Triplet Superfluidity in Quasi-one-dimensional Conductors and Ultra-cold Fermi Gases AThesis Presented to The Academic Faculty by Wei Zhang In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of Physics Georgia Institute of Technology December 2006 Triplet Superfluidity in Quasi-one-dimensional Conductors and Ultra-cold Fermi Gases Approved by: Professor Carlos S´adeMelo,Advisor Professor T. A. Brian Kennedy School of Physics School of Physics Georgia Institute of Technology Georgia Institute of Technology Professor Dragomir Davidovic Professor Chandra Raman School of Physics School of Physics Georgia Institute of Technology Georgia Institute of Technology Professor Stavros Garoufalidis School of Mathematics Georgia Institute of Technology Date Approved: September 7, 2006 To my beloved wife Liu, and my parents, Fan and Ping, for their constant encouragement and support over the years. iii ACKNOWLEDGEMENTS First, I would like to thank my advisor Prof. Carlos S´a de Melo, and other members in my thesis committee: Prof. Dragomir Davidovic, Prof. T. A. Brian Kennedy, Prof. Chandra Raman in School of Physics, and Prof. Stavros Garoufalidis in School of Mathematics. It is always my pleasure to pursue my graduate studies in Georgia Tech, where I had the opportunity to work directly with and learn from a group of very talented scientists. In particular, Carlos provided me with an exceptional model of a theoretical physicist. Carlos is a great thinker. His abilities allow him to generate many interesting ideas and possible pathways towards them almost simultaneously, as well as to debug some technical difficulties in a mess of formulae. Carlos is also a very good advisor. He provides me time to clear my mind and room for creativity such that I always had a say in what I was working on. As a graduate student of Carlos, I shall never forget the time he spent to teach me how to think, work, and talk as a physicist. In addition, my master advisor in mathematics Prof. Stavros Garoufalidis helps me a lot about mathematics, which is invaluable for a theoretical physicist. Most importantly, his rigorous working style shows me a perfect example of how to organize ideas, works, as well as my office. I would like to thank the other two graduate students in our group, Sergio Botelho and Menderes Iskin. Although we did not collaborate in a same research topic, the numerous discussions with them are always delightful and helpful. In addition, Sergio helped me a lot on computer and programming problems. I still feel sorry for knocking his office door three times in a same day to ask the same command in Linux just because I kept forgetting it. I also thank Miles Stoudemire, Robert Cherng and Philip Powell for their collaboration and discussion. They are very gifted students and I am sure they will do great work in their graduate studies. Georgia Tech is full of so many other people that I have come to know well, I could not list all their names. However I would like to single out Jiang Xiao. Jiang is my classmate iv since 1997. We are very good friends and had a lot of discussions about physical and non- physical problems. Jiang is also a very good listener of my presentation. After explaining my research to him and answering all his questions, I finally realized an invited talk is not so hard. I would also like to express my thanks to the ones who are important to me, but are not directly related to my graduate studies. My parents and my wife constantly encourage me to pursue my goals and support my own decisions. They shared with all my experience in the past five years. I always feel their generous love. Most responsible for stimulating my interests about physics was Long Han, my physics teacher in Yaohua High School. I always appreciate his effort of delicately planning curriculum, designing experiments, and setting up observatory. I would also like to thank Han Zhang, my undergraduate advisor in Peking University. He taught me a lot about condensed matter experiments, which are proved to be very useful for a theorist to build a common sense about what can be done in the real world. Finally, I would like to thank Gilbert F. Amelio for his generosity to provide me financial support. Research discussed in this thesis are supported by National Science Foundation (Grant No. DMR-0304380). v TABLE OF CONTENTS DEDICATION ...................................... iii ACKNOWLEDGEMENTS .............................. iv LIST OF FIGURES .................................. viii SUMMARY ........................................ xii IINTRODUCTION................................. 1 1.1BriefHistoryofSuperfluidity.......................... 1 1.2 Triplet Superconductors . .................... 4 1.3SuperfluidityinUltra-ColdAtomicGases................... 