TVE 14 023 juni Examensarbete 15 hp Juni 2014 Efficient Computational Procedure for the Analytic Continuation of Eliashberg Equations Joakim Johansson Fredrik Lauren Abstract Efficient Computational Procedure for the Analytic Continuation of Eliashberg Equations Joakim Johansson & Fredrik Lauren Teknisk- naturvetenskaplig fakultet UTH-enheten The superconducting order parameter and the mass renormalization function can be solved either at Besöksadress: discrete frequencies along the imaginary axis, or as a Ångströmlaboratoriet Lägerhyddsvägen 1 function of continuous real frequencies. The latter is Hus 4, Plan 0 done with a method called analytic continuation. The analytic continuation can conveniently be done by Postadress: approximating a power series to the functions, the Box 536 751 21 Uppsala Padè approximation. Studied in this project is the difference between the Padè approximation, and a Telefon: formally exact analytic continuation of the functions. 018 – 471 30 03 Telefax: As it turns out, the Padè approximant is applicable to 018 – 471 30 00 calculate the superconducting order parameter at temperatures sufficiently below the critical Hemsida: temperature. However close to the critical temperature http://www.teknat.uu.se/student the approximation fails, while the solution presented in this report remains reliable. Handledare: Alexandros Aperis Ämnesgranskare: Henrik Olssson Examinator: Martin Sjödin ISSN: 1401-5757, TVE 14 023 juni Popul¨arvetenskaplig Sammanfattning Fenomenet supraledning ¨arn¨arett material leder str¨omutan n˚agon resistans. Det sker n¨arvissa material kyls ned till temperaturer n¨araden absoluta nollpunken, 0 Kelvin, d˚ade ¨overg˚arfr˚anden fasta fasen till den supraledande fasen. Man kan t¨anka sig denna fas¨overg˚angp˚asamma s¨attsom att till exempel vatten ¨overg˚arfr˚anflytande till fast form under 0◦C eller att j¨arn¨overg˚artill att vara magnetiskt under 770◦C. F¨oratt unders¨oka om en s˚adanfas¨overg˚anghar ¨agtrum studeras vanligen n˚agonparameter som ¨arnoll innan ¨overg˚angen,men antar n˚agot¨andligtv¨ardeefter ¨overg˚angen. I fallet supraledning kallas denna parameter ∆. Supraledare uppt¨acktes ˚ar1911 d˚aH. Kamerlingh Onnes1 skulle studera resistansen hos kvick- silver i fast form under f¨orh˚allandend¨artemperaturen n¨armadesig den absoluta nollpunkten. Detta hade nyligen blivit m¨ojligtatt studera eftersom en teknik f¨oratt f˚afram flytande helium precis uppfunnits och d˚akunde anv¨andassom kylmedel. Det han uppt¨ackte var att n¨artempera- turen kommit ned till cirka 4 K (-269◦C) f¨orsvann resistansen helt. Det dr¨ojdesedan mer ¨an40 ˚arav experimenterande innan en teori som kunde beskriva fenomenet presenterades. J. Bardeen, L.N. Cooper och J.R. Schrieffer ¨arnamnen bakom BCS- teorin2 som f¨orklararvarf¨oroch hur ett material kan leda str¨omutan n˚agotelektriskt motst˚and. Grunden i denna teori ¨aratt valenselektronerna i materialt paras ihop tv˚a-och-tv˚a. Dessa elektronpar konstituerar en ny slags "superkanal" f¨orelektrisk str¨om,och till˚atero¨andlig led- ningsf¨orm˚aga.Det vill s¨agamaterialet till˚atsleda str¨omhelt utan elektriskt motst˚and.∆ ¨arett m˚attp˚adenisiteten av s˚adanaelektron-par, och kan allts˚astuderas f¨oratt p˚avisaom ett material ¨ari den supraledande fasen eller i den normal fasta fasen. BCS-teorin har dock sina brister, och en mer fullst¨andigteori har presenterats av Eliashberg. H¨arinf¨orsett tidsberoende som tidigare f¨orbis˚agsav BCS-teorin. Detta g¨oratt ∆ blir en tids- beroende variabel, eller genom Fouriertransform frekvensberoende. Ekvationerna som beskriver tillst˚andethos det supraledande materialet l¨osesrelativt enkelt numeriskt med frekvenser fr˚anden imagin¨araaxeln. Tillst˚andsekvationerna f¨orm˚angaav de supraledande materialen l¨osesrelativit enkelt med imagin¨arafrekvenser. F¨oratt resultatet ska vara till nytta och ge anv¨andbarfysikalisk information m˚astel¨osningen¨overs¨attasmed v¨ardenp˚afrekvenser fr˚anden reella axeln. Tekniken f¨oratt ut¨oka definitiosm¨angdenf¨oren analytisk funktion kallas analytisk forts¨attning. Det kan anv¨andasf¨or att g˚afr˚andet komplexa talplanet till den reella axeln. Den teknik som vanligen anv¨andsf¨oratt g¨oraanalytiska forts¨attningkallas f¨orPad`eapproximationen, och ¨arsom namnet antyder en approximation. Det som studeras i den h¨arrapporten ¨aremellertid en metod som genomf¨ordenna analytiska forts¨attning analytiskt, och kan d¨armedses som en exakt l¨osningav tillst˚andsekvationerna p˚aden reella axeln. I rapporten j¨amf¨ors¨aven denna analytiska metod med Pad`eapproximationen. Resultatet fr˚anprojektet visar att Pad`eapproximationen ¨aren bra l¨osnigf¨oratt l¨osaden ana- lytiska forts¨attningenmen att den exakta l¨osningen¨arstabil under f¨orh˚allandend˚aPad`eapproximationen blir instabil. 