Susceptibility at the superfluid-insulator transition for one-dimensional disordered bosons

Abstract

A pair of recent Monte Carlo studies have reported evidence for and against a crossover from weak-to strongdisorder criticality in the one-dimensional dirty boson problem. The Monte Carlo analyses rely on measurement of two observables: the effective Luttinger parameter K eff and the superfluid susceptibility χ . The former quantity was previously calculated analytically, using the strong-disorder renormalization group (SDRG), by Altman, Kafri, Polkovnikov, and Refael. Here, we use an extension of the SDRG framework to find a nonuniversal anomalous dimension η sd characterizing the divergence of the susceptibility with system size, χ ∼ L 2−η sd . We show that η sd obeys the hyperscaling relation η sd = 1/2K eff . We also identify an important obstacle to measuring this exponent on finite-size systems and comment on the implications for numerics and experiments. Disordered bosonic systems pose theoretical challenges because of the unique pathologies of their noninteracting limits: At low temperatures, bosons condense into a localized single-particle state, forming a configuration that is intrinsically unstable to interactions. Therefore, Giamarchi and Schulz pioneered the study of the so-called "dirty boson problem" by perturbing a strongly interacting one-dimensional system with weak disorder. They identified a superfluidinsulator transition, belonging to the Kosterlitz-Thouless (KT) universality class, at which disorder is perturbatively irrelevant, 1 see also Refs. 2,3. It was long believed that this universality always characterizes the one-dimensional transition. In the past decade, the possibility has emerged that a novel criticality, also of KT type but with certain nonuniversal disorder-dependent features, takes over at sufficiently strongdisorder strength. This "strong-disorder criticality," first proposed by Altman, Kafri, Polkovnikov, and Refael, 4-6 remains unconfirmed.7 Recent Monte Carlo results by Hrahsheh and Vojta may provide evidence of the crossover between the two types of universality, 8 while calculations by Pielawa and Altman support the existence of a strong-disorder fixed point. 9 Meanwhile, experimental advances in various contexts, including cold atoms, spin systems, and dirty superconductors, have made it especially urgent to gain a better theoretical understanding of the seemingly universal properties of the dirty boson problem. [10][11][12][13][14][15] In this Rapid Communication, we extend the analysis of the universal aspects of the one-dimensional (1D) superfluidinsulator transition in the strong-disorder regime. In particular, we analytically calculate the superfluid susceptibility near the transition. This affords us a different perspective on recent numerical developments and allows us to clarify their relationship with the theoretical strong-disorder renormalization group (SDRG) framework. The model that we concentrate on is the particle-hole symmetric rotor modelThis model can describe a 1D array of superconducting islands connected by Josephson junctions, and we assume stron...

Strong disorder RG approach – a short review of recent developments

“…The SDRG treatment of this model is described in Refs. [8,[77][78][79][80][81] and has been recently reviewed in [82].…”

Random transverse field Ising model in dimensiond> 1: scaling analysis in the disordered phase from the directed polymer model

“…A possible explanation for this discrepancy could be that all these Strong Disorder RG numerical results are based on the approximation of the 'maximum rule' introduced in the very first paper concerning d > 1 [5], that allows to study much bigger sizes. This 'maximum rule' approximation has been recently questionned [33] for another quantum model, namely the superfluid transition for random bosons (see the discussions in section III B and Appendix A of [33]). For our present model, we believe that the use of the approximated 'maximum rules' instead of the full 'sum rules' could be a problem for the following reasons : (i) the full 'sum rule' corresponds for the Directed Polymer model to the computation of the partition function as a sum over all paths, so that what shows up is the fluctuation exponent ω DP < 1/2 of the free-energy, that we have discussed in detail in the present paper.…”

“…The remaining possibility could be to develop other types of approximation, like the cut-off approximation introduced recently in [33].…”

Numerical evidence for strong randomness scaling at a superfluid-insulator transition of one-dimensional bosons

“…In the course of renormalization to longer scales, the effective distribution of weak links flows toward a universal structure, which determines the universal behavior at the critical point. The value of the Luttinger parameter at the transition is, however, non-universal and depends on where the phase boundary is crossed [16][17][18][19] (i.e. on the strength of the bare disorder and interactions).…”