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Three dimensional quantum electrodynamics and cuprate superconductivity Davis, Thomas P. 2002

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Three Dimensional Quantum Electrodynamics and Cuprate Superconductivity by Thomas P. Davis B.Sc, University of Guelph, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E REQUIREMENTS FOR T H E DEGREE OF M A S T E R OF SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Physics and Astronomy)  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A October 8, 2002 © Thomas P. Davis, 2002  1  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy  The University Of British Columbia Vancouver, Canada  Date  ii  Abstract  Abstract The ubiquitous phase diagram of the cuprate superconductors has a phase where the density of states still has a gap above the superconducting transition temperature. Due to the presence of a gap without any other superconducting characteristics, this phase is known as the pseudogap (PG) state. Emery and Kivelson proposed that in this state the superconducting order parameter still has a finite amplitude, however the quantal phase is disordered in such a way that it does not possess long range order. Using these ideas, Franz and Tesanovic developed a theory to explain the pseudogap phase that is identical to three dimensional quantum electrodynamics ( Q E D ) apart from an intrinsic anisotropy. Amazingly, this theory can easily explain the antiferromagnetic region. The main purpose of this thesis is twofold. First, to explore some finite temperature properties of Q E D . We do this by calculating a Bosonic polarization bubble at finite temperature in the Matsubara formalism. Secondly we wish to derive some experimentally testable predictions of Q E D . To this end we calculate the modification to the Patrick Lee universal conductivity in the pseudogap state. 3  3  3  Contents  iii  Contents Abstract  ii  Contents  iii  List of Figures  v  Acknowledgements 1  Introduction 1.1 1.2 1.3  2  vi 1  E a r l y Superconductivity Discovery of Cuprate Superconductivity Introduction to Q E D  '.  3  Formal Developments 2.1 T h e B C S Theory of Superconductivity 2.1.1 T h e Cooper Instability 2.1.2 T h e Hubbard-Stratonovich Transformation 2.1.3 P a i r i n g Symmetries i n the B C S Theory 2.2 M a t h e m a t i c a l Description of Q E D 2.3 Zero Temperature Berryon Polarization 2.4 Zero Temperature Topological Fermion Self Energy 2.5 F i n i t e Temperature Berryon Polarization 2.6 Antiferromagnetic Order i n Q E D  7 7 8 9 10 12 16 19 22 28  Transport Properties in Q E D 3.1 Universal Conductance i n T w o Dimensional d-wave Superconductors 3.2 Modification of the Universal Conductance i n Q E D 3.3 Self-Consistent Scattering 3.3.1 Scattering w i t h i n Q E D 3.4 E x p e r i m e n t a l Tests of Q E D  33  3  3  3  1 2 5  3  3  3  3  33 37 39 41 42  Contents 4  Conclusions  iv 45  Bibliography  47  A  Dimensional Regularization  50  B  Fermionic M a t s u b a r a Summations  52  C  Manipulations of Hypergeometric Functions  54  List of Figures  v  List of Figures 1.1  The Schematic Phase Diagram for d-wave Superconductors . .  4  2.1 2.2 2.3  A Schematic Representation of the d 2_ 2 Order Parameter . . Feynman Diagram Representing the Berryon Polarization . . . Feynman Diagram Representing the Self-Energy of the Topological Fermion Diagrammatical Representation of Dyson's Equation for Topological Fermion Propagator Detailed k-space outline for constructing 4-component Fermionic spinors  12 16  2.4 2.5  3.1 3.2 3.3  x  y  Graphical Representation of the Self Consistent T-Matrix Approximation Graphical depiction of the result 3.47 The three 1/J\f contributions to the bubble diagram with arbitrary vertex  20 21 30  39 43 44  Acknowledgements  vi  Acknowledgements First off I would like to thank my supervisor, Marcel Franz. I am grateful for him for invigorating discussions and for providing all the ideas that generated the research presented in this thesis. There was also a supportive group of people whom I spent a great deal of time discussing ideas and issues. Dan Sheehy, for listening to my questions and telling me how to get the answers - "Recall those books!". Tamar Pereg-Barnea, without her I would probably still not have an adequate grasp on Q E D , and for discussions on life in general. Jose Rodriguez, who could answer all the computer questions I could generate. Also my office mates Kirk Buckley and Mark Laidlaw and my flat mate Ifiaki Olibarrieta for showing me interesting things in other fields of physics and generally broadening my knowledge. This thesis represents the culmination of many years studying physics. Throughout those years I have been taught by many professors and friends. Of this group of people, one stands out from the rest. Elisabeth Nicol has always been extremely supportive of my studies in physics. She has become a friend and a role model and I would be remiss if I did not thank her whole heartedly. I would also like to acknowledge my family for their support. Finally, I would like to thank Heather Nicholson for the unending love and support she has shown me throughout the past year. 3  Thank you all.  Chapter  1.  Introduction  1  Chapter 1 Introduction 1.1  Early Superconductivity  Research into low temperature physics began in 1911 when H . Kammerlingh Onnes succeeded in liquefying helium. He observed that at 4.2 Kelvin, mercury undergoes a transition where it loses all electrical resistance [1]. Onnes described this material as "superconducting". Subsequently, superconductors were shown to exhibit another equally remarkable phenomenon, the Meissner effect. If a material generates a magnetic field to partially cancel out an externally applied field, it is said to be diamagnetic. The Meissner effect is the total expulsion of a magnetic field from within the material [2]. The Meissner effect could be described as perfect diamagnetism, but it is actually something more subtle and profound. If a material was cooled below its hypothetical "diamagnetic transition temperature" in an external field, it would actually produce fields in such a way to retain the magnetic field within the sample even after the applied field is turned off. Superconductors cooled below Tc in the presence of a magnetic field will actually expel the magnetic field. This is a direct consequence of the loss of electrical resistance, since without resistance the electrons within a superconductor can set up a supercurrent to perfectly screen the applied field. The first theoretical attempts at describing superconductivity were all phenomelogical, as superconductivity is a quantum mechanical phenomenon and quantum theory was still in its infancy. It was not described microscopically until Bardeen, Cooper and Schrieffer published the now famous BCS theory in 1957 [3], nearly 50 years after the original discovery. The BCS theory of superconductivity was built on the ideas of Frolich who postulated that the electrons responsible for superconductivity interacted strongly with the lattice vibrations, known as phonons. This postulate came from the experimental "isotope" effect, where the superconducting transition temperature T has a strong dependence on isotope, and therefore the mass, c  Chapter  1.  Introduction  2  of the nuclei in the lattice. This was not the whole picture. Cooper showed that if there is an attractive interaction between electrons - no matter how weak - then the normal many electron ground state is unstable. He showed that in the presence of this attractive interaction the free energy could be minimized if electrons "paired" and condensed into a state similar to a BoseEinstein condensate. This condensate of electrons would then display the desired superconducting properties. The origin of the attractive potential lay in the electron-phonon interaction proposed by Frolich. For a certain range of phonon frequencies Bardeen showed that the screened Coulombic repulsion is actually overcome by the electron-phonon-electron interaction and the overall interaction is attractive. Therefore the Cooper instability exists and the electrons form pairs and condense into a phase coherent many body wavefunction, first written down by Schrieffer. The condensation of paired electrons lowers the total free energy of the system. This introduces an energy gap, a minimum amount of energy required to break the Cooper pairs. This is the essence of the BCS theory of superconductivity, and the three physicists won the Nobel prize in 1972 for their monumental achievement. Even though the BCS theory met with success, there was still some empirical data not accounted for. The theory was further improved upon by Eliashberg [4]. The full Eliashberg theory is incredibly successful in describing a large range of superconductors such as pure elements and simple compounds. Superconductors described by the Eliashberg theory are known as "conventional superconductors".  1.2  Discovery of Cuprate Superconductivity  With the mechanism of superconductivity explained, the race was on to find the material with the highest superconducting transition temperature. In 1986 Bednorz and Muller [5] discovered a copper oxide material with a transition temperature that was 12 degrees higher than any other material previously recorded. The brilliance of Bednorz and Muller came in abandoning the conventional materials already known to superconduct. They made materials out of copper, oxygen and a rare earth metal (originally barium). These materials can be made to superconduct as high as 100 Kelvin, much higher than can be explained using the Eliashberg theory. Bednorz and Muller were awarded the Nobel prize in 1987 as a result of their discovery.  