Typicality of the orthogonal dynamics (TOD) is established as a generic feature of temporal relaxation processes in isolated many-body quantum systems. The basic idea in the simplest case is that the transient non-equilibrium behavior is mainly governed by the component of the time-evolved system state parallel to the initial state, while the orthogonal component appears as equilibrated right from the beginning. The main emphasis is laid on the largely unexplored and particularly challenging case that one energy level exhibits a much larger population than all the others. Important examples are gapped many-body systems at low energies, for instance due to a quantum quench. A general analytical prediction is derived and is found to compare very well with various numerically exact results.
We study the dynamics of a quench-prepared domain wall state released into a system whose unitary time evolution is dictated by the Hamiltonian of the Heisenberg spin-1/2 gapped antiferromagnetic chain. Using exact wavefunctions and their overlaps with the domain wall state allows us to describe the release dynamics to high accuracy, up to the long-time limit, for finite as well as infinite systems. The results for the infinite system allow us to rigorously prove that the system in the gapped regime ($Δ>1$) cannot thermalize in the strict sense.
The NMR relaxation rates of (quasi-)one-dimensional spin gapped systems such as the $S=1/2$ spin ladder with a diagonal interaction and the Haldane-gap system are investigated at low temperatures in magnetic fields. We show that in the regime where the gap is collapsed by magnetic fields, the NMR relaxation rate diverges with decreasing temperature and its exponent shows a behavior characteristic of the model as a function of magnetic field. We discuss our results in comparison with recent experimental findings for the spin ladder system ${\rm Cu_{2}(C_{5}H_{12}N_{2})_{2}Cl_{4}}$ and for the H…
We study the decay mechanism of the gapped lowest-lying excitation of a quasi-pure box-trapped atomic Bose-Einstein condensate. Owing to the absence of lower-energy modes, or direct coupling to an external bath, this excitation is protected against one-body (linear) decay and the damping mechanism is exclusively nonlinear. We develop a universal theoretical model that explains this fundamental nonlinear damping as a process whereby two quanta of the gapped lowest excitation mode couple to a higher-energy mode, which subsequently decays into a continuum. We find quantitative agreement between o…
We discuss the emergence of nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional quantum many-body system subject to a linear ramp of one of its parameters. While performing a ramp within a gapped phase seems to be the most favorable situation for adiabaticity, we show that such a change leads eventually to the disruption of the order, no matter how slowly the ramp is performed. We show this in detail by studying the dynamics of the one-dimensional quantum Ising model subject to linear variation of the transverse magnetic field within the ferromagnetic phase, and t…
We compute the spin structure factor of XXZ spin chains in the Heisenberg and gapped (Ising) regimes in the high-temperature limit for nonzero magnetization, within the framework of generalized hydrodynamics including diffusive corrections. The structure factor shows a hierarchy of timescales in the gapped phase, owing to $s$-spin magnon bound states (`strings') of various sizes. Although short strings move ballistically, long strings move primarily diffusively as a result of their collisions with short strings. The interplay between these effects gives rise to anomalous power-law decay of the…
We consider isolated many-body quantum systems which do not thermalize, i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion …
We investigate the onset of thermalization and quantum chaos in finite one-dimensional gapped systems of hard-core bosons. Integrability in these systems is broken by next-nearest-neighbor repulsive interactions, which also generate a superfluid to insulator transition. By employing full exact diagonalization, we study chaos indicators and few-body observables. We show that with increasing system size, chaotic behavior is seen over a broader range of parameters and, in particular, deeper into the insulating phase. Concomitantly, we observe that, as the system size increases, the eigenstate the…
The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. In contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with non-small eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literatu…
We provide an overview of our numerical and analytical studies of isolated interacting quantum systems that are quenched out of equilibrium instantaneously. We describe the relaxation process to a new equilibrium and obtain lower bounds for the relaxation time of full random matrices and realistic systems with two-body interactions. We show that the size of the time fluctuations after relaxation decays exponentially with system size for systems without too many degeneracies. We also discuss the conditions for thermalization and demonstrate that it can happen after local and global quenches in …
We demonstrate the role of interactions in driving the relaxation of an isolated integrable quantum system following a sudden quench. We consider a family of integrable hard-core lattice anyon models that continuously interpolates between noninteracting spinless fermions and strongly interacting hard-core bosons. A generalized Jordan-Wigner transformation maps the entire family to noninteracting fermions. We find that, aside from the singular free-fermion limit, the entire single-particle density matrix and therefore all one-body observables relax to the predictions of the generalized Gibbs en…
An analytical prediction is established of how an isolated many-body quantum system relaxes towards its thermal long-time limit under the action of a time-independent perturbation, but still remaining sufficiently close to a reference case whose temporal relaxation is known. This is achieved within the conceptual framework of a typicality approach by showing and exploiting that the time-dependent expectation values behave very similarly for most members of a suitably chosen ensemble of perturbations. The predictions are validated by comparison with various numerical and experimental results fr…
