Inhomogeneous Superconducting States of Mesoscopic Thin-Walled Cylinders in External Magnetic Fields
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Inhomogeneous Superconducting States of Mesoscopic Thin-Walled Cylinders inExternal Magnetic Fields
K. Aoyama , , , R. Beaird , D. E. Sheehy , and I. Vekhter The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: October 9, 2018)We theoretically investigate the appearance of spatially modulated superconducting states inmesoscopic superconducting thin-wall cylinders in a magnetic field at low temperatures. Quanti-zation of the electron motion around the circumference of the cylinder leads to a discontinuousevolution of the spatial modulation of the superconducting order parameter along the transitionline T c ( H ). We show that this discontinuity leads to the non-monotonic behavior of the specificheat jump at the onset of superconductivity as a function of temperature and field. We argue thatthis geometry provides an excellent opportunity to directly and unambiguously detect distinctivesignatures of the Fulde-Ferrell-Larkin-Ovchinnikov modulation of the superconducting order. PACS numbers: xxx
Mesoscopic systems both serve as a platform to inves-tigate fundamental quantum physics of solids and are atesting ground for potentially transformative future de-vices [1–5]. Of special interest in this context are in-teracting systems exhibiting interplay of the collectiveemergent properties with the quantum physics of singleparticles. Small superconducting samples of nontrivialtopology provide an example of such interplay since theglobal phase of the pair condensate and the phases ofsingle particle wave functions respond differently to theapplied magnetic field.In mesoscopic superconducting rings of radius R ∼ ξ ,where ξ ≡ v F / πT c ( H = 0) is the superconducting co-herence length, and v F is the Fermi velocity, this leads toa doubling of the period of the oscillations of the transi-tion temperature, T c , as a function of the magnetic flux,Φ, through the ring, relative to the well-known Little-Parks (LP) effect [5, 6, 8]. The small ring radius, R ,implies that each single electron state can be labeled byits angular momentum, n , in units of ~ , and each par-ticle acquires an additional phase due to the magneticflux, Φ = πR H when circling around the ring. In theabsence of the field the wave function of the Cooper pairhas a net zero angular momentum, as the time-reversedstates with n = − n form a bound state. In contrast,under the applied field the non-zero quantum number l = n + n partially compensates the net flux and max-imizes the transition temperature, T c . Therefore, for asmall ring [7, 8], T c is a periodic function of φ = Φ / (2Φ )with the flux quantum Φ = hc/ e , while for a large ringthe periodicity is solely due to the flux experienced by anelectron pair with charge 2 e , i.e. T c is a periodic functionof Φ / Φ (LP effect).Under these assumptions, there is no overall suppres-sion of the maximal T c as the magnetic field increases:at integer values of φ the orbital coupling of the field tothe individual electrons can be fully compensated by the finite angular momentum l of the Cooper pair. The de-struction of superconductivity in such a geometry mustthen occur via paramagnetic (Zeeman) coupling of theelectron spins to the field, which raises the energy of thesinglet bound state: inclusion of this coupling is essen-tial for developing a complete picture. Consequently inthis Letter we consider the combined effect of the or-bital and Zeeman effect on superconductivity, and an-alyze a mesoscopic thin-walled cylinder with the fieldalong the axis. The cylinder geometry allows the for-mation of a spatially-modulated [9, 10] (Fulde-Ferrell-Larkin-Ovchinnikov, FFLO) superconducting state thatenables pairing under high Zeeman field. We show be-low that a) this state occurs even if the cylinder is madeout of materials where superconductivity is not param-agnetically limited in the bulk; b) signatures of such astate are much more