We propose and analyze a new approach to the coherent control and manipulation of quantum degrees of freedom in disordered, interacting systems in the many-body localized phase. Our approach leverages a number of unique features of many-body localization: a lack of thermalization, a locally gapped spectrum, and slow dephasing. Using the technique of quantum phase estimation, we demonstrate a protocol that enables the local preparation of a many-body system into an effective eigenstate. This leads to the ability to encode information and control interactions without full microscopic knowledge of the underlying Hamiltonian. Finally, we analyze the effects of weak coupling to an external bath and provide an estimate for the fidelity of our protocol.
We investigate dynamical quantum phase transitions in disordered quantum many-body models that can support many-body localized phases. Employing $l$-bits formalism, we lay out the conditions for which singularities indicative of the transitions appear in the context of many-body localization. Using the combination of the mapping onto $l$-bits and exact diagonalization results, we explicitly demonstrate the presence of these singularities for a candidate model that features many-body localization. Our work paves the way for understanding dynamical quantum phase transitions in the context of man…
Dynamical localization is the analog of Anderson localization in momentum space, where the system's energy saturates and the single-particle wave-functions are exponentially localized in momentum space. In the presence of interactions, in the context of a periodically kicked Bose gas, it has been argued that dynamical localization persists. Focusing on the Tonks (strongly interacting) regime, we show that the many-body dynamically localized phase is effectively thermal, a clear deviation from the breaking of ergodicity observed in standard many-body localized systems. We relate the effective t…
Tensor networks are a powerful formalism for transforming one set of degrees of freedom to another. They have been heavily used in analyzing the geometry of bulk/boundary correspondence in conformal field theories. Here we develop a tensor-network version of the Wilson-Wegner Renormalization Group Flow equations to efficiently generate a unitary tensor network which diagonalizes many-body localized Hamiltonians. Treating this unitary tensor network as a bulk geometry, we find this emergent geometry corresponds to the shredded horizon picture: the circumference of the network shrinks exponentia…
Adding interactions to many-body Hamiltonians of geometrically frustrated lattices often leads to diminished subspaces of localized states. In this paper, we show how to construct interacting many-body Hamiltonians, starting from the non-interacting tight-binding Hamiltonians, that preserve or even expand these subspaces. The methods presented involve modifications in the one-body network representation of the many-body Hamiltonians which generate new interacting terms in these Hamiltonians. The subspace of many-particle localized states can be preserved in the interacting Hamiltonian, by proj…
We show that the presence of a harmonic trap may in itself lead to many-body localization for cold atoms confined in that trap in a quasi-one-dimensional geometry. Specifically, the coexistence of delocalized phase in the center of the trap with localized region closer to the edges is predicted with the borderline dependent on the curvature of the trap. The phenomenon, similar in its origin to Stark localization, should be directly observed with cold atomic species. We discuss both the spinless and the spinful fermions, for the latter we address Stark localization at the same time as it has no…
We compute and compare the decay lengths of several correlation functions and effective coupling constants in the many-body localized (MBL) phase. To this end, we consider the distribution of the logarithms of these couplings and correlators: in each case the log-coupling follows a normal distribution with mean and variance that grow linearly with separation. Thus, a localization length is asymptotically sharply defined for each of these quantities. These localization lengths differ numerically from one another, but all of them remain short up to the numerically observed MBL transition, indica…
Many-body localization is a striking mechanism that prevents interacting quantum systems from thermalizing. The absence of thermalization behaviour manifests itself, for example, in a remanence of local particle number configurations, a quantity that is robust over a parameter range. Local particle numbers are directly accessible in programmable quantum simulators, in systems of cold atoms even in two spatial dimensions. Yet, the classical simulation aimed at building trust in quantum simulations is highly challenging. In this work, we present a comprehensive tensor network simulation of a man…
Many-body localized systems in which interactions and disorder come together defy the expectations of quantum statistical mechanics: In contrast to ergodic systems, they do not thermalize when undergoing nonequilibrium dynamics. What is less clear, however, is how topological features interplay with many-body localized phases as well as the nature of the transition between a topological and a trivial state within the latter. In this work, we numerically address these questions, using a combination of extensive tensor network calculations, specifically DMRG-X, as well as exact diagonalization, …
