Interface dynamics in the two-dimensional quantum Ising model

Balducci F., Gambassi A., Lerose A., In a recent paper [Phys. Rev. Lett. 129, 120601 (2022)0031-900710.1103/PhysRevLett.129.120601], we have shown that the dynamics of interfaces, in the symmetry-broken phase of the two-dimensional ferromagnetic quantum Ising model, displays a robust form of ergodicity breaking. In this paper, we elaborate more on the issue. First, we discuss two classes of initial states on the square lattice, the dynamics of which is driven by complementary terms in the effective Hamiltonian and may be solved exactly: (a) Strips of consecutive neighboring spins aligned in the opposite direction of the surrounding spins and (b) a large class of initial states, characterized by the presence of a well-defined "smooth"interface separating two infinitely extended regions with oppositely aligned spins. The evolution of the latter states can be mapped onto that of an effective one-dimensional fermionic chain, which is integrable in the infinite-coupling limit. In this case, deep connections with noteworthy results in mathematics emerge, as well as with similar problems in classical statistical physics. We present a detailed analysis of the evolution of these interfaces both on the lattice and in a suitable continuum limit, including the interface fluctuations and the dynamics of entanglement entropy. Second, we provide analytical and numerical evidence supporting the conclusion that the observed nonergodicity - arising from Stark localization of the effective fermionic excitations - persists away from the infinite-Ising-coupling limit, and we highlight the presence of a timescale T∼ecLlnL for the decay of a region of large linear size L. The implications of our work for the classic problem of the decay of a false vacuum are also discussed.

Slow melting of a disordered quantum crystal

Balducci F., The melting of the corner of a crystal is a classical, real-world, nonequilibrium statistical mechanics problem which has shown several connections with other branches of physics and mathematics. For a perfect, classical crystal in two and three dimensions the solution is known: The crystal melts reaching a certain asymptotic shape, which keeps expanding ballistically. In this paper, we move onto the quantum realm and show that the presence of quenched disorder slows down severely the melting process. Nevertheless, we show that there is no many-body localization transition, which could impede the crystal to be completely eroded. We prove such claim both by a perturbative argument, using the forward approximation, and via numerical simulations. At the same time we show how, despite the lack of localization, the erosion dynamics is slowed from ballistic to logarithmic, therefore pushing the complete melting of the crystal to extremely long timescales.

Localization and Melting of Interfaces in the Two-Dimensional Quantum Ising Model

Balducci F., Gambassi A., Lerose A., We study the nonequilibrium evolution of coexisting ferromagnetic domains in the two-dimensional quantum Ising model - a setup relevant in several contexts, from quantum nucleation dynamics and false-vacuum decay scenarios to recent experiments with Rydberg-atom arrays. We demonstrate that the quantum-fluctuating interface delimiting a large bubble can be studied as an effective one-dimensional system through a "holographic"mapping. For the considered model, the emergent interface excitations map to an integrable chain of fermionic particles. We discuss how this integrability is broken by geometric features of the bubbles and by corrections in inverse powers of the ferromagnetic coupling, and provide a lower bound to the timescale after which the bubble is ultimately expected to melt. Remarkably, we demonstrate that a symmetry-breaking longitudinal field gives rise to a robust ergodicity breaking in two dimensions, a phenomenon underpinned by Stark many-body localization of the emergent fermionic excitations of the interface.

Spatiotemporal heterogeneity of entanglement in many-body localized systems

Artiaco C., Balducci F., Heyl M., Russomanno A., We propose a spatiotemporal characterization of the entanglement dynamics in many-body localized (MBL) systems, which exhibits a striking resemblance to dynamical heterogeneity in classical glasses. Specifically, we find that the relaxation times of local entanglement, as measured by the concurrence, are spatially correlated yielding a dynamical length scale for quantum entanglement. As a consequence of this spatiotemporal analysis, we observe that the considered MBL system is made up of dynamically correlated clusters with a size set by this entanglement length scale. The system decomposes into compartments of different activity such as active regions with fast quantum entanglement dynamics and inactive regions where the dynamics is slow. We further find that the relaxation times of the on-site concurrence become broadly distributed and more spatially correlated, as disorder increases or the energy of the initial state decreases. Through this spatiotemporal characterization of entanglement, our work unravels a previously unrecognized connection between the behavior of classical glasses and the genuine quantum dynamics of MBL systems.

