In this chapter, we have presented a detailed calculation of the superfluid suscep- tibility at the critical point of the one-dimensional disordered rotor model, within the strong-disorder renormalization framework. Our calculation implies that the suscepti- bility diverges with system size asχ∼L2−ηsd(yi), where the strong-disorder anomalous dimensionηsd(yi) depends upon the bare disorder strength. This prediction contrasts sharply with the situation at the weak-disorder transition of Giamarchi and Schulz, where the anomalous dimension is always the universal Kosterlitz-Thouless value of
1 4.
In addition to our principal result, we found several other interesting properties of the strong-disorder critical point. For instance, we were able to find an expression,
¯
s(ζ,Γ), for the mean size of clusters with any charging energy. Furthermore, we found evidence that the variance of ¯s(0,Γ) vanishes at late RG time, suggesting a well- defined relationship between the size of superfluid islands and their local charging gap. We were similarly able to find a well-defined relationship between the gap and the superfluid stiffness, at least for the subleading superfluid islands that “dress” the largest cluster.
On the other hand, we also found an important qualification of our principal result:
the “true” thermodynamic value of ηsd may not be observed on systems that are accessible to numerics. The observed value ofη is polluted by a finite sampling error which dies off very slowly due to the wide distribution of internal J1. This distribution gets wider and wider as we tune the transition progressively closer to the unstable fixed point16. Hence, this effect must be kept in mind when interpreting Monte Carlo results. Since experiments on ultracold atoms, for instance, work with only moderately sized systems, this difficult could affect interpretation of these experiments as well.
Chapter 3
Mott Glass to Superfluid
Transition for Dirty Bosons in Two Dimensions
3.1 Introduction
3.1.1 Motivation
As we noted in Chapter 2, the strong-disorder renormalization group (SDRG) of Altman, Kafri, Polkovnikov, and Refael has raised the exciting possibility of a strong-disorder transition in the one-dimensional dirty boson problem [7, 9], and recent numerical simulations may be capturing the emergence of this novel criticality [70]. This naturally motivates the question of what the SDRG can tell us about the superfluid-insulator in higher dimensions.
This problem is tantalizing for several reasons: first, experiments are obviously not restricted to one-dimensional systems, so it is imperative to refine our understand- ing of higher-dimensional bosonic systems. Second, when implemented numerically, SDRG can access larger system sizes than many other methods. Therefore, it can
be a valuable tool in extracting universal properties of the superfluid-insulator tran- sition. Finally, suppose we consider the two-dimensional version of the rotor model that we studied in Chapter 2. The clean version of this model is characterized by an exponent ν ≈ 0.663 that violates the Harris criterion (1.18) [59]. Hence, in contrast to the one-dimensional case, we expect even weak disorder to be relevant, perhaps indicating the presence of strong-disorder physics.
In this chapter, we numerically extend the SDRG of Altman et al. to study the disordered rotor model in two dimensions. Our work is aimed at resolving two physical questions. Most importantly, we want to understand the character and universal properties of the superfluid-insulator transition. Here, “universal” refers to those properties that are independent of the structure of the microscopic distributions of couplings in the model. We also want to identify the glassy phase that intervenes between the superfluid and the Mott insulator. Intuition from the one-dimensional problem suggests that, in this particle-hole symmetric model, we ought to expect the incompressible Mott glass to appear in place of the Bose glass [7]. We would like to confirm this through the SDRG analysis1.
Application of the SDRG in d > 1 has historically been rare2. In part, this is because analytical approaches are usually not tractable. More importantly, there are few known transitions that exhibit infinite randomness, the property that ensures that
1Note that the numerical SDRG is a very ideal method for examining the effects of rare regions of superfluid ordering, because it explicitly constructs these regions in real space. Furthermore, because the method can reach larger systems than other numerical techniques, Griffiths effects can be more easily observed.
2Examples which do apply the method ind >1 include papers by Motrunich et al. and Kov´acs and Igl´oi, which treat the transverse field Ising model [104, 84].
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Figure 3.1: A schematization of the universal features of the proposed flow diagram.
