Germán Sierra Instituto de Física Teórica CSIC-UAM, Madrid, Spain

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1 Gemán Siea Instituto de Física Teóica CSIC-UAM, Madid, Spain Wo in pogess done in collaboation with J. Lins and S. Y. Zhao (Univ. Queensland, Austalia) and M. Ibañez (IFT, Madid) Taled pesented at the INSTANTS Confeence at the Galileo Galilei Institute, Fienze, Septembe 2008.

2 - Recent inteest in p-wave supeconductivity is motivated by its applications to He3 films, supeconductos as S2RuO4, supefluids of femi cold atoms in optical taps, etc. - p+ip paiing symmety in the BCS model gives ise to the pfaffian state, which is closely elated to the Mooe and Read state fo the Factional Quantum Hall state fo the filling faction 5/2. When consideing the votices in the BCS model one gets non abelians anyons simila to those of the Mooe-Read model (Geen-Read 2000). Thus the p+ip supeconductos allows fo Topological Quantum Computation, although non univesal. - So fa the studies of the BCS model with p+ip symmety ae based on a mean-field analysis using the BdG Hamiltonian. The coesponding phase diagam contains thee egions:

3 Wea coupling -> standad Coope pais (BCS type) Wea paiing -> Mooe Read pais Stong paiing -> localized Coope pais (BEC type) The wea and stong paiing egions ae sepaated by a second ode phase tansition whee the gap vanishes. The mean field wave function also expeiences a topological phase tansition acoss these two egions (Volovi). The bounday between the wea paiing and the wea coupling egions has not been well chaacteized. It is thus of geat inteest to have an exactly solvable BCS model with p+ip symmety to analyze in detail the natue of the Mooe-Read Pfaffian state and the diffeent phases boundaies of the model.

4 This model is the so called educed BCS model with p+ip wave symmety and it is analogous to the educed BCS model with s-wave symmety. The latte model was solved by Richadson in 1963 and it is closely elated to the Gaudin spin Hamiltonians. The Richadson model was extensively used to study ultasmall supeconducting gains made of Al in the 1990s. The integability of the Richadson-Gaudin models can be poved using the standad Quantum Invese Scatteing Methods. These ae the methods that we apply to the p+ip model.

5 Outline - The pfaffian state in the BCS model - Mean-field appoach to the p + i p model - Exact Bethe ansatz solution - Numeical solution of the BAEs - Themodynamic limit: electostatic analogy. Connection between the mean-field and the exact solution - Puzzles in the wea paiing egion

6 The BCS state (Poyected BCS state) fo p-wave % BCS "exp' 1 & 2 * c g $ g c * * c # ( % * 0 + PBCS "' $ g ) & c * * c # Opeato that ceates a polaised electon with momenta Wave function of the Coope pai in momentum space The poyected state has 2N electons with wave function "( 1,K, 2N ) = A #( 1 $ 2 )#( "( ) = ( * ) 3 $ 4 )K#( 2N$1 $ 2N ) [ ] $ g( )exp(i # ) This wave function is a pfaffian = sqt(deteminant) Mooe-Read coesponds to the long distance/small momenta behaviou: "( ) #1/(x + i y), $ %, g( ) #1/( x + i y ), N 0 $ 0

7 The educed BCS Hamiltonian with p+ip wave symmety eads: 2 H = " 2m c * c # G " 4m ( x + i y )($ x # i $ y ) c * * c # c # c $ $ % ' To be compaed with the s-wave symmety (Richadson model) 2 H = 2m c * * * # c $ G,"," # c c,& $ c c,' $ (,' (,&," % ' In the standad mean-field appoximation: H = $ " = ( x # i y ) c # # 2 2m " µ & )% ( c * c " G $ 2' 2m ) x >0, y c * ( x " i y )c " c ( + h.c. + + ) Which can be diagonalized by a Bogoliubov tansfomation

8 The gap " and chemical potential µ ae solution of the eqs (m=1) µ % x $0, y % x $0, y 2 ( 2 " µ ) # " µ ( ) # 2 = 1 G = 2N " L + 1 G L is the numbe of enegy levels and N is the numbe of pais The themodynamic limit is defined by L " #, N " #, G " 0 Such that g = G L, x = N L ( filling facto) ae constant The solution of the gap and chemical potential eqs yield µ = µ(g,x), " = "(g,x)

9 % BCS "exp' 1 & 2 The mean field wave function: $ g c * * c # g( ) = v u ( % * 0 + PBCS "' $ g ) & = E( ) " 2 + µ x + i y ( )# * c * * c # ( * ) N 0 The behaviou as -> 0 depends cucially on the sign of µ < 0 " g( ) # x $ i y, %( ) # e $ / 0 µ > 0 " g( ) # At whee µ = 0 E( ) = ( 2 " µ ) # 2 1, %( ) # x + i y is the enegy of the quasipaticles 1 x + i y µ : Stong paiing phase :Wea paiing phase thee is a second ode phase tansition (Read-Geen line) E( ) " # 0 but thee is also a topological tansition

10 Topological natue of the wea-stong tansition (Volovi) Momentum space " " # " Wave function g( ) " 0 " 0 " 0 " µ > 0 µ < # +1 µ > 0 (wea paiing) Winding numbe of the map S 2 " S 2 : $ % 0 µ < 0 (stong paiing)