Localization in the Discrete Non-linear Schrödinger Equation and Geometric Properties of the Microcanonical Surface

Arezzo C., Balducci F., Piergallini R., It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schrödinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.

Optimal quantum control of a spin qubit in diamond for biosensing

Hernandez-Gomez S., Balducci F., Cappellaro P., The nitrogen-vacancy (NV) center in diamond is a quantum defect in diamond with unique properties for use in high-sensitive, high-resolution quantum sensors of magnetic fields. One of the most interesting and challenging application of NV quantum sensors is nanoscale magnetic resonance imaging (nano-MRI), which would enable to address single biomolecules. To this goal, improving the sensitivity of the NV sensor is a crucial task. Here, we present a quantum optimal control method that optimizes the sensitivity of NV sensor to specific weak magnetic signals with biologically-relevant, complex spectrum. The method, based on the mapping of the sensing problem on a problem of energy optimization of an Ising chain, allows us to improve sensitivity by three orders of magnitude compared to standard control sequences.

Entropic barriers as a reason for hardness in both classical and quantum algorithms

Bellitti M., Ricci-Tersenghi F., We study both classical and quantum algorithms to solve a hard optimization problem, namely 3-XORSAT on 3-regular random graphs. By introducing a new quasi-greedy algorithm that is not allowed to jump over large energy barriers, we show that the problem hardness is mainly due to entropic barriers. We study, both analytically and numerically, several optimization algorithms, finding that entropic barriers affect in a similar way classical local algorithms and quantum annealing. For the adiabatic algorithm, the difficulty we identify is distinct from that of tunneling under large barriers, but does, nonetheless, give rise to exponential running (annealing) times.

Constraint-Induced Delocalization

Sierant P., Lazo E.G., Dalmonte M., We study the impact of quenched disorder on the dynamics of locally constrained quantum spin chains, that describe 1D arrays of Rydberg atoms in both the frozen (Ising-type) and dressed (XY-type) regime. Performing large-scale numerical experiments, we observe no trace of many-body localization even at large disorder. Analyzing the role of quenched disorder terms in constrained systems we show that they act in two, distinct and competing ways: as an on-site disorder term for the basic excitations of the system, and as an interaction between excitations. The two contributions are of the same order, and as they compete (one towards localization, the other against it), one does never enter a truly strong disorder, weak interaction limit, where many-body localization occurs. Such a mechanism is further clarified in the case of XY-type constrained models: there, a term which would represent a bona fide local quenched disorder term acting on the excitations of the clean model must be written as a series of nonlocal terms in the unconstrained variables. Our observations provide a simple picture to interpret the role of quenched disorder that could be immediately extended to other constrained models or quenched gauge theories.

Signatures of many-body localization in the dynamics of two-level systems in glasses

Artiaco C., Balducci F., We investigate the quantum dynamics of two-level systems (TLS) in glasses at low temperatures (1 K and below). We study an ensemble of TLSs coupled to phonons. By integrating out the phonons within the framework of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, we derive analytically the explicit form of the interactions among TLSs, and of the dissipation terms. We find that the unitary dynamics of the system shows clear signatures of many-body localization physics. We study numerically the time behavior of the concurrence, which measures pairwise entanglement also in nonisolated systems, and show that it presents a power-law decay both in the absence and in the presence of dissipation, if the latter is not too large. These features can be ascribed to the strong, long-tailed disorder characterizing the distributions of the model parameters. Our findings show that assuming ergodicity when discussing TLS physics might not be justified for all kinds of experiments on low-temperature glasses.

Subdiffusion in a one-dimensional Anderson insulator with random dephasing: Finite-size scaling, Griffiths effects, and possible implications for many-body localization

Taylor S.R., We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by Žnidarič et al. [Phys. Rev. Lett. 117, 040601 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.040601]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.160401], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of "Griffiths effects."We show that the finite-size scaling comes from the interplay of three characteristic lengths: One associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.

