The superconductor-metal quantum phase transition in ultra-narrow wires Adissertationpresented by Adrian Giuseppe Del Maestro to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2008 c 2008 - Adrian Giuseppe Del Maestro ! All rights reserved. Thesis advisor Author Subir Sachdev Adrian Giuseppe Del Maestro The superconductor-metal quantum phase transition in ultra- narrow wires Abstract We present a complete description of a zero temperature phasetransitionbetween superconducting and diffusive metallic states in very thin wires due to a Cooper pair breaking mechanism originating from a number of possible sources. These include impurities localized to the surface of the wire, a magnetic field orientated parallel to the wire or, disorder in an unconventional superconductor. The order parameter describing pairing is strongly overdamped by its coupling toaneffectivelyinfinite bath of unpaired electrons imagined to reside in the transverse conduction channels of the wire. The dissipative critical theory thus contains current reducing fluctuations in the guise of both quantum and thermally activated phase slips. A full cross-over phase diagram is computed via an expansion in the inverse number of complex com- ponents of the superconducting order parameter (equal to oneinthephysicalcase). The fluctuation corrections to the electrical and thermal conductivities are deter- mined, and we find that the zero frequency electrical transport has a non-monotonic temperature dependence when moving from the quantum critical to low tempera- ture metallic phase, which may be consistent with recent experimental results on ultra-narrow MoGe wires. Near criticality, the ratio of the thermal to electrical con- ductivity displays a linear temperature dependence and thustheWiedemann-Franz law is obeyed. We compute the constant of proportionality in asystematicexpansion and find a universal and experimentally verifiable fluctuationcorrectiontotheLorenz number. In the presence of quenched disorder, a novel algorithm is developed to solve the self-consistency condition arising when the number of complex order parameter com- ponents is taken to be large. In this limit, we find striking evidence for the flow to infinite randomness, and observe dynamically activated scaling consistent with predictions from the strong disorder renormalization group. Moreover, the infinite randomness fixed point of the pair-breaking superconductor-metal quantum phase transition is found to be in the same universality class as theonsetofferromagnetism in the one dimensional quantum Ising model in a random transverse field. This discov- ery may lead to the first calculations of real electrical transport in an experimentally relevant system exhibiting infinite randomness. iii Contents CitationstoPreviouslyPublishedWork . .. vii Acknowledgments................................ ix Dedication.................................... xi 1Introduction 1 1.1 Superconductivity............................. 2 1.1.1 BCStheory ............................ 4 1.2 Fluctuations in low dimensional superconductors . ....... 8 1.2.1 LAMHtheory........................... 9 1.3 Ultranarrowwires ............................ 14 1.3.1 Evidence for quantum phase slips . 14 1.3.2 Suspended molecular templating . 14 1.4 Pair-breaking theory . 17 1.4.1 Magnetic fields and impurities . 18 1.4.2 Experimental manifestations . 21 1.5 Quantum phase transitions . 26 1.5.1 Landau theory . 26 1.5.2 The scaling hypothesis . 28 1.5.3 Quantum statistical mechanics . 29 1.5.4 Quantum critical phenomena . 33 1.5.5 Finite temperature crossovers . 35 1.6 Disordered critical phenomena . .38 1.6.1 The Harris criterion . 39 1.6.2 Spin Glasses . 41 1.6.3 Rare region effects ........................ 43 1.7 Organization . 46 2DissipativeTheoryoftheSuperconductor-MetalTransition 47 2.1 Dissipativemodel............................. 50 2.2 Scaling analysis . 52 2.3 Particle-hole asymmetry . 55 2.4 Phase fluctuations . 57 2.5 Connection to microscopic BCS theory . .59 iv Contents v 2.5.1 Pair-breaking in quasi-one dimensional wires . .... 59 2.5.2 Microscopic parameters in the clean and dirty limits . ..... 60 2.6 Universality in the quantum critical regime . ..... 62 2.7 The role of disorder . 66 3ThermoelectricTransportintheLarge-N Limit 68 3.1 Previous transport results . 68 3.1.1 LAMH theory . 69 3.1.2 Microscopic theory . 70 3.2 Finite temperature dynamics . 70 3.2.1 Effective classical theory . 72 3.2.2 Classical conductivity . 78 3.3 The ordered phase . 81 3.3.1 Zero temperature effective potential . .82 3.3.2 Construction of a Ginzburg-Landau potential . .. 86 3.3.3 Free energy barrier height and LAMH theory . 90 3.4 Large-N expansion ............................ 