Strong-disorder renormalization-group study of the one-dimensional tight-binding model

We formulate a strong-disorder renormalization-group (SDRG) approach to study the $\beta$ function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The $\beta$ function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the $\beta$ function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional nonuniversal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model, while the generic case corresponds to the Griffiths paramagnetic phase.

Figure 1

Schematic decimation procedure for (a) bond and (b) site transformations.

Figure 2

The $\beta$ function for the particle-hole symmetric case (${\varepsilon }_i=0$, continuous, dashed and dotted lines) and for the more conventional case with diagonal disorder only ($t_i=t_0$, symbols). The SDRG (dashed red line), exact (continuous black line), and SDRG2 (dotted blue line) results are discussed in Secs. 4a, 4b, and 4d, respectively.

Figure 3

The $\beta$ function for different disorder strengths, highlighting its nonuniversality when particle-hole symmetry (${\varepsilon }_i=0$) is broken.

Figure 4

The $\beta$ function calculated using $g={\left(T\right)}_{\mathrm{geo}}/[1-{\left(T\right)}_{\mathrm{geo}}]$ for both diagonal and off-diagonal disorder. For comparison, we plot the scaling form $-(1+g)ln(1+g^{-1})$ derived in Ref. [5]. Dotted lines are guides to the eyes.