Effect of impurities on the Josephson current through helical metals: Exploiting a neutrino paradigm
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Effect of impurities on the Josephson current through helical metals: Exploiting aneutrino paradigm
Pouyan Ghaemi and V. P. Nair
Physics Department, City College of the City University of New York, New York, NY 10031
In this letter we study the effect of time-reversal symmetric impurities on the Josephson super-current through two dimensional helical metals such as on topological insulator surface state. Weshow that contrary to the usual superconducting-normal metal-superconducting junctions, the sup-pression of supercurrent in superconducting-helical metal-superconducting junction is mainly dueto fluctuations of impurities in the junctions. Our results, which is a condensed matter realizationof a part of the MSW effect for neutrinos, shows that the relationship between normal state conduc-tance and critical current of Josephson junctions is significantly modified for Josephson junctionson the surface of topological insulators. We also study the temperature-dependence of supercurrentand present a two fluid model which can explain some of recent experimental results in Josephsonjunctions on the edge of topological insulators.
Critical currents in the mesoscopic superconducting-normal-superconducting (SNS) Josephson junctions havebeen studied widely[1] as an important property whichcan identify important physical properties of SNS Joseph-son junctions. Different regimes of coherent length, meanfree path in the normal region and length of the normalregion have been considered both experimentally and the-oretically. One of the important results in this regard isthe relationship between the normal state conductanceand the critical current of the Josephson junctions [2–5]which is well established both theoretically and experi-mentally. The discovery of topological insulators (TIs)and the novel electronic band structure on their surfacehave led to many investigations on the unique features ofelectronic transport, such as quantum anti-localization,in this new class of materials[6–8]. The experimentalstudy of transport properties of the TI edge states hasbeen quite challenging though as the bulk carriers con-tribute significantly to the transport in most of the exper-imentally accessible topological insulators. Superconduc-tivity in TIs have been another widely studied area of re-search. The realization of superconductivity in doped TIs[9, 10], as well as the possibility of making heterostruc-tures of TIs with superconductors [11, 12] are importantdevelopments in this regard. Theoretical predictions suchas topological superconductivity in doped TIs [13] andpresence of Majorana zero-mode[14] in Josephson junc-tions through TI surface states [15, 16] motivated manyexperimental studies.The presence of helical edge states on the boundary ofTIs is by now well established. The surface state bandstructure which resembles massless fermions, provides aunique platform to realize phenomena previously stud-ied in high energy physics, such as axions[17] and super-symmetry[18, 19], in much more easily realizable con-densed matter system. An important property of suchstates is the absence of back-scattering from electrostaticimpurities, which is enforced by the strong correlationbetween spin and momentum. A reversal of momentumneeds a reversal of spin and since electrostatic fields can- not flip the spin, this helicity conservation forbids back-scattering. This, in turn, makes electronic transportthrough surface states of TIs insensitive to non-magneticimpurities. Given this situation, it might seem that suchimpurities will not affect the supercurrent through sur-face states of TIs as well. In this letter we show that con-trary to the situation for the normal state, non-magneticimpurities will affect the supercurrent carried by the sur-face states of TIs. This dynamical effect resembles yet an-other