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Quantum Phase Transitions in Disordered Superconductors and Detection of Modulated Superfluidity in Imbalanced Fermi Gases

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Quantum phase transitions in disordered superconductors and detection of modulated superfluidity in imbalanced Fermi gases DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Mason Swanson, B.S. Graduate Program in Physics The Ohio State University 2014 Dissertation Committee: Professor Nandini Trivedi, Advisor Professor Eric Braaten Professor Thomas Lemberger Professor Mohit Randeria c Copyright by Mason Swanson 2014 Abstract Ultracold atomic gas experiments have emerged as a new testing ground for finding elusive, exotic states of matter. One such state that has eluded detection is the Larkin-Ovchinnikov (LO) phase predicted to exist in a system with unequal populations of up and down fermions. This phase is characterized by periodic domain walls across which the order parameter changes sign and the excess polarization is localized Despite fifty years of theoretical and experimental work, there has so far been no unambiguous observation of an LO phase. In this thesis, we propose an experiment in which two fermion clouds, prepared with unequal population imbalances, are allowed to expand and interfere. We show that a pattern of staggered fringes in the interference is unequivocal evidence of LO physics. Finally, we study the superconductor-insulator quantum phase transition. Both super- conductivity and localization stand on the shoulders of giants { the BCS theory of super- conductivity and the Anderson theory of localization. Yet, when their combined effects are considered, both paradigms break down, even for s-wave superconductors. In this work, we calculate the dynamical quantities that help guide present and future experiments. Specif- ically, we calculate the conductivity σ(!) and the bosonic (pair) spectral function P (!) from quantum Monte Carlo simulations across clean and disorder-driven superconductor- insulator transitions (SIT). Using these quantities, we identify characteristic energy scales in both the superconducting and insulating phases that vanish at the transition due to en- hanced quantum fluctuations, despite the persistence of a robust fermionic gap across the SIT. The clean and disordered transition are compared throughout, and we find that disor- der leads to enhanced absorption in σ(!) at low frequencies and a change in the universality class, although the underlying T = 0 quantum critical point remains in both transitions. ii Vita January 27, 1986 . .Born|Minot ND May 2009 . B.S.|North Dakota State University, Fargo, ND September 2010 - present . Graduate Research Fellow|The Ohio State University, Columbus, OH Publications Spectral properties and duality across the superconductor-insulator transition, M. Swanson, N. Trivedi, and M. Randeria, in preparation Observation of the Higgs Mode in Disordered Superconductors Close to a Quantum Phase Transition, D. Sherman, U. S. Pracht, B.Gorshunov, S. Poran, J. Jesudasan, P. Raychaud- huri,5 M. Swanson, A.Auerbach, Nandini Trivedi, M. Scheffler, A. Frydman, and M. Dres- sel2, in preparation Dynamical conductivity across the disorder-tuned superconductor-insulator transition, M. Swanson, Y. L. Loh, N. Trivedi, and M. Randeria, Phys. Rev. X, 4 021007 (2014) Proposal for interferometric detection of the topological character of modulated superfluidity in ultracold Fermi gases, M. Swanson, Y. L. Loh, and N. Trivedi, New Journal of Physics, 14 033036 (2012) Spin-wave instabilities and non-collinear phases of a frustrated triangular-lattice antiferro- magnet, J.T. Haraldsen, M. Swanson, R.S. Fishman, and G. Alvarez, Phys. Rev. Lett. 102 237204 (2009) Critical anisotropies of a geometrically frustrated triangular lattice antiferromagnet, M. Swanson, J.T. Haraldsen, and R.S. Fishman, Phys. Rev. B 79 184413 (2009) iii Fields of Study Major Field: Physics Studies in theoretical condensed matter physics: Nandini Trivedi iv Table of Contents Page Abstract . ii Vita . iii List of Figures ...................................... vii List of Tables ....................................... xiii Chapters 1 Introduction 1 1.1 Detection of the elusive FFLO state of matter . 2 1.2 Superconductor-insulator quantum phase transition . 4 2 Interferometric detection of the topological character of modulated su- perfluidity in ultracold Fermi gases 6 2.1 Introduction . 6 2.2 Experimental Proposal . 8 2.3 Interference between coupled tubes . 10 2.4 Time-of-Flight Simulation . 12 2.5 Discussion . 15 3 Superconductor-insulator transition 20 3.1 Introduction . 20 3.1.1 Superconductor-insulator transition in 2D thin films . 21 3.1.2 Nature of the transition . 23 3.2 Conductivity across the SIT . 26 3.2.1 Bosonic model of SIT and methods . 26 3.2.2 Conductivity across the charge-tuned SIT . 29 3.3 Spectral function P (!) across the SIT . 