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Fermi Liquids Near Pomeranchuk Instabilities A FERMI LIQUIDS NEAR POMERANCHUK INSTABILITIES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Kelly Elizabeth Reidy August 2014 Dissertation written by Kelly Elizabeth Reidy B.S., American University, 2006 Ph.D., Kent State University, 2014 Approved by Dr. Khandker Quader, Chair, Doctoral Dissertation Committee Dr. David Allender, Members, Doctoral Dissertation Committee Dr. Almut Schroeder, Dr. Chuck Gartland, Dr. Kevin Bedell, Dr. Qi-Huo Wei, Accepted by Dr. James Gleeson, Chair, Department of Physics Dr. James Blank, Dean, College of Arts and Sciences ii TABLE OF CONTENTS LIST OF FIGURES . vi LIST OF TABLES . xv Acknowledgements . xxi Dedication . xxiii 1 Introduction . 1 2 Fermi liquid theory . 7 2.1 Landau Fermi liquid theory: a phenomenological perspective . 7 2.2 Landau Fermi liquid theory: a microscopic perspective . 9 2.3 Ferromagnetic Fermi liquid theory . 14 3 Pomeranchuk Instabilities . 17 3.1 Pomeranchuk instabilities in 3D . 17 3.2 Pomeranchuk instabilities in 2D . 22 3.3 Nematic instabilities . 24 3.4 Generalized Pomeranchuk instabilities . 27 3.5 Quantum criticality . 28 iii 4 Tractable Crossing-Symmetric Equation Method . 32 5 Model . 42 6 Solution Techniques . 45 6.1 Parameter space near GPIs . 45 6.2 Graphical and numerical methods . 46 6.3 Calculation of other quantities . 48 6.4 Beyond the local limit . 52 7 Results: 3D . 59 7.1 Repulsive interaction . 59 7.2 Attractive interaction . 63 7.3 q-dependence of FL parameters . 65 7.4 Nematic instability: approach to GPIs . 68 7.4.1 Approach to GPIs . 69 7.4.2 Approach to q = 0 PIs . 70 7.5 Competing quantum fluctuations . 71 8 Tractable crossing-symmetric equations in two dimensions . 84 8.1 TCSE method formulated in 2D . 84 8.2 2D Model . 86 8.3 2D Results . 88 iv 9 Application to ferromagnetic superconductors . 97 9.1 Phase transitions in FMSCs . 99 9.2 Transition temperatures . 99 10 Conclusions and future directions . 104 Appendices . 108 A Solution details . 109 A.1 3D Integrations . 109 A.2 2D Integrations . 117 B Tables of 3D Results . 121 BIBLIOGRAPHY . 154 v LIST OF FIGURES 1 Three channels of two-body interaction vertex Γ. Shown are the particle- particle (S) channel, particle-hole (T) channel, and exchange particle- hole (U) channel with their respective momentum transfers. 13 2 Figures from Quintanilla et al [27]. (a) Unpolarized, undeformed isotropic Fermi surface, (b) FS with ` = 2 Pomeranchuk deformation, (c) FS with ` = 3 Pomeranchuk deformation. 25 3 Figures from Fradkin et al [28]. A quantum nematic phase can be formed via a melting of a stripe phase or via a distortion of the FS due to a Pomeranchuk instability. 26 4 Generic phase diagram involving a quantum phase transition [36]. The associated quantum critical regime extends to T > 0, and so the effects of the QCP can be observed experimentally. 29 5 Three channels of the 2-body vertex function Γ: particle-particle (S), particle-hole (T), exchange particle-hole (U). The particle-hole and exchange particle-hole channels are topologically equivalent, and any channel can be transformed into any other channel by way of exchang- ing external lines. Shown also are the momentum transfers in each channel (K, q, and q', respectively). 33 vi 6 Generic examples of reducible and irreducible diagrams. Reducible di- agrams can be broken into independent parts by drawing a line through an intermediate state without crossing an interaction line. Irreducible diagrams cannot be split further into independent parts. 34 7 Shown here are the diagrammatic representations of Eqns. 67. Parquet equations for the particle-particle (S), particle-hole (T), and exchange particle-hole (U) channels are given. Notice that the S,T, and U chan- nels are all coupled to each other. When describing real many-body systems, treatment of all three coupled channels is needed. 35 8 Schematic form of tractable crossing-symmetric equations: F (Landau interaction function), A (scattering amplitude), D (direct term). 40 9 Two types of contribution to self-energy: (a) Hartree-type, (b) Ex- change of collective excitations . 41 a s 10 Regions of F0 ;F0 parameter space and generalized Pomeranchuk in- stabilities. 46 s 11 Sample 3D plot for U=30; solution is intersection point at F0 = a −0:676;F0 = −9:210: ........................... 48 s 12 Sample 2D plot for U=30; solution is intersection point at F0 = a −0:676;F0 = −9:210: ........................... 49 vii 13 Schematic representation of closed path used for numerical contour integration around a singularity marked by the dot at the origin. The path C is comprised of parts C1;C2;C3, and C4. Path C2 has a radius R, and path C4 has a radius ....................... 