Using projected entangled-pair states (PEPS) we analyze the localization properties of two-dimensional systems on a square lattice. We compare the dynamics found for three different disorder types: (i) quenched disorder, (ii) sum of two quasi-periodic potentials along both spatial dimensions and (iii) a single quasi-periodic potential rotated with respect to the underlying lattice by a given angle. We establish the rate of loss of information, a quantity measuring the error made while simulating the dynamics, as a good hallmark of localization physics by comparing to entanglement build-up as well as the inverse participation ratio in exactly solvable limits. We find that the disorder strength needed to localize the system increases both with the dimensionality of as well as the interaction strength in the system. The first two cases of potential (i) and (ii) behave similar, while case (iii) requires larger disorder strength to localize.
Using projected entangled-pair states (PEPS) we analyze the localization properties of two-dimensional systems on a square lattice. We compare the dynamics found for three different disorder types: (i) quenched disorder, (ii) sum of two quasi-periodic potentials along both spatial dimensions and (iii) a single quasi-periodic potential rotated with respect to the underlying lattice by a given angle. We establish the rate of loss of information, a quantity measuring the error made while simulating the dynamics, as a good hallmark of localization physics by comparing to entanglement build-up as well as the inverse participation ratio in exactly solvable limits. We find that the disorder strength needed to localize the system increases both with the dimensionality of as well as the interaction strength in the system. The first two cases of potential (i) and (ii) behave similar, while case (iii) requires larger disorder strength to localize.
Many-body localization is a striking mechanism that prevents interacting quantum systems from thermalizing. The absence of thermalization behavior 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 many-body localized systems in two spatial dimensions using a variant of an infinite projected entangled pair states algorithm. The required translational invariance can be restored by implementing the disorder into an auxiliary spin system, providing an exact disorder average under dynamics. We can quantitatively assess signatures of many-body localization for the infinite system: Our methods are powerful enough to provide crude dynamical estimates for the transition between localized and ergodic phases. Interestingly, in this setting of finitely many disorder values, which we also compare with simulations involving noninteracting fermions and for which we discuss the emergent physics, localization emerges in the interacting regime, for which we provide an intuitive argument, while Anderson localization is absent.
Physics
Disordered interacting quantum many-body systems can become hopelessly localized. This so-called many-body localization has been studied in one-dimensional systems. In two dimensions, experiments indicate that the localization persists, but because it is difficult to tell the difference between no thermalization and slow thermalization, more theoretical work is needed. Relying on an approximate numerical method, Wahl et al. undertook large-scale simulations of a two-dimensional lattice of bosonic atoms in the presence of interactions and disorder. At intermediate disorder strengths, the on-site entanglement entropies exhibited a bimodal distribution, indicating a coexistence of localized and thermalized states; as disorder was increased, localized states took over. The authors were able to extract a critical disorder strength and set a benchmark for future experiments.
Nat. Phys. 10.1038/s41567-018-0339-x (2018).
Infinite projected entangled pair states (iPEPS), the tensor network ansatz for two-dimensional systems in the thermodynamic limit, already provide excellent results on ground-state quantities using either imaginary-time evolution or variational optimisation. Here, we show (i) the feasibility of real-time evolution in iPEPS to simulate the dynamics of an infinite system after a global quench and (ii) the application of disorder-averaging to obtain translationally invariant systems in the presence of disorder. To illustrate the approach, we study the short-time dynamics of the square lattice Heisenberg model in the presence of a bi-valued disorder field.
We study quantum quenches in two-dimensional lattice gauge theories with fermions coupled to dynamical Z2 gauge fields. Through the identification of an extensive set of conserved quantities, we propose a generic mechanism of charge localization in the absence of quenched disorder both in the Hamiltonian and in the initial states. We provide diagnostics of this localization through a set of experimentally relevant dynamical measures, entanglement measures, as well as spectral properties of the model. One of the defining features of the models that we study is a binary nature of emergent disorder, related to Z2 degrees of freedom. This results in a qualitatively different behavior in the strong disorder limit compared to typically studied models of localization. For example, it gives rise to a possibility of a delocalization transition via a mechanism of quantum percolation in dimensions higher than 1D. We highlight the importance of our general phenomenology to questions related to dynamics of defects in Kitaev's toric code, and to quantum quenches in Hubbard models. While the simplest models we consider are effectively noninteracting, we also include interactions leading to many-body localizationlike logarithmic entanglement growth. Finally, we consider effects of interactions that generate dynamics for conserved charges, which gives rise to only transient localization behavior, or quasi-many-body localization.
We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with projected entangled pair states (PEPS) as a function of the bond dimension (D), temperature (${\ensuremath{\beta}}^{\ensuremath{-}1}$), and system size (N). First, we introduce a compression method in which the bond dimension scales as $D={e}^{O({log}_{2}^{2}(N/\ensuremath{\epsilon}))}$ if $\ensuremath{\beta}<O({log}_{2}N)$. Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D={(N/\ensuremath{\epsilon})}^{O(\ensuremath{\beta})}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with D=NO(log2N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.