31 3.3.1 Bosonic spectral function in the phase model . 32 3.3.2 Frequency and momentum resolved spectral-function . 33 3.3.3 P (!) across the clean SIT . 33 3.3.4 Compressibility and dispersing sound mode in SC . 35 3.4 Dynamical response across the disorder-tuned SIT . 37 3.4.1 Implementation of disorder . 37 3.4.2 Dynamical quantities σ(!) and P (!).................. 38 3.4.3 Quantum critical region . 40 v 3.4.4 Universal conductivity . 41 3.4.5 Nature of the dirty insulating phase . 46 3.5 Experimental comparison . 47 3.6 Concluding remarks . 50 Appendices A Quantum to classical mapping of phase models 58 A.1 Quantum classical correspondence . 58 A.2 Particle-hole symmetry and the quantum rotor model . 60 A.3 Equivalence of 2D JJA to (2+1)D XY model . 61 A.3.1 Mapping to XY phase model . 63 A.3.2 Mapping to current model . 64 A.3.3 Summary and duality of models . 66 B Observable quantities and analytic continuation 68 B.1 Kubo formula for the XY model . 68 B.2 Current-current correlation functions in electronic systems . 71 B.3 Sum rule derivation and comparison to Maximum Entropy results . 75 B.3.1 Sum rules on σ(!)............................ 75 B.3.2 Sum rules on Im P (!).......................... 76 vi List of Figures Figure Page 1.1 Schematic of a fully paired superfluid, an FFLO state, exists in a field range hc1 < h < hc2, with excess fermions in domain walls, and a polarized Fermi liquid. The superfluid pairing amplitude changes sign at each domain wall, a feature crucial to the experiment proposed in this thesis. 3 2.1 (Top) Schematic of a fully paired superfluid, an LO state with excess fermions in domain walls, and a polarized Fermi liquid. (Center) Pairing amplitude ∆ and magnetization m as a function of the Zeeman field h, which is the difference between the chemical potentials of up and down spins. The LO phase exists in a field range hc1 < h < hc2. (Bottom) Real space behavior of ∆(x) and m(x) in each phase. 7 2.2 Principle of our cold atom interference experiment. Two cigar-shaped condensates are allowed to expand. After a suitable time-of-flight, the shadow of a probe laser in the y-direction gives the interference pattern projected onto the x-z plane. If both clouds are in the uniform SF phase, the interference pattern is similar to the familiar double-slit experiment (left). In contrast, interference between an LO phase and a SF phase gives staggered fringes (right). 9 2.3 Proposed experimental geometry. The fermion gas is confined in a harmonic trap. An optical lattice with a large spacing in the z-direction is used to separate the gas into two independent quasi-2D layers. A second optical lattice in the y-direction cuts each layer into a series of weakly coupled tubes { the optimal geometry for LO physics. The trap and lattices are turned off abruptly, allowing the two layers to expand and interfere with one another, generating fringes as in Fig. 2.2. 11 vii 2.4 Interference patterns for three different configurations: fully paired SF state (top left), locked LO state (top right), and unlocked LO state (bottom). In each case we consider a 2 5 array of coupled tubes as shown in Fig. 2.3, × using a fully paired SF as a reference phase in the bottom layer. The pairing amplitude ∆(x) in each of the five top layer tubes is shown to the left of the interference pattern. The locked LO states were taken to be ∆(x) = −x2=2σ2 e sn(x/λLO k), for k 1 with period λLO, where sn(x k) is the Jacobi j . j elliptic function. In the unlocked case we added a random displacement of −x2=2σ2 the domain walls ∆(x) = e sn((x + δ)/λLO k) where δ [ λLO; λLO]. j 2 − This interference pattern contains signatures of the LO phase even when the domain wall locations fluctuate between tubes. In the bottom right, we show fixed-z cuts of the interference pattern for the fully locked LO state (top) and unlocked LO state (bottom). For the locked domain walls, the original domain wall spacing λLO is the same as the peak-to-peak distance of the horizontal interference fringes (since there is no expansion in the x- direction). While the visibility is reduced in the unlocked case, from the peak-to-peak distance we can still identify the domain wall spacing. 13 2.5 Interference patterns I ∆(x; z) 2 obtained from time-of-flight simulations. / j j The initial configuration (left) consists of a fully paired superfluid in the upper tube and an LO state in the lower tube. The domain walls character- izing the LO state can be seen in the figure. After a suitable intermediate time-of-flight expansion, the interference pattern develops with no significant expansion in the longitudinal (x) direction (center) consistent with the anal- ysis above, and staggered interference fringes are clearly visible. For longer times, expansion along the x-axis occurs, but LO signatures remain (right). 14 2.6 Results of BdG simulations on a 50 7 lattice in a harmonic potential Vr.
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