Quantum Field Theory in Condensed Matter Physics
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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. -
The Union of Quantum Field Theory and Non-Equilibrium Thermodynamics
The Union of Quantum Field Theory and Non-equilibrium Thermodynamics Thesis by Anthony Bartolotta In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended May 24, 2018 ii c 2018 Anthony Bartolotta ORCID: 0000-0003-4971-9545 All rights reserved iii Acknowledgments My time as a graduate student at Caltech has been a journey for me, both professionally and personally. This journey would not have been possible without the support of many individuals. First, I would like to thank my advisors, Sean Carroll and Mark Wise. Without their support, this thesis would not have been written. Despite entering Caltech with weaker technical skills than many of my fellow graduate students, Mark took me on as a student and gave me my first project. Mark also granted me the freedom to pursue my own interests, which proved instrumental in my decision to work on non-equilibrium thermodynamics. I am deeply grateful for being provided this priviledge and for his con- tinued input on my research direction. Sean has been an incredibly effective research advisor, despite being a newcomer to the field of non-equilibrium thermodynamics. Sean was the organizing force behind our first paper on this topic and connected me with other scientists in the broader community; at every step Sean has tried to smoothly transition me from the world of particle physics to that of non-equilibrium thermody- namics. My research would not have been nearly as fruitful without his support. I would also like to thank the other two members of my thesis and candidacy com- mittees, John Preskill and Keith Schwab. -
Electron-Phonon Interaction in Conventional and Unconventional Superconductors
Electron-Phonon Interaction in Conventional and Unconventional Superconductors Pegor Aynajian Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 Electron-Phonon Interaction in Conventional and Unconventional Superconductors Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der W¨urde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Pegor Aynajian aus Beirut (Libanon) Hauptberichter: Prof. Dr. Bernhard Keimer Mitberichter: Prof. Dr. Harald Giessen Tag der m¨undlichen Pr¨ufung: 12. M¨arz 2009 Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 2 Deutsche Zusammenfassung Die Frage, ob ein genaueres Studium der Phononen-Spektren klassischer Supraleiter wie Niob und Blei mittels inelastischer Neutronenstreuung der M¨uhe wert w¨are, w¨urde sicher von den meisten Wissenschaftlern verneint werden. Erstens erk¨art die ber¨uhmte mikroskopische Theorie von Bardeen, Cooper und Schrieffer (1957), bekannt als BCS Theorie, nahezu alle Aspekte der klassischen Supraleitung. Zweitens ist das aktuelle Interesse sehr stark auf die Hochtemperatur-Supraleitung in Kupraten und Schwere- Fermionen Systemen fokussiert. Daher waren die ersten Experimente dieser Arbeit, die sich mit der Bestimmung der Phononen-Lebensdauern in supraleitendem Niob und Blei befaßten, nur als ein kurzer Test der Aufl¨osung eines neuen hochaufl¨osenden Neutronen- spektrometers am Forschungsreaktor FRM II geplant. Dieses neuartige Spektrometer TRISP (triple axis spin echo) erm¨oglicht die Bestimmung von Phononen-Linienbreiten uber¨ große Bereiche des Impulsraumes mit einer Energieaufl¨osung im μeV Bereich, d.h. zwei Gr¨oßenordnungen besser als an klassische Dreiachsen-Spektrometern. Philip Allen hat erstmals dargelegt, daß die Linienbreite eines Phonons proportional zum Elektron-Phonon Kopplungsparameter λ ist. -
BCS Thermal Vacuum of Fermionic Superfluids and Its Perturbation Theory
www.nature.com/scientificreports OPEN BCS thermal vacuum of fermionic superfuids and its perturbation theory Received: 14 June 2018 Xu-Yang Hou1, Ziwen Huang1,4, Hao Guo1, Yan He2 & Chih-Chun Chien 3 Accepted: 30 July 2018 The thermal feld theory is applied to fermionic superfuids by doubling the degrees of freedom of the Published: xx xx xxxx BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon, where the pairing persists when the order parameter vanishes, emerges from the perturbation theory. The correspondence between the thermal vacuum and purifcation of the density matrix allows a unitary transformation, and we found the geometric phase associated with the transformation in the parameter space. Quantum many-body systems can be described by quantum feld theories1–4. Some available frameworks for sys- tems at fnite temperatures include the Matsubara formalism using the imaginary time for equilibrium systems1,5 and the Keldysh formalism of time-contour path integrals3,6 for non-equilibrium systems. Tere are also alterna- tive formalisms. For instance, the thermal feld theory7–9 is built on the concept of thermal vacuum. -