prominent and easily identified inthis geometry with R ∼ ξ than either in bulk materialsor flat thin films, and therefore mesoscopic systems of-fer a unique chance to detect the FFLO state that hasremained elusive for nearly 50 years since it was first pre-dicted.Our main results are shown in Fig. 1. While at lowfields the variation of the transition temperature T c ( H )is well described by the Little-Parks (LP) periodicity,at higher fields there is an extended region in whichthe superconducting order parameter is modulated alongthe cylinder axis. Near the phase boundary T c ( H ) inthis regime, the superconducting phase exhibits alter-nating regions of phase-modulated (FF) and amplitude-modulated (LO) order; however, the LO phase becomesstable at lower T . The wavevector, q z , of this modula-tion has non-analytic dependence on H , due to the in-terplay between the finite size effects and the LP oscilla-tions. The heat capacity jump at T c ( H ) varies dramat-ically along this sequence of transitions (in contrast tothe smooth evolution at temperatures where the FFLO HLO modulation qz ^ z R Δ z H T l = l = l = l = l = normalhomogeneous SC FIG. 1: (Color online) Superconducting thin-wall cylinderwith a small radius R in a magnetic field parallel to the cylin-der axis ( z -axis). Sketch shows the spatial modulation of theorder parameter in the LO phase. Right-panel: phase diagramin the H - T plane, showing regions of the normal phase, ho-mogeneous ( q z = 0) superconductor, and spatially-modulatedLO phase, with phase modulated FF states indicated by theshaded regions. modulation is absent), enabling a direct identification ofthe modulated states.Although the possibility of FFLO states has been dis-cussed in bulk materials such as the heavy-fermion su-perconductor CeCoIn [11–13] and organic superconduc-tors [14–17], it is difficult to design a “smoking gun”experiment that unequivocally points towards such astate. In real bulk systems both orbital and paramag-netic coupling suppress superconductivity, and inhomo-geneity arises due to both. The former effect leads toproliferation of the vortices. Each vortex contains ex-actly one flux quantum for the Cooper pairs, Φ , whichcorresponds to a 2 π -phase winding of the superconduct-ing order parameter around the vortex core. In that sensea thin-walled ring or cylinder can be viewed as a core-less “supervortex” of multiple flux quanta, with a phasewinding 2 πl . In contrast, paramagnetic pairbreaking fa-vors FFLO states. Recall that in a singlet superconduc-tor in the absence of Zeeman splitting the Cooper paircomprises electrons in time-reversed states, which haveequal energies, and therefore are unstable towards for-mation of a bound state. With paramagnetic couplingthe states with opposite spins have equal energies if theyhave a field-dependent momentum mismatch q , and themodulation of the FFLO state originates from this finitecenter-of-mass momentum (CMM) of the Cooper pairs.On a 1D ring the pair CMM is equivalent to the netphase winding, so that a different geometry is needed todistinguish the modulated states.We consider a long hollow superconducting cylinder ofradius R and thickness d ≪ ξ , which, in the absence ofa magnetic field, is described by the Hamiltonian H = X σ, p ξ ( p )ˆ c † p ,σ ˆ c p ,σ − λ X q ˆ B † ( q ) ˆ B ( q ) , ˆ B ( q ) = 12 X p ,α,β ( − i σ y ) α,β ˆ c − p + q ,α ˆ c p + q ,β . (1) Here ˆ c p ,α is the annihilation operator for an electron withmomentum p and spin projection α , λ is the strengthof the pairing interaction, and we assumed singlet s -wave superconductivity. For small R the motion aroundthe circumference of the cylinder is quantized, while themomentum along the axis ( z ) is continuous, so that p = ( m/R, p z ) with m integer, and the quasiparticle en-ergy takes the form ξ ( p ) = 12 M h p z + (cid:16) mR (cid:17) i − µ , (2)where M is the electron mass and µ is a chemical po-tential. Here the summation over the momenta means P p = (cid:2) (2 π ) R (cid:3) − P m ∈ Z R dp z .The magnetic field threading the cylinder along its axisleads to a Zeeman splitting of the single particle energylevels by H Z = P σ, p σ h ˆ c † p ,σ ˆ c p ,σ , where h = µ B gH , µ B is the Bohr magneton, and g is the g -factor of the quasi-particles in the crystal. At the same time the momen-tum operator has to be replaced by its gauge-invariantcounterpart, b p → b p + | e | A . For our model of a thin-walled cylinder the vector potential A = − ˆ ϕ H R/
