibbs state. We also find parameter values where this correspondence does not hold.

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary U(1)-invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a U(1)-invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of U(1)-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a S = 1 Heisenberg chain.

Quantum systems upon interaction with the environment loses its coherence properties and tends towards classical states. In general, the decay of quantum coherence is exponentially fast. Several theoretical and experimental works had explored the possibility of slowing down the rate of decoherence. In our work, we use a modified kicked rotor model by introducing a waiting time, $\tau$ drawn from Lévy distribution $w(\tau) \sim \tau^{-1-\alpha}$, between two consecutive kicks. We show both theoretically and through experiment that for $\alpha < 1$ the decay of quantum coherence can be made slower than exponential rate, i.e., the decoherence rate becomes non-exponential form. The slower coherence decay manifests in the form of quantum sub-diffusion that can be controlled by tuning the Lévy exponent, $\alpha$. The experimental results are in good agreement with the analytical estimates and numerical simulations for the mean energy growth and momentum profiles of an atom-optics kicked rotor.

One of the recent developements in condensed matter is the idea that there can exist new phases of matter in periodically driven Floquet systems that are absent in equilibrium. In the presence of global symmetries, it was shown by von Keyserlingk et al. that we could have new symmetry-protected-topological (SPT) phases characterized by the existence of robust ‘pumped’ boundary modes that are absent in equilibrium. Roy and Harper provided an alternative perspective on these phases in terms of a classification of loop-unitaries (time evolution operators of the form U(t)| U(0) = U(T) = 1). From this point of view, the Floquet-SPT phases that are unique to the driven setting are those that correspond to loop-unitaries that cannot be deformed to a trivial one (U(t) = 1) without breaking symmetry. In this work, we use Roy and Harper's scheme to study how the classification of driven one dimensional free-fermion phases for the different Altland-Zirnbauer symmetry classes changes in the presence of interactions. We find our results to be in agreement with those obtained by von Keyserlingk et al.

Periodic drives are a common tool to control physical systems, but have a limited applicability because time-dependent drives generically lead to heating. How to prevent the heating is a fundamental question with important practical implications. We address this question by analyzing a chain of coupled kicked rotors, and find two situations in which the heating rate can be arbitrarily small: (i) linear stability, for initial conditions leading to an effective integrability, and (ii) marginal localization, for drives with large frequencies and small amplitudes. In both cases, we find that the dynamics shows universal scaling laws that allow us to distinguish localized, diffusive, and sub-diffusive regimes. The marginally localized phase has common traits with recently discovered pre-thermalized phases of many-body quantum-Hamiltonian systems, but does not require quantum coherence.

Out-of-time-ordered correlators $\langle[\hat{V},\hat{W}(t)]^\dagger[\hat{V},\hat{W}(t)]\rangle$ have been found to be highly suitable tools to identify the onset of chaos in many-body quantum systems[1]. Contrary to already known indicators, the unusual time ordering of the operators is able to directly capture the local hyperbolic nature of the classical counterpart. One expects an exponential increase at short times with a rate related to classical Lyapunov exponents. Numerical studies in chaotic systems[2] indicate a saturation after the time scale for the classical-to-quantum-crossover, known as Ehrenfest or scrambling time. Our numerical studies show, that many-body criticality mimics this behaviour of chaotic systems, however with an exponent given by the local instability rate. We provide insight into the physical origin of the exponential growth and the saturation by using semiclassical methods based on the Van-Vleck-propagator for single- and many-body systems[3]. We show that the notion of interfering classical trajectories is well suited to provide a quantitative picture and we explicitly discuss the emergence of the Lyapunov exponent, resp.~instability rates and the relevant time scales. [1] J. Maldacena \it{et} \it{al.}, JHEP 2016:106 (2016) [2] E. B. Rozenbaum \it{et} \it{al.}, PRL \bf{118}, 086801 (2017) [3] T. Engl, J. Dujardin, A. Argüelles \it{et} \it{al.}, PRL \bf{112}, 140403 (2014)