Topological Superconductors, Majorana Fermions and Topological Quantum Computation
Topological Superconductors, Majorana Fermions and Topological Quantum Computation 0. … from last time: The surface of a topological insulator 1. Bogoliubov de Gennes Theory 2. Majorana bound states, Kitaev model 3. Topological superconductor 4. Periodic Table of topological insulators and superconductors 5. Topological quantum computation 6. Proximity effect devices Unique Properties of Topological Insulator Surface States “Half” an ordinary 2DEG ; ¼ Graphene EF Spin polarized Fermi surface • Charge Current ~ Spin Density • Spin Current ~ Charge Density Berry’s phase • Robust to disorder • Weak Antilocalization • Impossible to localize Exotic States when broken symmetry leads to surface energy gap: • Quantum Hall state, topological magnetoelectric effect • Superconducting state Even more exotic states if surface is gapped without breaking symmetry • Requires intrinsic topological order like non-Abelian FQHE Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE 2 2 e xy 1 2h B 2 0 e 1 xy n -1 h 2 2 -2 e n=1 chiral edge state xy 2h Anomalous QHE : Induce a surface gap by depositing magnetic material † 2 2 Hi0 ( - v - DM z ) e e - Mass due to Exchange field 2h 2h M↑ M↓ e2 xyDsgn( M ) EF 2h TI Egap = 2|DM| Chiral Edge State at Domain Wall : DM ↔ -DM Topological Magnetoelectric Effect Qi, Hughes, Zhang ’08; Essin, Moore, Vanderbilt ‘09 Consider a solid cylinder of TI with a magnetically gapped surface M 2 e 1 J xy E n E M h 2 J Magnetoelectric Polarizability topological “q term” 2 DL EB E e 1 ME n e2 h 2 q 2 h TR sym. -
How the Electron-Phonon Coupling Mechanism Work in Metal Superconductor
How the electron-phonon coupling mechanism work in metal superconductor Qiankai Yao1,2 1College of Science, Henan University of Technology, Zhengzhou450001, China 2School of physics and Engineering, Zhengzhou University, Zhengzhou450001, China Abstract Superconductivity in some metals at low temperature is known to arise from an electron-phonon coupling mechanism. Such the mechanism enables an effective attraction to bind two mobile electrons together, and even form a kind of pairing system(called Cooper pair) to be physically responsible for superconductivity. But, is it possible by an analogy with the electrodynamics to describe the electron-phonon coupling as a resistivity-dependent attraction? Actually so, it will help us to explore a more operational quantum model for the formation of Cooper pair. In particularly, by the calculation of quantum state of Cooper pair, the explored model can provide a more explicit explanation for the fundamental properties of metal superconductor, and answer: 1) How the transition temperature of metal superconductor is determined? 2) Which metals can realize the superconducting transition at low temperature? PACS numbers: 74.20.Fg; 74.20.-z; 74.25.-q; 74.20.De ne is the mobile electron density, η the damping coefficient 1. Introduction that is determined by the collision time τ . In the BCS theory[1], superconductivity is attributed to a In metal environment, mobile electrons are usually phonon-mediated attraction between mobile electrons near modeled to be a kind of classical particles like gas molecules, Fermi surface(called Fermi electrons). The attraction is each of which performs a Brown-like motion and satisfies the sometimes referred to as a residual Coulomb interaction[2] that Langevin equation can glue Cooper pair together to cause superconductivity. -
Theory of Superconductivity
Theory of Superconductivity Kwon Park KIAS-SNU Physics Winter Camp Camp Winter KIAS-SNU Physics Phoenix Park Jan 20 – 27, 2013 Outline • Why care about superconductivity? • BCS theory as a trial wave function method • BCS theory as a mean-field theory • High-temperature superconductivity and strong correlation • Effective field theory for superconductivity: Ginzburg- Landau theory Family tree of strongly correlated electron systems Quantum magnetism Topological Mott insulator FQHE insulator HTSC Quantum Hall effect Superconductivity Wigner crystal Breakdown of the Landau-Fermi liquid Collective behavior of a staring crowd Disordered State Ordered State Superconductivity as an emergent phenomenon Superconducting phase coherence: Josephson effect • Cooper-pair box: An artificial two-level system composed of many superconducting electron pairs in a “box.” reservoir box - - - - + + + + gate Josephson effect in the Cooper-pair box Nakamura, Pashkin, Tsai, Nature 