P(0) = 0
A seminal work by Ma, Halperin and Lee back in 1986 [9] showed that the Mean- Field theory breaks down for the Superfluid - Insulator transition since it results in a superfluid density that only vanishes for infinitely large disorder, when considering the box distribution, while experiments on superfluid Helium 4 adsorbed on porous media showed otherwise [7]. They claimed that the key ingredient for such a break- down is the fact that the disorder distribution has a non-zero weight at 0, meaning that there are always almost disorder-free sites which can be tilted in the XY plane with little energy cost, i.e. it is energetically favourable for coherence (or XY order) to be established at these sites which then act as nucleation centers for macroscopic coherence (long-range XY order). Our results for both the bimodal and the box distribution are in full agreement with such a scenario (cf. figures II.3 and II.4 sec- tions 2.1 and 2.2) since for the bimodal disorder, with P (0) = 0, the system is in an insulating state as soon as the disorder bound lies outside the band-width (W > 4t) and for the box disorder (with P (0) 6= 0) the superfluid density and the BEC density (i.e. the XY order parameter) only vanish for infinite disorder.
In order to check if the P (0) 6= 0 condition holds in other cases, we study the case of a uniform disorder distribution with zero weight around zero, i.e. a uniform distribution for values in [−W ; −α] and [α; W ]. This can be done with the same expressions as for the box distribution for the consistency equations (II.33) and the ground-state energy both with and without the twist (cf. section 2.2). The distribution is still symmetric with respect to 0, meaning that we are still studying the physics at half-filling. The results for the superfluid and BEC densities as a
10 W/t 10-5 10-4 10-3 10-2 10-1 100 α=0.5 10 100 W/t 10-4 10-3 10-2 10-1 100 sf L=8 sf L=12 sf L=16 sf L=20 sf L=24 sf L=28 sf L=32 sf L=48 bec L=8 bec L=12 bec L=16 bec L=20 bec L=24 bec L=28 bec L=32 bec L=48 α=0.1 S F an d BE C d en si ti es
W/t
W/t
↵ = 0.1 ⇢⇢sfsfL = 8L = 12 ↵ = 0.5 ⇢sfL = 16 ⇢sfL = 20 ⇢sfL = 24 ⇢sfL = 28 ⇢sfL = 32 ⇢sfL = 48 ⇢0L = 8 ⇢0L = 12 ⇢0L = 16 ⇢0L = 20 ⇢0L = 24 ⇢0L = 28 ⇢0L = 32 ⇢0L = 48Figure II.7: Evolution of the Bose-Einstein condensate and superfluid densities as a
function of disorder for the uniform disorder distribution in [−W ; −α]S[α; W ] for two different values of α = 0.1 and 0.5, and system sizes ranging from L = 8 to L= 48. Disorder averages were performed over 3000 for the smallest sizes and 700 samples for L= 48. For both values of α the behaviour is very similar to that of the box disorder α = 0: ρsf < ρ0 and both quantities drop with disorder strength W as
either a power-law or an exponential form. There is no critical disorder strength.
function of the disorder bound W for three values of α = 0.1, 0.5 are shown in figure II.7, for system sizes up to N = 48 × 48. Disorder averages were performed over 3000 samples for L = 8 down to 700 samples for L = 48 . For both values of
α the behavior is very similar to the one of the box disordered case with no hole.
Indeed, we find again that even at the Mean-Field level ρsf < ρ0. Once again, the
data is well fitted either by a power-law or an exponential form, indicating that the
XY order will only vanish in the limit of infinite disorder strength W → ∞ .
As a conclusion, the energetic argument by Ma et al. seems to be valid also for disorder distributions with P (0) = 0. In other words, it is still energetically favourable to tilt a site which has small local filed (or chemical potential) onto the XY plane and it being a nucleation center for long-range order if the disorder distribution allows for such small local fields. It would be very interesting, nonetheless, to carry out a more thorough study of the effect of the α parameter to see if this effect remains for all values of α inside the band-width. We can already predict that if α is larger than the the band-width the scenario will be identical to the bimodal disorder case studied in section 2.1 for which there is a disordered insulating phase in competition with the superfluid phase.
