Perturbative approach to tunneling and quantum interferences in spin clusters

Collective tunneling is a ubiquitous phenomenon in finite-size spin clusters that shows up in systems as diverse as molecular magnets or spin clusters adsorbed at surfaces. The basic problem we explore is to understand how small flipping terms can cooperate to flip a large spin to the opposite direction or a cluster of interacting elementary Ising spins into the time-reversed state. These high-order processes will involve at least two channels, a single spin-flip channel due to a transverse field and a two-spin flip channel due to exchange or other pairwise interactions or due to single-ion anisotropies. In view of the complexity of high-order perturbation theory, nonperturbative approaches based on large-spin path integrals were developed when this problem was first addressed in the context of single spin models. In the present paper, we show that high-order perturbation theory can in fact be formulated and evaluated with the help of simple recurrence relations, leading to a compact theory of tunneling in macroscopic spins, in one-dimensional clusters, as well as in small higher-dimensional clusters. This is demonstrated explicitly in the case of the Ising model with a transverse field and transverse exchange, and in the case of macroscopic spins with uniaxial anisotropy. Our approach provides a transparent theory of level crossings, where the tunneling between time-reversed configurations vanishes as a function of the external field. Those crossings result from the destructive quantum interferences between competing flipping channels. Destructive interferences are expected to be present as soon as the two-spin flip channels have an overall positive amplitude and thus compete with the intrinsically negative second-order processes due to the transverse field. Our theory consistently predicts N crossings in chains of N Ising spins and 2S crossings in single spins of magnitude S and yields explicit analytical formulas for the level crossings of open chains and macroscopic spins. Disorder can be easily implemented in this perturbative formalism. Leading disorder effects can be treated analytically for spin rings. We find that at the smallest transverse field crossing the suppression of tunneling is most robust with respect to disorder and fluctuations in the parameters. We briefly discuss the implications of our findings for the use of realistic spin clusters on surfaces to store information.

Level crossings and chiral transitions in transverse-field Ising models of adatom and Rydberg chains

Ivo Aguiar Maceira

This thesis is motivated by recent experiments on systems described by extensions of the one-dimensional transverse-field Ising (TFI) model where (1) finite-size properties of Ising-ordered phases -- specifically, ground state level crossings -- were observed and (2) continuous phase transitions related to the p-state chiral clock model were probed, with interesting but only partially conclusive results regarding the nature of the phase transitions and the possible existence of a chiral universality class.For (1), the relation of the level crossings to topologically protected edge modes of Majorana fermion models is discussed, and the implication is made explicit that the auto-correlation time of the edge spins may be infinite even for non-integrable albeit finite systems, and independently of temperature. The level crossings are then reinterpreted in the context of degenerate perturbation theory as being a consequence of destructive quantum interference between tunneling processes involving different numbers of spin-flip operations. It is shown that this phenomenon is independent of the lattice geometry, as long as this one is not geometrically frustrated, and is ubiquitous to TFI-like models, being also found in single spin-S systems, the latter having already been observed in the field of magnetic molecules. The effect of disorder on the crossings is studied in lowest order.For (2), we perform density matrix renormalization group simulations on open chains to investigate the experimentally observed quantum phase transitions and we conclude that isolated conformal critical points exist along the p=3 and p=4 critical boundaries: we accurately locate such points and characterize their universality classes by determining critical exponents numerically, where we find that the p=3 agrees with a 3-state Potts universality class and the p=4 point agrees with an Ashkin-Teller universality class with Îœ â 0.80 (Î» â 0.5). Our results are in favor of the existence of chiral transition lines surrounding the conformal points, beyond which a gapless intermediate phase is expected.

EPFL2022

Learning Reach-to-Grasp Motions From Human Demonstrations

Ashwini Shukla

Reaching over to grasp an item is arguably the most commonly used motor skill by humans. Even under sudden perturbations, humans seem to react rapidly and adapt their motion to guarantee success. Despite the apparent ease and frequency with which we use this ability, a complete understanding of the underlying mechanisms cannot be claimed. It is partly due to such incomplete knowledge that adaptive robot motion for reaching and grasping under perturbations is not perfectly achieved. In this thesis, we take the discriminative approach for modelling trajectories of reach-to-grasp motion from expert demonstrations. Throughout this thesis, we will employ time-independent (autonomous) flow based representations to learn reactive motion controllers which can then be ported onto robots. This thesis is divided into three main parts. The first part is dedicated to biologically inspired modelling of reach-to-grasp motions with respect to the hand-arm coupling. We build upon previous work in motion modelling using autonomous dynamical systems (DS) and present a coupled dynamical system (CDS) model of these two subsystems. The coupled model ensures satisfaction of the constraints between the hand and the arm subsystems which are critical to the success of a reach-to-grasp task. Moreover, it reduces the complexity of the overall motion planning problem as compared to considering a combined problem for the hand and the arm motion. In the second part we extend the CDS approach to incorporate multiple grasping points. Such a model is beneficial due to the fact that many daily life objects afford multiple grasping locations on their surface. We combine a DS based approach with energy-function learning to learn a multiple attractor dynamical system where the attractors are mapped to the desired grasping points. We present the Augmented-SVM (ASVM) model that combines the classical SVM formulation with gradient constraints arising from the energy function to learn the desired dynamical function for motion generation. In the last part of this thesis, we address the problem of inverse-kinematics and obstacle avoidance by combining our flow-based motion generator with global configuration-space planners. We claim that the two techniques complement each other. On one hand, the fast reactive nature of our flow based motion generator can used to guide the search of a randomly exploring random tree (RRT) based global planner. On the other hand, global planners can efficiently handle arbitrary obstacles and avoid local minima present in the dynamical function learned from demonstrations. We show that combining the information from demonstrations with global planning in the form of a energy-map considerably decreases the computational complexity of state-of-the-art sampling based planners. We believe that this thesis has the following contributions to Robotics and Machine Learning. First, we have developed algorithms for fast and adaptive motion generation for reach-grasp motions. Second, we formulated an extension to the classical SVM formulation that takes into account the gradient information from data. We showed that instead of being limited as a classifier or a regressor, the SVM framework can be used as a more general function approximation technique. Lastly, we have combined our local methods with global approaches for planning to achieve arbitrary obstacle avoidance and considerable reduction in the computation complexity of the global planners.

EPFL2014

Level crossings and chiral transitions in transverse-field Ising models of adatom and Rydberg chains

Ivo Aguiar Maceira

EPFL2022

Learning Reach-to-Grasp Motions From Human Demonstrations

Ashwini Shukla

EPFL2014