Chapter 1. Introduction  3  The class of materials containing copper oxygen planes and rare earth metals are known as "cuprate superconductors". The transition temperature of the cuprate superconductors can be changed by changing the number of oxygen atoms in the copper oxygen planes (known as doping). Remarkably, most cuprate superconductors have strikingly similar characteristics as a function of doping. They achieve a maximum transition temperature at "optimal doping" nearing 100 Kelvin. Since the discovery of the cuprate superconductors much theoretical effort has been devoted to understanding the superconducting mechanism. The next challenge in the field was to develop the art of crystal growing. Crystals of the highest purity were needed to separate extrinsic and intrinsic effects. Furthermore, controlling the oxygen doping was of the highest importance in order to map out the complete phase diagram. Once these technical feats were accomplished [6], a surprising revelation was made. The parent compound (cuprate materials at zero oxygen doping) were seen to be antiferromagnetic Mott-Hubbard insulators. A Mott-Hubbard insulator is a material that is not a traditional insulator in the sense that the Fermi energy lies within a band gap. It is a material that has exactly one electron per lattice site and Coulomb repulsion forbids electrons from hopping from one site to another. Traditionally it was thought that any magnetic behavior was detrimental to superconductivity. One of the most important theoretical challenges was to be able to describe both antiferromagnetic and superconducting behaviour in the same material. A number of unexpected and interesting phenomena were discovered once the whole phase diagram was mapped. First, a gap was seen in the energy spectrum above the superconducting transition temperature! The gapped energy spectrum in the absence of any superconducting behaviour is known as the "pseudogap" and is the main focus of this thesis. The characteristic temperature where the pseudogap disappears is known as T* and it can be as high as room temperature. Figure 1.1 shows a general sketch of the phase diagram. Secondly, the energy gap was a function of momentum, and it actually vanished along certain directions in the crystal. The vanishing of the energy gap gives rise to gapless excitations in the superconductor and greatly modifies its low energy properties. The microscopic physics underlying this unconventional symmetry has so far eluded physicists, and has proved to be one of the most challenging pursuits of condensed matter physics.  Chapter 1. Introduction  Figure 1.1: The Schematic Phase Diagram for d-wave Superconductors  4  Chapter 1. Introduction  1.3  Introduction to Q E D  5  3  The incredibly successful BCS-Eliashberg theory of superconductivity describes the superconducting phase as an instability of a well known and well understood parent system, namely the Fermi liquid. Fermi liquid theory is arguably one of the most successful theories in all of physics for its breadth of applicability. In attempting to follow this route in the cuprates, one quickly runs into trouble; the parent compound is itself a mystery. The antiferromagnetic Mott-Hubbard insulator is an example of a strongly correlated electron system which is currently the source of a large amount of research. Furthermore, the pseudogap phase that separates the A F M Mott-Hubbard insulator from the superconducting phase is itself poorly understood. Therefore we adopt a new paradigm for the investigation of cuprate superconductivity. We attempt to describe the complete phase diagram by taking the d-wave superconducting phase to be the most well-understood phase. The reason we believe this to be appropriate is that the d-wave region is remarkably well described by a BCS-like theoretical model with unconventional gap symmetry [7]There are two temperature scales associated with superconductivity, T i and T h . The former is the temperature scale where Cooper pairing becomes locally significant, and the latter is the temperature scale where these local regions of Cooper pairs become phase coherent over macroscopic distances. In the BCS-Eliashberg theory the wavefunctions have long range phase coherence all the way up to T ~ T . In this theory, T h is infinitesimally smaller than T g so the loss of phase coherence does not play a role in determining T . However, there could be a situation where T hase << T i i g, then the material would still display some of the properties associated with Cooper pairing, however would not display the infinite conductance and the Meissner effect that are the hallmark of a superconductor. This is precisely what is seen in the pseudogap state. The possible connection between the loss of phase coherence and the pseudogap state was first elucidated by Emery and Kivelson [8]. Further evidence suggested that the mechanism for the loss of phase coherence was related to the "topological defects", or "vortices" present in superconductors [9]. Externally applied magnetic fields are expelled from the interior of a superconductor due to the Meissner effect. However, in some circumstances it can be shown that it is energetically favourable to allow the penetration of the magnetic field into the sample. This is due to interplay pair  p  ase  c  pairin  c  p  pa r n  pairing  p  ase  ng  Chapter  1.  Introduction  6  between the energy cost of the "normal" state within the superconductor and the energy gained by the interaction of the internal and external magnetic fields, as can be seen by a simple Landau-Ginzberg model (see [10] chapter 6). When the penetration of the magnetic fields into the bulk of the superconductor occurs, it does so in a very regular manner. A regular lattice of flux tubes, known as the Abrikosov flux lattice [11], is created. The tubes of flux are surrounded by a circulation of supercurrent. For this reason the flux tubes are known as vortices. Vortex/anti-vortex pairs are also present in the cuprates in the absence of an externally applied magnetic field due to thermal and quantum mechanical fluctuations. In the superconducting state, these pairs are bound together and therefore have a negligible effect on the low energy properties. However, as the material is brought into the pseudogap state, the situation changes drastically. A transition takes place where the vortex/anti-vortex pairs unbind (known as the Kosterlitz-Thouless transition [12]). The vortices can be found to be separated from the anti-vortices by arbitrarily large distances. In this state the low energy properties of the system are going to be dramatically altered by the presence of these unbound fluctuations.  Chapter 2. Formal Developments  7  Chapter 2 Formal Developments This chapter is devoted to the mathematical description of the ideas presented in the introduction. It begins with a mathematical description of the BCS theory and proceeds to introduce Q E D and calculate a number of important quantities such as the polarization of the gauge field and the Fermion self energy. It continues to generalize the polarization to finite temperature and ends by describing the antiferromagnetic instability inherent in Q E D . 3  3  2.1  The BCS Theory of Superconductivity  The BCS reduced Hamiltonian H  = Y1 (p) l* p* ~ 7^ XI U - 'l -pi pt a  €  a  a  a  a  P  per  a  (2- ) 1  p'p  can be derived using various techniques. The most straight forward is to consider the electron-electron interaction mediated by phonons in the second Born approximation. This gives a functional form for the interaction potential  V  -  V Q  I  "  M  (22)  where w is the Debye frequency of the material. The first term represents the screened Coulomb electron-electron interaction. The second term represents phonon mediated electron-electron interaction, that can dominate and produce an overall negative (attractive) potential for frequencies just below D  oo D  •  Using the renormalization group we can actually show that repulsive interactions grow weaker and attractive interactions grow stronger in the momentum region responsible for superconductivity [13]. This implies we can  Chapter 2. Formal Developments  8  simplify the theory without losing its salient features by replacing the interaction potential 2.2 with the simplified interaction  in equation (2.1).  2.1.1  The Cooper Instability  One of the first important results of superconductivity was derived by Cooper in 1956 [14]. By variational methods he showed that an attractive interaction between electrons in a filled Fermi sea will lead to a pairing instability, no matter how weak the attraction. The variational state he chose was  |V >=  P U -pi\ >>  a  a  a  F  (-) 2  4  p,e >tp p  where \F > is the many body wavefunction describing the filled Fermi sea. This wavefunction corresponds to a formation of a pair with equal and opposite momentum and opposite spin. To see if this trial wavefunction results in a lowering of energy the quantity to calculate is E-E =<ip\H\ip>  - <F\H\F>,  0  (2.5)  where the H is the BCS reduced Hamiltonian 2.1. Assuming that the density of states times the electron-electron coupling constant is small gv «  1 the  result is E — E = -hu e~ . 1/9,/  Q  D  (2.6)  The important feature of equation 2.6 is the presence of the minus sign. This implies that the variational ansatz 2.4 results in a lowering of the ground state energy. Therefore for any positive value of g it is favourable to form "Cooper pairs". This result shows that the filled Fermi sea is not the correct ground state when there exists an attractive interaction between electrons.  Chapter  2.1.2  2.  Formal  Developments  9  The Hubbard-Stratonovich Transformation  In a system exhibiting the Cooper instability, the filled Fermi sea is no longer the appropriate description of the ground state. A new ground state must be found. To ease the search for this new ground state we perform a H u b b a r d Stratonovich transformation on the H a m i l t o n i a n 2.1. T h i s transformation makes use of the identity  Z(H) = j  VA^VAZ(H')  (2.7)  to define the new "transformed" H a m i l t o n i a n as H' = J2 e(p)a  PCT  V a . + - AtA -  {tia^a  P  9  per  p i  + A a J / ^ } .  (2.8)  p  T h e beauty of the new H a m i l t o n i a n is that the four-Fermion interaction has been decoupled into two two-Fermion interactions, which can be handled analytically. Using the mean-field saddle point approximation, A can be written as ^2  <  a  (2.9)  *,t°-*,4. > •  T h e variable A introduced by the Hubbard-Stratonovich transformation goes by many names: order parameter, pair potential or the superconducting gap function. A l l three names w i l l be used interchangeably throughout this thesis. T h e transformed H a m i l t o n i a n 2.8 can now be diagonalized by performing a Bogoliubov transformation  ua-i + va\. c | = —v*ai + u*a\.  (2.10)  B y imposing canonical anti-commutation relations on the new operators c± and c | we find equations relating the magnitude of the so-called "coherence factors"  u  1 ' 2 1 2  1 +  1 -  Ve V^  2  2  + |A|  + |A|  2  (2-11)  (2.12) 2  2. Formal  Chapter  Developments  10  and the Hamiltonian 2.8 becomes  H = Y  \Ve (p)  + WY, * °  2  c  + (?) - Vz (p) + W\  c  e  2  We can read off the ground state energy as  + -|A| .(2.13) 2  |e(p) — \/e (p) + | A | | 2  +  2  y | A | and the energy of the excited states E(p) = y/e (p) + | A | . By noting that the energy in the Hamiltonian 2.13 is measured relative to the Fermi surface the limit A —> 0 gives the correct answer. We can now write down the new many-body ground state wavefunction from the Hamiltonian 2.13 by demanding that c|\& >= 0 2  2  2  i*>=n[«+™iXt] i°>  (- ) 2 14  which represents a coherent superposition of the vacuum and a paired state. An important feature of this wavefunction is that this ground state does not have a definite number of particles. This implies that the ground state has long range phase coherence, as quantum mechanical phase is canonically conjugate to particle number.  2.1.3  Pairing Symmetries in the B C S Theory  Any pairing symmetry is allowed by the BCS theory if the interaction parameter g is allowed to depend on momentum. The BCS reduced Hamiltonian would now read H  =  Yl (P) l<T pcr e  a  a  - ^  Y  9vV' \<Acr< P° P°'l a  a  a  (  2  -  1  5  )  where in full generality ^  =  £jWy/i(p)7*(pO  (2-16)  ij and the fi(p)'s are some complete set of functions. Since g must be Hermitian we can perform a unitary transformation to diagonalize Mjj. Further simplification can be gained by noting that the /j(p)'s don't vary much over  Chapter  2. Formal  11  Developments  the momentum range involved in superconductivity (\p — pp\ « WD/VF), but they will still depend on the direction of the momentum p. One can now choose the /i(p)'s to be the spherical harmonics that carry an angular momentum index I. Even / correspond to symmetric functions and odd I correspond to anti-symmetric functions. Therefore the total Cooper pairs will be spin singlet and triplet states respectively. It can be shown in the BCS theory that it is energetically favourable to have spin singlet Cooper pairs. However odd angular momenta can still appear as sub dominant contributions to the order parameter. Conventional superconductors were shown to have predominantly I = 0 contributions, and were therefore called s-wave superconductors. Initially it was unclear which pairing symmetry existed in cuprate superconductors. After the crystals were grown to a high enough purity, it was conclusively shown that they have predominantly / = 2, or d-wave, symmetry (see [15], for example). Mathematically the gap looks like A .l 2_ 2 d  x  y  <XPI-P y 2  (2.17)  = C0S(2¥>),  or schematically depicted in figure 2.1. One of the most remarkable properties of this function is that it changes sign twice as <f> goes from 0 —>• 2n around the Fermi surface. This implies that there are four points on the Fermi surface where A = 0, called "nodes". Therefore along these specific directions in the material, the quasiparticles are not gapped and exhibit non-superconducting properties. The existence of nodes leads to a finite density of low energy Fermionic excitations which dominate the low energy physics of cuprate superconductors. The Hamiltonian derived earlier (2.13) assumed s-wave symmetry of the pair potential. The Hamiltonian that includes arbitrary pairing potential is known as the Bogoliubov-de Gennes (BdG) Hamiltonian and takes the form  H =  A(r) A*(r)  ^(V-fA) + e  (2.18)  2  F  The B d G Hamiltonian represents a fully microscopic treatment of the cuprate superconductors.  Chapter  2. Formal  Figure 2.1: A Schematic Representation of the  2.2  12  Developments  Order Parameter  d 2_ 2 x  y  Mathematical Description of Q E D  3  Our starting point into the investigation of the cuprate superconductors begins with the Lagrangian density associated with the B d G Hamiltonian 2.18 I  £ = &d * + &H* T  + -\A\ .  (2.19)  2  9 Here H is the B d G Hamiltonian 2.18 and the \I>'s are Nambu spinor's containing Grassmanian Fermionic creation and annihilation operators = The important quantity to investigate in condensed matter systems is the partition function. Fundamentally it is a weighted trace of the Hamiltonian. Equivalently it can be written as a path integral over all dynamical variables [13]. By writing the order parameter as A ( r , r ) = \A \e ^ ' \ the partition function for our system is l  r T  0  \~ I I  T><f>(r, r)exp  dT  ) ^ ' °^  d2rC  The Lagrangian density 2.19 is written in terms of second quantized Fermionic creation and annihilation operators. For a complete introduction to these techniques, see [13]. 1  2 2  Chapter  2.  Formal  Developments  13  T h e effect of vortex fluctuations is manifest i n the partition function 2.20 by virtue of the path integral over the phase variable <f>(r,r). In the absence of vortices, this integral is over smooth functions that represent spin waves. However the presence of vortices complicates this integral, we are now required to integrate over singular functions that encode the position of the vortices. In order to simplify calculations it would be useful to eliminate the 0(r, r ) terms from the off diagonal elements of the B d G H a m i l t o n i a n 2.18 v i a a gauge transformation H' = UHU~ .  (2.21)  l  The most obvious choice (  e^l  «=(%  2  0  \  «-*»)  (2  -  22)  does not encompass a l l the physics of the situation. T h e gauge transformation 2.22 neglects branch cuts i n the quasiparticle wavefunctions ip. The branch cuts result from the fact that the vortices are topological defects i n the Cooper pair field and have flux quantum of hc/2e, exactly half of the "natural" flux quantum associates w i t h the quasiparticles. Therefore, as the quasiparticles wind around the vortex they acquire a phase of only e and therefore acquire a minus sign. Franz and Tesanovic developed a gauge transformation that circumvents this problem, in  (2.23) where 4>A + (j>B = <f>- T h i s gauge transformation separates the vortices into two classes A and B , and <f>A(B) is the singular part of the phase due to the A(B) vortex defect V x V<f>  A{B)  = 2 7 ^ ^ (r - r f i  ( B )  (r)) ,  (2.24)  where = +1 for vortices and = — 1 for anti-vortices. T h e particle and hole part of the quasiparticles are only affected by the A and B vortices respectively. T h e separation of vortices into A and B is arbitrary, and one must  Chapter  2. Formal  14  Developments  perform a summation over the 2^ distinct vortex configurations , iVj being the number of vortex defects. By performing this summation we change the fundamental excitations of the theory from the aforementioned quasiparticles ^ to "topological Fermions" ^f. The beauty of the F T transformation is that the effect of the branch cuts in the quasiparticle wavefunctions are elevated to the level of the Hamiltonian. They now appear as a pair of gauge fields of the form ^  = ^(^A(r) + ^ (r)),  (2.25)  B  a = M  ^(cV^(r)-cVMr)).  (2.26)  The term is known as the "Doppler" gauge field because it represents the classical effect of Doppler shifting of the energy levels due to the circulation of the supercurrent about the vortices. The a term is a purely quantum mechanical manifestation of the winding of the phase due to the non-trivial geometry created by the vortices frustrating the topology. Since a describes a geometrical phase effect it is known as the "Berry" field [16]. Quantized excitations of the Berry gauge field are termed "Berryons". M  M  It can be shown that both and are both massive in the superconducting state [17]. However this situation changes dramatically in the pseudogap state. The Doppler gauge field still remains massive, but the Berry gauge field becomes massless. The Berry gauge field is therefore responsible for much of the low energy physics in the pseudogap state. After performing the F T transformation and the summation over vortex configurations, the effective low energy Lagrangian takes the form  C  =  J2¥ [D a  iv D a  T  F  x  2  - iv^DyCJ^ *  Q  a=l,l +  *a [D - ivpDyO-3 - iu D (7i] * T  A  s  a  + ^  M  {d x a)l (2.27)  a=2,2  where D^ is the Berryon covariant derivative defined by D^ = <9 — ia^. For explicit details of the calculations leading to the Lagrangian 2.29 see [18]. M  At this point the Lagrangian 2.27 is reminiscent of the Dirac Lagrangian describing quantum electrodynamics. The analogy is not exact because the matrix associated with the temporal derivative is the unit matrix, which commutes with the C T J ' S .  