phenomenon familiar from high-energy physics inthe context of neutrino oscillations known as Mikheyev-Smirnov-Wolfenstein (MSW) effect in topological insula-tors. We will show that the fluctuations of the impuritieslead to a a renormalization of the Fermi velocity. This inturn means that the optical length of the junction (de-fined via the phase of the wave function) is larger thanthe geometric length. The modification of the phase alsomodifies (via matching conditions) the energy eigenval-ues of the Andreev states. This is the essence of ourresult. In fact it has been noticed before that oscilla-tions of impurities might affect the critical temperaturein superconductors[20], but effect studied here as the solemechanism for impurities to change the supercurrent inhelical metals has not been considered before. Our resultscan be used to interpret the measurements on TI Joseph-son junctions which are currently the focus of many ex-perimental studies [21–27].The low-energy effective Hamiltonian of TI surfacestates reads as H s = v F σ · k (1)where σ = ( σ x , σ y ) are the Pauli matrices in the basis ( ψ ↑ , ψ ↓ ) , with ψ σ being the electronic state with spin σ localized on the surface of the TI. The low-energy effec-tive Hamiltonian describing a Josephson junction on thesurface of the TI, with supercurrent along ˆ x , is given by[28] H = ( − iv F ∇ · σ − µ ) τ + ∆ R ( x ) τ + ∆ I ( x ) τ , (2)taken to act on the electronic state of the form (cid:16) ψ ↑ , ψ ↓ , ψ †↓ , − ψ †↑ (cid:17) T . Here ∆ R ( x ) and ∆ I ( x ) are real andimaginary parts, respectively, of the induced supercon-ducting gap ∆ = ∆ R + i ∆ I . In (1) and (2), v F is theFermi velocity. As for the matrix structure, σ i act onphysical spin space whereas the τ i act on the supercon-ducting particle-hole space. As the Hamiltonian in (2)is invariant under translation along ˆ y , the momentum k y in this direction is conserved. The low-energy An-dreev states in the junction thus correspond to k y = 0 and k x close to the two fermi wave vectors, k x = µv F for σ x = 1 and k x = − µv F for σ x = − . Notice that since σ x commutes with Hamiltonian (2) we can decouple thelow-energy effective Hamiltonians into two independentsectors corresponding to electron-hole states close to rightor left fermi points. Here we will focus on one of the ef-fective Hamiltonians, but the other independent one canbe similarly studied. Since we are aiming for the effectof temporal fluctuations it is more efficient to use thecorresponding -dimensional action given by S = ˆ dxdt ¯Φ ( x, t ) h iτ D t + τ D x + ˜ M ( x ) i Φ ( x, t ) (3)where Φ T ( x, t ) = (cid:16) φ ↑ ( x, t ) , − φ †↓ ( x, t ) (cid:17) and φ σ ( x, t ) = ψ σ ( x, t ) e ∓ ik F x , with σ = ↑ , ↓ , are the fermionic field op-erators for excitations close to the right or left Fermipoints. D µ = ∂ µ − i e τ A µ is the covariant derivative.Notice that (3) is of the standard form of a Dirac action ¯Φ( iγ µ D µ + ˜ M )Φ if we identify γ = τ , γ = − iτ , γ = τ . Further, in (3), ¯Φ ( x, t ) = Φ † ( x, t ) γ = Φ † ( x, t ) τ and ˜ M ( x ) = ∆ R ( x ) τ + i ∆ I ( x ) τ . The Fermi velocityhas been set to by scaling x , or equivalently, the mo-mentum k x . The effect of charged impurities is capturedby A = V ( x − a ( t )) where a ( t ) identifies instantaneousposition of the impurity. As we will see below, in orderto capture the effect of impurities on the supercurrentin the junction, we should consider the natural fluctu-ations in the position of the impurity. For small fluc-tuations a ( t ) = a + ξ ( t ) , the impurity potential readsas V ( x − a ( t )) ≈ V ( x − a ) + ∂ x V ( x − a ) ξ ( t ) . Aswe will show below, the impurities can only affect thesupercurrent as a result of their temporal fluctuations.We would like to note that such treatment of impuri-ties and its effect on superconductivity in normal metalshave been considered long before [20]. But here we showthat as a result of helical band structure of the surfacestates of