2, and ϕ is the azimuthal angle around the cylinder. Hence | A | = const on the cylinder.In the superconducting (SC) phase the pair field∆( q ) = λ h ˆ B ( q ) i , where h . . . i denotes the thermal av-erage, acquires a non-zero value. Due to the cylindricalgeometry, ∆( r ) can be expanded in the Fourier series,∆( r ) = | ∆ | X q z X l ∈ Z C l,q z e i lϕ e i q z z . (3)The uniform SC state at H = 0 only has C , = 0, whilea single component C l, with flux-dependent l = 0 char-acterizes the SC transition under orbital coupling to thefield and gives rise to the Little-Parks effect. If the cylin-der were to unfold into a two-dimensional (2D) plane, l and q z would become components of a 2D vector q , andin response to a Zeeman field a state with q = 0 would berealized. The cylindrical geometry is unique since l = 0gives the magnetic flux through the cylinder, and there-fore the FFLO modulation is only along the axis, q z = 0.Near the transition the linearized gap equations for dif-ferent l , | q z | decouple [18, 19], and hence superconductingstates appear either with a single C l, q z (phase modulated,∆( r ) = ∆ e ilϕ e i q z z , FF), or with C l,q z = C l, − q z (ampli-tude modulated, ∆( r ) = ∆ e ilϕ cos( q z z ), LO, see Fig. 1).Thus, this setting has an advantage over other ways toachieve FFLO states (such as bulk paramagnetically-limited superconductors, imbalanced fermionic atomicgases [32] or thin SC films), in which modulation direc-tion is arbitrary, and therefore complex states may befavored [20–25].To study the transition between the normal andSC states we use the Ginzburg-Landau (GL) expan-sion of the free energy, F GL = a (2) ( l, q z , T, H ) | ∆ | + a (4) ( l, q z , T, H ) | ∆ | , At each H, l, q z , the temperature T where a (2) ( l, q z , T, H ) becomes negative (if it exists) indi-cates a putative second order transition from the normalinto the SC state with given values of l, q z . The highest ofthese temperatures is the physical transition point T c ( H )into a state with the corresponding l, q z . The necessarycondition for the continuous transition is that the quar-tic coefficient a (4) ( l, q z , T c ( H ) , H ) remains positive at thetransition point.We evaluate the coefficients a (2) and a (4) using theGreen’s function formalism for the Hamiltonian, Eq. (1)with the Zeeman and orbital coupling terms. We notehere that a similar setup was considered in Ref. 27 using aphenomenological GL expansion that is valid only in thelong-wavelength modulation limit. Due to the neglect ofthe field dependence of the coefficients of the GL expan-sion, lack of connection with a microscopic model Hamil-tonian, and the assumption of a small modulation wavevector, that approach failed to capture any of the physicsfound in this Letter, and led the authors of Ref. 27 to fo-cus on the fluctuation contribution to the specific heatas the main observable. Our analysis below shows thatthe “mean field” features of the transition, when analyzedproperly, strongly reflect the interplay of the quantizationof single electron motion and spatial modulation of the SC order. We use a quasi-classical approximation for thenormal state Green’s function −h T τ ˆ ψ σ ( r , τ ) ˆ ψ † σ ( r ′ , i ≃ T P ε n e − ε n τ G ε n ,σ ( r − r ′ ) e i | e | R r ′ r d s · A ( s ) , where the inte-gral in the exponent is evaluated along a straight line.This approximation smears out the even-odd flux peri-odicity for a 1D ring, but this periodicity is already bro-ken by Zeeman coupling, and hence the approximation isadequate for our goals. We obtain for the quadratic term a (2) | ∆ | = Z d r ∆ ∗ ( r ) (cid:16) λ − T X ε n ,σ X p ˆ K ( ε n , σ ) (cid:17) ∆( r ) , ˆ K ( ε n , σ ) = G ε n ,σ ( p ) G − ε n , − σ ( − p + Π ) , (4)where G ε n ,σ ( p ) = ( iε n − ξ ( p )+ σh ) − is the Fourier trans-form of G ε n ,σ ( r ), ε n is a fermionic Matsubara frequency,and Π = − i ∇ +2 | e | A . The full expression for the quarticterm is given in the supplementary information [26].The effects due to small ring size R ∼ ξ are con-tained in the discrete sum over integers m in P p . Weuse the Poisson summation formula [8], P m ∈ Z δ ( x − m ) = P k ∈ Z e i πk x to elucidate these effects: k = 0 gives thecontinuum result for a 2D superconductor, and higher or-der terms, k ≥
1, account for the finite size corrections.After a straightforward calculation, we find a (2) ( l, q z , T, H ) ≃ M π " ln (cid:16) TT c (cid:17) + ψ (cid:16) (cid:17) − X s ε ,σ = ± Z π dϕ p π ψ (cid:16) − i s ε πT (cid:2) σh − v F · Q (cid:3)(cid:17) (5)+ 2 X k> Z π dϕ p π cos (cid:16) π k R p F cos( ϕ p ) (cid:17) X n> (cid:16) e − Rξ π (2 n +1) k | cos( ϕ p ) | n + − X s ε ,σ e − Rξ TTc π (2 n +1) k | cos( ϕ p ) | n + − i s ε πT [2 σh − v F · Q ] (cid:17) , where we defined the product v F · Q = 2 πT c (cid:16) ξ q z sin( ϕ p )+ h l − ΦΦ i ξ R cos( ϕ p ) (cid:17) . (6)Here, ψ ( z ) is the digamma function and the Fermi mo-mentum is p F = √ M µ . In the second line of Eq. (5) weneglected terms of order T c /µ . Because of the exponen-tial decay of the terms with increasing k > k = 1. We checked that incorporating higher k does not qualitatively change our results.Fig. 2 shows the upper critical field and the parameters l, q z of the modulation of the superconducting order pa-rameter at transition. Hereafter we consider ξ p F = 100,and R ≃ ξ . In the cylindrical geometry the inhomoge-neous superconducting states emerge even for the mate-rials that do not support FFLO modulation in the bulk:we present the results for the paramagnetic parameter α M = gµ B Φ / ( πξ T c ) = 0 .
6, which corresponds to the Pauli limiting field H P ≈ H orb c , so that the bulk ma-terial is a conventional orbital-limited type-II supercon-ductor. The scalloped shape of the boundary of the su-perconducting region is the consequence of the LP effect,and the overall suppression of T c with increased field isdue to the paramagnetic pairbreaking. Below a charac-teristic temperature, which is non-universal and differentfrom the T ⋆ = 0 . T c for bulk Pauli limited supercon-ductors, the inhomogeneous pairing along the cylinderaxis ( q z = 0) becomes advantageous, and the FFLO stateappears.Fig. 2 shows that the modulation wave vector, q z , alongthe transition line exhibits a “sawtooth” pattern, quitedistinct from the uniform increase in q in the standardpicture of Pauli-limited superconductors. This feature isdue to the effective discretization of the modulation inEq. (6). For a 2D sheet the role of the winding number l is taken by a continuous variable q x , and it is the net q = p q z + q x that ensures matching of the energy of thetwo electrons in a Cooper pair with the center of massmomentum q . In contrast, in the cylindrical geometry thechoice of l is determined by the net flux, Φ, and thereforethe momentum q z adjusts to this selection, and exhibitsdiscontinuities at the points where transitions betweenwinding numbers l and l + 1 occur. The detailed balancebetween q z and l depends on the finite size quantum cor-rection term, second line in Eq. (5). Note that the prefac-tor of the first non-vanishing k term, cos[2 πR p F cos( ϕ p )],has the same angle-dependence in the momentum spaceas the the LP term, | l − Φ / Φ | cos( ϕ p ) /R , and is outof phase with the FFLO modulation that enters withsin( ϕ p ) in Eq. (6). Consequently the details of the switch-ing between values of l and q z depend on the value of Rp F (close to integer vs. half-integer), but the qualitative pic- l q ( l ) FF LO z q z H
16 17181920 21222324 (a)(b) c2 H c2 T/ T c (0) FIG. 2: (Color online) Structure of the modulated state for
R/ξ = 3 and α = 0 .
