The Superconductor-Metal Quantum Phase Transition in Ultra-Narrow Wires
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Lecture Notes: BCS Theory of Superconductivity
Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner. -
Phase-Transition Phenomena in Colloidal Systems with Attractive and Repulsive Particle Interactions Agienus Vrij," Marcel H
Faraday Discuss. Chem. SOC.,1990,90, 31-40 Phase-transition Phenomena in Colloidal Systems with Attractive and Repulsive Particle Interactions Agienus Vrij," Marcel H. G. M. Penders, Piet W. ROUW,Cornelis G. de Kruif, Jan K. G. Dhont, Carla Smits and Henk N. W. Lekkerkerker Van't Hof laboratory, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands We discuss certain aspects of phase transitions in colloidal systems with attractive or repulsive particle interactions. The colloidal systems studied are dispersions of spherical particles consisting of an amorphous silica core, coated with a variety of stabilizing layers, in organic solvents. The interaction may be varied from (steeply) repulsive to (deeply) attractive, by an appropri- ate choice of the stabilizing coating, the temperature and the solvent. In systems with an attractive interaction potential, a separation into two liquid- like phases which differ in concentration is observed. The location of the spinodal associated with this demining process is measured with pulse- induced critical light scattering. If the interaction potential is repulsive, crystallization is observed. The rate of formation of crystallites as a function of the concentration of the colloidal particles is studied by means of time- resolved light scattering. Colloidal systems exhibit phase transitions which are also known for molecular/ atomic systems. In systems consisting of spherical Brownian particles, liquid-liquid phase separation and crystallization may occur. Also gel and glass transitions are found. Moreover, in systems containing rod-like Brownian particles, nematic, smectic and crystalline phases are observed. A major advantage for the experimental study of phase equilibria and phase-separation kinetics in colloidal systems over molecular systems is the length- and time-scales that are involved. -
Phase Transitions in Quantum Condensed Matter
Diss. ETH No. 15104 Phase Transitions in Quantum Condensed Matter A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH¨ (ETH Zuric¨ h) for the degree of Doctor of Natural Science presented by HANS PETER BUCHLER¨ Dipl. Phys. ETH born December 5, 1973 Swiss citizien accepted on the recommendation of Prof. Dr. J. W. Blatter, examiner Prof. Dr. W. Zwerger, co-examiner PD. Dr. V. B. Geshkenbein, co-examiner 2003 Abstract In this thesis, phase transitions in superconducting metals and ultra-cold atomic gases (Bose-Einstein condensates) are studied. Both systems are examples of quantum condensed matter, where quantum effects operate on a macroscopic level. Their main characteristics are the condensation of a macroscopic number of particles into the same quantum state and their ability to sustain a particle current at a constant velocity without any driving force. Pushing these materials to extreme conditions, such as reducing their dimensionality or enhancing the interactions between the particles, thermal and quantum fluctuations start to play a crucial role and entail a rich phase diagram. It is the subject of this thesis to study some of the most intriguing phase transitions in these systems. Reducing the dimensionality of a superconductor one finds that fluctuations and disorder strongly influence the superconducting transition temperature and eventually drive a superconductor to insulator quantum phase transition. In one-dimensional wires, the fluctuations of Cooper pairs appearing below the mean-field critical temperature Tc0 define a finite resistance via the nucleation of thermally activated phase slips, removing the finite temperature phase tran- sition. Superconductivity possibly survives only at zero temperature. -
Density Fluctuations Across the Superfluid-Supersolid Phase Transition in a Dipolar Quantum Gas
