ARTICLESPUBLISHED ONLINE: 31 JULY 2011 | DOI: 10.1038/NPHYS2037
Single- and two-particle energy gaps across thedisorder-driven superconductor–insulator transitionKarim Bouadim, Yen Lee Loh, Mohit Randeria and Nandini Trivedi*The competition between superconductivity and localization raises profound questions in condensed-matter physics. In spiteof decades of research, the mechanism of the superconductor–insulator transition and the nature of the insulator are notunderstood. We use quantum Monte Carlo simulations that treat, on an equal footing, inhomogeneous amplitude variations andphase fluctuations, a major advance over previous theories. We gain new microscopic insights and make testable predictionsfor local spectroscopic probes. The energy gap in the density of states survives across the transition, but coherence peaks existonly in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrastto conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the superconductor–insulatortransition, despite a robust single-particle gap.
Attractive interactions between electrons lead to super-conductivity, a spectacular example of long-range order inphysics, whereas disorder leads to localization of electronic
states. One of the most fascinating examples of the interplaybetween the effects of interactions and localization is the destructionof superconductivity in thin films with increasing disorder and theresulting superconductor–insulator transition (SIT; refs 1–9).
It was recognized decades ago that s-wave superconductivity(SC) is remarkably robust against weak disorder10,11. It was laterargued12 that SC can survive even when disorder localizes thesingle-particle states. Thus the SIT must occur in a strong disorderregime that is difficult to treat theoretically in an interacting system.Critical phenomena at the SIT have been described in terms ofdisordered bosons13, which model fermion pairs and describephase fluctuations of the SC order parameter. A more microscopicdescription must necessarily start with the fermionic degrees offreedom. A Bogoliubov–de Gennes (BdG) treatment of attractiveelectrons in a randompotential shows that the SCpairing amplitudebecomes spatially inhomogeneous with strong disorder14–16. Thisleads to a robust energy gap and a large suppression of the superfluiddensity14,15. However, the phase fluctuations ultimately responsiblefor the SIT are beyond the BdG approach and are treated in anapproximate manner14,15,17.
In this paper we make a major advance using quantum MonteCarlo (QMC) simulations on a fermionic model that includesthermal and quantum fluctuations of the SC phase and thespatially inhomogeneous amplitude on an equal footing. As wellas confirming the bosonic mechanism for the SIT, our work alsogives new insights into the experimentally observable electronicspectral functions.Our results provide uswith a detailed descriptionof the phases, the transition, and the quantum critical region atfinite temperature.
Our main results are as follows:(1) Single-particle gap: at T = 0 the gap in the single-particle
density of states (DOS) survives through the SIT, so that one goesfrom a gapped superconductor to a gapped insulator. Although thelocal gap extracted from the local density of states (LDOS) is highlyinhomogeneous, it is nevertheless finite at every site.
(2) Coherence peaks: These characteristic pile-ups in the DOSat the gap edges are directly correlated with superconducting order
Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA. *e-mail: [email protected].
and vanish as the temperature is raised above Tc, or as the disorderis increased across the SIT.
(3) Pseudogap: Near the SIT, a pseudogap—a suppression inthe low-energy DOS—persists well above the superconducting Tcup to a crossover temperature scale T ∗, in marked deviation fromBCS theory. This disorder-driven pseudogap also exists at finitetemperatures in the insulating state and growswith disorder.
(4) Two-particle gap: There is a characteristic energy scale ωpairto insert a pair in the insulator that collapses on approaching theSIT from the insulating side. In addition, the two-particle spectralfunction may also have a very small spectral weight at low energiescoming from rare regions.
Our predictions for the local tunnelling DOS and the dynamicalpair susceptibility as a function of temperature and disorder havethe potential to guide future experiments using scanning tunnellingspectroscopy18–21 and other dynamical probes22.
Model and methods. To model the competition between super-conductivity and localization that leads to the SIT in quench-condensed films with thicknesses less than the coherence length, wetake the simplest lattice Hamiltonian that has an s-wave supercon-ducting ground state in the absence of disorder (V =0) and exhibitsAnderson localization when the attractive interaction is turned off(U = 0). Thus, we study the two-dimensional attractive Hubbardmodel in a random potential:
H = −t∑〈RR′〉σ
(c †Rσ cR′σ + c
†R′σ cRσ )
−
∑Rσ
(µ−VR)nRσ −|U |∑R
nR↑nR↓ (1)
with lattice sites R and R′, spin indices σ = ↑ or ↓, fermioncreation and annihilation operators c †
Rσ and cRσ , number operatorsnRσ = c †
Rσ cRσ , hopping t between neighbouring sites 〈RR′〉, and achemical potentialµ chosen such that the average density is 〈n〉 6=1.VR is a random potential at each site drawn from the uniformdistribution on [−V ,+V ], and |U | is the on-site attraction leadingto s-wave SC.Wewill measure all energies in units of t .
We use the determinantal QMC method23, which is free of thefermion sign problem for the Hamiltonian of equation (1). We
Figure 1 | Energy and temperature scales across SIT. The superconductingTc (blue dots) decreases to zero at the critical disorder strength Vc. Thesingle-particle gap ωdos (black diamonds), obtained from the DOS shown inFig. 2, is large and finite in all states. The two-particle energy scale ωpair
(red squares), obtained from the dynamical pair susceptibility shown inFig. 3, is non-zero in the insulator but vanishes at the SIT. The dashedcurves are guides to the eye; extracting critical exponents requiresfinite-size scaling beyond the scope of this paper. The statistical error barsin all the figures are dominated by disorder averaging and not from theQMC. These results are obtained at fixed attraction |U| =4 and averagedensity 〈n〉≈0.87 on 10 disorder realizations on 8×8 lattices. ωpair andωdos are calculated at the lowest accessible temperature, T=0.1. Forspecific parameter values, we have run extensive simulations that averageover 100 disorder realizations.
choose |U | = 4, so that the coherence length is within the systemsize, and 〈n〉 = 0.875. We have made extensive comparisons of theQMC results with self-consistent BdG calculations, which take intoaccount only the spatial amplitude variations; see SupplementaryInformation. These comparisons permit us to separate the effects ofamplitude inhomogeneity and phase fluctuations.
We compute frequency-dependent observables across the SITfor the first time. The single-particle DOS, LDOS and the pairsusceptibility are obtained using the maximum entropy method(MEM) for analytic continuation24,25. We have verified that theseresults obey various sum rules to high precision, and that theMEM correctly reproduces the low-energy structure of test spectraas shown in the Supplementary Information. What gives usconfidence is that our central results on the single- and two-particle gaps can be equally well estimated directly from theexponential decay of the imaginary-time QMC data, withoutrecourse to the MEM.
Phase diagram. In Fig. 1 we summarize our key results for thedisorder dependence of various temperature and energy scales. Asthe finite temperature transition is expected to be in the Berezinskii–Kosterlitz–Thouless universality class, we estimate the criticaltemperature Tc from the superfluid density ρs, calculated from thetransverse current correlator26,27. We note that this procedure onfinite systems provides an upper bound on the actual Tc in thethermodynamic limit. As disorder strength V increases, Tc falls andfinally vanishes at the critical disorder Vc, which defines the SIT.The single-particle energy gapωdos remains non-zero across the SIT,whereas the two-particle energy scale ωpair is finite in the insulatorand goes to zero at the transition. These gap scales are extractedfrom the DOS and the dynamical pair susceptibility discussed indetail below. Figure 1 can be interpreted as a phase diagram:Tc is thesuperconducting transition temperature, ωpair is a crossover scalebetween the insulator and the quantum critical region, and ωdos isrelated to the pseudogap crossover scale described below.