11 Solution of the gap and chemical potential equations Paameteize the dispesion elation as E( ) = ( 2 " µ ) # 2 = ( 2 " a) 2 " b ( ) The paametes a and b have a meaning in the electostatic solution of the exact model (see late) Wea coupling a,b = " 0 ± i# 0 µ > # 2 4 Mooe $ Re ad line a = b = $µ µ = # 2 4 x > x MR % x MR = 1$ 1 ( ' * & g) Wea paiing a < b < 0 0 < µ < #2 x 4 RG < x < x MR Re ad $ Geen line a < b = 0 µ = 0 x RG = 1 % 2 1$ 1 ( ' * & g) Stong paiing a < b < 0 µ < 0 x < x RG

12 Phase diagam of the p + ip wave model

13 Duality between wea and stong paiing phase Given two points in the phase diagam (g,x I ) " wea paiing phase (µ > 0) (g,x II ) " stong paiing phase (µ < 0) If x I + x II =1" 1 g # E I = E II, $ I = $ II, µ I = "µ II The Read-Geen line is selfdual The Mooe-Read line is dual to the empty state x = 0 (in paticula the GS enegy on this line is zeo) This duality also appeas in the exact solution and plays an impotant ole.

14 Recall the Hamiltonian: 2 H = " 2m c * c # G " 4m ( x + i y )($ x # i $ y ) c Setting z 2 = 2 /m % ' * * c # c # $ c $ Define the had coe boson opeatos: b * = " i x y Then the Hamiltonian can be bought into the fom c * * c " H = % x &0, y z 2 N " G % $ # z z b * # b # And can be solved using the Quantum Invese Scatteing Method stating fom the XXZ R-matix and taing a quasi-classical limit

15 The Schoedinge equation: H " = E " is solved exactly by the states (m=1) N " = # C(y m ) 0, C(y) = m=1 & x %0, y x $ i 2 $ y c * * c $ whee the apidities y m satisfy the Bethe ansatz eqs " G"1 " L + 2N "1 y m " L $ =1 N 1/2 2 y m " z + 1 $ y = 0 (m =1,K,N) j#m m " y j N " The total enegy is E = (1+ G) y m m=1 2 lim G "0 y m = z Fo G # 0 " y m : eal o complex Complex solutions always appea in conjugate pais

16 Roots of the p + i p model (numeical solution with L=12,N=6, x=1/2) Im y m Re y m

17 Roots in the complex y-plane (p + i p model) g

18 Roots fo the exactly solvable s-wave model g

19 L, N " #, G " 0, with x = N L, g = G L finite y m Let us assume that the oots fom an ac in the complex plane with a density (y) 2 The enegies " = z fom anothe ac " with density "(#) The BAEs become $ # %(") d" " & y & q 0 y & P q 0 = 1 2G & L 2 + N $ ( dy' N = $ dy (y), E = dy y (y) ( $ ( " (y') y' & y = 0, y ) ( Thee is a analytic solution of these equations which agee with the mean-field solution to leading ode in L and N

20 Stuctue of the acs fomed by the oots y

21 The equation of the complex ac can be detemined and compaed with the numeical esults: Example with x = 1/2 and g = 1.99 (wea coupling egion)

22 The Mooe-Read line x " x MR =1# 1 g $ y m " 0 %m Recall N " = # C(y m ) 0, C(y) = m=1 & x %0, y x $ i y 2 $ y c * * c $ C(0) = N 1 c * * $ c x + i " % & = C(0) 0 y x #0, y ', MR state m=1

23 At x MR =1" 1 g all the oots collapse to y = 0 and thee is no ac as we assumed above. This implies that the mean field appoximation is not valid on the Mooe-Read line lim x "x MR lim L "# $ lim L "# lim x "x MR This suggests the existence of a phase tansition on the MR-line, whose natue is not clea

24 The collapse of oots to zeo occus in the whole wea paiing phase fo fixed values of the coupling

25 Wea-stong paiing duality: dessing opeation N W N 0 N S : numbe of oots in the wea paiing phase : numbe of zeo oots (y=0) : numbe of non zeo oots N W = N 0 + N S The collapse of oots happens iff N 0 L + 2 N S L =1" 1 g Moeove the non zeo oots satisfy the BAE in the stong paiing phase N 0 + N S L + N S L =1" 1 g # x I + x II =1" 1 g Wea-stong duality An eigenstate S> in the stong paiing phase can be dessed by Mooe-Read pais obtaining an eigenstate W> in the wea paiing phase with the same enegy N 0 DRESSING : H S = E S " H W = H [ C(0) ] N 0 S = E W

26 Discontinuity of the GS enegy on the MR line Tae one pai fo g >1. Its enegy in the L>> 1 limit is finite y 1 " log # % 1+ " $ y 1 & ( =1) 1 ' g Dess this pai with N 0 MR pais x I + x II = x I + 1 L =1" 1 g # x I =1" 1 g " 1 L $1" 1 g = x MR lim x "x MR E 0 ( g,x) = y 1 # E 0 (g,x MR ) = 0 One may call this a zeo ode phase tansition

27 Questions and suggestions: - What s the themodynamic limit of this model in the wea paiing phase? It seems that one have to distinguish between ational and iational values of the coupling g. Rational g s -> collapse of oots (non mean-field desciption) Iational g s -> smooth acs (mean-field desciption) - What ae the elementay excitations: is thee a gap in the wea paiing phase? is thee any signatue of non abelian anyons? - The Mooe-Read line is a cossove o a tue phase tansition (pehaps topological)?