The x-axis gives the ratio of the mean of the renormalized Josephson coupling dis- tribution to the mean of the renormalized charging energy distribution. The y-axis gives the ratio of the standard deviation of the Josephson coupling distribution to its mean. In this context, the Josephson coupling distribution only includes the domi- nant 2 ˜N couplings in the renormalized J distribution, where ˜N is the number of sites remaining in an effective, renormalized lattice. See the text of Section 3.2 for the reasoning behind the exclusion of weaker Josephson couplings from statistics.
the SDRG becomes asymptotically exact near criticality3. The superfluid-insulator transition for dirty bosons is not expected to be of the infinite randomness class:
indeed, the one-dimensional transition of Altman et al. occurs at finite disorder [7], and as we will see below, the same is true of the transition that we find in two dimensions. Hence, in addition to the physical questions that we posed above, our work also aims to address a methodological question: might the SDRG give useful information about a model, even when confronted with the twin difficulties of higher dimensionality and the absence of infinite randomness?
3.1.2 Preview of the Results
Our main results are as follows: we present numerical evidence for the existence of an unstable finite-disorder fixed point of the RG flow, near which the distributions of Josephson couplings and charging energies in the rotor model flow to universal forms.
A schematic picture of this unstable fixed point and the flows in its vicinity is given in Figure 3.1.
To the left of the diagram, flows propagate towards a regime in which the ratio of J, the mean of the Josephson couplings, to ¯¯ U, the mean of the charging energies, van- ishes; meanwhile, the ratio of ∆J, the width of the Josephson coupling distribution, to ¯J grows very large. These flows terminate in one of two insulating phases. The first is a conventional Mott insulator, in which it is energetically unfavorable for the particle number to fluctuate from the large filling at any site. The other is a glassy phase, in which there exist rare Griffiths regions of superfluid ordering. As the ther- modynamic limit is approached, arbitrarily large rare regions appear, driving the gap for charging the system to zero. However, the density of the largest clusters decays exponentially in their size, and the size of the largest cluster in a typical sample does not scale extensively in the size of the system. Moreover, the largest clusters are so rare that they cannot generate a finite compressibility. Thus, the phase is a Mott glass.
This insulating phase gives way to global superfluidity when the rare regions of superfluid ordering percolate, producing a macroscopic cluster of superfluid ordering.
3See 1.4.1.3 for a discussion of infinite-disorder fixed points.
The appearance of the macroscopic cluster is associated with flows that propagate towards the lower right of Figure 3.1, indicating that the unstable fixed point governs the glass-superfluid transition. Our numerical implementation of the SDRG allows us to extract estimates for the critical exponents that characterize this transition. We are thus able to construct a compelling picture of the superfluid-insulator transition:
a picture that must, however, be checked by other methods because of the perils of employing the SDRG method in the vicinity of a finite-disorder fixed point.
3.1.3 Organization of the Chapter
In Chapter 2, we have already introduced the disordered rotor model and the SDRG method. As such, we begin our two-dimensional study in Section 3.2 by de- scribing the modifications that are necessary to apply the SDRG in 2D. Next, in Sec- tion 3.3, we present data collected from our numerical implementation of the SDRG.
Our data is suggestive of striking universality in the superfluid-insulator transition, and in Section 3.4, we proceed to use the data to characterize the phase transition and the phases it separates. We then conclude in Section 3.5 by summarizing the results, making connections to experiments, and giving an outlook.
Appendices 3.A-3.D report supplementary material. In Appendix 3.A, we elabo- rate on an important difference between our numerical implementation of the SDRG and those used in previous studies 4. Then, in Appendix 3.B, we provide technical details regarding our calculation of physical properties in the SDRG. Appendix 3.C is devoted to details regarding a choice of microscopic distributions that we use exten-
4The significance of this difference will become more apparent in Section 3.2
sively in the body of the paper. Finally, in Appendix 3.D, we take up the important tasks of identifying and testing potential weaknesses of the SDRG and comparing to exact diagonalization of small systems.