A crossover between different power-law relaxation behaviors of many-body periodically driven integrable systems has come to light in recent years. We demonstrate using integrable quantum systems, that similar kinds of dynamical transitions may also occur in the relaxation of such systems following a sudden quench. Particularly, we observe two distinct power-law relaxation behaviors following a sudden quench in the integrable XY model, depending upon whether the quenched Hamiltonian lies in the commensurate or the incommensurate phase. The relaxation behavior for quenches at and near the bound…
We establish a setting - atoms in optical superlattices with period 2 - in which one can experimentally probe signatures of the process of local relaxation and apparent thermalization in non-equilibrium dynamics without the need of addressing single sites. This opens up a way to explore the convergence of subsystems to maximum entropy states in quenched quantum many-body systems with present technology. Remarkably, the emergence of thermal states does not follow from a coupling to an environment, but is a result of the complex non-equilibrium dynamics in closed systems. We explore ways of meas…
We study the thermalization of a small $XX$ chain coupled to long, gapped $XXZ$ leads at either side by observing the relaxation dynamics of the whole system. Using extensive tensor network simulations, we show that such systems, although not integrable, appear to show either extremely slow thermalization or even lack thereof since the two can not be distinguished within the accuracy of our numerics. We show that the persistent oscillations observed in the spin current in the middle of the $XX$ chain are related to eigenstates of the entire system located within the gap of the boundary chains.…
It is "conventional wisdom" that the uncertainty of local temperature measurements on equilibrium systems diverges exponentially fast as their temperature $T$ drops to zero. In contrast, some exactly solvable models showcase a more benign power-law-like scaling, when only a small non-equilibrium fragment of the equilibrium system is measured. Does this mean that a part may contain more information about the global temperature than the whole? Certainly not. Here, we resolve this apparent paradox. First, we prove that local quantum thermometry at low $T$ is exponentially inefficient in non-criti…
We substantially extend our relaxation theory for perturbed many-body quantum systems from [Phys. Rev. Lett. 124, 120602 (2020)] by establishing an analytical prediction for the time-dependent observable expectation values which depends on only two characteristic parameters of the perturbation operator: its overall strength and its range or band width. Compared to the previous theory, a significantly larger range of perturbation strengths is covered. The results are obtained within a typicality framework by solving the pertinent random matrix problem exactly for a certain class of banded pertu…
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 w…
We investigate how the observable relaxation behavior of an isolated quantum many-body system is modified in response to weak-to-moderate perturbations within a nonperturbative typicality framework. A key role is played by the so-called perturbation profile, which characterizes the dependence of the perturbation matrix elements in the eigenbasis of the unperturbed Hamiltonian on the difference of the corresponding energy eigenvalues. In particular, a banded matrix structure is quantitatively captured by a perturbation profile which approaches zero for large energy differences. The temporal mod…
In arXiv:2001.03419 we introduce a universal error bound for constrained unitary dynamics within a well-gapped energy band of an isolated quantum system. Here, we provide the full details on the derivation of the bound. In addition, we generalize the result to isolated quantum many-body systems by employing the local Schrieffer-Wolff transformation, obtaining an error bound that grows polynomially in time. We also generalize the result to Markovian open quantum systems and quantitatively explain the quantum Zeno effect.
Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems
Universidade de Lisboa（里斯本大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Technische Universität Dresden（德累斯顿工业大学）；University of Cambridge（剑桥大学）
It is shown that dipolar and weak superexchange interactions between the spin systems of single-molecule magnets (SMM) play an important role in the relaxation of magnetization. These interactions can reduce or increase resonant tunneling. The one-body tunnel picture of SMMs is not always sufficient to explain the measured tunnel transitions. We propose to improve the picture by including also two-body tunnel transitions such as spin-spin cross-relaxation (SSCR). A Mn4 SMM is used as a model system to study the SSCR which plays also an important role for other SMMs like Mn12 or Fe8. At certain…
Through an exact method we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime. We observe the localized state collapsing into a flat profile in equilibrium for a large number of particles such as N=1000. The relaxation time increases as the coupling constant decreas…
The ground state of the quantum spin system kappa-(BEDT-TTF)2Cu2(CN)3 in which antiferromagnetically-interacting S=1/2 spins are located on a nearly equilateral triangular lattice attracts considerable interest both from experimental and theoretical aspects, because a simple antiferromagnetic order may be inhibited because of the geometrical frustration and hence an exotic ground state is expected. Furthermore, recent two reports on the ground state of this system have made it further intriguing by showing completely controversial results; one indicates the gapless state and the other gapped. …
This paper presents a unified perspective on the results of two recent works (C. Buragohain and S. Sachdev cond-mat/9811083 and S. Sachdev cond-mat/9810399) along with additional background. We describe the low frequency, non-zero temperature, order parameter relaxational dynamics of a number of systems in the vicinity of a quantum critical point. The dynamical correlations are properties of the high temperature limit of renormalizable quantum field theories in spatial dimensions d=1,2. We study, as a function of d and the number of order parameter components, n, the crossover from the finite …
The relaxation of Fourier modes of hamiltonian chains close to equilibrium is studied in the framework of a simple mode-coupling theory. Explicit estimates of the dependence of relevant time scales on the energy density (or temperature) and on the wavenumber of the initial excitation are given. They are in agreement with previous numerical findings on the approach to equilibrium and turn out to be also useful in the qualitative interpretation of them. The theory is compared with molecular dynamics results in the case of the quartic Fermi-Pasta-Ulam potential.
