Universal High-Frequency Behavior of Periodically Driven Systems: from Dynamical Stabilization to Floquet Engineering

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer-Wolff transformation extending the latter to driven systems.Comment: 84 pages, 25 figures, 4 appendice

AFIT School of Engineering Contributions to Air Force Research and Technology. Calendar Year 1971

This report contains abstracts of Master of Science theses and Doctoral Dissertations completed during the 1971 calendar year at the School of Engineering, Air Force Institute of Technology

Solution of the Schrodinger equation for quasi-one-dimensional materials using helical waves

We formulate and implement a spectral method for solving the Schrodinger equation, as it applies to quasi-one-dimensional materials and structures. This allows for computation of the electronic structure of important technological materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons, chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and systematic manner. Our work is motivated by the observation that one of the most successful methods for carrying out electronic structure calculations of bulk/crystalline systems -- the plane-wave method -- is a spectral method based on eigenfunction expansion. Our scheme avoids computationally onerous approximations involving periodic supercells often employed in conventional plane-wave calculations of quasi-one-dimensional materials, and also overcomes several limitations of other discretization strategies, e.g., those based on finite differences and atomic orbitals. We describe the setup of fast transforms to carry out discretization of the governing equations using our basis set, and the use of matrix-free iterative diagonalization to obtain the electronic eigenstates. Miscellaneous computational details, including the choice of eigensolvers, use of a preconditioning scheme, evaluation of oscillatory radial integrals and the imposition of a kinetic energy cutoff are discussed. We have implemented these strategies into a computational package called HelicES (Helical Electronic Structure). We demonstrate the utility of our method in carrying out systematic electronic structure calculations of various quasi-one-dimensional materials through numerous examples involving nanotubes, nanoribbons and nanowires. We also explore the convergence, accuracy and efficiency of our method. We anticipate that our method will find numerous applications in computational nanomechanics and materials science

Synchrotron Radiation Micro-CT Imaging of Bone Tissue

International audienc

Tensor network methods for low-dimensional quantum systems

This thesis contributes to developing and applying tensor network methods to simulate correlated many-body quantum systems. Numerical simulations of correlated quantum many-body systems are challenging. To describe a many-body wavefunction, the required number of parameters grows exponentially with respect to the system size. This exponential wall fundamentally limits our progress on correlated quantum systems in low dimensions. Tensor network methods in recent years have proven to be a useful framework to understand, control and possibly reduce this intrinsic complexity. The basic idea of tensor network methods is to decompose a many-body wave function as a network of small, multi-index tensors. A one-dimensional (1D) tensor network factorizes a wave function into a train of three-index tensors. This 1D tensor network ansatz is called a matrix product state (MPS) or a tensor train. A two-dimensional (2D) tensor network state is called a projected entangled-pair state (PEPS). This peculiar name PEPS comes from a quantum information perspective, where each local tensor is interpreted as a projector and correlates with the rest of the tensor network through (auxiliary) maximally entangled pairs. In the first part, we consider MPSs to study 1D and quasi-2D quantum systems. The key parameter of an MPS is its bond dimension, which controls the numerical accuracy. How large a bond dimension can be reached highly depends on the algorithms employed. The contemporary algorithms, although widely used, have to limit the bond dimension due to their high numerical costs. We develop a controlled bond expansions (CBE) scheme that allows us to grow the bond dimensions with marginal computational efforts. This CBE scheme stems from a geometric point of view to parametrize the variational space of an MPS and can be applied in various contexts. Here, we focus on applying the CBE scheme to two types of problems. The first are optimization problems, like solving the extremal eigenvalue problem. This is relevant for the ground state search, and we show that CBE can accelerate the convergence of MPS in terms of CPU time. The second is to solve ordinary differential equations, such as the time-dependant Schrödinger equation. With the help of CBE, it becomes feasible to use MPS to simulate long-time dynamics that could not be accurately computed hitherto. In the second part, we employ PEPS to simulate 2D quantum systems. PEPS is an expensive but powerful tool to simulate 2D lattices directly in the thermodynamic limit. The PEPS on infinite lattices is acronymed iPEPS. For completeness, a pedagogical review of iPEPS based on Benedikt Bruognolo’s PhD work, which I helped polsih for publication in Scipost, is included to cover the algorithmic details. Using iPEPS methods, we study the two-dimensional t-J model on square lattices at the small doping. In this work, we uncover the importance of spin rotational symmetry. Our numerics suggest that by allowing spontaneous spin-symmetry breaking or not, we can supress or permit the emergence of superconducting order in the thermodynamic limit. This finding provides useful insight to cuprate materials. Also, we use iPEPS to investigate the ground state nature of the honeycomb Kitaev-Γ model. Through a joint effort of classical and iPEPS simulations, we identify an exotic magnetic order in the parameter regime relevant to α-RuCl3 materials. In the third and final part, we study the parton construction of tensor network states. Here, we do not simulate the ground state of a given many-body Hamiltonian. Instead, we take an indirect route that first constructs a parton state in an enlarged Hilbert space, and then applies the Gutzwiller projection to return to the original physical Hilbert space. Such a parton approach has been an important theoretical technique to treat electron-electron correlations nonperturbatively in condensed matter physics. Its marriage with tensor network methods furthers its influence. Various properties of parton wave functions, which are difficult to compute previously, can now be easily accessed. We first use the parton approach to construct MPSs that harbor SU(N) chiral topological orders. The MPS representation of these Gutzwiller projected parton states allows us to compute entanglement spectra, which hold crucial information to characterize different chiral topological orders. We also develop a method to construct parton states using PEPSs. In this project, we use PEPS to approximate parton states of the π-flux models that host U(1)-Dirac spin liquids. Our approach enables us to compute the critical exponent of the spin-spin correlations for the spin-half system, whose value is still currently under debate