398, 786 (99) • Phase vs. number uncertainty relationship: When the phase gets coherent, the Cooper-pair number becomes uncertain, which is nothing but the Josephson effect. Devoret and Schoelkopf, Nature 406, 1039 (00) Meissner effect • mv-momentum = p-momentum − e/c × vector potential • current density operator = 2e × Cooper pair density × velocity operator • θ=0 for a coherent Cooper-pair condensate in a singly connected region.region London equation • The magnetic field is expelled from the inside of a superconductor: Meissner effect. London equation where Electromagnetic field, or wave are attenuated inside a superconductor, which means that, in quantum limit, photons become massive while Maxwell’s equations remain gauge-invariant! Anderson-Higgs mechanism • Quantum field theories should be renormalizable in order to produce physically meaningful predictions via systematic elimination of inherent divergences. -
Keldysh Field Theory for Dissipation-Induced States of Fermions
CORE Metadata, citation and similar papers at core.ac.uk Provided by Electronic Thesis and Dissertation Archive - Università di Pisa Department of Physics Master Degree in Physics Curriculum in Theoretical Physics Keldysh Field Theory for dissipation-induced states of Fermions Master Thesis Federico Tonielli Candidate: Supervisor: Federico Tonielli Prof. Dr. Sebastian Diehl University of Koln¨ Graduation Session May 26th, 2016 Academic Year 2015/2016 UNIVERSITY OF PISA Abstract Department of Physics \E. Fermi" Keldysh Field Theory for dissipation-induced states of Fermions by Federico Tonielli The recent experimental progress in manipulation and control of quantum systems, together with the achievement of the many-body regime in some settings like cold atoms and trapped ions, has given access to new scenarios where many-body coherent and dissipative dynamics can occur on an equal footing and the generators of both can be tuned externally. Such control is often guaranteed by the toolbox of quantum optics, hence a description of dynamics in terms of a Markovian Quantum Master Equation with the corresponding Liouvillian generator is sufficient. This led to a new state preparation paradigm where the target quantum state is the unique steady state of the engineered Liouvillian (i.e. the system evolves towards it irrespective of initial conditions). Recent research addressed the possibility of preparing topological fermionic states by means of such dissipative protocol: on one hand, it could overcome a well-known difficulty in cooling systems of fermionic atoms, making easier to induce exotic (also paired) fermionic states; on the other hand dissipatively preparing a topological state allows us to discuss the concept and explore the phenomenology of topological order in the non-equilibrium context. -
Introduction to Unconventional Superconductivity Manfred Sigrist
Introduction to Unconventional Superconductivity Manfred Sigrist Theoretische Physik, ETH-Hönggerberg, 8093 Zürich, Switzerland Abstract. This lecture gives a basic introduction into some aspects of the unconventionalsupercon- ductivity. First we analyze the conditions to realized unconventional superconductivity in strongly correlated electron systems. Then an introduction of the generalized BCS theory is given and sev- eral key properties of unconventional pairing states are discussed. The phenomenological treatment based on the Ginzburg-Landau formulations provides a view on unconventional superconductivity based on the conceptof symmetry breaking.Finally some aspects of two examples will be discussed: high-temperature superconductivity and spin-triplet superconductivity in Sr2RuO4. Keywords: Unconventional superconductivity, high-temperature superconductivity, Sr2RuO4 INTRODUCTION Superconductivity remains to be one of the most fascinating and intriguing phases of matter even nearly hundred years after its first observation. Owing to the breakthrough in 1957 by Bardeen, Cooper and Schrieffer we understand superconductivity as a conden- sate of electron pairs, so-called Cooper pairs, which form due to an attractive interaction among electrons. In the superconducting materials known until the mid-seventies this interaction is mediated by electron-phonon coupling which gises rise to Cooper pairs in the most symmetric form, i.e. vanishing relative orbital angular momentum and spin sin- glet configuration (nowadays called s-wave pairing). After the introduction of the BCS concept, also studies of alternative pairing forms started. Early on Anderson and Morel [1] as well as Balian and Werthamer [2] investigated superconducting phases which later would be identified as the A- and the B-phase of superfluid 3He [3]. In contrast to the s-wave superconductors the A- and B-phase are characterized by Cooper pairs with an- gular momentum 1 and spin-triplet configuration. -