3
Alternative Mean-Field approach: The cavity
mean-field
We have presented in the previous sections of this chapter the results of two different fairly simple Mean-Field methods which do not capture the phase transi- tion from the Superfluid to the Bose glass although they do provide some interest- ing information concerning the transport properties. However, more sophisticated Mean-Field approaches exist. One example is the cavity Mean-Field approach. First developed to study classical frustrated spin systems, i.e. spin glasses [231], it is a Mean-Filed theory that solved the fully connected Sherrington-Kirkpatrick model of spin glasses [232] in a much more transparent way than the previously found Replica Symmetry Breaking ansatz in the replica approach by Parisi [233–235]. It uses a probabilistic approach based on three assumptions: ultrametricity [236], the exis- tence of many pure states [237, 238] and the exponential distribution of their free energies [239].
This method can be extended to the quantum cavity-method and be used in the context of quantum phase transitions in quantum disordered systems [194, 195]. In the following we will briefly present this method treated in detail in Ref. [240]. To make things more concrete, let us consider the random transverse-field ferromagnet described by the Hamiltonian on the Bethe lattice with connectivity z = K + 1, at an inverse temperature β: HRT F IM = − X i ξiσiz− X (ij) Jijσixσ x j (II.41)
where σx/z are the Pauli matrices, the second sum runs over all bonds (ij), and Jij is the couplings on bond ()ij. Using the Suzuki-Trotter representation with M
imaginary time steps we can describe the system by the classical time trajectory of each spin.
The resolution in the Replica Symmetric case for this problem is exactly the same as for the classical method with the only difference of treating spin trajectories instead of Ising spins [240, 241]. The natural order parameter for this problem is a distribution of distributions, a very complicated object, and this resolution is too numerically costly.
At this point, the necessity of developing an approximate version of this exact mapping becomes evident. One possible approximation, called the projected cavity mapping, can be thought of as studying the properties of a spin 0 in the Bethe lattice where one of its neighbours has been deleted from the graph. Under a few assump- tions, the local Hamiltonian describing the system of spin 0 and its K neighbours
is [240]: H0 = −ξ0σ0z− K X i=1 (ξiσiz+ Biσxi + J0iσ0xσxi) . (II.42)
where Bi is a variable that parametrizes the problem. This 2K+1×2K+1 Hamiltonian
can be diagonalized so as to compute the magnetisation of spin 0. This projected cavity mapping uses a single number Bi to describe the trajectory distribution and
gives a self-consistent equation for the distribution of the Bi.
However, this resolution is also very costly numerically for it involves diagonaliz- ing a 2K+1×2K+1matrix at each iteration step. One further approximation allowing
to obtain an explicit mapping [194,195] similar to that of the classical problem [231], is a Mean-Field approximation to compute the magnetisation m0 from the cavity
Hamiltonian (II.42). This is the cavity mean-field approximation consisting in writ- ing the cavity Hamiltonian acting on spin 0 by
H0cav−M F = −ξ0σz0− σ x 0 K X i=1 J0ihσxii. (II.43)
And a recursion relation between the Bi field follows: B0 = K X i=1 J0i Bi q ξ2 i + Bi2 tanhβqξ2 i + Bi2 (II.44) Finally, a self-consistent equation for the distribution of the B fields P (B) is obtained. By solving it one has access to the order parameter for the quantum phase transition from a ferromagnetic to a paramagnetic phase [240] since in the paramagnetic phase
P(B) = δ(B) and the distribution is non trivial in the ferromagnetic phase.
It is capital to notice that the cavity mean-field approach does not neglect quantum fluctuations. Therefore, an adapted version of this method to the two- dimensional case [205] of the square lattice with the same Hamiltonian enabled Lemarié et al. to study a Superfluid - Insulator transition and find a transition even at the Mean-Field level [30], as discussed in section 2.5 of chapter I.
To conclude, the cavity Mean-Field is a much more sophisticated and power- ful method than the Mean-Field approaches used in this chapter. However, since quantum fluctuations are already present in its framework, including them by linear spin-wave theory (LSWT) is unnecessary. On the contrary, the simpler Mean-Field approaches previously developed in this chapter, which neglect quantum fluctuations, are perfectly suited for the LSWT formalism as will be shown in the next chapter.
Beyond Mean-Field: Semi-classical
approach
Contents
1 Linear spin-wave theory in real space . . . . 50
1.1 Linear Spin-Wave Theory for Hard-core bosons . . . 51