10 1.3.1 BoseSystems.............................. 10 1.3.2 FermiSystems.............................. 16 1.4Summary..................................... 18 II BACKGROUND .................................. 20 2.1CooperPairing................................. 20 2.2SpinStructureofPairedStates........................ 24 2.2.1 SingletPairing.............................. 25 2.2.2 TripletPairing.............................. 26 2.2.3 Pseudo-spinSpace........................... 27 2.3SuperfluidityasaMany-bodyProblem.................... 28 2.3.1 HamiltonianandEffectiveTheory................... 29 2.3.2 Saddle-pointEquations......................... 34 2.3.3 Ginzburg–LandauTheory....................... 38 III COEXISTENCE OF SPIN DENSITY WAVES AND TRIPLET SUPER- CONDUCTIVITY IN QUASI-ONE-DIMENSIONAL CONDUCTORS 43 3.1Background................................... 44 3.1.1 CrystalStructureandElectronicDispersion............. 45 3.1.2 Upper Critical Fields of (TMTSF)2PF6 ................ 47 77 3.1.3 Se Knight Shift and NMR Measurements in (TMSTF)2PF6 ... 53 3.1.4 Triplet Superconducting State in (TMTSF)2PF6 ........... 55 vi 3.2 Experiments Suggesting Coexistence of Spin Density Waves and Supercon- ductivity in (TMTSF)2PF6 ........................... 57 3.3TheMicroscopicHamiltonian......................... 60 3.4Ginzburg–LandauTheory........................... 66 3.5PhaseDiagram................................. 76 3.6MagneticFieldEffect.............................. 79 3.7Summary..................................... 84 IV TIME EVOLUTION AND MATTER-WAVE INTERFERENCE IN P - WAV E F E RM I C O N D E N S AT E S ....................... 87 4.1Background................................... 88 4.1.1 FeshbachResonance.......................... 88 4.1.2 Experimental Observations of p-wave Feshbach Resonances . 93 4.2HamiltonianandEffectiveAction....................... 95 4.3EquationofMotionandTimeEvolutionofaSingleCloud......... 102 4.4Matter-waveInterferenceandPolarizationEffect.............. 107 4.5AnisotropicFreeExpansion.......................... 110 4.6Summary..................................... 113 V CONCLUSIONS AND FUTURE TOPICS ................. 115 APPENDIX A — SUMMATION OVER SPIN INDICES ......... 118 APPENDIX B — MATSUBARA FREQUENCY SUMMATION .... 120 REFERENCES ..................................... 123 VITA ............................................ 138 vii LIST OF FIGURES 1.1 The energies of different hyperfine states as a function of magnetic field. 14 2.1 Feynman diagrams corresponding to (a) the first, and (b) the second terms of the quadratic effective action S2,respectively................ 40 2.2 Feynman diagrams corresponding to the quartic effective action S4...... 41 3.1 Generalized phase diagram for (TMTSF)2X and (TMTTF)2X compounds adapted from Ref. [157]. The notation CL, SP, SDW and SC refers to charge-localized, spin-Peierls, spin density wave, and superconducting states, respectively. The lower-case letters designate compounds and indicate their location at ambient pressure in the generalized diagram. (a) (TMTTF)2PF6, (b) (TMTTF)2Br, (c) (TMTSF)2PF6, (d) (TMTSF)2ClO4. ......... 45 ∗ 3.2 View of the crystal structure of (TMTSF)2PF6 showing the a and c axes. 46 3.3 View of the crystal structure of (TMTSF)2PF6 showing the principal direc- tions a, b and c∗. Adapted from Ref. [168]. ............. 47 3.4 The Fermi surface of quasi-1D systems with transfer integrals tx ty tz > 0isopen................................... 48 3.5 The magnetic field versus temperature (H–T ) phase diagram for (TMTSF)2PF6 for magnetic fields aligned along the three principal axis a, b,andc∗. AdaptedfromRef.[41].............................. 51 77 3.6 NMR Se absorption spectra collected at temperatures below and above Tc for a magnetic field H =2.38 T applied along the b axis. The solid line indicates the measured first moment, and the shaded region indicates the expected first moment for a singlet ground state. Adapted from Ref. [51]. 55 3.7 P –T phase diagram of (TMTSF)2PF6. SDW/M denotes the region where metallic and SDW phases coexist inhomogeneously, below TSDW line (large dots). Below TSC =1.20 ± 0.01 K line (small dots), this coexistence switches into a coexistence of SC and SDW phases, due to Metal–Superconductivity phase transition. A gradient in shading (SDW/SC region) below TSC denotes the increase of SC volume. The solid curve separating the M and SDW phases is a fit to the data using the empirical formula TSDW(P )=T1 −[(T1 − 3 TSC)(P/Pc) ].AdaptedfromRef.[68]...................... 58 3.8 Simultaneous resistivity and proton NMR measurements. Shown here are the temperature dependence of interlayer resistance (top panel), proton spin- lattice relaxation rate (middle panel), and local field variations at the proton site (bottom