We study the quantum many-body dynamics and the entropy production triggered by an interaction quench in a system of $N=10$ interacting identical bosons in an external one-dimensional harmonic trap. The multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is used for solving the time-dependent Schrödinger equation at a high level of accuracy. We consider many-body entropy measures such as the Shannon information entropy, number of principal components, and occupation entropy that are computed from the time-dependent many-body basis set used in MCTDHB. These measures quantify …
We study the real-time dynamics of a translationally invariant quantum spin chain, based on the East kinetically constrained glass model, in search for evidence of many-body localisation in the absence of disorder. Numerical simulations indicate a change, controlled by a coupling parameter, from a regime of fast relaxation---corresponding to thermalisation---to a regime of very slow relaxation. This slowly relaxing regime is characterised by dynamical features usually associated with non-ergodicity and many-body localisation (MBL): memory of initial conditions, logarithmic growth of entangleme…
In a many-body localized (MBL) quantum system, the ergodic hypothesis breaks down completely, giving rise to a fundamentally new many-body phase. Whether and under which conditions MBL can occur in higher dimensions remains an outstanding challenge both for experiments and theory. Here, we experimentally explore the relaxation dynamics of an interacting gas of fermionic potassium atoms loaded in a two-dimensional optical lattice with different quasi-periodic potentials along the two directions. We observe a dramatic slowing down of the relaxation for intermediate disorder strengths and attribu…
In a recent Letter [Dabelow and Reimann, Phys. Rev. Lett. 124, 120602 (2020)], a perturbative relaxation theory is applied to the Bose-Hubbard model and compared to data from a quantum simulator experiment [Trotzky et al. Nature Phys 8, 325 (2012)]. In this Comment, I argue that the discrepancy found is due to the quasi-local observable measured in the experiment being affected by a harmonic trapping potential that is unaccounted for in the analytic calculation. I support my claim with quasi-exact numerics computed using a parallel time-evolving block decimation algorithm (TEBD), and show that…
A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles. We demonstrate that long time scales can be analytically found by analyzing dynamical manifestations of spectral correlations. Using this approach, we show that the Thouless time, $t_{\text{Th}}$, and the relaxation time, $t_{\text{R}}$, increase exponentially with system size. We define $t_{\text{Th}}$ as the time at which the spread of the initial state in the many-body Hilbert space i…
In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such state are. We confirm the relaxation hypothesis through a thorough ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more me…
We investigate the relaxation dynamics in an isolated quantum many-body system. The stationary value of an observable after relaxation is a topic of researches in the field of quantum thermalization, since thermalization is a relaxation phenomena where this stationary value coincides with the equilibrium value. Therefore, computing the stationary value in quantum many-body systems is regarded as an important problem. We, however, prove that the stationary value in quantum many-body systems is incomputable. More precisely, we show that whether the stationary value is in the vicinity of a given …
I consider a self-gravitating, N-body system assuming that the N constituents follow regular orbits about the center of mass of the cluster, where a central massive object may be present. I calculate the average over a characteristic timescale of the full, N-body Hamiltonian including all kinetic and potential energy terms. The resulting effective system allows for the identification of the orbital planes with N rigid, disk-shaped tops, that can rotate about their fixed common centre and are subject to mutual gravitational torques. The time-averaging imposes boundaries on the canonical general…
To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size $\ell$ with up to $n$-body Lindblad operators, which are $n$-local in the complexity-theory sense. Without locality ($n=\ell$), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403;arXiv…
The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g. in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation $\varepsilon$ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this letter, we prove a version of the adiabatic theorem for gapped ground states of quantum…