PHYSICAL REVIEW X 11, 011037 (2021) Density Fluctuations across the Superfluid-Supersolid Phase Transition in a Dipolar Quantum Gas J. Hertkorn ,1,* J.-N. Schmidt,1,* F. Böttcher ,1 M. Guo,1 M. Schmidt,1 K. S. H. Ng,1 S. D. Graham,1 † H. P. Büchler,2 T. Langen ,1 M. Zwierlein,3 and T. Pfau1, 15. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 2Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 3MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 15 October 2020; accepted 8 January 2021; published 23 February 2021) Phase transitions share the universal feature of enhanced fluctuations near the transition point. Here, we show that density fluctuations reveal how a Bose-Einstein condensate of dipolar atoms spontaneously breaks its translation symmetry and enters the supersolid state of matter—a phase that combines superfluidity with crystalline order. We report on the first direct in situ measurement of density fluctuations across the superfluid-supersolid phase transition. This measurement allows us to introduce a general and straightforward way to extract the static structure factor, estimate the spectrum of elementary excitations, and image the dominant fluctuation patterns. We observe a strong response in the static structure factor and infer a distinct roton minimum in the dispersion relation. Furthermore, we show that the characteristic fluctuations correspond to elementary excitations such as the roton modes, which are theoretically predicted to be dominant at the quantum critical point, and that the supersolid state supports both superfluid as well as crystal phonons. -
Equation of State and Phase Transitions in the Nuclear
National Academy of Sciences of Ukraine Bogolyubov Institute for Theoretical Physics Has the rights of a manuscript Bugaev Kyrill Alekseevich UDC: 532.51; 533.77; 539.125/126; 544.586.6 Equation of State and Phase Transitions in the Nuclear and Hadronic Systems Speciality 01.04.02 - theoretical physics DISSERTATION to receive a scientific degree of the Doctor of Science in physics and mathematics arXiv:1012.3400v1 [nucl-th] 15 Dec 2010 Kiev - 2009 2 Abstract An investigation of strongly interacting matter equation of state remains one of the major tasks of modern high energy nuclear physics for almost a quarter of century. The present work is my doctor of science thesis which contains my contribution (42 works) to this field made between 1993 and 2008. Inhere I mainly discuss the common physical and mathematical features of several exactly solvable statistical models which describe the nuclear liquid-gas phase transition and the deconfinement phase transition. Luckily, in some cases it was possible to rigorously extend the solutions found in thermodynamic limit to finite volumes and to formulate the finite volume analogs of phases directly from the grand canonical partition. It turns out that finite volume (surface) of a system generates also the temporal constraints, i.e. the finite formation/decay time of possible states in this finite system. Among other results I would like to mention the calculation of upper and lower bounds for the surface entropy of physical clusters within the Hills and Dales model; evaluation of the second virial coefficient which accounts for the Lorentz contraction of the hard core repulsing potential between hadrons; inclusion of large width of heavy quark-gluon bags into statistical description. -
The Development of the Science of Superconductivity and Superfluidity
Universal Journal of Physics and Application 1(4): 392-407, 2013 DOI: 10.13189/ujpa.2013.010405 http://www.hrpub.org Superconductivity and Superfluidity-Part I: The development of the science of superconductivity and superfluidity in the 20th century Boris V.Vasiliev ∗Corresponding Author: [email protected] Copyright ⃝c 2013 Horizon Research Publishing All rights reserved. Abstract Currently there is a common belief that the explanation of superconductivity phenomenon lies in understanding the mechanism of the formation of electron pairs. Paired electrons, however, cannot form a super- conducting condensate spontaneously. These paired electrons perform disorderly zero-point oscillations and there are no force of attraction in their ensemble. In order to create a unified ensemble of particles, the pairs must order their zero-point fluctuations so that an attraction between the particles appears. As a result of this ordering of zero-point oscillations in the electron gas, superconductivity arises. This model of condensation of zero-point oscillations creates the possibility of being able to obtain estimates for the critical parameters of elementary super- conductors, which are in satisfactory agreement with the measured data. On the another hand, the phenomenon of superfluidity in He-4 and He-3 can be similarly explained, due to the ordering of zero-point fluctuations. It is therefore established that both related phenomena are based on the same physical mechanism. Keywords superconductivity superfluidity zero-point oscillations 1 Introduction 1.1 Superconductivity and public Superconductivity is a beautiful and unique natural phenomenon that was discovered in the early 20th century. Its unique nature comes from the fact that superconductivity is the result of quantum laws that act on a macroscopic ensemble of particles as a whole. -