6. (a) The upper critical fields H c nor-malized by Φ /πξ . Inset and dashed lines show H c ( T ) fora given angular momentum l pairing state as indicated. Solidsymbols denote the physical transition. Note the switchingbetween the FF (triangles) and LO (circles) states along thetransition line. (b) FFLO modulation wave vector q z nor-malized by 1 /ξ for each l (dotted lines), and at the physicaltransition as in panel (a). The circles (triangles) denote thestability regions of the LO (FF) state. Note the non-analyticbehavior of q z exhibiting kinks and discontinuous jumps attemperatures denoted by arrows. T c (H)/ T c (0) Δ C ( T c ( H ) ) / T c ( H ) T c (H)/ T c (0) Δ C ( T c ( H ) ) / T c ( H ) q z = z ≠ FIG. 3: (Color online) The specific heat jump at the onsetof the modulated SC order in field, ∆ C ( T c ( H )) /T c ( H ). Thenotations are the same as in Fig.2. Inset: the same over theentire temperature range of the superconducting transition.The non-monotonous behavior appears only for the FFLOstate, q z = 0. ture remains unchanged.The discontinuous behavior of the modulation q z ( T )is reflected in the experimental properties that allow un-ambiguous determination of the modulated state. Fig. 3shows the specific heat jump at the superconducting tran-sition for different fields. We verified that the quarticterm, a (4) ( l, q z , T c ( H ) , H ), remains positive along the en-tire transition line, and therefore the transition is alwaysof second order. The favored state is determined by com-paring the magnitude of the quartic term for the FF andthe LO states: A smaller value corresponds to the greatercondensation energy and a more stable phase. We findthat in the vicinity of the discontinuous drop of the mod-ulation q z the FF state is favored, and is superseded bythe LO state as q z increases within the realm of eachfixed winding number l . The specific heat jump at thetransition is given by (we omit full labels for brevity)∆ C/T c ( H ) = ([ a (2) ] ′ ) / a (4) evaluated at T c ( H ), where[ a (2) ] ′ = ( ∂a (2) /∂T ).The heat capacity exhibits signifi-cant enhancement on transitions between different wind-ing numbers. It is important to note that this non-monotonous behavior of the specific heat jump only ap-pears when the transition is into FFLO state, at low tem-peratures and high fields. At higher T , when the tran-sition is into the superconducting state with q z = 0, thespecific heat jump at the transition varies smoothly, seethe inset of Fig. 3. The enhancement of the specific heatjump in the hollow cylinder geometry can be detected,for example, by the ac calorimetry technique, and there-fore can serve as experimental proof of the existence ofthe FFLO-like modulations of the superconducting orderin mesoscopic cylinders.It is likely that in the experimental realization of theproposed geometry the superconductor will be disor-dered. We checked that the modulated states, and thenon-monotonic behavior of q z ( T ) are robust against mod-erate impurity scattering [26] The former result is consis-tent with the conclusions of Refs. 3, 5. FFLO modulationdisappears at strong disorder when transport becomesdiffusive [2].To conclude, we find novel spatially-inhomogeneoussuperconducting states, exhibiting