1Kamerling Onnes tilldelades Nobelpriset i fysik 1913 f¨orsin forskning p˚amaterial vid l˚agatemperaturer 2J. Bardeen, L.N. Cooper och J.R. Schieffer tilldelades ˚ar1972 nobelpriset f¨ordenna teori iii Contents 1 Introduction 1 1.1 Description . .1 1.1.1 Goals . .2 2 Theory 3 2.1 Fourier transform properties . .3 2.2 Complex analysis . .3 2.2.1 Analytic functions . .3 2.2.2 Analytic continuation . .4 2.3 Solid state physics . .4 2.3.1 Solids . .5 2.3.2 Superconductivity . .6 2.4 Numerical methods . .7 2.4.1 Numerical integration . .7 2.4.2 Iterative methods . .7 3 Methods 8 3.1 The equations . .8 3.1.1 Properties of the Fourier transform . .8 3.1.2 Interval for the calculations . .9 3.1.3 The α2F (!)-distribution . .9 3.2 Dirac delta function . .9 3.3 The Lorentzian . 11 3.4 Smearing factor . 12 3.5 Solving the system of equations . 13 4 Result 14 4.1 Results with delta Dirac function . 14 4.2 Results with the Lorenztian . 18 5 Discussion 21 5.1 The Dirac delta approach . 21 5.2 The Lorentzian approach . 21 5.3 Other approaches for solving the equations . 21 5.3.1 Convolution . 22 6 Conclusions 24 A Complex analysis 25 A.1 Analytic functions and power series . 25 B Additional plots 26 B.1 Additional plots for the Dirac delta approach . 26 B.2 Additional plots for the Lorentzian approach . 29 iv 1. Introduction 1.1 Description When cooled down below critical temperature, a material may possess the ability to exhibit zero electrical resistance; the material is then said to be in a state of superconductivity. For a specific group of these superconducting materials, the superconducting state can be described with a system of coupled equations, known as the Eliashberg equations. These equations calculates the mass renormalization function, Z(t), and the superconducting order parameter ∆(t). The analytic solution to this system of equation is unfortunately not known. However, when Z(t) and ∆(t) are Fourier transformed, a discrete numerical solution can be obtained for the complex frequencies i!n = i(2n + 1)πT for Z(i!n) and ∆(i!n). The Eliashbergs equations are j!n0 j<!c 1 X !n0 Z(i! ) = 1 + λ(! − ! 0 ) (1.1) n n n p 2 2n + 1 0 2 n0 !n0 + ∆(i!n ) j!n0 j<!c X ∗ ∆(i!n0 ) Z(i! )∆(i! ) = πT λ(! − ! 0 ) − µ (! ) : (1.2) n n n n c p 2 0 2 n0 !n0 + ∆(i!n ) Where !n = (2n+1)πT are the Matsubara frequencies with n 2 Z , T is the temperature, λ and µ∗ are the electron-phonon and Coulomb coupling strengths. The summation for n' is truncated for some cut off frequency, !c. To get a useful solution and to be able to calculate real physical quantities the solutions need to be functions of real frequencies !. The technique to expand the domain of an analytic function is called analytic continuation and it can be used to go from the complex plane to the real frequency axis. A commonly used technique to do the analytic continuation is with an approximation called Pad`eapproximation method. The disadvantages of this method is that it is an approximation and becomes unreliable under certain circumstances. A relatively recent technique provides a formally exact solution for the analytic continuation and the associated equations becomes: X ∗ ∆(i!n0 ) ∆(!)Z(!) = πT λ(! − i!n0 ) − µ (!c) p 0 n0 R(i!n ) 1 (1.3) Z Z(! − !0)∆(! − !0) + iπ d!0Γ(!; !0)α2F (!0) pZ(! − !0)R(! − !0) −∞ πT X !n0 Z(!) = 1 + i p )λ(! − i!n0 ) ! 0 n0 R(i!n 1 0 (1.4) πT Z (! − ! 0 )Z(! − ! ) + i d!0Γ(!; !0)α2F (!0) n ! pZ2(! − !0)R(! − !0) −∞ where 2 2 R(i!n) = !n + ∆ (i!n) (1.5) 1 1 ! − !0 !0 Γ(!; !0) = tanh + coth (1.6) 2 2T 2T Z 1 2 0 0 α F (! ) λ(! − i!n) = − d! 0 (1.7) −∞ ! − i!n − ! and α2F (!) varies for different approaches. 1.1.1 Goals The goal is to write a section of code in Matlab that solves (1.3) and (1.4), taking fixed parameters and values of Z(i!) and ∆(i!) from equation (1.1) and (1.2) as input and produces values of ∆ and Z for real frequencies. These results will also be compared to the known Pad`eapproximation of the solution to the equations. 2 2. Theory 2.1 Fourier transform properties The physical quantities in this project is mainly treated in their frequency domain. The following properties of the Fourier transform are important for the method later in this report. Theorem 2.1.1 (Fourier transform symmetry). Consider a real function f of some real variable x and its fourier transform f^(!). Then the following holds: if f is an even function then f^(!) is also an even function. If f is an odd function then f^(!) is also an odd function. Proof. Only the proof of the even function relationship is presented here, the proof for the odd function relationship can easily be done in a similar way. Substituting f(−x) with f(x) (assuming f is an even function) in the Fourier transform yields: 1 Z f^(!) = f(−x)e−i2π!xdx (2.1) −∞ Substitute −x with u, and dx with −du yields: −∞ Z f^(!) = −f(u)e−i2π!(−u)du (2.2) 1 1 Z = f(u)e−i2π(−!)udu (2.3) −∞ = f^(−!) (2.4) 2.2 Complex analysis This section includes necessary results from complex analysis and the idea is to make the underlying mathematics to the methods in the report more understandable.