Quantum jamming: Critical properties of a quantum mechanical perceptron

Artiaco C., Balducci F., Parisi G., In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrained on an N-dimensional sphere, with N→∞, and subjected to a set of randomly placed hard-wall potentials. This model has several applications, ranging from learning protocols to the effective description of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space. We find that the jamming transition with quantum dynamics shows critical exponents different from the classical case. We also find that the quantum jamming transition, unlike the typical quantum critical points, is not confined to the zero-temperature axis, and the classical results are recovered only at T=∞. Our findings have implications for the theory of glasses at ultralow temperatures and for the study of quantum machine-learning algorithms.

Relaxation to equilibrium in controlled- not quantum networks

Novotný J., Mariano A., Pascazio S., The approach to equilibrium of quantum mechanical systems is a topic as old as quantum mechanics itself, but has recently seen a surge of interest due to applications in quantum technologies, including, but not limited to, quantum computation and sensing. The mechanisms by which a quantum system approaches its long-time, limiting stationary state are fascinating and, sometimes, quite different from their classical counterparts. In this respect, quantum networks represent mesoscopic quantum systems of interest. In such a case, the graph encodes the elementary quantum systems (say qubits) at its vertices, while the links define the interactions between them. We study here the relaxation to equilibrium for a fully connected quantum network with controlled-not (cnot) gates representing the interaction between the constituting qubits. We give a number of results for the equilibration in these systems, including analytic estimates. The results are checked using numerical methods for systems with up to 15-16 qubits. It is emphasized in which way the size of the network controls the convergency.

Breakdown of ergodicity in disordered U(1) lattice gauge theories

Giudici G., Surace F.M., Ebot J.E., We show how U(1) lattice gauge theories display key signatures of ergodicity breaking in the presence of a random charge background. We argue that, in such gauge theories, there is a cooperative effect of disorder and interactions in favoring ergodicity breaking: This is due to the confining nature of the Coulomb potential, which suppresses the number of available energy resonances at all distances. Such a cooperative mechanism reflects into very modest finite-volume effects: This allows us to draw a sharp boundary for the ergodic regime, and thus the breakdown of quantum chaos for sufficiently strong gauge couplings, at system sizes accessible via exact diagonalization. Our conclusions are independent on the value of a background topological angle, and are contrasted with a gauge theory with truncated Hilbert space, where instead we observe very strong finite-volume effects akin to those observed in spin chains.

Non-Abelian Symmetries and Disorder: A Broad Nonergodic Regime and Anomalous Thermalization

Protopopov I.V., Panda R.K., Parolini T., Previous studies reveal a crucial effect of symmetries on the properties of a single particle moving in a disorder potential. More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest. MBL systems are characterized by the emergence of quasilocal integrals of motion and by the area-law entanglement entropy scaling of its eigenstates. In this paper, we investigate the effect of a non-Abelian SU(2) symmetry on the dynamical properties of a disordered Heisenberg chain. While SU(2) symmetry is inconsistent with conventional MBL, a new nonergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still strongly subthermal, scaling of the entanglement entropy. Using extensive exact diagonalization simulations, we establish that this nonergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use the real-space renormalization group (RSRG) to construct approximate excited eigenstates by tree tensor networks and demonstrate the accuracy of this procedure for systems of sizes up to L=26. As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by the RSRG. This approach is capable of describing systems of size L≈2000. We characterize the resonances that arise due to such processes, finding that they involve an ever-growing number of spins as the system size is increased. Crucially, the probability of finding resonances grows with the system's size. Even at strong disorder, we can identify a large length scale beyond which resonances proliferate. Presumably, this proliferation would eventually drive the system to a thermalizing phase. However, the extremely long thermalization timescales indicate that a broad nonergodic regime will be observable experimentally. Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems, well beyond sizes accessible in previous studies.

Subdiffusion in the Anderson model on the random regular graph

De Tomasi G., Bera S., We study the finite-Time dynamics of an initially localized wave packet in the Anderson model on the random regular graph (RRG) and show the presence of a subdiffusion phase coexisting both with ergodic and putative nonergodic phases. The full probability distribution Π(x,t) of a particle to be at some distance x from the initial state at time t is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on Zd lattices, d>2, which is subdiffusive only at the critical point implies that the limit d→∞ is highly singular in terms of the dynamics. A detailed analysis of the propagation of Π(x,t) in space-Time (x,t) domain identifies four different regimes determined by the position of a wave front Xfront(t), which moves subdiffusively to the most distant sites Xfront(t)∼tβ with an exponent β<1. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.