Local and Bulk Measurements in Novel Magnetically Frustrated Materials:

Thesis advisor: Michael J. GrafQuantum spin liquids (QSL)’s have been one of the most hotly researched areas ofcondensed matter physics for the past decade. Yet, science has yet to unconditionally identify any one system as harboring a QSL state. This is because QSL’s are largely defined as systems whose electronic spins do not undergo a thermodynamic transition as T→0. Quantum spin liquids remain fully paramagnetic, including dynamical spin fluctuations, at T=0. As a result, distinguishing a QSL system from a conventionally disordered system remains an outstanding challenge. If a system spin freezes or magnetically orders, it cannot be a QSL. In this thesis I present published experiments I have performed on QSL candidate materials. By using muon spin rotation (μSR) and AC magnetic susceptibility I have evaluated the ground states of several candidates for the absence of long-range magnetic disorder and low-temperature spin-fluctuations. For the systems which order or spin-freeze, my research provided key knowledge to the field of frustrated magnetism. The systems I studied are as follows: The geometrically frustrated systems NaYbO2 and LiYbO2; the Kitaev honeycomb systems Cu2IrO3 and Ag3LiIr2O6; and the metallic kagome system KV3Sb5. Each of these systems brought new physics to the field of frustrated magnetism. NaYbO2 is a promising QSL candidate. LiYbO2 harbors an usual form of spiral incommensurate order that has a staggered transition. Cu2IrO3 has charge state disorder that results in a magnetically inhonogenious state. Ag3LiIr2O6 illustrates the role structural disorder plays in disguising long-range magnetic order. And finally, KV3Sb5 isn’t conventionally magnetic at all; our measurements ruled out ionic magnetism and uncovered a type-II superconductor. Our measurements on KV3Sb5 stimulated further research into KV3Sb5 and it’s unconventional electronic states.Thesis (PhD) — Boston College, 2022.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Physics

Applications of finite reflection groups in Fourier analysis and symmetry breaking of polytopes

Cette thèse présente une étude des applications des groupes de réflexion finis aux problems liés aux réseaux bidimensionnels et aux polytopes tridimensionnels. Plusieurs familles de fonctions orbitales, appelées fonctions orbitales de Weyl, sont associées aux groupes de réflexion cristallographique. Les propriétés exceptionnelles de ces fonctions, telles que l’orthogonalité continue et discrète, permettent une analyse de type Fourier sur le domaine fondamental d’un groupe de Weyl affine correspondant. Dans cette considération, les fonctions d’orbite de Weyl constituent des outils efficaces pour les transformées discrètes de type Fourier correspondantes connues sous le nom de transformées de Fourier–Weyl. Cette recherche limite notre attention aux fonctions d’orbite de Weyl symétriques et antisymétriques à deux variables du groupe de réflexion cristallographique A2. L’objectif principal est de décomposer deux types de transformations de Fourier–Weyl du réseau de poids correspondant en transformées plus petites en utilisant la technique de division centrale. Pour les cas non cristallographiques, nous définissons les indices de degré pair et impair pour les orbites des groupes de réflexion non cristallographique avec une symétrie quintuple en utilisant un remplacement de représentation-orbite. De plus, nous formulons l’algorithme qui permet de déterminer les structures de polytopes imbriquées. Par ailleurs, compte tenu de la pertinence de la symétrie icosaédrique pour la description de diverses molécules sphériques et virus, nous étudions la brisure de symétrie des polytopes doubles de type non cristallographique et des structures tubulaires associées. De plus, nous appliquons une procédure de stellation à la famille des polytopes considérés. Puisque cette recherche se concentre en partie sur les fullerènes icosaédriques, nous présentons la construction des nanotubes de carbone correspondants. De plus, l’approche considérée pour les cas non cristallographiques est appliquée aux structures cristallographiques. Nous considérons un mécanisme de brisure de symétrie appliqué aux polytopes obtenus en utilisant les groupes Weyl tridimensionnels pour déterminer leurs extensions structurelles possibles en nanotubes.This thesis presents a study of applications of finite reflection groups to the problems related to two-dimensional lattices and three-dimensional polytopes. Several families of orbit functions, known as Weyl orbit functions, are associated with the crystallographic reflection groups. The exceptional properties of these functions, such as continuous and discrete orthogonality, permit Fourier-like analysis on the fundamental domain of a corresponding affine Weyl group. In this consideration, Weyl orbit functions constitute efficient tools for corresponding Fourier-like discrete transforms known as Fourier–Weyl transforms. This research restricts our attention to the two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2. The main goal is to decompose two types of the corresponding weight lattice Fourier–Weyl transforms into smaller transforms using the central splitting technique. For the non-crystallographic cases, we define the even- and odd-degree indices for orbits of the non-crystallographic reflection groups with 5-fold symmetry by using a representation-orbit replacement. Besides, we formulate the algorithm that allows determining the structures of nested polytopes. Moreover, in light of the relevance of the icosahedral symmetry to the description of various spherical molecules and viruses, we study symmetry breaking of the dual polytopes of non-crystallographic type and related tube-like structures. As well, we apply a stellation procedure to the family of considered polytopes. Since this research partly focuses on the icosahedral fullerenes, we present the construction of the corresponding carbon nanotubes. Furthermore, the approach considered for the non-crystallographic cases is applied to crystallographic structures. We consider a symmetry-breaking mechanism applied to the polytopes obtained using the three-dimensional Weyl groups to determine their possible structural extensions into nanotubes

NASA Tech Briefs, December 1989

Topics include: Electronic Components and Circuits. Electronic Systems, Physical Sciences, Materials, Computer Programs, Mechanics, Machinery, Fabrication Technology, Mathematics and Information Sciences, and Life Sciences