both the Little-Parksand FFLO phenomenology, can emerge due to the vectorpotential and Zeeman coupling induced by a magneticfield threading a thin hollow cylinder. Our principal mo-tivation was conventional superconductors, for which thecoherence length can be well in excess of 1000˚A. In thissetting, the relevant sample sizes are experimentally ac-cessible and we believe that the predicted variations inthe specific heat jump can be found under realistic con-ditions, providing a possible “smoking-gun” experimentfor detecting the FFLO state. In principle, cold atomicgases in a cylindrical geometry and coupled to a light-induced artificial magnetic field could realize a similarphase diagram and FFLO state [31]. However, given thedifficulty of directly measuring the heat-capacity jump ina trapped cold atomic gas, it may currently be easier todetect the effects in small-size superconducting systems.This work is supported by NSF via Grant No. DMR-1105339 (K. A. and I. V.) and Grant No. DMR-1151717(D.E.S.). Portions of this research were conducted withhigh performance computing resources provided by theCenter for Computation and Technology at LSU andLouisiana Optical Network Initiative. [1] R. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer, New York, 1987).[2] S. M. Reimann and M. Manninen, Rev. Mod. Phys. ,1283 (2002).[3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[4] K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. ,237001 (2009).[5] W. A. Little and R. D. Parks, Phys. Rev. Lett. , 9 (1962).[6] Y. Liu, Yu. Zadorozhny, M. M. Rosario, B. Y. Rock, P. T.Carrigan, and H. Wang, Science , 2332 (2001).[7] E. N. Bogachek, G. A. Gogadze, and I. O. Kulik, Phys.Status Solidi B , 287 (1975).[8] T.-C. Wei and P. M. Goldbart, Phys. Rev. B , 224512(2008).[9] P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964). [10] A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP , 762 (1965).[11] A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso,and J. L. Sarrao, Phys. Rev. Lett. , 187004 (2003).[12] T. Watanabe, Y. Kasahara, K. Izawa, T. Sakakibara, Y.Matsuda, C. J. van der Beek, T. Hanaguri, H. Shishido, R.Settai, and Y. Onuki, Phys. Rev. B , 020506(R) (2004).[13] K. Kumagai, M. Saitoh, T. Oyaizu, Y. Furukawa, S.Takashima, M. Nohara, H. Takagi, and Y. Matsuda, Phys.Rev. Lett. , 227002 (2006); K. Kumagai, H. Shishido,T. Shibauchi, Y. Matsuda, Phys. Rev. Lett. , 137004(2011).[14] J. Singleton, J. A. Symington, M.-S. Nam, A. Ardavan,M. Kurmoo, and P. Day, J. Phys. Condens. Matter ,L641 (2000).[15] S. Uji, T. Terashima, M. Nishimura, Y. Takahide,T. Konoike, K. Enomoto, H. Cui, H. Kobayashi, A.Kobayashi, H. Tanaka, M. Tokumoto, E. S. Choi, T. Toku-moto, D. Graf, and J. S. Brooks, Phys. Rev. Lett. ,157001 (2006).[16] K. Cho, B. E. Smith, W. A. Coniglio, L. E. Winter, C. C.Agosta, and J. A. Schlueter, Phys. Rev. B , 220507(R)(2009).[17] S. Yonezawa, S. Kusaba, Y. Maeno, P. Auban-Senzier, C.Pasquier, K. Bechgaard, and D. Jerome, Phys. Rev. Lett. ,051005 (2007).[19] Under special assumptions about pairing interactions,multiple Fourier components of l may coexist at the tran-sition, see, for example, F. Loder, A. P. Kampf, and T.Kopp, arXiv:1212.2429 (unpublished).[20] H. Shimahara, J. Phys. Soc. Jpn.