We report the presence of multiple flat bands in a class of two-dimensional (2D) lattices formed by Sierpinski gasket (SPG) fractal geometries as the basic unit cells. Solving the tight-binding Hamiltonian for such lattices with different generations of a SPG network, we find multiple degenerate and non-degenerate completely flat bands, depending on the configuration of parameters of the Hamiltonian. Moreover, we establish a generic formula to determine the number of such bands as a function of generation index, l of the fractal geometry. We show that the flat bands and their neighboring dispersive bands have remarkable features, the most interesting one being the spin-1 conical-type spectrum at the band center without any staggered magnetic flux, in contrast to the Kagome lattice. We furthermore investigate the effect of magnetic flux in these lattice settings and show that different combinations of fluxes through such fractal unit cells lead to richer spectrum with a single isolated flat band or gapless electron- or hole-like flat bands. Finally, we discuss a possible experimental setup to engineer such fractal flat band network using single-mode laser-induced photonic waveguides.

We develop an effective large N Keldysh field theory for studying dissipative non-equilibrium transport in a regular one-dimensional Mott insulator system subjected to a uniform electric field. The Mott insulator system that we consider is a lattice of mesoscopic metallic grains. The grains are characterized by a capacitive charging cost $E_{C}$. The presence of multiple energy levels in a grain gives rise to an effective dissipative mechanism. The large number of transverse channels, N, leads to a tractable Keldysh phase action, analogous to the famous Ambegaokar-Eckern-Schon phase action in the equilibrium case. We calculate the electric current as a function of the applied electric field. The current at small fields is governed by large distance co-tunneling, absent in the equilibrium counterpart, and qualitatively differs from the Landau-Zener expression for the non-dissipative case. The breakdown of perturbation theory in the Mott phase signals a non-equilibrium phase transition to a metallic phase and from this we construct the phase diagram for this transition in the dissipation-drive parameter space.

Irreversibility, despite being a necessary condition for thermalization, still lacks a sound understanding in the context of quantum many-body systems. In our work we approach this question by studying the behavior of generic many-body systems under imperfect effective time reversal, where the imperfection is introduced as a perturbation of the many-body state at the point of time reversal. The resulting echo dynamics is closely related to out-of-time-order correlators. Based on numerical simulations of the full quantum dynamics we demonstrate that observable echoes occurring in this setting decay exponentially with a rate that is intrinsic to the system meaning that the dynamics is effectively irreversible.

In classical systems, chaos can be characterized by the sensitivity of the particles’ trajectories with respect to small deviations in the initial state. An exponential growth in time then marks chaotic behav-ior. Recently, it has been proposed that certain out-of-time-ordered correlation functions (OTOCs) may be a suitable extension to characterize chaos in quantum many-body systems. OTOCs also probe the scrambling of quantum information across the system and thereby provide a direct connection between an information theoretic measure and chaotic dynamics. We calculate the time evolution of OTOCs in an O(N) symmetric scalar field theory at high temperatures using non-perturbative expansion techniques. Apart from the Lyapunov exponent quantifying a potential exponential growth of chaos, we are also interested in the emergent time scales of information propagation, including light-cone and butterfly velocities.

"Hydrodynamics and subdiffusion in open spin chains" Maxi Schulz, Scott Taylor, Chris Hooley and Antonello Scardicchio One recent strand of work on many-body localisation has concentrated on the question of whether there is an intermediate subdiffusive region between the ergodic phase and the many-body localised one. Recent results [1] indicate that, in the XXZ spin chain, such a subdiffusive regime does exist, though the finite-size effects are severe. In this work, we present results of a time-evolving block decimation study of the disordered anisotropic XYZ spin chain in a configuration similar to that used in [1]. The significance of our choice of model is that its clean hydrodynamics involves only an energy current, whereas that of the XXZ spin chain has both energy and spin currents. This allows us to explore the question of how the nature of this subdiffusive phase depends on the hydrodynamic properties of the underlying clean model. We also discuss connections of our model to superconductivity. [1] M. Žnidarič, A. Scardicchio, and V.K. Varma, Phys. Rev. Lett. 117, 040601 (2016)."