0.5
1.0
1.5
2.0
2.5
¬2 ¬1 0 1 2
¬2 ¬1 0 1 2
0.5
1.0
1.5
2.0
2.5
¬2 ¬1 0 1 2¬2 ¬1 0 1 2
0.2
0.3
0.4
05
0.6
0
0.1
0.2
0.3
¬2 ¬1 0 1 20
0.1
0.2
0.3
¬2 ¬1 0 1 20
0.1
0.2
0.3
V T
Vc
Tc
T = 0.1 V = 0.5 T V = 3
ωω
ω ω
ω
ω
T = 0.1 V = 0.1V = 0.5V = 1V = 2V = 3
V = 0.5 T = 0.1T = 0.125T = 0.2T = 0.25T = 0.5
V = 3 T = 0.1T = 0.25T = 0.5T = 1.5T = 2.5
0
0.35 0
0.28
0
0.11
a c e
b d fN(ω)ω N(ω)ω N(ω)ω
Figure 2 | The single-particle DOS. N(ω) (upper panels) and representative spectra (lower panels) along three different cuts through thetemperature–disorder plane. a,b, Disorder dependence of N(ω) at a fixed low temperature. A hard gap (black region) persists for all V above and below theSIT (Vc≈ 1.6), but the coherence peaks (red) exist only in the SC state and not in the insulator. c,d, T-dependence of the N(ω) for the superconductor(V<Vc). The coherence peaks (red) visible in the SC state, vanish for T ∼> Tc≈0.14. A disorder-induced pseudogap, with loss of low-energy spectralweight, persists well above Tc. e,f, T-dependence of N(ω) for the insulator (V>Vc). The hard insulating gap at low T evolves into a pseudogap at higher T.No coherence peaks are observed at any T. All panels show data averaged over 10–100 disorder realizations.
Single-particle spectra. We show in Fig. 2 the disorder andtemperature dependence of the DOS N (ω). Figure 2a,b shows theevolution with disorder at a very low temperature T = 0.1. The gapωdos clearly remains finite in both superconducting and insulatingstates, a counterintuitive observation that agrees qualitatively withBdG results14,15. In contrast, the coherence peaks diminish withincreasingV and disappear near the SIT atVc≈1.6.
Figure 2c,d shows the temperature evolution of N (ω) at weakdisorder V <Vc. Unlike in BCS theory, the hard SC gap does notclose with increasing T . Instead, the coherence peaks graduallydisappear as the temperature increases across Tc. Above Tc, the gapgradually fills up, with a pseudogap persistingwell aboveTc.
The temperature evolution of N (ω) at strong disorder V > Vcis shown in Fig. 2e,f. Here the ground state is an insulator with ahard gap and little evidence for coherence peaks, and the pseudogappersists up to an even higher temperature.
Two-particle spectra. Given that we find an insulator with asingle-particle gap, what is the energy scale that vanishes onapproaching the quantum critical point from the insulating side?We propose that it is the typical energy for a two-particleexcitation in the insulator. To access this scale, we examine thepair susceptibility P(ω) obtained by analytical continuation ofthe correlation function P(τ ) =
∑R〈TτF(R;τ )F †(R;0)〉, where
F(R,τ )= cR↓(τ )cR↑(τ ). ThusP(τ ) is the amplitude for a pair createdat a site R at τ = 0 to be found at the same site at a later timeτ . We find that in the insulating phase P(τ ) decays exponentially,which allows us to define ωpair, the characteristic energy scale fortwo-particle excitations.
In Fig. 3 we show the imaginary part of the pair susceptibilityP ′′(ω)/ω for three disorder strengths. At weak disorder P ′′(ω)/ω isvery large at lowω, whereas at strong disorder it has a clear two-peakstructure with a characteristic energy scale ωpair. This dominantscale represents the typical energy required to insert a pair into thesystem. We find that ωpair collapses to zero at the SIT because thereis no cost for inserting a pair into a condensate.
At sufficiently small energies our insulating state is similarto a Bose glass, in which rare regions28 give rise to a verysmall but non-zero spectral weight in P ′′(ω)/ω at low energies.Such Griffiths–McCoy–Wu singularities can be very difficult topin down in numerical simulations and even experimentally.Nevertheless, we do indeed see some signs of low-energy spectralweight in, for example, Fig. 3b. In this paper, however, we focuson the most salient features in P ′′(ω)/ω. These are the peaksat ±ωpair, which imply that the typical energy cost to insert apair is finite.
Local probes. In Fig. 4 we track the behaviour of various localquantities with increasing disorder strength V . We show the LDOSN (R,ω) at representative points, maps of the spatial variation ofthe density n(R), and the BdG pairing amplitude1op(R)=〈cR↓cR↑〉(which cannot be computed in QMC). We see that the systembecomes increasingly inhomogeneous with increasing disorder aswe move from left to right in Fig. 4. The SIT occurs owing to lossof phase coherence between superconducting islands, seen as bluepatches in themap of the pairing amplitude1op(R).