the TIs, the temporal fluctuations are the solemechanism through which impurities can affect the su-percurrent. The action, including the effect of fluctuatingimpurities, read as S = S + S int + S osc S = ˆ dxdt h ¯Φ ( x, t ) (cid:16) iτ ∂ t + τ ∂ x + ˜ M ( x ) (cid:17) Φ ( x, t ) − V ( x − a ) ¯Φ ( x, t ) iτ Φ ( x, t ) i S int = ˆ dxdt ∂ x V ( x − a ) ¯Φ ( x, t ) iτ Φ ( x, t ) ξ ( t ) S osc = 12 M I ˆ dt ¯ ξ ( t ) (cid:18) − ∂ ∂t − ω (cid:19) ξ ( t ) (4)Since we are interested in localized impurities, both theimpurity potential V ( x − a ) and the resulting electricfield E ( x ) = ∂ x V ( x − a ) are localized in space and thedynamics of the impurities are captured by the harmonicoscillator action S osc ( M I and ω are the mass and theharmonic oscillation frequency of the impurities). Cou-pling of Andreev states and the harmonic oscillations ofimpurities are captured in S int and will lead to a self-energy correction in S (see supplementary materials),leading to an effective action of the form ¯ S = ˆ dxdt h (1 + Σ ( x )) ¯Φ (cid:16) iτ ∂ t + ˜ M ( x ) (cid:17) Φ+ ¯Φ τ ∂ x Φ + ¯Φ ˜ M ( x )Σ ( x ) Φ i + ˆ dxdt V ( x − a ) ¯Φ iτ Φ (5)To the lowest nontrivial order in perturbation theory, theself-energies can be calculated as Σ ( x ) ≈ e E ( x )2 πω M I R and Σ ( x ) ≈ e E ( x )2 πω M I R h log( ωM I ) − i where R is the lengthscale over which E ( x ) = − ∂ x V ( x − a ) is non-zero (seesupplementary materials). What is important for us isnot so much the specific formulae for these self-energies,but that the general form of the effective action is as givenin (5), with the self-energies as corrections concentratedaround the impurities.Notice that the modification due to Σ does not af-fect the spatial derivative term for the electrons. This isbecause the oscillator variable ξ ( t ) does not have a spa-tial dependence. Thus although the free fermion action ¯Φ( iγ µ ∂ µ + ˜ M )Φ has a Lorentz-type symmetry (albeit with v F in place of the speed of light c ) the interactions withimpurities, and hence corrections, do not respect thissymmetry. The temporal and spatial derivative termscan be renormalized differently, and then with a possiblescaling of the field, the action can be brought to the form(5). The general result is that the effect of fluctuating im-purities appear as renormalization of the effective Fermivelocity and of the size of superconducting gap in theregion where the electric field of the impurity potentialsare present. The change of Fermi velocity can be viewedas a “refractive index" for the electron. There is then anadditional phase change acquired for the wave functionsas the effective “optical length" is modified. This will bethe essence of how the Andreev states are modified.It is well known that for massless particles, propagat-ing at the speed of light in vacuum, the primary effect ofinteractions is to generate a refractive index rather than amass (which is usually forbidden for symmetry reasons).A similar situation is obtained even for massive particlesin the ultrarelativistic limit. Our argument is that, forthe surface states in a TI which have a Lorentzian sym-metry (with c → v F ), again a refractive index is preciselywhat we should expect as the primary effect of interac-tions.The situation here is closely related, conceptually, tohow interactions with matter modify neutrino oscillationsas in the MSW effect [29–32]. There are two ingredientsto this. First, the neutrinos acquire a refractive indexwhich can be calculated in terms of the forward scatteringamplitude, a calculation which mirrors, mutatis mutan-dis , what is given in the supplementary material. Therefractive index for electron-neutrinos is different fromthat for other flavors due to charged current interactionswith matter. The resulting difference in the phase of thewave functions modifies the neutrino mass-eigenstates inmatter, and hence the oscillations between different fla-vors of neutrinos. The second part is a resonance effectwhich can enhance