Introduction to Superconductivity Theory
Ecole GDR MICO June, 2010 Introduction to Superconductivity Theory Indranil Paul [email protected] www.neel.cnrs.fr Free Electron System εεε k 2 µµµ Hamiltonian H = ∑∑∑i pi /(2m) - N, i=1,...,N . µµµ is the chemical potential. N is the total particle number. 0 k k Momentum k and spin σσσ are good quantum numbers. F εεε H = 2 ∑∑∑k k nk . The factor 2 is due to spin degeneracy. εεε 2 µµµ k = ( k) /(2m) - , is the single particle spectrum. nk T=0 εεεk/(k BT) nk is the Fermi-Dirac distribution. nk = 1/[e + 1] 1 T ≠≠≠ 0 The ground state is a filled Fermi sea (Pauli exclusion). µµµ 1/2 Fermi wave-vector kF = (2 m ) . 0 kF k Particle- and hole- excitations around kF have vanishingly 2 2 low energies. Excitation spectrum EK = |k – kF |/(2m). Finite density of states at the Fermi level. Ek 0 k kF Fermi Sea hole excitation particle excitation Metals: Nearly Free “Electrons” The electrons in a metal interact with one another with a short range repulsive potential (screened Coulomb). The phenomenological theory for metals was developed by L. Landau in 1956 (Landau Fermi liquid theory). This system of interacting electrons is adiabatically connected to a system of free electrons. There is a one-to-one correspondence between the energy eigenstates and the energy eigenfunctions of the two systems. Thus, for all practical purposes we will think of the electrons in a metal as non-interacting fermions with renormalized parameters, such as m →→→ m*. (remember Thierry’s lecture) CV (i) Specific heat (C V): At finite-T the volume of πππ 2∆∆∆ 2 ∆∆∆ excitations ∼∼∼ 4 kF k, where Ek ∼∼∼ ( kF/m) k ∼∼∼ kBT. -
ANTIMATTER a Review of Its Role in the Universe and Its Applications
A review of its role in the ANTIMATTER universe and its applications THE DISCOVERY OF NATURE’S SYMMETRIES ntimatter plays an intrinsic role in our Aunderstanding of the subatomic world THE UNIVERSE THROUGH THE LOOKING-GLASS C.D. Anderson, Anderson, Emilio VisualSegrè Archives C.D. The beginning of the 20th century or vice versa, it absorbed or emitted saw a cascade of brilliant insights into quanta of electromagnetic radiation the nature of matter and energy. The of definite energy, giving rise to a first was Max Planck’s realisation that characteristic spectrum of bright or energy (in the form of electromagnetic dark lines at specific wavelengths. radiation i.e. light) had discrete values The Austrian physicist, Erwin – it was quantised. The second was Schrödinger laid down a more precise that energy and mass were equivalent, mathematical formulation of this as described by Einstein’s special behaviour based on wave theory and theory of relativity and his iconic probability – quantum mechanics. The first image of a positron track found in cosmic rays equation, E = mc2, where c is the The Schrödinger wave equation could speed of light in a vacuum; the theory predict the spectrum of the simplest or positron; when an electron also predicted that objects behave atom, hydrogen, which consists of met a positron, they would annihilate somewhat differently when moving a single electron orbiting a positive according to Einstein’s equation, proton. However, the spectrum generating two gamma rays in the featured additional lines that were not process. The concept of antimatter explained. In 1928, the British physicist was born. -
3: Applications of Superconductivity
Chapter 3 Applications of Superconductivity CONTENTS 31 31 31 32 37 41 45 49 52 54 55 56 56 56 56 57 57 57 57 57 58 Page 3-A. Magnetic Separation of Impurities From Kaolin Clay . 43 3-B. Magnetic Resonance Imaging . .. ... 47 Tables Page 3-1. Applications in the Electric Power Sector . 34 3-2. Applications in the Transportation Sector . 38 3-3. Applications in the Industrial Sector . 42 3-4. Applications in the Medical Sector . 45 3-5. Applications in the Electronics and Communications Sectors . 50 3-6. Applications in the Defense and Space Sectors . 53 Chapter 3 Applications of Superconductivity INTRODUCTION particle accelerator magnets for high-energy physics (HEP) research.l The purpose of this chapter is to assess the significance of high-temperature superconductors Accelerators require huge amounts of supercon- (HTS) to the U.S. economy and to forecast the ducting wire. The Superconducting Super Collider timing of potential markets. Accordingly, it exam- will require an estimated 2,000 tons of NbTi wire, ines the major present and potential applications of worth several hundred million dollars.2 Accelerators superconductors in seven different sectors: high- represent by far the largest market for supercon- energy physics, electric power, transportation, indus- ducting wire, dwarfing commercial markets such as trial equipment, medicine, electronics/communica- MRI, tions, and defense/space. Superconductors are used in magnets that bend OTA has made no attempt to carry out an and focus the particle beam, as well as in detectors independent analysis of the feasibility of using that separate the collision fragments in the target superconductors in various applications. -
