The authors construct an efficient algorithm for extracting the entanglement structure of wave functions in the form of a cluster decomposition. They apply this to the many-body localization problem by analyzing entanglement clusters in exact eigenstates. Strongly entangled ergodic clusters are connected. Their size is distributed according to a power law at the transition, while at higher disorder the distribution is a stretched exponential. This constrains possible phenomenological renormalization group descriptions of the transition, and is consistent with a Kosterlitz-Thouless type scaling.
The authors construct an efficient algorithm for extracting the entanglement structure of wave functions in the form of a cluster decomposition. They apply this to the many-body localization problem by analyzing entanglement clusters in exact eigenstates. Strongly entangled ergodic clusters are connected. Their size is distributed according to a power law at the transition, while at higher disorder the distribution is a stretched exponential. This constrains possible phenomenological renormalization group descriptions of the transition, and is consistent with a Kosterlitz-Thouless type scaling.
We introduce techniques for analyzing the structure of quantum states of many-body localized (MBL) spin chains by identifying correlation clusters from pairwise correlations. These techniques proceed by interpreting pairwise correlations in the state as a weighted graph, which we analyze using an established graph theoretic clustering algorithm. We validate our approach by studying the eigenstates of a disordered XXZ spin chain across the MBL to ergodic transition, as well as the nonequilibrium dynamics in the MBL phase following a global quantum quench. We successfully reproduce theoretical predictions about the MBL transition obtained from renormalization group schemes. Furthermore, we identify a clear signature of many-body dynamics analogous to the logarithmic growth of entanglement. The techniques that we introduce are computationally inexpensive and, in combination with matrix product state methods, allow for the study of large-scale localized systems. Moreover, the correlation functions we use are directly accessible in a range of experimental settings, including cold atoms.
We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized (MBL) and thermal phases. We develop an improved real space renormalization group approach that enables numerical simulation of large system sizes and systematic extrapolation to the infinite system size limit. For systems smaller than the correlation length, the average entanglement follows a subthermal volume law, whose coefficient is a universal scaling function. The full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump at the transition from fully thermal volume law on the thermal side, to pure area law on the MBL side.
We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.
Closed, interacting, quantum systems have the potential to transition to a many-body localized (MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failing to thermalize. In this work we study the distribution of correlations throughout the ergodic-MBL phase diagram. We find the typical correlations in the MBL phase decay as a stretched exponential with range r eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as e−Ar, a decay that is reminiscent of the random singlet phase. While the standard deviation of the log(QMI) has a range dependence, the log(QMI) converges to a range-invariant distribution on all other moments (i.e., the skewness and higher) at the transition. The universal nature of these distributions provides distinct phenomenology of the transition different from both the ergodic and MBL phenomenologies. In addition to the typical correlations, we study the extreme correlations in the system, finding that the probability of strong long-range correlations is maximal at the transition, suggesting the proliferation of resonances there. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, finding that this probability is non-zero deep in the MBL phase but vanishes at moderate disorder well above the transition.
The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.
Drawing motivation from the fact that eigenstates in many-body localized and ergodic phases behave differently with regard to their support on the Fock space, the authors study a classical percolation problem in Fock space that captures certain aspects of the many-body localization transition. In particular, it has two classical phases that provide simplified pictures of the extended and many-body localized states. At weak disorder, the entire Fock space is contained in a single cluster, whereas for strong disorder, the typical cluster size scales with the Fock-space dimension in a subvolume law fashion.