Single- and two-particle energy gaps across the disorder-driven superconductor–insulator transition

Journal name:
Nature Physics
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The competition between superconductivity and localization raises profound questions in condensed-matter physics. In spite of decades of research, the mechanism of the superconductor–insulator transition and the nature of the insulator are not understood. We use quantum Monte Carlo simulations that treat, on an equal footing, inhomogeneous amplitude variations and phase fluctuations, a major advance over previous theories. We gain new microscopic insights and make testable predictions for local spectroscopic probes. The energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the superconductor–insulator transition, despite a robust single-particle gap.

At a glance


  1. Energy and temperature scales across SIT.
    Figure 1: Energy and temperature scales across SIT.

    The superconducting Tc (blue dots) decreases to zero at the critical disorder strength V c. The single-particle gap ωdos (black diamonds), obtained from the DOS shown in Fig. 2, is large and finite in all states. The two-particle energy scale ωpair (red squares), obtained from the dynamical pair susceptibility shown in Fig. 3, is non-zero in the insulator but vanishes at the SIT. The dashed curves are guides to the eye; extracting critical exponents requires finite-size scaling beyond the scope of this paper. The statistical error bars in all the figures are dominated by disorder averaging and not from the QMC. These results are obtained at fixed attraction |U|=4 and average density left fencenright fence0.87on 10 disorder realizations on 8×8 lattices. ωpair and ωdos are calculated at the lowest accessible temperature, T=0.1. For specific parameter values, we have run extensive simulations that average over 100 disorder realizations.

  2. The single-particle DOS.
    Figure 2: The single-particle DOS.

    N(ω) (upper panels) and representative spectra (lower panels) along three different cuts through the temperature–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 the SIT (V c1.6), 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 <V c). The coherence peaks (red) visible in the SC state, vanish for TTc0.14. A disorder-induced pseudogap, with loss of low-energy spectral weight, persists well above Tc. e,f, T-dependence of N(ω) for the insulator (V >V c). The hard insulating gap at low Tevolves into a pseudogap at higher T. No coherence peaks are observed at any T. All panels show data averaged over 10–100 disorder realizations.

  3. Imaginary part of the dynamical pair susceptibility.
    Figure 3: Imaginary part of the dynamical pair susceptibility.

    P′′(ω)/ω at T=0.1t, averaged over 10 disorder realizations at three disorder strengths. Error bars represent variations between disorder realizations. For V <V c, there is a large peak at ω=0, indicating zero energy cost to insert a pair into the SC. For V >V c, there is a gap-like structure with an energy scale ωpair, the typical energy required to insert a pair into the insulator, which increases with V.

  4. Local density of states (LDOS)
N(R,[omega]), density
n(R), and BdG pairing amplitude
[Delta]op(R) as a function of disorder strength for a montage of nine disorder realizations of 8[times]8lattices.
    Figure 4: Local density of states (LDOS) N(R,ω), density n(R), and BdG pairing amplitude Δop(R) as a function of disorder strength for a montage of nine disorder realizations of 8×8lattices.

    ac, Correspond to V =0.1,1,2 respectively. The LDOS is plotted at three representative sites Ri. At moderate and 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 the characteristic asymmetries in the LDOS in the centre and right columns for R1 and R3. The small local pairing amplitude Δop(R) at these two sites is reflected 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 pairing amplitude, a much more symmetrical LDOS, and coherence peaks that persist even at strong disorder.

  5. Emergent granularity.
    Figure 5: Emergent granularity.

    a, Disorder realization V (R) on a 36×36 lattice at V =3t. b, Local pairing amplitude Δop(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 the scale 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 the smallest gaps occur on the SC islands defined by the largest pairing amplitude.


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  1. Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

    • Karim Bouadim,
    • Yen Lee Loh,
    • Mohit Randeria &
    • Nandini Trivedi


K.B. and Y.L.L. performed the numerical calculations; M.R. and N.T. were responsible for the project planning; all authors contributed to the data analysis, discussions and writing.

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