We predict experimentally measurable signatures of the localdensity and pairing amplitude in the LDOS N (R,ω). Let us focuson three representative sites R1, R2, and R3. At moderate andstrong disorder, R1 is located on a potential hill, with a low densityn(R1)≈ 0 and a negligible pairing amplitude1op(R1)≈ 0. Thus theLDOS atR1 is highly asymmetric, withmost of the spectral weight atω> 0, for adding an electron. In contrast, R3 is in a potential well,with a high density n(R3)≈ 2 and a negligible pairing amplitude1op(R3)≈ 0. Thus R3 also has a highly asymmetric LDOS, but withmost of the spectral weight at ω < 0, for removing an electron.
1
¬1 0 1
2
ω
P''(
)/ω
ωP'
'()/
ωω
P''(
)/ω
ω
V = 3.2
2 pairω
2 pairω
1
1
2
3
4
5
¬1 0 1
2
ω
¬1 0 1ω
V = 1.8
V = 0.1
a
b
c
Figure 3 | Imaginary part of the dynamical pair susceptibility. P′′(ω)/ω atT=0.1t, averaged over 10 disorder realizations at three disorder strengths.Error bars represent variations between disorder realizations. For V<Vc,there is a large peak at ω=0, indicating zero energy cost to insert a pairinto the SC. For V>Vc, there is a gap-like structure with an energy scaleωpair, the typical energy required to insert a pair into the insulator, whichincreases with V.
We believe that MEM correctly captures the gap, coherence peaks,and integrated spectral asymmetry (tested by sum rules); it is muchless reliable for high-energy spectral features, which are in any caseirrelevant for our purposes.
Finally, R2 lies in a superconducting island close to half-filling,n(R2) ≈ 1, which permits particle–hole mixing, and therefore alarge pairing amplitude 1op(R2). The LDOS at R2 is much moresymmetrical, with large coherence peaks that persist across the SITand even in the insulating state. Note that all the LDOS curveshave a clear gap. We thus find that symmetrical coherence peaksin the LDOS, and not the local energy gap, are a clear experimentalsignature of a local pairing amplitude, which is difficult to probe byother means.
DiscussionWe now discuss our results in light of existing theories. Wehave ignored the renormalization of the effective interaction
Figure 4 | Local density of states (LDOS) N(R,ω), density n(R), and BdG pairing amplitude 1op(R) as a function of disorder strength for a montage ofnine disorder realizations of 8×8 lattices. a–c, Correspond to V=0.1,1,2 respectively. The LDOS is plotted at three representative sites Ri. At moderateand strong disorder, site R1 is on a high potential hill that is nearly empty, and R3 is in a deep valley that is almost doubly occupied. This leads to thecharacteristic asymmetries in the LDOS in the centre and right columns for R1 and R3. The small local pairing amplitude1op(R) at these two sites isreflected in the absence of coherence peaks in the LDOS. In contrast, site R2 has a density closer to half-filling, leading to a significant local pairingamplitude, a much more symmetrical LDOS, and coherence peaks that persist even at strong disorder.
between electrons arising from changes in screening with increasingdisorder29. Our point of view is that electronic inhomogeneity(which we focus on) is much more important in the vicinity ofthe SIT than the disorder dependence of the effective |U | (whichwe neglect), so long as the latter is not driven to zero. Thisassumption is validated by experiments that have found a non-zerogap across the SIT (ref. 18).
Our results are consistent with the absence of a fermionicor bosonic metal phase in between the superconductor and theinsulator. Although we have not computed transport here (seeref. 27 for an approximate calculation of the resistivity in the
same model), we do not find any extended low-energy excitationscharacteristic of a metallic phase.
