Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.
The many-body localized phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalize and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body localized regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalization or lack thereof. We consider the minimal case, namely, the second R\'enyi entropy of entanglement, of an initial product state as well as that of the time-evolution operator, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energy scales. We show that the hierarchy of these timescales along with their nontrivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localized regime. The underlying spatial structures in the set of special quartets also provides a microscopic understanding of the spacetime picture of the entanglement growth. The theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the $\ensuremath{\ell}$-bit picture.
We have recently shown that the logarithmic growth of the entanglement entropy following a quantum quench in a many-body localized phase is accompanied by a slow growth of the number entropy ${S}_{N}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}ln\phantom{\rule{0.16em}{0ex}}t$. Here we provide an in-depth numerical study of
SN(t) for the disordered Heisenberg chain and show that this behavior is not transient and persists even for very strong disorder. Calculating the truncated R\'enyi number entropy ${S}_{N}^{(\ensuremath{\alpha})}(t)={(1\ensuremath{-}\ensuremath{\alpha})}^{\ensuremath{-}1}ln{\ensuremath{\sum}}_{n}{p}^{\ensuremath{\alpha}}(n)$ for $\ensuremath{\alpha}\ensuremath{\ll}1$ and
p(n)>pc---which is sensitive to large number fluctuations occurring with low probability---we demonstrate that the particle number distribution
p(n) in one half of the system has a continuously growing tail. This indicates a slow but steady increase in the number of particles crossing between the partitions in the interacting case and is in sharp contrast to Anderson localization for which we show that ${S}_{N}^{(\ensuremath{\alpha}\ensuremath{\rightarrow}0)}(t)$ saturates for any cutoff
pc>0. We show, furthermore, that the growth of
SN is not the consequence of rare states or rare regions but rather represents typical behavior. These findings indicate that the interacting system is never fully localized even for very strong but finite disorder.
An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied numerically for a system of interacting spinless fermions in one dimension described by the random-field XXZ Hamiltonian. Interactions induce a dramatic change in the propagation of entanglement and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.
We study entanglement dynamics in a diagonal dephasing model in which the strength of interaction decays exponentially with distance---the so-called l-bit model of many-body localization. We calculate the exact expression for entanglement growth with time, finding in addition to a logarithmic growth, a sublogarithmic correction. Provided the l-bit picture correctly describes the many-body localized phase this implies that the entanglement in such systems does not grow (just) as a logarithm of time, as believed so far.
We show that von Neumann entropy grows algebraically in time for many-body localized systems with long-range interactions. The dephasing time of far-apart localized degrees of freedom is studied first. For a disentangled high-energy initial state, our analysis indicates that the entropy of one half of the system evolves as
t1/(p+1), where
p controls the spatial decay of interactions. Numerical simulations of the entropy evolution in the Coulomb and dipolar cases also exhibit an algebraic growth. We find that the saturation value of von Neumann entropy is extensive but small for an ergodic state, and it does not depend on the shape of the interactions. Due to the algebraic entanglement spread, the signature of an extensive entropy should be much easier to observe in experiments with nonlocal interactions than in those with local ones.
We study the growth of the number entropy
SN in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of
SN has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying
SN in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of
SN. However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained
SN: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of
SN with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of
SN also heavily varies depending on the initial state, and therefore
SN and von Neumann entropy can behave rather differently.