the mixing of flavors in matter, evenup to the maximal mixing. For us, we have only one fla-vor to consider, so the situation is simpler; there is noresonance part, but the phase enhancement due to therefractive index is similar and can modify the matchingconditions (and energies) for the Andreev states.We will now use the effective action (5) to study theeffect of impurities on the supercurrent. The Andreeveigenstates are determined by the effective Schrödingerequation which follows from (5), − [( i ¯ v ∂ x + V ( x − a )) τ + ∆ R τ + ∆ I τ ]Ψ( x )= E Ψ( x ) (6)where ¯ v ( x ) = v F / (1 + Σ ( x )) is the renormalized Fermivelocity. For constant ∆ , there are two independenteigenstates with energy E given by Ψ ± E ( x ) = e iW ± ( x ) η ± ( E, ∆) W ± ( x ) = ˆ x du V ( u − a ) ± p E − | ∆ | ¯ v ( u ) (7) η + ( E, ∆) = 1 √ E q E + p E − | ∆ | − ∆ / q E + p E − | ∆ | (8) η − ( E, ∆) = 1 √ E − ∆ ∗ / q E + p E − | ∆ | q E + p E − | ∆ | (9)To model the Josephson Junction, we consider thestepwise variation of ∆( x ) in three regions x < (re-gion I), < x < x p (region II) and x > x p (region III)given by ∆( x ) = ∆ (I)0 (II)∆ e iφ (III) (10) The eigenstates can be expressed in each region as thesuperposition of Ψ ± E ( x ) as Ψ i ( x ) = A i Ψ + E ( x ) + B i Ψ − E ( x ) where i = I, II and III, corresponding to the three regions.We define the transfer matrices: ˆ T B ( E, ∆) = 1 √ E (cid:20) √ E + κv − ∆ ∗ / √ E + κv − ∆ / √ E + κv √ E + κv (cid:21) ˆ T n ( x ) = e iφ I ( x ) (cid:20) e i h k i x x e − i h k i x x (cid:21) (11) ˆ T s ( x ) = (cid:20) e iκx e − iκx (cid:21) where κ = p E − ∆ /v is the wave vector in the su-perconducting region, h k i x = Ex ´ x du ¯ v ( u ) is the averagedwave vector in the normal (TI) region of the junctionand φ I ( x ) = ´ x V ( u − a )¯ v ( u ) du is the phase resulting fromthe static impurity. In terms of these matrices, the eigen-states in the three regions can be written as Ψ I ( x ) = ˆ T B ( E, ∆ ) ˆ T s ( x ) (cid:20) A I B I (cid:21) (12) Ψ II ( x ) = ˆ T n ( x ) (cid:20) A II B II (cid:21) (13) Ψ III ( x ) = ˆ T B ( E, ∆ e iφ ) ˆ T s ( x − x p ) (cid:20) A III B III (cid:21) (14)The boundary conditions which determine the spectrumof the states in the junction are Ψ I (0 − ) = Ψ II (0 + ) and Ψ II ( x − p ) = Ψ III ( x + p ) . These continuity equations reduceto (cid:20) A I B I (cid:21) = ˆ T J ( E, ∆ , φ ) (cid:20) A III B III (cid:21) (15)where the wave function on the two ends of theJosephson junctions are connected by ˆ T J ( E, ∆ , φ ) =ˆ T B ( E, ∆ ) − ˆ T n ( x p ) − ˆ T B ( E, ∆ e iφ ) . The S -matrix forthe junction must relate the incoming and outgoing statesas (cid:20) A III B I (cid:21) = S (cid:20) A I B III (cid:21) ; this S -matrix can be written interms of ˆ T J ( E, ∆ , φ ) as S = 1ˆ T J ( E, ∆ , φ ) (cid:20) − ˆ T J ( E, ∆ , φ ) ˆ T J ( E, ∆ , φ ) e − iφ I (cid:21) (16)The supercurrent in the junction can be derived us-ing the well-known relationship between the Josephsoncurrent of the junction and the spectrum [3], namely, I = I + I + I (17) I = − e ~ X p tanh ( E p / k B T ) dE p dφ (18) I = − e ~ ˆ ∞ ∆ dE ln [2 cosh ( E/ k B T )] ∂ρ ( E, φ ) ∂φ (19) I = e ~ ddφ ˆ dx | ∆( x ) | /g (20) I is the contribution form the discrete spectrum of in-gap states and I is the contribution form continuum ofstates with energy above the gap with density of states ρ ( E, φ ) for the one spin state at each fermi point as wehave for the helical metal. The third them I vanishesfor the phase independent gap and will be ignored inthis letter. In (20), g is the interaction constant of BCStheory.For the states with energy E < ∆ , κ is imaginary andas a result Ψ + E ( x ) in region I and Ψ − E ( x ) in III are notnormalizable. Equation (15) then simplifies to (cid:20) B I (cid:21) = ˆ T B ( E, ∆ ) − ˆ T n ( x p ) − ˆ T B ( E, ∆ e iφ ) (cid:20) A III (cid:21) (21)and leads to the following equation determining the in-gap energies: cos − ( E n / ∆ ) + E n v L + φ n π, n ∈ Z (22)where L = ´ x