The existence of gapped fermions implies a phase-fluctuation-dominated ‘bosonic’ picture for the SIT13,28. However, we mustemphasize that we did not assume such a bosonic picture fromthe outset. A nontrivial aspect of our results is that even thoughwe started with a model of interacting fermions in a randompotential and could have, in principle, obtained (localized) gaplessfermions in the insulator, we did not find such excitations. Thereason all fermionic excitations are gapped is intimately relatedto the structure of the inhomogeneous local pairing amplitude
Figure 5 | Emergent granularity. a, Disorder realization V(R) on a 36×36 lattice at V= 3t. b, Local pairing amplitude1op(R) from a BdG calculation at|U| = 1.5t, T=0, and n=0.875. Note the emergent ‘granular’ structure where the pairing amplitude ‘self-organizes’ into superconducting islands on thescale of the coherence length, even though the ‘homogeneous’ disorder potential in a varies on the scale of a lattice spacing. c, Local energy gap ωdos(R)from BdG, defined as the smallest energy at which the local DOS is non-zero (N(R,ω)>0.004). Note that this gap is finite everywhere and that thesmallest gaps occur on the SC islands defined by the largest pairing amplitude.
1op(R)=〈cR↓cR↑〉 generated in the presence of large disorder, as wenow explain.
We show in Fig. 5 that even for ‘homogeneous’ disorder, thatis, an uncorrelated random potential V (R) (Fig. 5a), the pairingamplitude1op(R) exhibits an emergent ‘granular’ structure (shownin Fig. 5b). The system self-organizes into superconducting islands,on the scale of the coherence length, with finite1op(R), interspersedwith insulating regions where 1op(R) is negligible. The spatialvariations of spectral features (asymmetry and coherence peaks)in this inhomogeneous state were already discussed above inconnection with Fig. 4.
The close connection between inhomogeneity and energy gapsis made clear in Fig. 5b,c, which demonstrates two striking facts.We see that (1) there is an energy gap in the LDOS at every site,and (2) small gaps ωdos(R) in the LDOS are spatially correlated withlarge1op(R) SC islands.
A simple way to understand these results is to use the pairing-of-exact-eigenstates approach generalized to highly disorderedsystems15. In the limit ofweak attraction, pairing leads to a gap in thelow-energy DOS in the underlying Anderson insulator and leads tothe islands with non-zero1op and a small energy gap. On the otherhand, the insulating sea corresponds to the higher-energy stronglylocalized states in the system.
From this perspective one can see that the gap ωdos, observedin the spatially average DOS, initially decreases with increasingdisorder owing to a reduction in the DOS near the chemicalpotential in our model. (In a real material, the coupling willalso decrease29 with disorder.) However, at high disorder, thegap grows (consistent with Fig. 1) like ωdos ≈ |U |/(2ξ 2loc), whereξloc is the single-particle localization length15. This is due to theenhanced effective attraction between fermions confined to asmaller localization volume ξ 2loc.
The phase stiffness (or superfluid density) ρs(T = 0), on theother hand, decreases monotonically with disorder as the SCislands become smaller and the Josephson coupling between islandsbecomes weaker. Thus, even if one starts with a weak-couplingBCS superconductor with ωdos � ρs, disorder will necessarilydrive it into the ωdos � ρs regime. Eventually, quantum phasefluctuations destroy long-range order at T = 0, leading to aninsulator with low-energy excitations that are pairs localized onSC islands.
The low-ρs regime on the SC side of the SIT leads to a finite-temperature transition driven by thermal phase fluctuations30 withTc ∼ ρs(0). The large energy gap then leads to a marked deviationfrom conventional BCS theory, with a pairing pseudogap in thethe temperature range Tc ∼
< T ∼<ωdos. This pseudogap exists evenin the weak-coupling regime, provided one is close enough to theSIT so that ρs�ωdos.
Comparison with experiments. We describe the connectionbetween our predictions and experiments on the disorder-tunedSIT in systems such as indium oxide, titanium nitride, and niobiumnitride films, forwhich our theory seems to be themost appropriate.First, let us discuss the insulating side of the SIT. The existence ofa gap in the insulator implies activated transport, consistent withearlymeasurements on amorphous InOx films5. Furthermore, thereis evidence for pairs on the insulating side of the transition8 inspecially patterned amorphous bismuth films.