736 (1998).[21] R. Combescot and C. Mora, Phys. Rev. B , 184521(2001).[23] H. Adachi and R. Ikeda, Phys. Rev. B , 184510 (2003).[24] N. Hiasa, T. Saiki, and R. Ikeda, Phys. Rev. B , 014501(2009).[25] L. Radzihovsky, Phys. Rev. A , 023611 (2011).[26] K. Aoyama et al., supplementary information.[27] A. A. Zyuzin and A. Yu. Zyuzin, Phys. Rev. B , 174514(2009).[28] L. G. Aslamazov, Zh. Eksp. Teor. Fiz. , 1477 (1968)[Sov. Phys. JETP , 773 (1969)].[29] D. F. Agterberg and K. Yang, J. Phys.: Condens. Matter , 9259 (2001).[30] A. B. Vorontsov, I. Vekhter, and M. J. Graf, Phys. Rev.B , 180505(R) (2008).[31] Y.-J. Lin, R.L. Compton, A.R. Perry, W.D. Phillips, J.V.Porto, and I.B. Spielman, Phys. Rev. Lett. , 130401(2009).[32] L. Radzihovsky and D.E. Sheehy, Rep. Prog. Phys. ,076501 (2010). SUPPLEMENTAL MATERIAL FOR ”INHOMOGENEOUS SUPERCONDUCTING STATES OFMESOSCOPIC THIN-WALLED CYLINDERS IN EXTERNAL MAGNETIC FIELDS”1. Quartic Term in the Ginzburg-Landau Expansion
The quartic term in the expansion of the free energy is given by a combination of four single-particle propagators a (4) ( l, q z , T, H ) | ∆ | = 12 Z d r ˆ K ( Π i )∆ ∗ ( s )∆( s )∆ ∗ ( s )∆( s ) (cid:12)(cid:12) s i → r , ˆ K ( Π i ) = T X ε n ,σ X p G ε n ,σ ( p ) G − ε n , − σ ( − p + Π † ) G − ε n , − σ ( − p + Π ) G ε n ,σ ( p + Π † − Π ) , (7)where Π i = − i ∇ s i + 2 | e | A acts on ∆( s i ) [1]. The summation over p is performed in the same manner as that usedin obtaining Eq. (5) in the text, and we find a (4) ( l, q z , T, H ) = 12 M π X q + q = q + q C ∗ l,q C l,q C ∗ l,q C l,q X k ∈ Z Z π d ϕ p π exp h i πkR p F cos( ϕ p ) i × Y j =1 Z ∞ dρ j πT cos (cid:0) h [ P i =1 ρ i ] (cid:1) sinh (cid:2) πT ([ P i =1 ρ i ] + Rξ | k cos( ϕ p ) | ) (cid:3) (8) × h cos (cid:0) v F · Q ( ρ + ρ ) − v F · Q ρ + v F · Q ( ρ + ρ ) (cid:1) + cos (cid:0) v F · Q ρ + v F · Q ρ + v F · Q ρ (cid:1)i , where Q i = (cid:0) R [ l − Φ / Φ ] , q i (cid:1) , the Fermi velocity is v F = 2 πT c ξ (cid:0) cos( ϕ p ) , sin( ϕ p ) (cid:1) , and we used the identity α − = R ∞ dρ exp[ − α ρ ] (Re α >
0) to exponentiate the operators. We evaluate Eq. (8) using the definitions of the FF andthe LO states, wherein C l,q i = δ q i ,q z (cid:0) ∆( r ) = | ∆ | e i lϕ e i q z z (cid:1) , (9) C l,q i = δ q i ,q z + δ q i , − q z √ (cid:0) ∆( r ) = | ∆ | e i lϕ √ q z z ) (cid:1) , (10)are used for the FF and LO states, respectively.