55 0 0 14 For 0 ≤ q ≤ 2kF , the Lindhard function is in the range 1 ≥ χ0(q ) ≥ 0:5. 57 15 Equation surface prior to evaluating divergences with a numerical con- tour integration. Each \spike" corresponds to a value of q in the di- a vergent range of −2 < F0 < −1 which was sampled by the Gaussian integration. 58 16 After treatment of poles . 58 17 U=12 Equation surfaces before and after numerical contour integrations. 58 18 In the multicritical branch of solutions, it is unclear which channel drives the multicritical behavior. The multicritical point is approached with increasing U. 68 0 a 19 With no q -dependence, F0 in the weak FM branch approaches -2, the q'=2 GPI; with the addition of q0-dependence parametrized by a c = 1; α = 0:2, to F0 , it can move into this region and approach a GPI in the middle of this original range of finite-q’divergences. With a a c = 1; α = 1 parametrization, F0 moves all the way through this region and can get arbitrarily close to -1, the q'=0 FM PI. 69 viii s 20 Scaled spin-symmetric Landau parameters (F` ) upon approach to charge density instability. Dashed line indicates position of q0 = 0 PI for any s s F2 s channel. Note that F2 crosses its instability ( 5 = −1) before F0 reaches -1 (the CD instability). 71 s 21 Scaled spin-symmetric Landau parameters (F` ) upon approach to FM instability. Dashed line indicates position of q0 = 0 PI for any channel. s s F2 a Note that F2 crosses its instability ( 5 = −1) before F0 reaches -2 (the GPI associated with the FM instability). 72 a 22 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to charge density instability. Dashed line indicates position of q0 = 0 a s PI for any channel. Note that no F` 's reach an instability before F0 reaches -1 (the CD instability). 73 a 23 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to FM instability. Dashed line indicates position of q0 = 0 PI for any a a channel. Note that no F` 's reach an instability before F0 reaches -2 (the GPI associated with the FM instability). 74 s 24 Scaled spin-symmetric Landau parameters (F` ) upon approach to charge 0 a density instability for the case of added q -dependence in F0 (c = 1; α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel. 75 ix s 25 Scaled spin-symmetric Landau parameters (F` ) upon approach to FM 0 a instability for the case of added q -dependence in F0 (c = 1; α = 1 parametrization). Dashed line indicates position of q0 = 0 PI for any channel. 76 s 26 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to 0 a charge density instability for the case of added q -dependence in F0 (c = 1; α = 1 parametrization). Dashed line indicates position of q0 = 0 s s F2 PI for any channel. Note that F2 crosses its instability ( 5 = −1) s before F0 reaches -1 (the CD instability). 76 s 27 Scaled spin-antisymmetric Landau parameters (F` ) upon approach to 0 a charge density instability for the case of added q -dependence in F0 (c = 1; α = 1 parametrization). Dashed line indicates position of q0 = 0 s s F2 PI for any channel. Note that F2 crosses its instability ( 5 = −1) s before F0 reaches -1 (the CD instability). 77 28 Quantum fluctuations in spin channel as function of U for the case 0 a of added q -dependence (c = 1; α = 1) in F0 . Here, fluctuations are shown including the prefactor and sign appropriate for the spin channel. 77 29 Quantum fluctuations in density channel as function of U for the case 0 a of added q -dependence (c = 1; α = 1) in F0 . Here, fluctuations are shown including the prefactor and sign appropriate for the density channel. 78 x 30 Quantum fluctuations in spin channel as function of U. Here, fluctu- ations are shown including the prefactor and sign appropriate for the spin channel (see (117)). 78 31 Quantum fluctuations in density channel as function of U. Here, fluc- tuations are shown including the prefactor and sign appropriate for the density channel (see (117)). 79 32 Spin fluctuations and density fluctuations in the weak FM branch with no included prefactors or signs (i.e. independent of channel) as function of U, where increasing U corresponds to approach to PI. 79 33 Spin fluctuations and density fluctuations in the strong FM branch with no prefactors or signs (i.e. independent of channel) as function of U, where increasing U corresponds to approach to PI. 80 a 34 Contributions of ` = 0 spin and density fluctuations to F1 with a 0 a q -dependent F0 . ............................. 80 s 35 Contributions of ` = 0 spin and density fluctuations to F1 with a 0 a q -dependent F0 .
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