Chapter  2. Formal  Developments  15  In 3 dimensions we make the following identifications  70 = cr  "I  2  71 = - a  72 =  J  0"3  in order to satisfy the Dirac algebra {7^, j } the Lagrangian becomes v  £=  (2.28)  \  x  *al»D«V  = 28^. Introducing \l> = 2  + ±K»(dxa)l.  a  i^j  0  (2.29)  a=l,T,2,2 Now the only difference between this Lagrangian 2.29 and the Q E D Lagrangian is the anisotropy present in the definition of the covariant derivatives. D  X  D  2  = D= T  [D ,v D , T  F  x  A  (2.30)  vpD \  (2.31)  A  = D* = [D , v D , T  v D] x  y  y  For the rest of this chapter we will be doing calculations in the isotropic limit where we set vp = v = 1. This is quite unphysical as Vp/vA ~ 10 — 20 as measured in the cuprates. However calculation is much more convenient in this limit and in most cases the results can be readily generalized to the physical anisotropic limit at the end of the calculation. A  The Lagrangian is now £ =  Y  ^ a l M - i a ^ a  + ^K^dxa) ^. 2  (2.32)  a=l,T,2,2 This Lagrangian is now identical to three dimensional quantum electrodynamics with four species of Fermions. The Lagrangian 2.32 is amenable to perturbation theory in the \/M expansion if we consider M species of Fermions. This is permissible, but we must remember to set M — 4 at the end of calculations. Therefore the Lagrangian which will be investigated for the remainder of this chapter is C = ^* 7M(3 q  m  - ia^a.  (2.33)  a=l  In regular QED, this is done to maintain Lorentz covariance. In QED3 it is merely a computational convenience. Therefore this transformation must be undone if one is interested in looking at measurable quantities. 2  Chapter  2. Formal  16  Developments  Figure 2.2: Feynman Diagram Representing the Berryon Polarization  2.3  Zero Temperature Berryon Polarization  Armed with Lagrangian 2.29 we are now ready to investigate some properties of Q E D . The tree level Green's function can be read directly from the Lagrangian 2.29 as 3  Go(k) = ^  (2.34)  which is exactly the massless free Fermion propagator. To first order the polarization of the Berryon is calculated by the insertion of a T F loop into the bare propagator, as seen in figure 2.2. The resulting Feynman integral is  dk 3  /  j^TRlGoikh.Goik  + q)>y ]. v  (2.35)  In order to perform the integral, we break it up into two parts, a standard Feynman integral I p(q) and a trace; a  n„„(g) = J (g)TR[7 7 7 Q/3  Q  /i  /37l/  ].  (2.36)  The trace can be evaluated using the anticommutation relation that defines the Dirac algebra { 7 , 7 „ } = 2<5 „ (keeping in mind we are in 3 dimensions). P  M  Chapter 2. Formal Developments  17  Therefore, the only task is to evaluate the Feynman integral  {2*yk*(k+qr  ^ - J  -  (2 37)  To perform this integral, one generally introduces Feynman parameters to put the denominator into a more manageable form. Implementing this parameterization, the integral becomes  f dk f j (2^yJ0 f d' r 3  l  a  ~  m  fc (fe  1  [kH  1  P  J  (2TT)8 y  + g)ff  Q  + (i-x)(k  ax  d  + qyY  E ^ ^ M l  x  . )  (2  38  0  Where i n the second line we have made a change of variable p = k + (1 — x)q and have identified q x(l — x) = A . B y symmetry, p pp w i l l integrate to zero unless a = B, so it must be proportional to 5 pp . A factor of | is out front since we are working i n 3 dimensions. Therefore the integral reduces to 2  2  a  2  a  To regularize the integral, we enforce that the Berryon excitation be massless, n,u/(<z —> 0) = 0 . Therefore the integral to calculate is I p{q) = I p(o) ~ l p(0). T h i s integral is now convergent, and the result is a  a  a  Ipil)  =^  (<W +  "-f)  (2.40)  Therefore, the self energy is IV(g)  =  7^(?)TR[7^7«7^] q  _q q  x  i * \  (2-41)  M  s  ~ - T )  {  2  A  2  )  T h i s result is true for a single node. T o make contact w i t h the large A/" expansion, we introduce M "species" of Fermions, corresponding to 2J\f nodes around the Fermi surface.  IVfa)  =  m^-**)  (2.43)  Chapter 2. Formal Developments  18  In general the full propagator solves Dyson's equation ^(Q)  = ^ ( Q ) + ^AQ)-  (2-44)  However we are concerned with the low energy properties of the theory. Therefore we can neglect the 0(q ) contribution from the bare propagator and the resulting full propagator is simply the inverse of the polarization 2  V^q)  = U^(q).  (2.45)  However, the polarization operator is proportional to a projection operator, which is singular and cannot be inverted. The underlying cause for this singular behaviour can be most easily seen by using functional methods. The action for the Berryon field is  S = jd x^(d z  x a) .  (2.46)  2  The functional integral over all a^x) fields is formally divergent because the integral  /  Vae ^  (2.47)  iS  integrates over all fields a (x), even those that are equivalent under a gauge transformation a^(x) = a^x) + dfj,a(x). This infinite double counting of the fields leads to this singular operator 2.43. Therefore we introduce a method to restrict the integration to only fields that are physically distinct. This method was first introduced by Fadeev and Popov in 1967 [19]. We proceed by expanding the action M  S  =  ^j  d xe  =  ^j  d x(8 S p  =  ^j  d x (-5^d  e p d a d ap  3  lil/<T  a  a  fi  v  3  2  afi  (2.48)  a  - StfSv^dpdvdaap  3  m  l/  + dMa .  (2.49) (2.50)  v  By inserting a delta function into the functional integral in a clever way we can enforce that we only integrate over physically distinct field configurations  J2>ae*M  =f j Va  Vadet  (^r)  *(G(a))e"M.  (2.51)  2. Formal  Chapter  Developments  19  The function G(a) is called the "gauge-fixing" term. In a general covariant gauge we can choose G(a) — d a^{x)—uj{x).  Secondly, we perform a Gaussian  tl  integration over UJ(X) with width £. Jvae  i  S  ^  =  C J VOJ J Vae- ^5(3^-u{x))e ^  (2.52)  =  Cj  (2.53)  iJd3x  iS  Vae-tf^-^-e^.  It now appears that we have added an extra term to the action. The new action takes the form S'[a]  =  j  (Pxa^-S^d  + (1 + j)dMa„  2  (2.54)  This extra term can be seen as unphysical Berryon degrees of freedom, namely longitudinal excitations with energy proportional to £. With this longitudinal term present, the polarization is now invertible and has the form v»M)  2.4  = y^7  - ^r(i  - o)  •  (2-55)  Zero Temperature Topological Fermion Self Energy  The basic excitation in this theory is the topological Fermion (TF). To first order the self energy is calculated by an insertion of a renormalized Berryon line to the T F propagator, see figure 2.3. J3  j ^ V ^ G ^ k  /  q(k + q) {'"  q  "  7  r  »  + q)^  W  2  /  (2-56)  q  2  1  V)  I*™" (2.57)  The integral simplifies using (<W - ^ r U  - 0)  7 , 7 a 7 , = -2(1 - 0 7 , ^  (2-58)  Chapter  2.  Formal  20  Developments  Figure 2.3: Feynman Diagram Representing the Self-Energy of the Topological Fermion  to  with  >  1  The vector integral I^k) I^{k)  = C(k)k /k . 2  li  W  =  J*^7&£>  (2  will only have components in the  C(k) is given by the scalar product  60)  direction,  k^I^k).  Once again we use the Feynman parameterization of the denominator and the integral becomes  x)k)  2  + k x{l 2  -  x)fl  2  (2.62) This integral is formally divergent. In condensed matter systems it is completely legitimate to introduce a momentum cutoff to regularize the integral,  Chapter  2.  Formal  21  Developments  -*  +  u  Figure 2.4: D i a g r a m m a t i c a l Representation of Dyson's E q u a t i o n for Topological Fermion Propagator  since the divergence is a consequence of the linearization of the momentum dispersion near the F e r m i surface. In our system, the momentum cutoff A w i l l be of the order of the superconducting gap. Performing the change of variables p = q + (1 — x)k and again identifying A = k x(l — x) 2  C{k)  = 2n ^  dx^x ^  dp'-  ^  2  /  A  2  ]  2  5  /  2  (2-63)  We focus on the most divergent part of the integral, which is logarithmic. The integral is now elementary and i n the large A limit is C(k)  (2.64)  = ~irk \n^. 2  We finally arrive at the one loop topological Fermion self energy  W  ~  TTW  7  "  k  2  '  =  (  2  -  6  5  )  Using this self energy we can renormalize the tree level propagator using Dyson's equation (see figure 2.4)  G-\k)  = =  Go\k) 7  + 2(k)  ^ ( l + 77ln^),  (2.66) (2.67)  where we have set r\ = 4(2 3£)/3ir J\f. Borrowing ideas from the renormalization group analysis we recognize this as the first order "leading log" contribution to the propagator. Therefore we 2  Chapter  2.  Formal  22  Developments  can anticipate and re-sum the complete series. function that lies at the heart of Q E D  The result is the Green's  3  G  i  -  k  )  =  ^{%)  •  J  (  2  -  6  8  )  The state of matter described by the Green's function 2.68 is known as "Algebraic Fermi Liquid" due to the anomalous dimension rj.  2.5  Finite Temperature Berryon Polarization  The knowledge of a Green's function at zero temperature allows for the calculation of a large number of observables. However experiments actually measure the temperature dependence of these observables. Therefore a very important quantity to determine is the Green's function at finite temperature. The finite temperature properties of Green's functions are readily determined by identifying temperature with "imaginary time" it. In the Heisenberg representation operators evolve in time according to 6{t)  = e 6(0)e- .  (2.69)  itA  it6  By defining an imaginary time evolution operation  <5(T) = e "6(0)e- ",  (2.70)  = e &(0)e-  (2.71)  T  6 (r) f  T  TA  TA  we can introduce a thermal Green's function G(r)  =  -<T A(r)At(0)> r  =  -{A{T)A\0)),T>0  =  T(A (0)A(r)),r<0  (2.72)  t  where the angular brackets denote thermal averaging (•••) = T R { e " ^ • • •}. In the last line, the minus sign is for commuting Bosonic operators and the plus sign is for anti-commuting Fermionic variables. A very important feature of the Green's function arises due to the cyclic property of the trace  Chapter  2.  Formal  23  Developments  = TR{CAB} = TR{BCA}, namely that the Green's function at finite temperature is periodic or anti-periodic in imaginary time for Bosonic and Fermionic operators with periodicity B. This periodicity is of tantamount importance when Fourier transforming TR{ABC}  g(iu )  = /  n  dre^" G(r), T  (2.73)  e-^ g(iu ).  (2.74)  Jo oo G(T)  =  £  T  n  n——oo  One now only has to sum over discrete frequencies. The periodicity G(0) = ±G(p) implies that Un = ^ -  (2.75)  = ^  (2.76)  for Bosonic operators and  for Fermionic operators. The cu are known as Matsubara frequencies after the physicist who developed this formalism [20]. In practice, this corresponds to replacing the temporal component of the momentum with a Matsubara frequency and replacing the integral over the corresponding momenta with a Matsubara sum. Performing this replacement on equation 2.35 the finite temperature Berryon polarization becomes n  IW*,*) = i ^ / ( g i j ^ f T  R  [  W  W  ],  (2.77)  where k = (2n+l)7r//5, n € Z , a Fermionic Matsubara frequency, is now a function of incoming momenta q and Bosonic frequency q = 2mir/B, m G 0  0  Z.  The calculation is identical to the zero temperature case up to the point where we introduce the Feynman parameterization of the denominator. The analog of 2.36 is n„ p  =  /a/3TR[7 7 7^7„], Q  M  (2.78)  Chapter  2.  Formal  24  Developments  where  J ^  Pv ' °  2  K + ( - » k  q  x  x  2+ A 2  ^ ' 2  A (g) = (q + ql)x{\ - x) + (w„ + g„(l - *)) 2  2  2  (2-80)  Normally we would simply change the variable of integration to p^ = k + q (l-x). li  (2.81)  ll  However the compactification of the temporal dimension breaks the fortuitous Lorentz symmetry that made the earlier calculation so elegant. The Matsubara frequencies can not be transformed, so we restrict the transformation 2.81 to spatial coordinates only. This change of variables will affect the I o, hi and i y integrals differently. Therefore we must calculate each of them separately. 0  First we tackle the I o term. Implementing the change of variable 2.81 the integral becomes 0  °° =  7  ^Jwv+wr  -  (2 82)  which is a standard integral with the result  ^  = - ^ T , h  d  x  A (g) 2  •  ( 2  '  8 3 )  Next we calculate 7oi + ho (since this symmetric combination is the one that appears in the polarization). This term is rather straight forward, and using the fact that everything linear in p will integrate to zero the result is  ' « ' ' =? ? s i ' ^ +  <- > 2 84  The remaining integral hj is the trickiest since it is formally divergent. We split the integral in two in order to focus on the divergent part  Chapter  2.  Formal  25  Developments  The latter integral is now amenable to dimensional regularization (see Appendix A). The result is  l  ?  "  W  V  &  W  ^  A  )  (  2  8  6  )  We are now ready to calculate the Matsubara sums. Defining  where a = |g| a;(l — x) and b = q {l — x) and / £ 0,1, 2. Using this definition, the integrals become 2  0  loo  = £ r ^{S(2) + E(l)}  1  1  o  /oi + Ao  9o  = i / dx{(2x-l)g E(l)-(l-a )^iS(0)} 1  0  i  ;  I  ( .88) 2  The Matsubara sums 2.87 are amenable to computations by the methods outlined in appendix B. By making the identification  =  fiM*n)  (  r  M  .  f^fr  •  (Vo + iu) + ib)(y/a  rr-y  (2-89)  - iu) - lb)  n  n  the sums become  l y _ M _ B ^  a  + (6 +  u)  y  =  H  ^-t  2  n  n  W  _ i  m  (e^  z  + I) {z + y/a + ib){z + y/a - ib)  Zj£{—,/a±ib}  (2.90) By this method, E(0) and E ( l ) are easily evaluated. E(0) = ) — n_ , 2 | q , ^ L  I  ( 2  .9i)  2(1) = - 2 & V ^ - - ^ n  where n  _  =  -sinh/3|q|v/rr(l-x) cos/3#o(l — x) + cosh/3|q| y/x(l — x) cosh/3|q| o;(l - x) +cos8q (l - x) + isin/3q (l — . cos/?go(l — x) + cosh/?|q|y :r(l — x) /  n  v  +  0  0  ( 2  - x)  Q  2  )  . (2.93)  Chapter  2.  Formal  26  Developments  Power counting tells us that E(2) is divergent. We need a method of regularization to sift out the unphysical divergent constant from the dependence of the sum on the physical quantities. To achieve this we add and subtract one from the sum  By doing this "trick", we find that the infinite contribution is simply the value of the Riemman zeta function with argument zero. Therefore we regularize by disregarding this portion of the sum, that does not depend on any physical quantities in the theory.  n  - ( a + 6 ) E ( 0 ) - 2 6 E ( l ) + - ^ ( l + 2C(0))  =  2  (2.95)  Lastly we must evaluate the logarithmic sum appearing in equation 2.88  X  L  = lY\n  j±  -.  (2.96)  a + (b + co )  2  n  To perform this sum we note that by taking the derivative with respect to the parameter a we reduce to a sum previously calculated  ^  =  -  ^  E  =  n -E(0).  a  +  (6  + ^ ) 2  ( 2  9 7  )  (2.98)  Therefore TIL  =  f da-^ , f f , lyfa cos (3b + cosh By/a h  J  =  /  (2.99)  3  r  ^ In ( c o s +  cosh/Va) +T {a L  v  = 0),  1  (2.100)  since the differetiation kills the terms in the sum independent of a.  *< > = £ 2 > ( ^ -  E  a= 0  (2101)  Chapter  2.  Formal  27  Developments  This part of the sum can be calculated by differentiating with respect to b. The result is E ( a = 0) = | l n ( l + exp(«/36)).  (2.102)  L  These two pieces give the full logarithmic sum E  £  = ^{ln(cos/S6 + cosh/3Va) - 2 1 n ( l +  exp(t06))} .  (2.103)  Inserting 2.91 into 2.88 we find the result for the "bare bubble" at finite temperature. This is an interesting quantity because it gives more information than simply the Berryon polarization. The final result is ho  i  ~r  =  47T  i  hi + ho  — 47T  i Iij  =  r  1  oto{E(2)-t-£(l)<7o}  Jo  r  (2.104)  1  / dx {(2x - l)fcE(l) - (1 - x)q Z(0)} Jo  (2.105)  oqi  r  1  dx {-q x(l  J  iqj  - x)T(0)  (2.106)  + Sij^L}  Performing the trace we can now calculate the polarization directly. n  = 2(2/oo - T R ( / ) )  00  ]  n =2(/ + / ) >, (2.107) % =2(2/^-^^(7)) J which give the final result for the finite temperature Berryon polarization bubble as oi  n  o  o  _  ^  /  [ \ x { 4in «JoJo {{ .q  q  l  0i  ~ * 2k| 2\q\  Jxjj^x)  2  v  i0  ^  vV  8inhB\q\y/x(l-x)  ccooss^^ ( (l l — x) + cosh^|g|y a;(l — a;) /  0 0  / cosh^|g|y a;(l - x) + cos/?g (l ~ ) ~ isin/3g (l - x) /  0  2 y In [cosBqo(l  x  0  cosBq (l 0  — x) + cosh B\q\y/x(l  - x) + cosh B\q\y/x(l  + |ln(l + exp(^ (l-x)))| 0  0  — x)  - x)^j (2.108)  Chapter  U Oi  4Wo  dx  2.  -\Jx(l  Formal  - x)  28  Developments  sinh B\q\\J'x(l cosBq (l Q  — x)  — x) + cosh B\q\y/x(l  ,q (cosh/3|g|A/a:(l — x) + cos/3go(l —  — x)  -Hsin/3gn(l — a;)  0  cos/3g (l — #) + cosh  sinh / % |  n  (2.109)  — x)  0  — x)  cos /3g (l — x) + cosh/3|g|^a;(l — x) 0  -A  —%-  \Jx(l  2? • Qo  - x)  sinh/3|g|^/a;(l — x) cos Bq (l — x) + cosh B\q\\Jx(l 0  cosh B\q\y/x(l  — x)  — x) + exp(i/3f/ (l — x)) 0  cosh /3|gj \ / ^ ( l — x) + cosBq (l 0  (2.110)  — x)  A striking feature of 2.108-2.110 the fact that n „ has a finite limit as q —> 0. This corresponds to a so-called "thermal mass". Taking this limit we obtain M  IUi(q->0)  =  0  n ( ->o) n (<7^0)  = =  0 ^ln2  oi  00  q  (2.111)  Further investigation of the finite temperature properties of Q E D is still a priority. However the equations 2.108-2.110 are rather cumbersome and therefore not amenable to further investigation. To continue we must either calculate a limiting form of the equations, or start on a new tack with clever insights into the finite temperature form of the self energy. 3  2.6  Antiferromagnetic Order in  QED3  One of the most difficult theoretical challenges facing physicists studying high temperature superconductors is the fact that the parent compounds of the cuprates are antiferromagnetic Mott-Hubbard insulators. There have been many attempts to incorporate the two phenomena, resulting with modest success. Conventional wisdom says that superconductivity and magnetism are competing orders and can not exist simultaneously. After the cuprates were discovered, this wisdom was seen to be incomplete. Since then it has been  Chapter  2.  Formal  29  Developments  strongly suggested empirically that superconductivity and magnetism i n fact coexist simultaneously. Perhaps antiferromagnetism and superconductivity are actually different manifestations of the same order. T h i s idea was fleshed out i n the SO(5) theory put forward by Zhang i n 1997 [21]. It described the three dimensional antiferromagnetic order vector and the two components of the complex superconducting order parameter as 5 components of a larger order parameter. T h i s theory stated that mathematically the superconducting order could be achieved from antiferromagnetic order by an internal SO(5) rotation. W h i l e this is a promising theory, it starts by assuming the coexistence of the two orders. One of the most remarkable and promising features of the Q E D theory is that the antiferromagnetic phase develops from an instability i n the theory. Therefore the coexistence of antiferromagnetism and superconductivity i n the same material arises naturally i n the theory. Three dimensional quantum electrodynamics has been studied by high energy theorists for some time [22], [23]. The reason it has been under scrutiny for so long is that Q E D displays a phenomenon known as dynamical mass generation. The Lagrangian is initially written down without a m^^t term, which would correspond to a massive Fermionic excitation. In these early investigations of Q E D the number of Fermionic excitations J\f was not kept fixed, but assumed to be large to use perturbation theory w i t h the parameter a = 8e /J\f. It was seen that below a critical number of Fermions fife a Fermionic mass was dynamically generated. In the present study, Q E D is not a toy model, but one derived earnestly to investigate the cuprate superconductors. Therefore the question naturally arises: W h a t does this dynamically generated mass correspond to? A m a z ingly, the dynamically generated mass term corresponds to antiferromagnetic order i n the cuprate superconductors! Antiferromagnetic order naturally arises as a consequence of Q E D [24]. We are again only interested i n the low energy excitations of the Fermionic degrees of freedom. We start by constructing Fermionic spinors which represent the low energy excitations located near the nodes of the order parameter. 3  3  3  2  3  3  Chapter  2.  Formal  Developments  30  Figure 2.5: Detailed k-space outline for constructing 4-component Fermionic spinors  Chapter  2.  