2. Impurity Effects on the FFLO State
It is well known that in a strongly disordered regime, when the single-particle propagation is diffusive, the FFLOstate does not survive [2]. However, in many realistic situations, even though disorder leads to a finite lifetimeof quasiparticles, the transport remain ballistic. We focus on this regime and explore how robust our conclusionsare against a finite concentration of non-magnetic impurities. Those are described by an additional term in theHamiltonian H imp = X σ X p , p ′ V ( p − p ′ ) ˆ c † p ,σ ˆ c p ′ ,σ , (11)where V ( q ) is the Fourier transform of the scattering potential of a random ensemble of impurities, V ( r ) = P i u ( r − R i ), located at positions R i with a net concentration n imp . We further assume that individual impurity potential isshort-ranged and isotropic, so that the s -wave scattering amplitude u is dominant, u ( q ) ≃ u , and assume a smallphase shift of scattering (Born limit). Within the Born approximation, computing the quadratic term in the Ginzburg-Landau expansion requires inclusion of the vertex corrections, and the result is obtained by replacing ˆ K ( ε n , σ ) withˆ K (˜ ε n , σ ) / [1 − | u | ˆ K (˜ ε n , σ )] in Eq. (4) in the main text in analogy to Ref. [3], where the renormalized frequency˜ ε n = ε n + sgn( ε n ) / τ , and the lifetime τ is12 τ = 2 πT c δ (cid:16) X k> Z π dϕ p π cos (cid:0) πR p F k cos( ϕ p ) (cid:1) exp h − π TT c Rξ (cid:12)(cid:12)(cid:12)(cid:16) n + 12 + δ T c T (cid:17) k cos( ϕ p ) (cid:12)(cid:12)(cid:12)i (cid:17) . (12)The dimensionless parameter δ = n imp M | u | / (4 πT c ) measures the strength of the impurity scattering. H c2 T / T c q z (b) q z q z T / T c H c2 H c2 (a) δ = 0 δ = 0.05 δ = 0.13 δ = 0.05 δ = 0.13 δ = 0 FIG. 4: Impurity effect on the modulated state for
R/ξ = 3 and α M = 0 .
6. (a) The upper critical field H c , and (b) the FFLOmodulation wave vector q z for δ = 0 (top), δ = 0 .
05 (middle), and δ = 0 .
13 (bottom). Black dots trace the FFLO stabilityregion (a) and the corresponding physical modulation (b). The notations and normalizations are the same as in Fig. 2 in thebody.
Figure 4 shows the upper critical field, H c ( T ), and the corresponding evolution of the modulation q z ( T ) for δ = 0 (clean limit), δ = 0 .
05 and δ = 0 .
13. At zero field, T c is not suppressed by nonmagnetic impurities asa manifestation of the Anderson’s theorem [4]. While the onset of the FFLO modulation is suppressed to lowertemperature by impurity scattering, the “sawtooth” behavior in the q z ( T ) curve survives. This indicates that even ina moderately disordered superconductor, the qualitative features of our main conclusions persist. In particular, thenon-monotonous evolution of the specific heat jump at the transition can still be observed in experiments, and serveas a strong evidence for the spatially inhomogeneous FFLO state. Note that, while we did not explicitly check at whatimpurity concentration the transition may become first order order, in known cases the nature of the FFLO-normaltransition is not altered by Born impurity scattering [3, 5]. The situation may be different in the strong scatteringlimit, but we leave this subject for future studies. Our analysis here established that the conclusions in the main textof the paper are robust against moderate impurity scattering. [1] H. Adachi and R. Ikeda, Phys. Rev. B , 184510 (2003); K. Aoyama and R. Ikeda, Phys. Rev. B , 1477 (1968) [Sov. Phys. JETP , 773 (1969)].[3] D. F. Agterberg and K. Yang, J. Phys.: Condens. Matter , 9259 (2001).[4] P. W. Anderson, J. Phys. Chem. Solids , 26 (1959).[5] A. B. Vorontsov, I. Vekhter, and M. J. Graf, Phys. Rev. B78