We show how tailored laser field pulses can selectively generate coherent oscillation of electronic orders such as charge density waves (CDW). After the pump, a macroscopic number of electrons start oscillating and coherence is built up through a dynamical synchronization transition described by an effective Kuramoto model. The oscillation may be used as a purely electronic way of realizing a Floquet state respecting space time crystalline symmetries.

The venerable phenomena of Anderson localization, along with the much more recent many-body localization, both depend crucially on the presence of disorder. The latter enters either in the form of quenched disorder in the parameters of the Hamiltonian, or through a special choice of a disordered initial state. Here, we present a family of very simple translationally invariant quantum models with only local interactions between spins and fermions. By identifying an extensive set of conserved quantities, we show that the system generates purely dynamically its own disorder, which gives rise to localization of fermionic degrees of freedom. Our work provides an answer to a decades old question whether quenched disorder is a necessary condition for localization. It also offers new insights into the physics of many-body localization, lattice gauge theories, and quantum disentangled liquids.

We investigate the role of disorder in a two-dimensional semi-Dirac material characterized by a linear dispersion in one, and a parabolic dispersion in the orthogonal, direction. Using the self-consistent Born approximation, we show that disorder can drive a topological Lifshitz transition from an insulator to a semi-metal, as it generates a momentum independent off-diagonal contribution to the self-energy. Breaking time-reversal symmetry enriches the topological phase diagram with three distinct regimes-- single-node trivial, two-node trivial and two-node Chern. We find that disorder can drive topological transitions from both the single- and two-node trivial to the two-node Chern regime. We further analyze these transitions in an appropriate tight-binding Hamiltonian of an anisotropic hexagonal lattice, by calculating the real-space Chern number. Additionally we compute the disorder-averaged entanglement entropy which signals both the topological Lifshitz and Chern transition as a function of the anisotropy of the hexagonal lattice. Finally, we discuss experimental aspects of our results.

For a chain of spins, we take a time evolution operator consisting of random unitaries coupling neighbouring sites. For a chain of free fermions, we analogously use random Gaussian operations coupling nearest neighbours. The random unitaries or orthogonals are distributed according to the invariant Haar measure. In these settings, we can show some properties of the final state averaged over all time evolution operators due to its composition of random matrices. By applying numerics, we find that the dynamics exhibit thermalisation and localisation, respectively. In the fermionic case, we find very different properties for a translation invariant time evolution compared to independently distributed couplings of neighbours. Further, by choosing a different distribution for the unitary building blocks of the time evolution operator that allows to tune the coupling strength, we observe an MBL phase transition.

Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra are modeled through the appropriate ensemble of random matrix theory. However, a small subset of states violate this principle and display eigenstate localization, a counter-intuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and this modifies the spectral statistics. In this work, a 3×3 random matrix model is used to obtain an exact result for the ratio of spacings between a generic and localized state. We consider time-reversal-invariant as well as non-invariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.

Certain wave functions of non-interacting quantum chaotic systems can exhibit “scars” in the fabric of their real-space density profile. Quantum scarred wave functions concentrate in the vicinity of unstable periodic classical trajectories. We introduce the notion of many-body quantum scars which reflect the existence of a subset of special many-body eigenstates concentrated in certain parts of the Hilbert space. We demonstrate the existence of scars in the Fibonacci chain – the one- dimensional model with a constrained local Hilbert space realized in the 51 Rydberg atom quantum simulator [H. Bernien et al., arXiv:1707.04344]. The quantum scarred eigenstates are embedded throughout the thermalizing many-body spectrum, but surprisingly lead to direct experimental signatures such as robust oscillations following a quench from a charge-density wave state found in experiment. We develop a model based on a single particle hopping on the Hilbert space graph, which quantitatively captures the scarred wave functions up to large systems of L = 32 atoms. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