Formal  31  Developments  To this end we introduce the 4-component spinors  V  /  + k,u)  OtiQa  \  a\{-Qa  ~ k,  Ot(-Qa  +  ~u)  (2.112)  k,u)  [(Qa  ~ k,  a  -U))  j  where |A;| << \Q \ and the Q 's are defined in figure 2.5. By recasting Q E D in such a way we have grouped the antipodal nodal Fermions together in one "species" of Fermions. For the rest of the thesis we are therefore dealing with Jf = 2 species of Fermions. a  a  3  In terms of the rotated coordinate system depicted in 2.5, we have = vpk and A& = v&k . Furthermore we have the symmetries = — £k-2Q and Afc = — A ^ _ Q to write the Lagrangian as x  a  y  2  a  j  C  dr j  dr { # i ( 7 d + v ^d  2  0  r  F  x  *2(7o<9 + f A T i d * + VF72dy)y } 2  r  +  v ^ d )^i+ A  2  y  •  (2.113)  Here we have introduced the 4 x 4 gamma matrices such that 7o  0  1  1  0 (2.114)  <g> 1 7i  0  - -03  os  0 (2.115)  o~2 <8> 0-3 =  72  -*  0  -01  0"!  0  0"i  -02  (2.116)  An interesting phenomenon now hidden in the Lagrangian 2.113. The Lagrangian is symmetric under the following three "rotations" (2.117) *  f  -»• * G f  (2.118)  where G E {a <8> 1, o\ <g> a , 1 <g> cr }. These three matrices correspond to internal symmetries present in the Lagrangian 2.113. Given a symmetry, it x  2  2  Chapter  2.  Formal  Developments  32  is natural to ask whether we can break this symmetry. The simplest way to break this symmetry is to introduce a Fermionic mass term (2.119) However our Lagrangian does not possess such a term. Through past studies of Q E D , it is well known that a Fermionic mass term of the form 2.119 can actually be generated dynamically via over screening of the gauge field a . The Fermionic mass term is generated if the number of Fermions J\f is less than some critical number Nc = 32/IT [25]. The Fermions mutually interact via the Berryon gauge field. For a large number of Fermions, the Fermions adequately screen the gauge field. However, if the number of Fermions is small enough, there aren't enough Fermions to effectively screen the gauge field. The gauge field is then a strong enough interaction to generate s Fermionic mass. 3  M  2  Given that the theory dynamically generates a Fermionic mass of the form 2.119, it is natural to ask what this mass corresponds to in this formulation of Q E D . By undoing the transformations it is straight forward to see that writing 2.119 in terms of electron operators we find 3  m Y  c  \i  k  + <7> w ) c ( f c , OJ) t  c[(k  + q, w ) c ( f c , u) ;  (2.120)  or in real space (2.121)  <5=±,i=l,2 which is an incommensurate spin density wave characteristic of antiferromagnetic order.  Chapter  3.  Transport  Properties  in  33  QED  3  Chapter 3 Transport Properties in  QED3  For a theory to be successful it must not only explain the experiments previously performed. It must also have testable predictions that provide a unique signature to confirm or disprove the theory. T h i s chapter is devoted to the calculation of a number of experimentally testable predictions of Q E D . It is suspected that the anomalous dimension 77 w i l l provide the testability of the theory by rearing itself i n some experimentally observable quantity. 3  3.1  Universal Conductance in Two Dimensional d-wave Superconductors  A very striking property of two dimensional d-wave superconductors was discovered i n 1993 by Patrick Lee [26]. B y adding i m p u r i t y scattering i n a self-consistent way Lee showed that the zero temperature dc conductivity reaches a constant l i m i t e (TO =  v  2  F  (3.1)  — 7T  VA  which is independent of the scattering strength and type. T h i s result was dubbed "universal", so named due to the independence of the result on the scattering type or strength present i n the sample. A n y two dimensional superconducting system should exhibit this constant scattering rate at low frequencies (the frequency cutoff is not universal and depends on the strength of the scatterers present i n the superconductor). Based on the universality of this result, the anisotropy ratio that characterizes high T superconductors V / V A can be determined experimentally. E m p i r i c a l l y it is seen to lie between 10 and 20 for most cuprate superconductors at o p t i m a l doping. To calculate this result, we start w i t h the B d G Green's function including a self energy m a t r i x £ c  F  Q (k, iui ) = [ioo l - e(k)cr - A(k)<7i] + S ( k , ioj ) l  n  n  3  n  (3.2)  Chapter  3.  Transport  Properties  in  34  QED  3  to calculate the electrical conductivity via a(n,T) = - = 5 g l W ,  (  3  3  )  where H t = TL(ifl —> Q, + i8) and U(iQ) is the photon polarization bubble. The fundamental starting point in the calculation of the conductivity is the electrical current-current correlation function in the Matsubara finite temperature formalism: re  n(tft) = - /  dTe  T j(r)j(0) > .  <  iQT  (3.4)  T  Jo  In any theory where the vector potential is minimally coupled to the Fermionic excitations, the current operator is given by j(x, r) =  ( ^ ( x , r) V ^ ( x , r) - W ( x , r ) ^ ( x , r)) . f  (3.5)  Taking the Fourier Transform of 3.5 j(q,r)  =  j d x j dte j(x,r), 2  e m* \ *  (3.6)  ikx  E  (  k  +  f  )  ^  ^  k  +  q  .  (3-7)  k,u>  The part of this current that goes into the correlation function is the q = 0 term j(q  = 0,r)  =  -  —  E  =  - e j ] v  ^  k  k  k  ^ k  (3.8)  (3.9)  which is the expected result, the current is a velocity times a number density. Inserting 3.9 into the correlation function 3.4 results in the following quantity to evaluate pp  U{iQ) = -  /  Jo  dre e inr  2  ^  < T j(w)j(w + « ) > T  (3.10)  Chapter  3.  Transport  Properties  in  QED3  35  iuj +  (3.11)  which reduces to  = lYl e v v TR[s(k, iu )g(k, 2  k  k  n  n  To calculate the low energy properties of d-wave superconductors we l i n earize this equation about the four nodes. The sum over all momenta turns into an integral over a small region about the four nodes  We can simplify even further by making the substitution p  x  =  v k ,p F  x  y  =  v k. A  y  The velocity-velocity tensor also reduces to 4  Y ^  {  j  W  j  )  =  (3.14)  2v l. 2  F  J'=I  C o m b i n i n g these simplifications, 3.11 becomes  2  p  i 1  / d TR[g(p,ico )g(p,iu  n(ift) = ^ - - V  + iQ)}.  2  P  n  n  (3.15)  n  The most straight forward way to perform the M a t s u b a r a sum i n 3.15 is to use the spectral representation, where we define the function A(p, ui) as the Hilbert transform of the Green's function g(p,iu) )= n  /  duoi-  .  (3.16)  B y complex analysis we have the identity  A(p ) = >w  -iG^ ( t  P > W  )  (3.17)  3.  Chapter  Transport  Properties  in  36  QED  3  where G'^(p,oj) is the imaginary part of the retarded Green's function. In this representation the Matsubara sums are straight forward (see appendix B). The result of the summation is  x  TR[C7';  (p, u)G" (p,  E T  OJ  (3.18)  + i^}  where n(ui) is the Fermi distribution function. According to equation 3.3 we must take the imaginary part of this function after setting i£l —> Q + i5. Doing so we achieve the integral that must be performed to calculate the conductance. m\  a(n,T)  e VF 2  = —  f dp  , n(co) — niui + Vt) ^  f°°  2  /  -f-  /  do; ^  r  ^TR G  V  .  r e t  .  ( p , w)G  .  ret  ( p , w + fi) 3.19)  To simplify the expression 3.19, we assume that all but the scalar parts of the self energy can be absorbed into the definition of e(p) and A(p) [27]. This assumption has been justified in both the Born and Unitary scattering limits [28]. We define the frequency dependent scattering rate as 7  M  = -ImE  r e t  >).  (3.20)  The linearized B d G Green's function becomes  f  1  oo - E(w) + p  p  x  y  Equation 3.19 simplifies in the universal limit (ti,T) —> 0 to  °  a  =  iv  J ^  &  A  T  R  Kt(P.  ° ) r e t ( P . 0)] . G  (3-22)  where denotes the imaginary part of the retarded Green's function. Furthermore, we define 70 = 7(0; —> 0), the value of the scattering rate at low frequency. The integral reduces to e v (J = -z— 2  0  F  f°° ^ /  ^ VAJ-OO 2  p —h~\2 *  2 7o  (7O+P ) 2  2  (3-23)  Chapter  3.  Transport  Properties  in  37  QED  3  By making the simple change of variables x = p/jo if can be easily seen that the dependence on 70 completely cancels, and we recover the universal result stated earlier (3.24) The universality of this result is the result of a detailed balance of the effects of scattering. Scattering gives the quasiparticle excitations a finite lifetime, as can be seen by the definition 3.20. The competing effect is that scattering gives rise to a finite density of states at the nodes of a d-wave superconductor. These competing effects, the enhancement of the quasiparticle density of states and the finite lifetime of these quasiparticles conspire to give the universal result. This result is strongly dependent on the two dimensional nature of the cuprates as well as the d-wave symmetry of the order parameter. Without these two crucial properties, this delicate balance could not be achieved.  3.2  Modification of the Universal Conductance in Q E D 3  The anomalous exponent 77 of Q E D ruins this delicate balance that gives rise to the universal conductance in two dimensional d-wave superconductors. This may give a signature of Q E D that can be checked with experiment. 3  3  The starting point is similar to the original calculation by Lee, however we use the Q E D Green's function with the anomalous dimension 77. However, we must be careful before blindly using the form given by 2.68. Recall that we had originally multiplied SP* by 70 in order to introduce a "Lorentz" invariance to make calculation easier. In order to calculate physical quantities, that transformation must be undone. By doing so (and recalling the definitions of the 7 matrices in terms of the Pauli spin matrices), we rewrite the Q E D Green's function in the form 3  3  Q (k, ioj ) = [iu l l  0  n  n  - v k o- - v k Ox F  x  3  A  y  A (iu )  2  n  2  +k  2  (3.25)  To calculate the low energy properties in the presence of impurities, we again add a self energy matrix to the Green's function 3.25 via Dyson's  3.  