Phenomenology of anomalous transport in disordered one-dimensional systems

Schulz M., Taylor S.R., We study anomalous transport arising in disordered one-dimensional spin chains, specifically focusing on the subdiffusive transport typically found in a phase preceding the many-body localization transition. Different types of transport can be distinguished by the scaling of the average resistance with system's length. We address the following question: What is the distribution of resistance over different disorder realizations, and how does it differ between transport types? In particular, an often evoked so-called Griffiths picture, that aims to explain slow transport as being due to rare regions of high disorder, would predict that the diverging resistivity is due to fat power-law tails in the resistance distribution. Studying many-particle systems with and without interactions we do not find any clear signs of fat tails. The data is compatible with distributions that decay faster than any power law required by the fat tails scenario. Among the distributions compatible with the data, a simple additivity argument suggests a Gaussian distribution for a fractional power of the resistance.

Anderson transition on the Bethe lattice: An approach with real energies

Parisi G., Pascazio S., Pietracaprina F., Ros V., We study the Anderson model on the Bethe lattice by working directly with propagators at real energies E. We introduce a novel criterion for the localization-delocalization transition based on the stability of the population of the propagators, and show that it is consistent with the one obtained through the study of the imaginary part of the self-energy. We present an accurate numerical estimate of the transition point, as well as a concise proof of the asymptotic formula for the critical disorder on lattices of large connectivity, as given in Anderson (1958 Phys. Rev. 109 1492-505). We discuss how the forward approximation used in analytic treatments of localization problems fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.

Phase diagram of bipartite entanglement

Facchi P., Parisi G., Pascazio S., We investigate the features of the entanglement spectrum (distribution of the eigenvalues of the reduced density matrix) of a large quantum system in a pure state. We consider all Rényi entropies and recover purity and von Neumann entropy as particular cases. We construct the phase diagram of the theory and unveil the presence of two critical lines.

Asymmetry in energy versus spin transport in certain interacting disordered systems

Mendoza-Arenas J., Žnidarič M., Varma V., Goold J., Clark S., We study energy transport in disordered XXZ spin-1/2 chains driven to nonequilibrium configurations by thermal reservoirs of different temperatures at the boundaries, using large-scale matrix product simulations. In particular we discuss the transition between diffusive and subdiffusive transport in sectors of zero and finite magnetization at high temperature. At large anisotropies we find that diffusive energy transport prevails over a large range of disorder strengths, which is in contrast to spin transport that is subdiffusive in the same regime for weak disorder. However, at finite magnetization both energy and spin currents decay as a function of the system size with the same exponent. We conclude that diffusion of energy is much more pervasive than that of magnetization in these disordered spin-1/2 systems, and occurs across a significant range of the interaction-disorder parameter phase space. We support the existence of this asymmetry, reminiscent of that in the clean limit, by an analytical estimation of diffusion constants for weak disorder.

Massively parallel implementation and approaches to simulate quantum dynamics using Krylov subspace techniques

Brenes M., Varma V., We have developed an application and implemented parallel algorithms in order to provide a computational framework suitable for massively parallel supercomputers to study the unitary dynamics of quantum systems. We use renowned parallel libraries such as PETSc/SLEPc combined with high-performance computing approaches in order to overcome the large memory requirements to be able to study systems whose Hilbert space dimension comprises over 9 billion independent quantum states. Moreover, we provide descriptions of the parallel approach used for the three most important stages of the simulation: handling the Hilbert subspace basis, constructing a matrix representation for a generic Hamiltonian operator and the time evolution of the system by means of the Krylov subspace methods. We employ our setup to study the evolution of quasidisordered and clean many-body systems, focussing on the return probability and related dynamical exponents: the large system sizes accessible provide novel insights into their thermalization properties. Program summary: Program Title: DSQMKryST Program Files doi: http://dx.doi.org/10.17632/f6vty3wkwj.1 Licensing provisions: BSD 3-clause Programming language: C++ Supplementary material: https://github.com/mbrenesn/DSQMKryST External routines/libraries: PETSc (https://www.mcs.anl.gov/petsc/), SLEPc (http://slepc.upv.es), Boost C++ (http://www.boost.org) Nature of problem: Unitary dynamics of quantum mechanical many-body systems Solution method: Krylov subspace techniques (Arnoldi procedure) with a massively parallel, distributed memory approach