Superfluid/Bose-glass transition in one dimension

We consider a one-dimensional system of interacting bosons in a random potential. At zero temperature, it can be either in the superfluid or in the insulating phase. We study the transition at weak disorder and moderate interaction. Using a systematic approach, we derive the renormalization group equations at two-loop order and discuss the phase diagram. We find the universal form of the correlation functions at the transitions and compute the logarithmic corrections to the main universal power-law behavior. In order to mimic large density fluctuations on a single site, we study a simplified model of disordered two-leg bosonic ladders with correlated disorder across the rung. Contrarily to the single-chain case, the latter system exhibits a transition between a superfluid and a localized phase where the exponents of the correlation functions at the transition do not take universal values.

Figure 1

Phase boundary between the Luttinger liquid phase and the disordered phase for the bosonic ladder system (112) as a function of the bare parameters $K$ and $\mathcal{D}$ is given for three initial values of the rung coupling $V$, corresponding to $y=0.01$, $0.2$, and $0.5$. The region of large $K$ corresponds to the Luttinger liquid phase, where the anharmonic coupling constants $\mathcal{D}$ and $y$ renormalize to zero at large scales.

Figure 2

Solid (dashed) lines represent fixed-point value $K_{cR}^{*}$ ($K_{sR}^{*}$) for the parameter $K_c$ ($K_s$) at the transition as a function of the bare disorder strength $\mathcal{D}$, for different values of the bare coupling $y$. At the transition we have $K_{cR}^{*}+K_{sR}^{*}=3$, however, each of the parameters depends on bare disorder and rung couplings, leading to the nonuniversal exponents in various correlation functions.

Figure 3

Possible phase diagram of a disordered one-dimensional Bose gas as a function of the boson repulsion $g$ and the disorder $\mathcal{D}$. The green dashed-dotted line schematically indicates the boundary below which a bosonization description of such a problem is guaranteed (namely, the disorder is smaller than the chemical potential). In this region, the superfluid/Bose-glass transition, denoted by the solid blue line, is described by universal exponents. The question of the nature and critical behavior of the transition in the regime for which bosonization cannot be directly applied is yet open (see text). One possibility (not shown on the figure) is that the exponent remain universal along the whole superfluid/Bose-glass line, leading to a single Bose-glass phase. The only singular line is then the noninteracting bosons ($g=0$) which are localized by rare events of the disorder and indicated by the blue box. The other possibility (shown in the figure) is that if at small interaction, there are nonuniversal exponents along the dashed blue line, then this forces the existence of a critical point on the superfluid/Bose-glass boundary between the nonuniversal and universal regimes. Consequences for the various phases (superfluid or Bose glass) are still unknown. For example, it could lead to the existence of two distinct Bose-glass phases.