Chapter  Transport  Properties  QED3  in  38  equation. G (k,iu )  + E(k,?w )l  1  0  n  n  ^ ( k ^ where E = E ( k , ^  + E),  (3.26)  ,1/2  n  ) ( ^ ± ^ ) '  We are now ready to determine how the 77, the anomalous exponent of Q E D affects the universal conductivity. Inserting the Green's function 3.26 into 3.24 results in 3  e  2  0"o  VF V  TC  2  A  e  2  TT  2  s  dk 2  2?r  VF  7Q  v  A " fJ  4.  A" 2  t.2-277  (3.28)  dy-  4  A  (3.27) JSL.  0  +  A" 2  Jo  y ~> 1  r  The resulting integral is only solved in closed form by a hypergeometric function. For the calculational details see appendix C. Inserting the result of the integration in the limit of large A from the appendix we finally arrive at the desired result  0"o  e  v  2  F  ll  1-17  V  r  A e v • ° which remarkably simplifies n v \ Ato J 7T  2  V  A  F  1-1  -  rjir  7  2  A  2  (3.29)  2  2  2  r  1+  fi_  i q  )2  s i n  ^^j  (3.30)  The result 3.30 gives the result of the electrical conductivity calculated with the Q E D propagator 3.25 in the universal limit. As anticipated, the anomalous dimension 77 ruins the delicate balance that conspired to produce the universal result 3.24. A check on the result 3.30 is the limit 77 —> 0. By taking this limit we recover the original universal conductivity 3.24 as required. 3  3.  Chapter  o  Transport  Properties  9  9  +  in  39  QED  3  9  ;  +  \  +  Figure 3.1: Graphical Representation of the Self Consistent T-Matrix Approximation  3.3  Self-Consistent Scattering  In the last section we saw that the anomalous exponent 77 of Q E D disrupts the delicate balance that lead to the universal result. Therefore the conductivity in the limit {u>,T) —> 0 depends on the scattering 70. In this section we will determine the scattering for both the normal d-wave superconductors and in the Q E D theory. In order to calculate a closed form expression for the scattering we employ the self consistent T-matrix approximation [28]. In this method the self energy is given by an infinite resummation of single site scatterers as depicted in figure 3.1. Furthermore, the off diagonal components of the self energy matrix can be absorbed into the redefinition of the energy and gap functions. Following Hirsch et al. [28] we finally arrive at the fundamental equation for the self energy 3  3  coV6  0  gZiiUn)  -  where T is the impurity density, So is the scattering phase shift, g {iu; ) 0  n  = -^-M ~2TRg(k,ico ), /  (3.32)  n  p is the density of states at the Fermi surface, J\f is the number of Dirac nodes and G(k,iu> ) is the Green's function in the Matsubara formalism. Since the Green's function also contains the self energy E(iw ), equation 3.31 needs to be solved self-consistently. 0  n  n  3. Transport Properties  Chapter  in  QED  40  3  There are two limits of interest in 3.31. The limit of strong scattering 5o'—t 7r/2 is known as the unitary limit. In this limit 3.31 becomes  - -^f-  E(tu/„) =  ^TRC7(M^))  2  ,  (3.33)  since c —>• 0 (we have now defined c = cot <5 for the rest of the thesis). Since we are only interested in the universal limit, we analytically continue 3.33 and take the limit u —¥ 0. By doing so we can make the replacement S(0) - » —i~/o (equation 3.20). Therefore we arrive at the self-consistent equation for the scattering rate in the unitary limit 0  2  To =  Tirpo (^  JV  1  \^il  + k  2  Replacing the sum by an integral via a transformation similar to 3.13 gives 2  1 b  _ =  2K YV V  (  2  F  _  f  pdp  A  ^  _  1  U^ ) '  aPo  A ? "  <33 5 )  which is easily solved for the self consistent scattering in the unitary limit  4-K rv v p  ( A [in^j  2  2 7 o  =  F  7  A  Q  7  2  \  _  1  .  (3.36)  The other limiting case of 3.31, the limit of weak scattering So —> 0 is known as the Born limit. In this limit the equation which determines the self consistent form of the scattering is  k  Similar manipulations as above take us to the equation TN c np v v  f  2  0  with result  F  A  fJ  pdp  A  i  A  0  + p  z  (3.38)  Jo  ^ A ' ^ f - ^ ) .  (3.39)  Even though the two limits give different functional forms, there is no significant difference in the physics of the two different limits. Both equations 3.36 and 3.39 admit an arbitrarily small solution of 70.  3.  Chapter  3.3.1  Transport  Properties  Scattering within Q E D  41  QED3  in  3  The results presented in the last section will be modified by the anomalous exponent 77 appearing in the Q E D Green's function 3.25. Insertion of the Q E D Green's function in the unitary limit of equation 3.31 leads to 3  3  Performing the transformation 3.13 brings us to the non trivial integral equation A2  _ 4n v v Tp  I r JO  2  2  F  A  0  dp_ -7*7-1- V '^ 1  In the Born limit, similar manipulations to equation 3.31 with 5 —¥ 0 gives 1  TJV  £ - 5 — ^ — »  jL 4.  A>^ 2r  TXOQC  2  ( 3  4 2  )  i-2-27,  performing the transformation 3.13 gives  1* C 2  PQ  [r  A  1  TJV  N*  J  2  2  dp  A2  jjL  0  pl-r,  +  and we find the same non trivial integral as in the unitary case 3.41. Following the manipulations in appendix C we find the results for the scattering in the Unitary limit  ^  1  ^ * AV  7  l -^ . ) s :i n ^ j^  +  nj  ^  =  *  N^  =  .  (3-44)  For the weak scattering Born limit we find A  A) 2  nn c v v 2  =  2  Po  F  V  A  _ 1\ s i n ^  Vj  7T  (3.45)  Chapter  3.  Transport  Properties  in  42  QED  3  We find an interesting result for the scattering within the Born limit that is not present in the unitary limit. The behaviour of the scattering changes dramatically depending on the scattering impurity concentration T. Below a critical impurity concentration T =  (3.46)  c  the scattering vanishes. In the Born limit of weak scattering, we can insert the result 3.45 into the conductivity within Q E D 3.30 with result 3  which represents the main result of this thesis. The consequences of the concise result 3.47 are explored in the following section.  3.4  Experimental Tests of Q E D  3  The main result of this last chapter is the apparent rj dependence of an experimentally testable quantity o given in equation 3.47. The result is displayed graphically in figure 3.2. It is interesting to note that in the limit of large impurity concentration, the conductivity reaches a limit 0  ao(  ^  )=  °(T^F  CT  (3  -48)  This limit is indicated on the graph by a dotted line. The interesting result apparent in 3.47 is the fact that there is an critical value of impurities T that qualitatively changes the behaviour of the sample. The sample will go from conducting to insulating below this critical number. This is a promising feature of the theory because of the fact that the behaviour of the cuprates goes from conducting to insulating as doping is lowered. c  The critical impurity density has been seen in other circumstances. It is seen in the superconducting state when there is a vortex lattice present (in the "vortex state"), where a small gap will open up near the nodes. In this case the critical impurity density will be proportional to the size of the aforementioned gap [29].  Chapter  3.  Transport  Properties  in  43  QED  3  Conductivity Versus Impurity Concentration  2  4  6  8  r/r 0 1 1 1  Figure 3.2: Graphical depiction of the result 3.47.  10  Chapter  3.  Transport  Properties  in  44  QED  5  Figure 3.3: The three 1/jV contributions to the bubble diagram with arbitrary vertex.  In principle, an experiment could test for the deviation of the universal conductance in the underdoped region as a signature of the algebraic Fermi liquid behaviour predicted in the Q E D theory. However, the predicted universality of the electrical conductivity is not even seen in the superconducting state. After further investigation and careful treatments of Fermi liquid and vertex corrections, the electrical conductivity does not retain its universality. It can be shown (see [27]) that thermal conductivity does retain its universality after these corrections are calculated, and it is this quantity that experimentalists use in practice to determine the anisotropy ratio v /v . Equation 3.47 could potentially be missing some important physics. There are actually three diagrams which contribute to order 1/J\f as seen in figure 3.3. The first two diagrams are included in this calculation by virtue of the fact that we used a renormalized electron propagator depicted in figure 2.4. We have not yet succeeded in calculating the third diagram in figure 3.3, which amounts to neglecting vertex corrections. A troubling feature of the result 3.47 is the fact that it is not gauge invariant, for 77 is itself gauge dependent. This is a direct result of not treating the vertex corrections. In fact it has been seen in other calculations that when all three bubbles are treated properly, the gauge dependent part exactly cancels [30]. We are confident that if all three bubbles are treated carefully there would still be a measurable exponent in the conductivity in the universal limit. 3  F  A  Chapter  4.  