Disorder-driven transition in a chain with power-law hopping

We study a one-dimensional (1D) system with a power-law quasiparticle dispersion ${\propto \left|k\right|}^{\alpha }sgnk$ in the presence of a short-range-correlated random potential, and demonstrate that for $\alpha <1/2$ it exhibits a disorder-driven quantum phase transition with critical properties similar to those of the localization transition near the edge of the band of a semiconductor in high dimensions, as studied recently [Phys. Rev. Lett. 114, 166601 (2015); Phys. Rev. B 91, 035133 (2015)]. Despite the absence of localization in the considered 1D system, the disorder-driven transition manifests itself, for example, in a critical form of the disorder-averaged density of states. We confirm the existence of the transition by numerical simulations and find the critical exponents and the critical disorder strength as a function of $\alpha$. The proposed system thus presents a convenient platform for numerical studies of the recently predicted unconventional high-dimensional localization effects and has the potential for experimental realizations in chains of ultracold atoms in optical traps.

Figure 1

The disorder-averaged density of states $\rho \left(E\right)$ vs energy $E$ for $\alpha =0.4$ and various disorder amplitudes $W$. For subcritical disorder strength, $W<W_c\approx 0.5$, the density of states vanishes at $E=0$, whereas for stronger disorder, $W>W_c$, the density of states is finite for all energies. At the critical disorder strength, $W=W_c\approx 0.5$, the density of states is linear in energy, in agreement with the analytical predictions based on one-loop RG calculations.

Figure 2

Energy $\left(E\right)$ vs disorder amplitude $\left(W\right)$ diagram for the low-energy density of states in a 1D chiral system with the quasiparticle spectrum ${\xi }_k=a{\left|k\right|}^{\alpha }sgnk$. The color shows the exponent $\theta$ of the density of states $\rho \left(E\right)\propto E^{\theta }$ [see the color bar and Eq. (7)]. For $\alpha <1/2$ the density of states has a critical point $(E=0$, $W=W_c)$ that separates the weak-disorder $\left({\theta }_{\alpha }={\alpha }^{-1}-1\right)$ and strong-disorder $\left(\theta =0\right)$ phases at $E=0$. The transition disappears for $\alpha >1/2$. The gray vertical line shows the analytical value of the critical disorder amplitude obtained assuming $0<1-2\alpha \ll 1$. The black crosses show the isoline $\theta =1$. The black solid lines show the theoretical crossover energies $E=\left(\right|W-W_c{|/W_c)}^{z\nu }$ between the critical region $\left(\theta =1/z-1\right)$ near $W\approx W_c$ and the weak- $\left({\theta }_{\alpha }={\alpha }^{-1}-1\right)$ and strong- $\left(\theta =0\right)$ disorder phases, with the exponents $\nu$ and $z$ given by Eqs. (3) and (4).