We investigate a spatial subsystem entropy extracted from the one-particle density matrix (OPDM) in one-dimensional disordered interacting fermions that host a many-body localized (MBL) phase. Deep in the putative MBL regime, this OPDM entropy exhibits the salient features of localization, despite not being a proper entanglement measure. We numerically show that the OPDM entropy of the eigenstates obeys an area law. Similar to the von-Neumann entropy, the OPDM entropy grows logarithmically with time after a quantum quench, albeit with a different prefactor. Both these features survive at moder…
Quantum coherence quantifies the amount of superposition a quantum state can have in a given basis. Since there is a difference in the structure of eigenstates of the ergodic and many-body localized systems, we expect them also to differ in terms of their coherences in a given basis. Here, we numerically calculate different measures of quantum coherence in the excited eigenstates of an interacting disordered Hamiltonian as a function of the disorder. We show that quantum coherence can be used as an order parameter to detect the well-studied ergodic to many-body-localized phase transition. We a…
Strongly disordered systems in the many-body localized (MBL) phase can exhibit ground state order in highly excited eigenstates. The interplay between localization, symmetry, and topology has led to the characterization of a broad landscape of MBL phases ranging from spin glasses and time crystals to symmetry protected topological phases. Understanding the nature of phase transitions between these different forms of eigenstate order remains an essential open question. Here, we conjecture that no direct transition between distinct MBL orders can occur; rather, a thermal phase always intervenes.…
The discrete time crystal (DTC) is a recently discovered phase of matter that spontaneously breaks time-translation symmetry. Disorder-induced many-body-localization is required to stabilize a DTC to arbitrary times, yet an experimental investigation of this localized regime has proven elusive. Here, we observe the hallmark signatures of a many-body-localized DTC using a novel quantum simulation platform based on individually controllable $^{13}$C nuclear spins in diamond. We demonstrate the characteristic long-lived spatiotemporal order and confirm that it is robust for generic initial states…
Many-body localized (MBL) systems are often described using their local integrals of motion, which, for spin systems, are commonly assumed to be a local unitary transform of the set of on-site spin-z operators. We show that this assumption cannot hold for topologically ordered MBL systems. Using a suitable definition to capture such systems in any spatial dimension, we demonstrate a number of features, including that MBL topological order, if present: (i) is the same for all eigenstates; (ii) is robust in character against any perturbation preserving MBL; (iii) implies that on topologically no…
We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we show that the (generalized) third cohomology class of the symmetry group is a topological invariant; however our approach leaves room for the existence of additional topological indices. We argue that our classification applies to quasi-periodic systems in two dimensions and systems with true random disorder within times which scale superexponentially with the inverse interaction strength. Our tech…
Area law in the exact solution of many-body localized systems
通过构建使哈密顿量对角化的准局部酉旋转，在现实假设下证明了多体局域化（Imbrie，2016）。一个自然的推广是考虑所有具有相似结构的酉。我们对由这种幺正产生的态进行了纠缠束缚，从而为多体局域系统本征态的面积定律提供了独立的证明。得到了用有限深度局部电路近似幺正的一个误差。我们将定义的幺正族与其他关于多体局域化的结果（Kim et al，2014）联系起来，特别是Lieb-Robinson束缚。最后，我们认为，任何哈密尔顿可以对角化这样一个酉，它有一个足够慢的对数光锥在其Lieb-Robinson界限。
The collective and quantum behavior of many-body systems may be harnessed to achieve fast charging of energy storage devices, which have been recently dubbed quantum batteries. In this paper, we present an extensive numerical analysis of energy flow in a quantum battery described by a disordered quantum Ising chain Hamiltonian, whose equilibrium phase diagram presents many-body localized (MBL), Anderson localized (AL), and ergodic phases. We demonstrate that i) the low amount of entanglement of the MBL phase guarantees much better work extraction capabilities than the ergodic phase and ii) int…
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 …
As disorder strength increases in quantum many-body systems a new phase of matter, the so-called anybody localization, emerges across the whole spectrum. This transition is energy dependent, a phenomenon known as mobility edge, such that the mid-spectrum eigenstates tend to localize at larger values of disorder in comparison to eigenstates near the edges of the spectrum. Many-body localization becomes more sophisticated in long-range interacting systems. Here, by focusing on several quantities, we draw the phase diagram as a function of disorder strength and energy spectrum, for a various rang…
Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called l-bits. These spin algebras define quasilocal l-bit measurement ($τ^z_i$) and l-bit flip ($τ^x_i$) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate l-bit flip operators by finding them exactly on small windows of systems and extending them onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates. We then describe a method to calculate products of local obser…