TBA

We show that many-body localized symmetry protected topological systems with time reversal symmetry define a topological phase at arbitrary energy scales. For concreteness, we consider the one-dimensional disordered cluster model with periodic boundary conditions and prove that all eigenstates in the fully many-body localized symmetry protected regime have exactly four-fold degenerate entanglement spectra in the thermodynamic limit. To that end, we employ unitary quantum circuits where the number of sites the gates act on grows linearly with the system size. We find that the corresponding matrix product operator representation has tensors invariant under similar local symmetries as translationally invariant matrix product ground states of symmetry protected topological phases. Those local symmetries give rise to a $Z_2$ topological index, which is robust against arbitrary perturbations so long as they do not break time reversal symmetry or drive the system out of the fully many-body localized phase.

Initially applied in 1980 by Ma and Dasgupta to random Heisenberg chains, the strong disorder renormalization group (SDRG) method has since been used to study the role of quenched disorder in ground state, excited state, thermal and dynamical properties of many random quantum models. The method is usually applied by looking for fixed points in the RG flow of disordered Hamiltonians; for certain classes of models, the flow is dominated by "infinite-randomness" fixed points where SDRG becomes an asymptotically exact description. However, the results are inconclusive for many other models where disorder is important but does not become infinite asymptotically. We instead take a wavefunction viewpoint; starting from the realization that the SDRG procedure produces tree tensor network ansatz wavefunctions and using input from current tensor network algorithms such as DMRG, we create a series of improvements on the SDRG method with increasing accuracy and analyze the complexity of these methods using the detailed structure of entanglement in the eigenstates.

The many-body localization (MBL) transition describes an exotic phase transition from an ergodic metal to a non-ergodic localized phase. So far the MBL transition has been mainly investigated with exact diagonalisation, limiting the studies to one dimension and to rather small systems. In light of these limitations, we aim at building a field-theoretic description of the disordered spinless Fermi-Hubbard model with nearest-neighbor interactions. Such an approach enables us to investigate the non-equilibrium dynamics of systems with up to 200 sites. Using a self-consistent small-coupling expansion we find, that the system delocalizes for weak disorder already at the Hartree-Fock level. Consistent with theoretical predictions for the Griffith regime [1], a staggered initial state decays with a subdiffusive power-law in the delocalized phase at intermediate disorder. Employing that time-dependent noise leads to delocalization [2], we develop a picture that describes why Hartree-Fock effects can delocalize the system dynamically. However, at strong disorder, the system reamains localized to extraordinarily long times. The field theory also gives us access to the non-equilibrium site resolved spectral function, which shows a broad spectrum in the case of weak disorder, while it exhibits sharp spikes in the localized phase, indicating a many-body localization transition.

We investigate quantum quenches in the antiferromagnetic $S=1$ Heisenberg chain. Due to the gap in the excitation spectrum, after a small quench, the initial state is supposed to be a dilute gas of magnons (with spin $S=1$). An effective approach is then the so called semi-classical description, where we describe the system as a classical gas of quasiparticles that collide totally reflectively with each other. In our work we make an attempt to test the predictions of the semi-classical description, by comparing them with large scale MPS simulations. Exploiting the $SU(2)$ symmetry of the model, we can extract the half-chain spin fluctuations from the non-abelian MPS, and find that at short times they are accurately described by semi-classics. However, at longer times the effects of fast (non reflected) quasiparticles make the semi-classical description untenable. This problem can be solved by taking the proper two-particle S-matrix into account at the collisions. This improved "semi-semi-classical" approach is also tested against MPS simulations, and it is indeed found to be remarkably more accurate at longer times.