Kosterlitz-Thouless scaling at many-body localization phase transitions

Abstract

We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a 'quantum avalanche'. We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the lengthscale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law distributed in size. This points to the existence of a second transition within the MBL phase, at which these power-laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent αc = 2, and continuously varying exponents in the MBL phase consistent with the KT picture. arXiv:1811.03103v3 [cond-mat.dis-nn]

Renormalization-group study of the many-body localization transition in one dimension

“…The result, shown in Figure 3(a), indicates that the MBL phase is a fixed line at f = 0 which can be parametrized byζ ∞ ∈ (0, 1), the global decay length as Λ → ∞. This flow is consistent with the KT-type scenario [2,39], where the critical fixed point is a terminus of the MBL fixed line located at f = 0, ζ = 1, and the transition is driven by the avalanche instability. However, finite-size effects limit the extent to which numerical simulations alone can confirm this.…”

“…This is an exact result; the flow is closed and stable within the subspace of exponentially distributed d. A useful flow equation for ρ T (Λ) is not similarly available, so we determine the behavior of ρ T (Λ) numerically. The data shown in Figure 3 Returning to the analysis of Equations (6) and (7), these equations and the flow of ρ T (Λ) and µ I (Λ) detailed in this section imply that in the MBL phase the flow lines approach a fixed line at f = 0 and 0 <ζ < 1, as suggested in [39]. Furthermore, the flow of f andζ are proportional to the factor 1 − (1 − f )ζ, implying a change of the stability of the fixed line at f = 0,ζ = 1 where the MBL phase ends and a global avalanche instability occurs.…”

“…Many approximate RG studies [2,36,37,39] and one recent ED study [31] of the MBL transition seem to agree on the prediction that physical lengths of thermal inclusions in critical systems are distributed according to a power law ∝ (L T ) −α , with the exponent taking an apparently universal value near α = 2. In addition, and consistent with a KT-type scenario, is the possibility of an intermediate critical MBL phase [39] where the lengths of thermal inclusions are power-law distributed with a continuously varying exponent [2,31,39].…”

“…As mentioned earlier, before a KT-like scenario for the MBL transition was proposed [2,39], studies using approximate RG treatments of the MBL transition assumed a one-parameter scaling ansatz [1,[34][35][36][37][38]. These works reported apparent correlation-length critical exponents in the range ν ∼ = 2.5 to ν ∼ = 3.5.…”

“…The proposed critical MBL phase of [2,39] contains rare large thermal inclusions with fractal dimension zero, whose lengths are distributed as a power law. Here we show that our model does not contain an intermediate MBL phase by showing that such an extended power law regime does not exist in our model, and arguing that the fractal dimension of large thermal inclusions decreases continuously in the MBL phase, reaching zero only at the single critical point in our RG, thus following "scenario (i)" of Ref.…”