p dx (1 + Σ ( x )) is the effective length ofthe junction which is modified by fluctuations of the im-purity. This is the new “optical length" of the junction.The phase factor φ I has cancelled out in (22) confirm-ing that static impurities have no effect on the energy ofin-gap Andreev states. The effect of impurities is onlythrough their dynamical fluctuations that leads to the fi-nite self-energy Σ ( x ) which modifies the effective Fermivelocity. The effect on the energy eigenvalues is mostvividly illustrated by considering states with E n ≪ ∆ ,in which case we get E n ≈ ( v/ L ) (cid:2) π ( n + ) − φ (cid:3) . Theincrease in L implies that E n and ∂E n /∂φ are decreasedrelative to the case with no impurities. More generally,defining Θ n = E n v L + φ / − nπ , the supercurrent asso-ciated with each in-gap state reads as I n ( φ ) = − e ~ ∂E n ∂φ = e | ∆ | ~ (cid:20) sin(Θ n )1 + sin(Θ n ) L| ∆ | /v (cid:21) (23)As a function of Θ n , this has a maximum at Θ n = π/ ,so that the critical current is I crit = ( e | ∆ | / ~ )(1 + L| ∆ | /v ) − . The condition Θ n ≈ π/ is actually ob-tained for modes of very low energy E n ≪ | ∆ | . It isimportant to note that by that the suppercurrent gen-erated by in gap states decreases by increasing L whichshows that impurities clearly affect the supercurrent.For the states above the gap, the density of statesis given by the Krein-Friedel-Lloyd formula ρ ( E ) = πi ∂∂E (ln det S ) [33]. Using (16) we get det S = e − iφ − β E cos (Θ E ) hp − β E cos(Θ E ) − i sin(Θ E ) i where Θ E = Ev L + φ / and β E = ∆ E . The supercurrentdue to above-the-gap states then simplifies to I = e π i ~ (cid:20) ˆ ∞ ∆ dE tanh ( E/ k B T ) ∂ (ln det S ) ∂φ (cid:21) − ek B T π ~ ln [2 cosh (∆ / k B T )]= e π ~ ˆ ∞ ∆ dE tanh ( E/ k B T ) " E p E − ∆ E − ∆ cos (Θ E ) − − ek B T π ~ ln [2 cosh (∆ / k B T )] (24)We would like to emphasize two important featuresof the supercurrent contribution from states with energyabove the superconducting gap:1. For low temperatures T ≪ ∆ , I is only weakly T -dependent through the temperature dependenceof superconducting gap ∆ .2. I is also only weakly dependent on L , i.e, onlyweakly sensitive to impurities.To elucidate the second point, we first note that thesecond term in (24) is independent of L . Assuming T ≪ ∆ , tanh ( E/ k B T ) ∼ . The integrand in thefirst term in (24) has two types of dependence on E .One is a periodic dependence, with period ~ v/ L dueto cos(Θ E ) , and the other is a decaying dependence, ofthe form ∆ /E for large E . For the effective junctionlength L larger than ~ v/ ∆ , the oscillatory dependenceis is much faster than the decay rate and so can be aver-aged over Θ E . (This may be viewed as an application ofthe Riemann-Lebesgue lemma.) As a result, the depen-dence on L will be eliminated and I will not be seriouslyaffected by impurities even when fluctuation effects areincluded.In conclusion we have shown that the supercurrent inJosephson junctions with helical metals, such as on thesurface of three-dimensional TIs, is affected by impuri-ties through their temporal fluctuations. However, thisapplies primarily to the supercurrent generated by in-gap Andreev states. The supercurrent carried by thestates above the gap will not be seriously affected byimpurities. Based on our results, the supercurrent inthe Josephson junctions on the surface of TIs can be in-terpreted as a superposition of two contributions, onewhich is strongly temperature-dependent and also sen-sitive to the impurities in the junction and one whichis only weakly temperature-dependent and not sensitiveto the impurities. Given new advances in controlling thelevel of disorder in TIs [34, 35], these results will be usefulin analyzing many of the experimental results on Joseph-son junctions made on TIs. For example, our analysisis consistent with the experimental results in [21, 22];whether different level of impurities could affect the crit-ical current in the Josephson junction on TI was the