Recent scanning tunnelling microscpy (STM) experiments aredirectly relevant to our predictions on the superconducting sideof the SIT. Experiments on homogeneously disordered TiN films18have shown that, whereas Tc goes to zero at the SIT, the STMgap ωdos remains finite, in agreement with Fig. 1. Furthermore, thegap in the LDOS shows marked inhomogeneity, which supportsour picture of emergent granularity (see Figs 4 and 5). After ourpaper was written, we became aware of new experiments thatcorroborate our predictions. STM experiments on InOx (ref. 31),TiN (ref. 32), and NbN films33 have all found a pseudogappersisting up to many times Tc. In particular, they observe amarked suppression of the low-energy DOS together with adestruction of coherence peaks above Tc, in complete agreementwith our predictions.
We hope that future STM experiments will study in detailthe anticorrelation that we predict between the height of thecoherence peaks (associated with large pairing amplitude) and thesmall energy gaps in the local DOS. The obvious quantum criticalscaling between Tc and ρs(0) at the SIT, well studied in ratherdifferent systems34, also remains to be tested experimentally ins-wave superconducting films.
ConclusionIn conclusion, we have obtained detailed insights and predictionsfor observable properties of the highly disordered superconductingand insulating states in 2D films, and of the transition betweenthese states. Although we focused on s-wave SC films, it hasnot escaped our attention that aspects of our results bear astriking resemblance to the completely different—and much lessunderstood—problem of the pseudogap in the d-wave high-Tcsuperconductors. Features such as the loss of low-energy spectralweight persisting across thermal or quantum phase transitions,even as coherence peaks are destroyed, may well be common toall systems where the small superfluid stiffness drives the loss ofphase coherence. The pseudogap in underdoped cuprates is drivenby the proximity to the Mott insulator and further complicatedby competing order parameters, with disorder probably playing asecondary role, unlike the disorder-induced pseudogap near the SITdiscussed in this paper.
NATURE PHYSICS DOI: 10.1038/NPHYS2037 ARTICLESReceived 2 December 2010; accepted 1 June 2011;published online 31 July 2011
References1. Goldman, A. & Markovic, N. Superconductor–insulator transitions in the
two-dimensional limit. Phys. Today 51, 39–44 (November, 1998).2. Gantmakher, V. F. & Dolgopolov, V. T. Superconductor–insulator quantum
phase transition. Phys. Usp. 53, 3–53 (2010).3. Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the
two-dimensional limit. Phys. Rev. Lett. 62, 2180–2183 (1989).4. Hebard, A. F. & Paalanen, M. A. Magnetic-field-tuned
superconductor–insulator transition in two-dimensional films. Phys.Rev. Lett. 65, 927–930 (1990).
5. Shahar, D. & Ovadyahu, Z. Superconductivity near the mobility edge.Phys. Rev. B 46, 10917–10922 (1992).
6. Adams, P. W. Field-induced spin mixing in ultra-thin superconducting Al andBe films in high parallel magnetic fields. Phys. Rev. Lett. 92, 067003 (2004).
7. Steiner, M. A., Boebinger, G. & Kapitulnik, A. Possible field-tunedsuperconductor–insulator transition in high-Tc superconductors: Implicationsfor pairing at high magnetic fields. Phys. Rev. Lett. 94, 107008 (2005).
8. Stewart, M. D., Yin, A., Xu, J. M. & Valles, J. M. Superconducting paircorrelations in an amorphous insulating nanohoneycomb film. Science 318,1273–1275 (2007).
9. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 1999).10. Anderson, P. W. Theory of dirty superconductors. J. Phys. Chem. Solids 11,
26–30 (1959).11. Abrikosov, A. A. &Gor’kov, L. P. Superconducting alloys at finite temperatures.
12. Ma, M. & Lee, P. A. Localized superconductors. Phys. Rev. B 32,5658–5667 (1985).
13. Fisher, M. P. A., Grinstein, G. & Girvin, S. M. Presence of quantum diffusionin two dimensions: Universal resistance at the superconductor–insulatortransition. Phys. Rev. Lett. 64, 587–590 (1990).
14. Ghosal, A., Randeria, M. & Trivedi, N. Role of spatial amplitudefluctuations in highly disordered s-wave superconductors. Phys. Rev.Lett. 81, 3940–3943 (1998).
15. Ghosal, A., Randeria, M. & Trivedi, N. Inhomogeneous pairing in highlydisordered s-wave superconductors. Phys. Rev. B 65, 014501 (2001).