45  Conclusions  Chapter 4 Conclusions Ever since its discovery, superconductivity has been at the forefront of physics research, both experimental and theoretical. In the decades following the discovery of high temperature superconductors, condensed matter physics has experienced a resurgence of interest i n superconductivity. A t the present no satisfactory microscopic description of high temperature superconductivity has been found. T h e Q E D theory espoused i n this thesis is an assimilation of many ideas and concepts concerning the cuprate superconductors. Even though the form of the theory is an effective Lagrangian that is mostly phenomelogical, we are confident that Q E D is an accurate theory and w i l l continue to provide important insights into the physics underlying the cuprate superconductors. The Q E D theory was first derived i n the context of superconductivity by Franz and Tesanovic i n reference [17] i n 2001. Since then it has been the subject of much scrutiny. A m a z i n g l y the antiferromagnetic phase present i n the cuprates parent compounds has a natural description w i t h i n the framework of Q E D . In order for the theory to be widely accepted it must both explain experimental data previously collected, but make testable predictions for future experiments. To this end we have calculated quantities amenable to experimental verification that w i l l display the anomalous dimension of the Q E D propagator, specifically the dc conductivity i n the pseudogap state. T h e i n teresting feature of this quantity is its dependence on the i m p u r i t y density T. A t large values of T the conductivity saturates at a constant value a / ( l —ry) , where a is the usual universal conductivity. A t low i m p u r i t y densities below a critical value Tc the conductivity is seen to actually vanish. T h e future directions of study i n Q E D are clear. T h e theory must be extended to finite temperature v i a means other than the usual M a t s u b a r a formalism, which quickly becomes too cumbersome. Further insight must be sought into the form of the Q E D propagator, or equivalently the form of the self energy at finite temperature. One promising avenue would be 3  3  3  3  3  2  0  0  3  3  Chapter  4.  Conclusions  46  to find some limiting form of the equations 2.108-2.110, and subsequently calculating the self energy in the corresponding limit. From this we may be able to develop an ansatz for the general finite temperature propagator.  47  Bibliography  Bibliography [1] H . K. Onnes. Further experiments with liquid helium. Commun. Lab.  Univ.  Leiden,  Phys.  120b:3, 1911.  [2] W . Meissner and R. Ochsenfelf. One new effect as to the beginning of superconductivity. Naturwissenschaften, 21:787, 1933. [3] J . Bardeen, L . N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Physical Review, (108):1175, 1957. [4] G . M . Eliashberg and Zh. Eksperim. and Theoretical  Physics,  Soveit  Journal  of  Experimental  11:696, 1960.  [5] J . G . Bednorz and K . A. Muller. Possible high tc superconductivity in the Ba-La-Cu-0 system. Z. Phys. B, 64:189, 1986. [6] R. Liang, D. A. Bonn, and W. N. Hardy. Growth of high quality crystals using BaZr03 crucibles. Physica C, 304:105, 1998.  YBCO  [7] E . Schanchinger and J. P. Carbotte. Application of an extended eliashberg theory to high-Tc cuprates. cond-mat/'0207668, 2002. [8] V . J . Emery and S. A . Kivelson. Importance of phase fluctuations in superconductors with small superfluid density. Nature, 374:30, 1995. [9] J . Corson, R. Mallozzi, J . Orenstein, J . M . Eckstein, and I. Bozovic. Vanishing of phase coherence in underdoped Bi Sr2CaCu208+<s. Nature, 398:221, 1999. 2  [10] P. G . deGennes. Superconductivity 1966. [11] A . A . Abrikosov. Zh. Eksp.  of Metals  and Alloys.  Perseus Books,  Teor. Fiz., 32, 1957.  [12] J . M . Kosterlitz and D. J . Thouless. Journal  of Physics,  C6, 1973.  48  Bibliography  [13] N. Nagaosa.  Quamtum  Field  Theory  in Condensed  Matter  Physics.  Springer, 1995. [14] L . N. Cooper. Bound electron paris in degenerate fermi gas. Review, 106:1189, 1956.  Physical  [15] W . N. Hardy, S. Kamal, and D. A . Bonn, volume 371, chapter Magnetic Penetration Depth in Cuprates: A Short Review of Measurement Techniques and Results, page 373. Plenum Press, 1998. [16] M . V . Berry. Quantal phase factors accompanying adiabatic changes. Proceedings Physical  of the Royal Society  Sciences,  of London.  Series A, Mathematical  and  392:45, 1982.  [17] M . Franz and Z. Tesanovic. Marginal fermi liquid from phase fluctuations: "topological" fermions, vortex "berryons" and QED3 theory of cuprate superconductors. Phys. Rev. Lett., 2001. [18] M . Franz, Z. Tesanovic, and O. Vafek. Q E D theory of pairing in cuprates: From d-wave superconductor to antiferromagnet via "algebraic" fermi liquid. Physical Review, 2002. 3  [19] L . D. Fadeev and V . N. Popov. Feynman diagrams for the yang-mills field. Physics Letters B, 25(1):29, 1967. [20] T . Matsubara. Prog.  Theor. Phys.  (Kyoto),  14:351, 1955.  [21] S. C. Zhang. A unified theory based on £ 0 ( 5 ) symmetry of superconductivity and antiferromagnetism. Science, 275:1089, 1997. [22] N. Dorey and N. E . Mavromatos. Three-dimensional Q E D at finite temperature. Physics Letters B, 266:163, 1991. [23] N. Dorey and N. E . Mavromatos. Q E D and two-dimensional superconductivity without parity violation. Nulcear Physics B, 386:614, 1992. 3  [24] I. F . Herbut. Antiferromagnetism from phase disordering of a d-wave superconductor. Phys. Rev. Lett., 88, 2002. [25] T . Applequist, D. Nash, and L . Wijewardhana. Critical behavior in (2+l)-dimensional Q E D . Phys. Rev. Lett., 60(25):2575, 1988.  49  Bibliography  [26] P. A . Lee. Localized states in a d-wave superconductor. Lett, 71(12):1887, 1993.  Phys.  Rev.  [27] A . C. Durst and P. A . Lee. Impurity-induced quasiparticle transport and universal limit wiedemann-franz violation in d-wave superconductors. cond-mat/9908182, 2000. [28] P. J. Hirschfield, P. Wolfe, and D. Einzel. Consequences of resonant impurity scattering in anisotropic superconductors: Thermal and spin relaxation properties. Phys. Rev. B, 37(83), 1988. [29] M . Franz and O. Vafek. Universal thermal conductivity in the vortex state of cuprate superconductors, cond-mat/0107521, 2001. [30] W. Ratner and X . G . Wen. Gauge invariant electron spectral function in underdoped cuprates. cond-mat/'0105540, May 2001. [31] E . B. Saff and A . D. Snider. Mathematics,  Science  Fundamentals  and Engineering.  of Complex  Analysis  Prentice Hall, 1993.  for  Appendix  A.  Dimensional  50  Regularization  Appendix A Dimensional Regularization In this appendix we perform dimensional regularization on the logarithm! cally divergent integral  /  „2  Dimensional regularization starts by performing the integral in arbitrary dimension d (not necessarily an integer) and specify that d = 2 at the end of the calculation. This isolates the non-physical singularity in the integral, and allows us to determine the important physical information contained in the integral.  f  J  d  * V. + A ] = p2  d  Pr  A91  2  [dQ  J  2 7 r  [  d  d/2  f f  p^dp.  7  oo  0 p  2  + A ] 2  dp  T(i)Jo  ' V  Jo  r (  d+l  [p  |)/  2  0  d  {  + A ] 2  p  )  W T ^ \  (  A  '  2  )  We make the transformation  which reduces A.2 to  Using the definition of the beta function B(a  + 1, B + 1)  =  f dxx {l-xf '0 Jo  (A.5)  a  r  (  q  +  i ) r ( f t  r ( a + /3 + 2)  +1) :  (A.6)  Appendix  A.  Dimensional  51  Regularization  the integral reduces to  /  .  P  2  T *  -L d  2  As the dimension approaches two, we see that the singularity arises from the pole of the Gamma function. We isolate this infinity by setting d = 2 + e.  / P P dd  J  2  J - ^ _ fJ-Y rm  (A 8)  /2  (2TT)V + A  2  =  47r(47r) / \A £  2  J  2  \2J '  {  }  Using the expansion of the Gamma function about zero 2  r  (fK"7  (A.9)  where 7 is the Euler-Mascheroni constant 7 « 0.57721. We arrive at the answer  J  (2ir) p d  2  + A  4?r I  2  =  2  2  -i- {--ln47r-lnA 47T I e  J \e 2  -7}  I  ' f  (A.10)  Appendix  B. Fermionic Matsubara  52  Summations  Appendix B Fermionic Matsubara Summations Generally, Matsubara sums have the form  n  where u = ( "+ ) for Fermionic sums. At first glance the task of evaluating these sums seems quite formidable. However, they are actually quite straight forward when viewed in the light of the Cauchy residue theorem borrowed from complex analysis [31]. 2  1  7r  n  Cauchy's Residue T h e o r e m 1 If T is a simple closed positively oriented contour and g is analytic inside and on T except at the points Z\, z , z ...z inside T, then 2  i d ? ^g(z)^.=2KiYRe (f(z), ).  3  n  S  (B.2)  Zj  To use this to our advantage we construct a function with poles at the Matsubara frequencies with residue ^. The function needed is  Integrating the product of n(z) and the original function f(z) over the entire complex plane gives jf f{z)n{z)dz = ~  + Y  n(zi)Rss{f(z),  Zi).  (B.4)  n  Appendix  B.  Fermionic  Matsubara  Summations  53  By choosing the contour of integration to be a circle whose radius tends to infinity and assuming f(z) tends to zero sufficiently fast as z —¥ oo the value of the integral tends to zero [31]. Therefore we have the identity ^ / ( 2 o ;  "  n  )  = ^n(^)Res(/(z), ). 2 j  (B.5)  j  We have achieved a great simplification. The infinite Matsubara sum turns into the algebraic problem of finding the poles of the function f(z).  Appendix  C.  Manipulations  of Hyper geometric  Functions  54  Appendix C Manipulations of Hypergeometric Functions Throughout the calculations presented in this thesis, we run across integrals of the form  f ,  d  \  v  -  ( C D  These integrals are non trivial and can only be solved by using the special functions. In order to easily perform algebraic manipulations we define a dimensionless parameter g = 7 o / A . By doing so, equation C . l becomes 2  dp  1  (C.2)  To make this integral simpler, we employ the change of variables x = g^^p  (C.3)  which makes C.2 -i  This integral is solved by the hypergeometric function / ' ' " J T — = Jo {l + x 1  ^-^Fx  (a,\ 1  ,1 + - 7 7  1 -77  , — . gj  C.5  While equation C.5 is analytic and exact, it is not very useful in illuminating the physical situation. Luckily there exists a series representation of the  Appendix  C. Manipulations  of Hyper geometric Functions  55  specific hypergeometric function about g = 0 (which corresponds to the physical limit A —> oo). The limiting power series is a F l  ( _1_ i + _L_ a  =  iV( i^) H i^) 1 +  q  +  *  r  +  i r ( i ^ ) r ( d  +  0  (  5  )  J  Luckily we need not consider the series past here, as we don't keep terms of order 0(g ). The situations cited in this thesis have the parameter a being 1 or 2. When a = 2 we only keep the first term of the series. 2  

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