Quantum Phase Transitions
INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS Rep. Prog. Phys. 66 (2003) 2069–2110 PII: S0034-4885(03)33852-7 Quantum phase transitions Matthias Vojta Institut fur¨ Theorie der Kondensierten Materie, Universitat¨ Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany E-mail: [email protected] Received 1 August 2003, in final form 8 October 2003 Published 3 November 2003 Online at stacks.iop.org/RoPP/66/2069 Abstract In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d-wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high- level theoretical presentations and research papers specialized in certain classes of materials. -
Thermodynamics of Quantum Phase Transitions of a Dirac Oscillator in a Homogenous Magnetic field
Thermodynamics of Quantum Phase Transitions of a Dirac oscillator in a homogenous magnetic field A. M. Frassino,1 D. Marinelli,2 O. Panella,3 and P. Roy4 1Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, D-60438 Frankfurt am Main, Germany 2Machine Learning and Optimization Lab., RIST, 400487 Cluj-Napoca, Romania∗ 3Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italyy 4Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India (Dated: July 24, 2017) The Dirac oscillator in a homogenous magnetic field exhibits a chirality phase transition at a par- ticular (critical) value of the magnetic field. Recently, this system has also been shown to be exactly solvable in the context of noncommutative quantum mechanics featuring the interesting phenomenon of re-entrant phase transitions. In this work we provide a detailed study of the thermodynamics of such quantum phase transitions (both in the standard and in the noncommutative case) within the Maxwell-Boltzmann statistics pointing out that the magnetization has discontinuities at critical values of the magnetic field even at finite temperatures. I. INTRODUCTION Quantum Phase Transitions (QPT) [1] are a class of phase transitions that can take place at zero tem- perature when the quantum fluctuations, required by the Heisenberg's uncertainty principle, cause an abrupt change in the phase of the system. The QPTs occur at a critical value of some parameters of the system such as pressure or magnetic field. In a QPT, the change is driven by the modification of particular couplings that characterise the interactions between the microscopic el- ements of the system and the dynamics of its phase near the quantum critical point. -
Heavy Fermions and Quantum Phase Transitions Qimiao Si and Frank Steglich
REVIEW non-Fermi liquid behavior (9), which goes beyond the standard theory of metals [Fermi- liquid theory (10)], is another phenomenon that is Heavy Fermions and broadly relevant to the physics of strongly cor- related systems (11, 12). Quantum Phase Transitions Quantum criticality has been implicated to one degree or another in a host of other heavy- fermion metals (4, 13, 14). These include CeCu2Si2, 1 2 Qimiao Si * and Frank Steglich * the first superconductor to be observed among heavy-fermion metals (15), and CeRhIn5 (Fig. Quantum phase transitions arise in many-body systems because of competing interactions that promote 1C) (16). Extensive theoretical studies have led – rivaling ground states. Recent years have seen the identification of continuous quantum phase transitions, to unconventional quantum criticality (17 20). or quantum critical points, in a host of antiferromagnetic heavy-fermion compounds. Studies of the More recently, a plethora of phases have been interplay between the various effects have revealed new classes of quantum critical points and are uncovered in heavy-fermion metals near a QCP b uncovering a plethora of new quantum phases. At the same time, quantum criticality has provided [such as in Ir-doped YbRh2Si2 (8)andin -YbAlB4 fresh insights into the electronic, magnetic, and superconducting properties of the heavy-fermion metals. (21)]. Together with the theoretical studies of the We review these developments, discuss the open issues, and outline some directions for future research. global phase diagram of the heavy-fermion metals (22, 2), these developments open up an entirely new frontier on the interplay between quantum critical- uantum mechanics not only governs the uent particles.