16. Feigel’man, M. V., Ioffe, L. B., Kravtsov, V. E. & Yuzbashyan, E. A.Eigenfunction fractality andpseudogap state near the superconductor–insulatortransition. Phys. Rev. Lett. 98, 027001 (2007).
17. Dubi, Y., Meir, Y. & Avishai, Y. Nature of the superconductor–insulatortransition in disordered superconductors. Nature 449, 876–880 (2007).
18. Sacépé, B., Chapelier, C., Baturina, T. I., Vinokur, V. M., Baklanov, M. R. &Sanquer, M. Disorder-induced inhomogeneities of the superconductingstate close to the superconductor–insulator transition. Phys. Rev. Lett. 101,157006 (2008).
19. Cren, T., Roditchev, D., Sacks, W., Klein, J., Moussy, J-B., Deville-Cavellin,C. & Laguës, M. Influence of disorder on the local density of states in high-Tc
superconducting thin films. Phys. Rev. Lett. 84, 147–150 (2000).20. Lang, K. M. et al. Imaging the granular structure of high-Tc superconductivity
in underdoped Bi2Sr2CaCu2O8+δ . Nature 415, 412–416 (2002).
21. Gomes, K., Pasupathy, A. N., Pushp, A., Ono, S., Ando, Y. & Yazdani, A.Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ . Nature 447, 569–572 (2007).
22. Crane, R. W., Armitage, N. P., Johansson, A., Sambandamurthy, G., Shahar, D.& Grüner, G. Fluctuations, dissipation, and nonuniversal superfluid jumps intwo-dimensional superconductors. Phys. Rev. B 75, 094506 (2007).
23. Blankenbecler, R., Scalapino, D. J. & Sugar, R. L. Monte Carlo calculations ofcoupled boson-fermion systems. I. Phys. Rev. D 24, 2278–2286 (1981).
24. Gubernatis, J. E., Jarrell, M., Silver, R. N. & Sivia, D. S. Quantum Monte Carlosimulations and maximum entropy: Dynamics from imaginary-time data.Phys. Rev. B 44, 6011–6029 (1991).
25. Sandvik, A. W. Stochastic method for analytic continuation of quantumMonteCarlo data. Phys. Rev. B 57, 10287–10290 (1998).
26. Scalapino, D. J., White, S. R. & Zhang, S. Insulator, metal, or superconductor:The criteria. Phys. Rev. B 47, 7995–8007 (1993).
27. Trivedi, N., Scalettar, R. T. & Randeria, M. Superconductor–insulatortransition in a disordered electronic system. Phys. Rev. B 54,R3756–R3759 (1996).
28. Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Bosonlocalization and the superfluid-insulator transition. Phys. Rev. B 40,546–570 (1989).
29. Finkel’stein, A. M. Suppression of superconductivity in homogenouslydisordered systems. Physica B 197, 636–648 (1994).
30. Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations insuperconductors with small superfluid density. Nature 374, 434–437 (1995).
31. Sacépé, B. et al. Localization of preformed Cooper pairs in disorderedsuperconductors. Nature Phys. 7, 239–244 (2011).
32. Sacépé, B. et al. Pseudogap in a thin film of a conventional superconductor.Nature Commun. 1, 140 (2010).
33. Mondal, M. et al. Phase fluctuations in a strongly disordered s-wave NbNsuperconductor close to the metal–insulator transition. Phys. Rev. Lett. 106,047001 (2011).
34. Hetel, I., Lemberger, T. R. & Randeria, M. Quantum critical behaviour inthe superfluid density of strongly underdoped ultrathin copper oxide films.Nature Phys. 3, 700–702 (2007).
AcknowledgementsWe gratefully acknowledge support from NSF DMR-0907275 (K.B.), US Department ofEnergy, Office of Basic Energy Sciences grant DOE DE-FG02-07ER46423 (N.T., Y.L.L.),NSF DMR-0706203 and NSF DMR-1006532 (M.R.), and computational support fromthe Ohio Supercomputing Center.
Author contributionsK.B. and Y.L.L. performed the numerical calculations; M.R. and N.T. were responsiblefor the project planning; all authors contributed to the data analysis, discussions andwriting.
Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturephysics. Reprints and permissionsinformation is available online at http://www.nature.com/reprints. Correspondence andrequests for materials should be addressed to N.T.
