Abstract Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, a phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them—and in the light of the observation that much of the phenomenology of many properties can be derived from them—it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, hard experimental evidence is still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.
Abstract Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, a phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them—and in the light of the observation that much of the phenomenology of many properties can be derived from them—it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, hard experimental evidence is still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.
Interacting quantum many-body systems are expected to thermalize, in the sense that the evolution of local expectation values approaches a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation (MBL). In stark contrast to the non-interacting case of Anderson localisation, the entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matter physics, capturing this phenomenon in the Heisenberg picture. We show that the mere existence of local constants of motion, often taken as the defining property of MBL, together with a generic spectrum of the Hamiltonian, is already sufficient to rigorously prove information propagation: these systems can be used to send a classical bit over arbitrary distances, in that the impact of a local perturbation can be detected arbitrarily far away. This counterintuitive result is compatible with and further corroborates the intuition of a slow entanglement growth following global quenches in MBL systems. We perform a detailed perturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our results provide a detailed and at the same time model-independent picture of information propagation in MBL systems.
The phenomenon of many-body localised (MBL) systems has attracted significant interest in recent years, for its intriguing implications from a perspective of both condensed-matter and statistical physics: they are insulators even at non-zero temperature and fail to thermalise, violating expectations from quantum statistical mechanics. What is more, recent seminal experimental developments with ultra-cold atoms in optical lattices constituting analog quantum simulators have pushed many-body localised systems into the realm of physical systems that can be measured with high accuracy. In this work, we introduce experimentally accessible witnesses that directly probe distinct features of MBL, distinguishing it from its Anderson counterpart. We insist on building our toolbox from techniques available in the laboratory, including on-site addressing, super-lattices, and time-of-flight measurements, identifying witnesses based on fluctuations, density-density correlators, densities, and entanglement. We build upon the theory of out of equilibrium quantum systems, in conjunction with tensor network and exact simulations, showing the effectiveness of the tools for realistic models.
A complete set of local integrals of motion (LIOM) is a key concept for describing many-body localization (MBL), which explains a variety of intriguing phenomena in MBL systems. For example, LIOM constrain the dynamics and result in ergodicity violation and breakdown of the eigenstate thermalization hypothesis. However, it is difficult to find a complete set of LIOM explicitly and accurately in practice, which impedes some quantitative structural characterizations of MBL systems. Here we propose an accurate numerical method for constructing LIOM, discover through the LIOM an interaction-induced characteristic length ξ + , and prove a ‘quasi-product-state’ structure of the eigenstates with that characteristic length ξ + for MBL systems. More specifically, we find that there are two characteristic lengths in the LIOM. The first one is governed by disorder and is of Anderson-localization nature. The second one is induced by interaction but shows a discontinuity at zero interaction, showing a nonperturbative nature. We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are the quasi-product states with such a localization length .
Interacting many-body quantum systems and their dynamics, while fundamental to modern science and technology, are formidable to simulate and understand. However, by discovering their symmetries, conservation laws, and integrability, one can unravel their intricacies. Here, using up to 124 qubits of a fully programmable quantum computer, we uncover local conservation laws and integrability in one- and two-dimensional periodically-driven spin lattices in a regime previously inaccessible to such detailed analysis. We focus on the paradigmatic example of disorder-induced ergodicity breaking, where we first benchmark the system crossover into a localized regime through anomalies in the one-particle-density-matrix spectrum and other hallmark signatures. We then demonstrate that this regime stems from hidden local integrals of motion by faithfully reconstructing their quantum operators, thus providing a more detailed portrait of the system’s integrable dynamics. Our results demonstrate a versatile strategy for extracting the hidden dynamical structure from noisy experiments on large-scale quantum computers. Local integrals of motion are useful for understanding emergent integrability in many-body localized systems. Here the authors use a large-scale superconducting quantum processor with up to 124 qubits to simulate many-body dynamics in 1D and 2D systems and demonstrate extraction of local integrals of motion.
Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body-localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin-flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, in both experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin-chain model. We show that the resulting LIOMs are quasilocal and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.
The characterizing feature of a many-body localized phase is the existence of an extensive set of quasi-local conserved quantities with an exponentially localized support. This structure endows the system with the signature logarithmic in time entanglement growth between spatial partitions. This feature differentiates the phase from Anderson localization, in a non-interacting model. Experimentally measuring the entanglement between large partitions of an interacting many-body system requires highly non-local measurements which are currently beyond the reach of experimental technology. In this work we demonstrate that the defining structure of many-body localization can be detected by the dynamics of a simple quantity from quantum information known as the total correlations which is connected to the local entropies. Central to our finding is the necessity to propagate specific initial states, drawn from the Hamiltonian unbiased basis (HUB). The dynamics of the local entropies and total correlations requires only local measurements in space and therefore is potentially experimentally accessible in a range of platforms.