mainmissing ingredient in the theoretical model used to inter-pret those results. In fact, our work may be consideredas further substantiating the interpretation, presented in[21], in terms of two types of supercurrent contributions.The detailed comparison with those experimental resultswill be subject of a following publication.We would like to thank J. D. Sau, D. J. Van Harling-nen and A. Bernevig for helpful discussions. This workwas supported in part by the U.S. National Science Foun-dation grant PHY-1213380 and by a PSC-CUNY award. [1] A. A. Golubov, M. Y. Kupriyanov, and E. Il’ichev,Rev. Mod. Phys. , 411 (2004).[2] I. O. Kulik, Sov, Phys. JETP , 944 (1970).[3] C. W. J. Beenakker, Phys. Rev. Lett. , 3836 (1991).[4] P. F. Bagwell, Phys. Rev. B , 12573 (1992).[5] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G.Williamson, L. P. Kouwenhoven, D. van der Marel, andC. T. Foxon, Phys. Rev. Lett. , 848 (1988).[6] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[8] M. 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I. EFFECTIVE ACTION FOR ELECTRONS FROM OSCILLATING ELECTROSTATIC IMPURITYA. The perturbative result
Here we present the calculation of the effective action for the electron field due to the interaction with impurities.The starting action is the one given in Eq.(3) in text. Since we will also be discussing how temperature affects thecalculation, it is more convenient to use a Euclidean space (imaginary time) formalism, so that thermal effects can beincluded via the Matsubara formalism.The Euclidean version of (3) is given by S = ˆ d x Φ † ( D − i τ D ) Φ + ˆ d x Φ † (cid:20) ∗ (cid:21) Φ (S1)Defining γ = γ = τ , γ = − τ , this becomes S = ˆ d x ¯Φ [ γ µ ( ∂ µ − iτ A µ ) + M ] Φ (S2)where ¯Φ = Φ † γ and M = (cid:20) ∆ ∗
00 ∆ (cid:21) (S3)The propagator for the electron is then given by S ( x, y ) ≡ h Φ( x ) ¯Φ( y ) i = (cid:18) γ · ∂ + M (cid:19) x,y = ˆ d p (2 π ) e − ip · ( x − y ) ( iγ · p + M ∗ )( p + M ∗ M ) (S4)The electrostatic field can be expanded as A ( x + ξ ) ≈ A ( x ) + ξ∂ A = A ( x ) + ξ F . The correction to the action,to quadratic order is then given by the Wick contractions of − S int S int ; we need one electron propagator and onecontraction for the ξ ’s. Thus ∆ S eff = e ˆ F ( x ) F ( y ) h ξ ( x ) ξ ( y ) i ¯Φ( x ) γ τ S ( x, y ) γ τ Φ( y ) (S5)For us, ξ depends only on time as it is the oscillating coordinate of the impurity relative to the mean position. Further,if we consider several impurity atoms, only the ξ ’s of the same impurity atom can have nonzero average h ξ ( x ) ξ ( y ) i .This means that the contribution to the integral is concentrated around x = y . We van encode these by wriitng h ξ ( x ) ξ ( y ) i = RM I ˆ d k (2 π ) e − ik · ( x − y ) k + ω (S6)The spatial part gives a delta function and integrating over x , we get the factor R . This shows that R may betaken as a rough measure of the extent over which this correlation exists. Using this in (S5) and changing variables p = q − k , we get ∆ S eff = ˆ d q (2 π ) e − iq · ( x − y ) ¯Φ( x ) Σ( q ) Φ( y ) (S7) Σ( q ) = e RF M I ˆ d k (2 π ) γ τ ( iγ · ( q − k ) + M ∗ ) γ τ ( k + ω )[( q − k ) + M ∗ M ]= e RF M I ˆ dα ˆ d K (2 π ) ( iγ q − M )(1 − α ) − αM [ K + αK + Q ] (S8) Q = α (1 − α ) q + (1 − α ) ω + αM ∗ M (S9)In the last line of (S8) we have combined the denominators of the previous expression using Feynman’s formula andalso defined K = k − αq , K = k − q . Notice that the self-energy Σ is naturally split as Σ = − Σ ( iγ q − M ) + Σ M Σ = − e RF M I ˆ dα ˆ d K (2 π ) (1 − α )[ K + αK + Q ] (S10) Σ = − e RF M I ˆ dα ˆ d K (2 π ) α [ K + αK + Q ] (S11)Evidently, the small q limit of Σ will give a term of the form ¯Φ( γ ∂ + M )Φ in the action, while the same limit of Σ will correct the mass M . Higher order powers of q in the expansion of Σ will give higher derivatives of the electronfield; these are not important since we are interested in the low energy modes of the electron. Explicitly, ∆ S eff ≈ Σ ( q = 0) ˆ ¯Φ( γ ∂ + M ) Φ + Σ ( q = 0) ˆ ¯Φ M Φ (S12)Carrying out the K integration, Σ ( q = 0) = − e RF M I ˆ dα (1 − α ) ˆ dK π αK + αM ∗ M + (1 − α ) ω ] ≈ e F RM I πω (S13)In the last line, we evaluated the remaining integral neglecting M ∗ M in comparison to ω , since the frequency ofvibration for impurity atom is much larger than the the possible gap. In any case, we set M =) for the region II of ourdiscussion in text to which this calculation is applied. We also made the continuation to real time by F → F = E .The result (S13) agrees with what is quoted in text.In a similar way, Σ ( q = 0) ≈ e E RM I πω [log(2 ω/ | ∆ | ) − (S14)We can include finite temperature effects by using Matsubara frequencies for k = 2 πn/β ≡ ω n and p = 2 π ( n + ) /β , β = 1 / ( k B T ) . The result is then ∆ S eff = 1 β X s ˆ dq π ¯Φ e − iq · ( x − y ) Σ( q ) Φ( y )Σ( q ) = e F RM I β X n ˆ dk π iγ ( q s − ω n ) + M ∗ + iγ k ( ω n + ω )[( q s − ω n ) + k + M ∗ M ] (S15)As usual, we carry out the summation via contour integration. Writing Q = k + M ∗ M , β X n iγ ( q s − ω n ) + M ∗ + iγ k ( ω n + ω )[( q s − ω n ) + k + M ∗ M ] = 12 π ˛ dz ( e iβz − iγ ( q s − z ) + iγ k + M ∗ ( z + ω )[( q s − z ) + Q ] (S16)where the contour encloses the poles of the ( e iβz − − factor. Folding the contours back to the upper and lowerhalf-planes and evaluating the residues at the other poles, we find some terms which are independent of β (and coincidewith what we have already done in (S8- S14)) and a set of terms which are β -dependent. The β -dependent part of Σ is Σ (cid:3) T =0 = e F RM I ˆ dk πω (cid:20) Q + ( q s − iω ) + 1 Q + ( q s + iω ) (cid:21) e βω − (S17)The key point is that, for us, ω ≫ k B T . As a result, the T -dependent correction is exponentially suppressed dueto the e βω factor in the denominator. A similar argument holds for Σ as well. Thus, in conclusion, the effectiveaction is of the form as in Eq.(5) of the text where we can take Σ and Σ to be independent of temperature. Anytemperature-dependence of the supercurrent would be due to the T -dependence of the gap ∆ and due to factors suchas tanh( E/ k B T ) . B. The more general argument
The form of the action in Eq.(5), which is all we need for the rest of the results in the paper, can be obtainedon general symmetry grounds. We will first consider a region of uniform distribution of impurities. The startingaction has a (1 + 1) -dimensional Lorentz symmetry with the Fermi velocity in place of the speed of light. If theinteractions respect this symmetry, the relative coefficients of the two terms in the combination ¯Φ iτ ∂ t Φ + ¯Φ τ ∂ x Φ must be preserved. However, the interactions do not preserve this Lorentz symmetry and so, on general symmetrygrounds, we expect the effective action to be of the form S eff = A ˆ ¯Φ iτ ∂ t Φ + B ˆ ¯Φ τ ∂ x Φ + ˆ ¯Φ C Φ+terms of higher order in derivatives of Φ (S18)The higher derivative terms are not important for the low energy modes which are of interest to us. A , B , C arecalculable constants. We can scale out one of them, say, B to write S eff = ( A/B ) ˆ ¯Φ iτ ∂ t Φ + ˆ ¯Φ τ ∂ x Φ + ˆ ¯Φ( C/B )Φ + · · · (S19)We now define Σ A/B = 1 + Σ (S20)With this, we see that the first term (S19) has the form in Eq.(5).As for the ¯Φ ˜ M Φ term, notice that the derivative terms of the starting action, Eq. (3) in text, has a chiralsymmetry, Φ → e iθτ Φ . This is true even with the electromagnetic interactions. Thus, if ˜ M is originally zero, itcannot be generated by perturbative corrections. Therefore, in the theory with nonzero, C must be such that itvanishes when ˜ M → . We therefore write C ∼ ˜ M , and define Σ in general by CB = (1 + Σ ) ˜ M + ˜ M Σ (S21)With the two definitions (S20) and (S21), we get S eff = ˆ dxdt h (1 + Σ ( x )) ¯Φ (cid:16) iτ ∂ t + ˜ M ( x ) (cid:17) Φ+ ¯Φ τ ∂ x Φ + ¯Φ ˜ M ( x )Σ ( x ) Φ i (S22)in agreement with what is given in text. The argument of the previous subsection was given to see how the perturbativecalculation of the coefficients A/B and