Spontaneous Symmetry Breaking and Mass Generation As Built-In Phenomena in Logarithmic Nonlinear Quantum Theory
Vol. 42 (2011) ACTA PHYSICA POLONICA B No 2 SPONTANEOUS SYMMETRY BREAKING AND MASS GENERATION AS BUILT-IN PHENOMENA IN LOGARITHMIC NONLINEAR QUANTUM THEORY Konstantin G. Zloshchastiev Department of Physics and Center for Theoretical Physics University of the Witwatersrand Johannesburg, 2050, South Africa (Received September 29, 2010; revised version received November 3, 2010; final version received December 7, 2010) Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry break- ing and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the funda- mental Bose–Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativis- tic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic pos- tulates, Mach’s principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon’s mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions. DOI:10.5506/APhysPolB.42.261 PACS numbers: 11.15.Ex, 11.30.Qc, 04.60.Bc, 03.65.Pm 1. Introduction Current observational data in astrophysics are probing a regime of de- partures from classical relativity with sensitivities that are relevant for the study of the quantum-gravity problem [1,2]. -
Lectures on Holographic Superfluidity and Superconductivity
PUPT-2297 Lectures on Holographic Superfluidity and Superconductivity C. P. Herzog Department of Physics, Princeton University, Princeton, NJ 08544, USA (Dated: March 26, 2009) Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems.1 The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated to a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture. I. INTRODUCTION: QUANTUM PHASE TRANSITIONS Spurred by the concrete proposal of refs. [1{3] for an AdS/CFT correspondence, there are some good reasons why holographic ideas have become so important in high energy theoretical physics over the last ten years. The first and perhaps most fundamental reason is that the AdS/CFT conjecture provides a definition of quantum gravity in a particular curved background space-time. The second is that AdS/CFT provides a tool for studying strongly interacting field theories. These lectures concern themselves with the second reason, but I will spend a paragraph on the first. Given the lack of alternative definitions of quantum gravity, the AdS/CFT conjecture is diffi- cult to prove, but the correspondence does give a definition of type IIB string theory in a fixed ten dimensional background and by extension of type IIB supergravity. Recall that the original 5 conjecture posits an equivalence between type IIB string theory in the space-time AdS5 × S and the maximally supersymmetric (SUSY) SU(N) Yang-Mills theory in 3+1 dimensions. -
Quantum-Zeno Fermi Polaron in the Strong Dissipation Limit
PHYSICAL REVIEW RESEARCH 3, 013086 (2021) Quantum-Zeno Fermi polaron in the strong dissipation limit Tomasz Wasak ,1 Richard Schmidt,2 and Francesco Piazza1 1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany 2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany (Received 20 January 2020; accepted 24 December 2020; published 28 January 2021) The interplay between measurement and quantum correlations in many-body systems can lead to novel types of collective phenomena which are not accessible in isolated systems. In this work, we merge the Zeno paradigm of quantum measurement theory with the concept of polarons in condensed-matter physics. The resulting quantum-Zeno Fermi polaron is a quasiparticle which emerges for lossy impurities interacting with a quantum-degenerate bath of fermions. For loss rates of the order of the impurity-fermion binding energy, the quasiparticle is short lived. However, we show that in the strongly dissipative regime of large loss rates a long-lived polaron branch reemerges. This quantum-Zeno Fermi polaron originates from the nontrivial interplay between the Fermi surface and the surface of the momentum region forbidden by the quantum-Zeno projection. The situation we consider here is realized naturally for polaritonic impurities in charge-tunable semiconductors and can be also implemented using dressed atomic states in ultracold gases. DOI: 10.1103/PhysRevResearch.3.013086 I. INTRODUCTION increasing the measurement rate of a closed quantum many- body system, a phase transition from a volume-law entangled The effect of measurement on the time evolution of a phase to a quantum Zeno phase with area-law entanglement system is one of the most puzzling aspects of quantum dy- can take place [11,13].