One of the defining features of many-body localization is the presence of extensively many quasi-local conserved quantities. These constants of motion constitute a corner-stone to an intuitive understanding of much of the phenomenology of many-body localized systems arising from effective Hamiltonians. They may be seen as local magnetization operators smeared out by a quasi-local unitary. However, accurately identifying such constants of motion remains a challenging problem. Current numerical constructions often capture the conserved operators only approximately restricting a conclusive unders…
The emergent integrability in a many-body localized (MBL) system can be well characterized by the existence of the complete set of local integrals of motion (LIOMs). Such exactly conserved and exponentially localized operators are often understood as quasiparticle operators which can be expanded in terms of single-particle operators dressed with different numbers of particle-hole pairs. Here, we consider a one-dimensional XXZ spin-$\frac12$ Heisenberg chain in the presence of a random field and try to quantify the corrections needed to be considered in the picture of quasiparticles associated …
When pushed out of equilibrium, generic interacting quantum systems equilibrate locally and are expected to evolve towards a locally thermal description despite their unitary time evolution. Systems in which disorder competes with interactions and transport can violate this expectation by exhibiting many-body localization. The strength of the disorder with respect to the other two parameters drives a transition from a thermalizing system towards a non-thermalizing one. The existence of this transition is well established both in experimental and numerical studies for finite systems. However, t…
We construct almost conserved local operators, that possess a minimal commutator with the Hamiltonian of the system, near the many-body localization transition of a one-dimensional disordered spin chain. We collect statistics of these slow operators for different support sizes and disorder strengths, both using exact diagonalization and tensor networks. Our results show that the scaling of the average of the smallest commutators with the support size is sensitive to Griffiths effects in the thermal phase and the onset of many-body localization. Furthermore, we demonstrate that the probability …
We consider a many-body localized system coupled globally to a central $d$-level system. Under an appropriate scaling of $d$ and $L$, we find evidence that the localized phase survives. We argue for two possible thermalizing phases, depending on whether the qudit becomes fully ergodic. This system provides one of the first examples of many-body localization in the presence of long-range (non-confining) interactions.
The combination of strong disorder and interactions in closed quantum systems can lead to many-body localization (MBL). However this quantum phase is not stable when the system is coupled to a thermal environment. We investigate how MBL is destroyed in systems that are weakly coupled to a dephasive Markovian environment by focusing on their entanglement dynamics. We numerically study the third Rényi negativity $R_3$, a recently proposed entanglement proxy based on the negativity that captures the unbounded logarithmic growth in the closed case and that can be computed efficiently with tensor n…
We show how second-order Floquet engineering can be employed to realize systems in which many-body localization coexists with topological properties in a driven system. This allows one to implement and dynamically control a topologically-protected qubit even at high energies. Floquet engineering - the idea that a periodically driven non-equilibrium system can effectively emulate the physics of a different Hamiltonian - is used to simulate an ffective three-body interaction among spins in one dimension, using time-dependent two-body interactions only. In the effective system emulated topology a…
Entanglement Growth from Structured Initial States in Many-Body Localized Systems
School of Physics, Renmin University of China（中国人民大学物理学院）；State Key Laboratory of Surface Physics & Department of Physics, Fudan University（复旦大学表面物理国家重点实验室及物理系）；Hefei National Laboratory（合肥国家实验室）
Efficient representation of fully many-body localized systems using tensor networks
我们提出了一个张量网络编码的一个完全多体局域系统在一维的所有本征态的集合。我们的构造在概念上基于Phys. Rev. B 94，041116（R）（2016）中引入的矩阵，由两层酉矩阵构建，这些酉矩阵作用于$\ell$连续位点的块。我们认为，这会产生一个指数级的减少计算时间和内存需求相比，所有以前的方法找到一个代表性的完整的本征谱的大型多体局域化系统具有给定的精度。具体地说，我们通过最小化近似运动积分和哈密顿量的交换子的大小来优化幺正，这可以在局部方式中完成。与以前的工作相比，这进一步降低了最小化过程中产生的张量网络的计算复杂性。我们通过比较16个地点的随机场海森堡模型的近似能谱与精确对角化结果来测试我们方法的准确性。我们发现，该技术是高度准确的本地化制度，并保持了令人惊讶的准确度，甚至在预测的动态相变附近的某些本地量预测。为了证明我们的技术的力量，我们研究了一个系统的72个网站，我们能够看到清晰的签名的相变。我们的工作为研究大系统中的多体局域化跃迁性质开辟了一条新的途径。
We focus on the many-body eigenstates across a localization-delocalization phase transition. To characterize the robustness of the eigenstates, we introduce the eigenstate overlaps $\mathcal{O}$ with respect to the different boundary conditions. In the ergodic phase, the average of eigenstate overlaps $\bar{\mathcal{O}}$ is exponential decay with the increase of the system size indicating the fragility of its eigenstates, and this can be considered as an eigenstate-version butterfly effect of the chaotic systems. For localized systems, $\bar{\mathcal{O}}$ is almost size-independent showing the…
