ductors
While the order-parameter for the Superfluid - Insulator transition (SIT) and the critical exponents governing it have been extensively studied, the local order- parameter distribution (OPD) is another important and very interesting feature of the SIT having received much less attention. On the theoretical side, the SIT of disordered superconducting films has been predicted to present glassy behaviour of the superconducting state close to the SIT [194, 195] by the cavity Mean-Field ap- proach. The emergence of a universal power-law decay of the probability distribution of the local order-parameter is hence brought about by the glassy physics. However, these results on the infinite-dimensionnal Cayley tree may not translate to finite di- mensions. On the experimental side, Scanning Tunneling Spectroscopy (STS) is a technique allowing for the measurement of the local density of states at atomic-scale resolutions [196, 197]. It has recently been used to scan the local order-parameter in homogeneously disordered supreconductors [28, 198–200] giving acces to its dis- tribution, but no quantitative connection between theory and experiment was found at that point, despite some efforts [198, 201].
Figure I.8: Panel a: Evolution of the typical local order-parameter Styp and of the
distribution width σS with increasing disorder (or decreasing Styp) obtained by 2D
cavity Mean-Field, 2D Mean-Field and cavity Mean-Field on the bosonic model and by two types of Bogolyubov-de Gennes theory for the fermionic model. For comparison, the experimental points of the three NbN films of figure I.9 are shown as blue left triangles. Panel b: Distribution of the rescaled variable RSof equation (I.20) obtained by the same methods. A perfect collapse of the data for all 2D models is found and the universal distribution is very well fitted by a Tracy-Widom distribution TW(−RS) (blue dashed line). Taken from Ref. [30].
A breakthrough came with the work of Lemarié et al. [30], who investigate theoretically the 2D SIT in both its fermionic and bosonic scenarios by prototype models and compare their results to STS mesarements on three disordered samples of NiN films. The fermionic model is studied using the Bogolyubov - de Gennes (BDG) Mean-Field theory [202–204] which does not describe the SIT but already captures several features of strongly disordered superconductors [203] The bosonic model is studied by the 2D cavity Mean-Field approach [205]. The local order-parameter for the fermionic model in BDG theory is connected to the local pairing amplitude ∆i
while in the 2D cavity Mean-Field of the bosonic model it is proportional to the local transverse fields Bi of the equivalent XY model [30, 205]. Let us call the normalized
local order parameter in both cases Si. The main result of their work is that in both
fermionic and bosonic models the numerically obtained distributions of Si (i.e. the
OPD) can be universally rescaled by considering the rescaled variable
RS =
ln S − ln Styp
σS
where S is the local order-parameter, Styp = exp(ln S) is the typical local order-
parameter with S denoting the average of S, and σ2
S = ln
2
S −ln S2 is the variance
of ln S which characterizes the width of the distribution of S. The distributions of this new variable collapse into a universal form describing both the fermionic and the bosonic 2D results as shown in panel b of figure I.8. The universal law is very well fitted by a Tracy-Widom distribution [206] with opposite asymmetry, i.e. TW(−RS) [30] as evidenced by the blue dashed curve. The fact that the simulations
were carried out for a low value of the pairing interaction in the fermionic model and that the fermionic data collapses onto the same behaviour as the bosonic one supports the bosonic scenario of the SIT for which Cooper pairs get localized at the SIT and not destroyed before that and electrons get localized afterwards.
Experimental STS measurements at 500 mK on three different different disordered NbN films of thickness ∼ 50nm with critical temperatures Tc ∼1.65, 2.9 and 6.5 K
are also performed. For these systems, the measured local order parameter is the average height of the coherence peaks at positive and negative bias over the normal state conductance, denoted by hi [30]. The corresponding normalized experimental
local order-parameter is
Siexp ≡ hi
Max[h], (I.21)
and the rescaled variable is defined as in equation (I.20).
Figure I.9: Panel a: Distribution of the normalized local order-parameter of three
disordered NbN films with critical temperatures Tc ∼1.65, 2.9 and 6.5 K. Panel b: The same data for the NbN films plotted in terms of the rescale variable RS (I.20). A very convincing collapse of the data onto the Tracy-Widom distribution with opposite asymmetry (orange curve) is found. Taken from Ref. [30].
Panel (a) of figure I.9 shows the distributions of the normalized local order- parameter P (S) for the three disordered NbN films. As expected, the weight of
S ≈ 0 increases with increasing disorder as Tc is smaller [199]. The distributions of
the rescaled variables P (RS) are plotted in panel (b) of figure I.9. They all collapse
into a single curve that is very well fitted by the same opposite-asymmetry Tracy- Widom distribution as the numerical data, as shown by the orange curve. This proves that such a rescaling of the (OPD) describes both the experimental and the theoretical data. The universal distribution has relevance in the insulating side of the transition as well, as discussed in Refs. [205, 207].
3
Objectives of this work
This thesis focuses on the Superfluid - Bose glass transition in two dimensions trying to answer some of the open questions raised in this introductory chapter. We will concentrate our efforts on the transition for hard-core bosons in 2D, i.e. the Bose-Hubbard model in the limit of diverging repulsive interactions U → ∞. Three different theoretical approaches will be used, namely a classical self-consistent Mean- Field approach, a semi-classical approach where the classical solution is used as a starting point to include quantum fluctuations in the form of non-interacting spin- waves and finally the use of extensive Quantum Monte Carlo simulations using the Stochastic Series expansion method. Each method has its degree of approximation and presents some advantages compared to one another, allowing for a comprehensive study of the properties of the Superfluid - Bose glass transition in 2D.
Mean-Field theory of the SF-BG
transition
Contents
1 Preliminary Studies . . . . 26
1.1 Heisenberg model with random exchange couplings . . . . 27