94 3.4.1 Thermoelectric transport . 96 3.5 Wiedemann-Franz ratio . 103 4 1/N Corrections to Transport 105 4.1 The critical theory . 105 4.1.1 Critical point at T =0...................... 107 4.1.2 Quantum critical propagator . 108 4.2 Criticalexponents............................. 111 4.3 Quantum transport at finite N . 116 4.3.1 Diagrammatic expansion . 116 4.3.2 Frequency summations . 118 4.3.3 Numerical evaluation . 121 4.3.4 Wiedemann-Franz law in the quantum critical regime . ... 124 5InfiniteRandomnessandActivatedScaling 126 5.1 Strong disorder renormalization group . ... 127 5.2 Lattice theory . 133 5.2.1 Infinite clean chain . 133 5.2.2 Finite disordered chain . 134 5.3 The solve-join-patch algorithm . .. 137 5.4 Evidence for infinite randomness . .139 5.4.1 Equal time correlation functions . 139 5.4.2 Energy gap statistics . 141 5.4.3 Dynamical Susceptibility . 144 5.4.4 Summary . 148 Contents vi 6Conclusions 151 AClassicaltransport 156 BTheFluctuationPropagator 159 B.1 T =0 ................................... 159 B.2 T>0...................................160 B.2.1 Numerical evaluation . 160 1 B.2.2 Re [ΠT (q, Ω,R)]− ......................... 161 CDetailsontheEvaluationofMatsubaraSums 165 DSusceptibilityScaling 171 D.1 δ =0.................................... 172 D.2 δ>0.................................... 173 References 174 Citations to Previously Published Work Chapters 2 to 4 describe the calculation of thermal and electrical transport near the quantum superconductor-metal transition in ultra narrow wires, and a brief account was published in a short paper that appeared in Physical Review B. “Universal thermal and electrical transport near the superconductor-metal quantum phase transition in nanowires” Adrian Del Maestro, Bernd Rosenow, Nayana Shah and Subir Sachdev, Physical Review B 77,180501(R)(2008),arXiv:0708.0687. The addition of disorder to the aforementioned model led to a study of infinite ran- domness and activated scaling with details given in Chapter 5. A summary of the important results have been submitted for publication in Physical Review Letters. “Infinite randomness fixed point of the superconductor-metalquantum phase transition” Adrian Del Maestro, Bernd Rosenow, Markus Mueller and Subir Sachdev, Submitted to Physical Review Letters, (2008), arXiv:0802.3900. During the last five years I have had the pleasure of working on anumberofextremely interesting projects on various topics that have not been included in this thesis for the aesthetic purpose of producing a self-contained document. The first includes studies of charge density wave ordering in both clean and disordered square lattices with application to the cuprate superconductors, “Thermal melting of density waves on the square lattice” Adrian Del Maestro and Subir Sachdev, Physical Review B 71,184511(2005),arXiv:cond-mat/0412498; “From stripe to checkerboard order on the square lattice in the presence of quenched disorder” Adrian Del Maestro, Bernd Rosenow, Subir Sachdev, Physical Review B 74,024520(2006),arXiv:cond-mat/0603029. Large scale numerical studies of supersolids on the triangular lattice with nearest and next-nearest neighbor interactions were performed “A striped supersolid phase and the search for deconfined quantum criti- cality in hard-core bosons on the triangular lattice” Roger G. Melko, Adrian Del Maestro and Anton A. Burkov, Physical Review B 74,214517(2006),arXiv:cond-mat/0607501, vii Contents viii and finally, I considered spin fluctuations and low temperature thermodynamic prop- erties in the geometrically frustrated pyrochlore gadolinium stanate, which lead to a prediction that was ultimately confirmed by experimental results “Low temperature specific heat and possible gap to magnetic excitations in the Heisenberg pyrochlore antiferromagnet Gd2Sn207” Adrian Del Maestro and Michel J.P. Gingras, Physical Review B 76,064418(2007),arXiv:cond-mat/0702661; “Evidence for gapped spin-wave excitations in the frustrated Gd2Sn2O7 pyrochlore antiferromagnet from low-temperature specific heat measure- ments” J.A. Quilliam, K.A. Ross, A. Del Maestro, M.J.P. Gingras, L.R. Corruc- cini and J.B. Kycia, Physical Review Letters 99,097201(2007),arXiv:0707.2072. Electronic preprints (shown in typewriter font)canbefoundonlineat http://arXiv.org Acknowledgments When choosing a path towards academia I had no idea of the importance of serendipity, but feel that I have been extraordinarily luckyinthisregard.Through determination, obstinance, and possibly prowess, I was taken on as student by Subir Sachdev. The scope and depth of his knowledge of condensed matter physics, as well as his analytical accuracy and passion for formalism and technique are awe inspiring. Under his mentorship I was