Localization transition in one dimension using Wegner flow equations

The flow-equation method was proposed by Wegner as a technique for studying interacting systems in one dimension. Here, we apply this method to a disordered one-dimensional model with power-law decaying hoppings. This model presents a transition as function of the decaying exponent $\alpha$. We derive the flow equations and the evolution of single-particle operators. The flow equation reveals the delocalized nature of the states for $\alpha <\frac{1}{2}$. Additionally, in the regime $\alpha >\frac{1}{2}$, we present a strong-bond renormalization group structure based on iterating the three-site clusters, where we solve the flow equations perturbatively. This renormalization group approach allows us to probe the critical point $\left(\alpha =1\right)$. This method correctly reproduces the critical level-spacing statistics and the fractal dimensionality of the eigenfunctions.

Figure 1

Phase diagram of PBRM model [Eq. (1)], with disordered onsite potential and random hoppings whose typical values decay with range as a power law $J_{ij}\sim \frac{1}{{|i-j|}^{\alpha }}$. For $\alpha <\frac{1}{2}$, the system is equivalent to the $\alpha =0$ Gaussian orthogonal ensemble (GOE). This region is studied in this work via the flow-equation technique. A strong-bond RG flow scheme based on the flow equations allows us to characterize the $\alpha >\frac{1}{2}$ phases. ThisRG scheme we propose does not eliminate any degrees of freedom, but consists of a sequence of unitaries. The critical point for the transition to a localized phase is at ${\alpha }_c=1$. The level-spacing statistics in this phase transitions to Poisson statistics.

Figure 2

Pictorial representation of the flow equations for the hoppings and fields as calculated in Eqs. (7) and (8). All the contri-butions are the product of three coupling constants. For the hop-pings, the first contribution comes from a sum of terms of the type $JJh$, that is the product of two hoppings and one field, while the second contribution comes from $Jhh$, the product of two fields and one hopping. For the renormalization of hoppings, all contributions are of type $JJh$.

Figure 3

Typical flow for the five-site problem. The initial fields and hoppings are random variables. The distribution of hoppings is Gaussian, with a power-law decay with distance ${\left|i-j\right|}^{\alpha },\alpha =1$. The distinct colors represent the different distances $\left|i-j\right|$ (red, blue, brown, and black curves, in order of increasing distance). Notice that one of the red curves, indicated by the arrow, flows more slowly to zero. This is due to the fact that the decay term in the $J$ flow is proportional to difference of the fields of the two sites connected by it [see Eq. (7) and the arrow in the inset curve]. Also shown in the inset is flow of fields (blue) and their asymptotic approach to the Hamiltonian eigenvalues (horizontal dashed orange lines).

Figure 4

The standard deviation of distributions of $J_i^j,\sigma \left(J_i^j\right)$, as a function of distance $l=|i-j|$ for different RG times $\mathrm{\Gamma }$. The simulations were run for system size $N=45$ and averaged over 100 realizations. The initial distribution of the bonds is Gaussian with standard deviation $\sigma {\left(J_i^j\right)}_{\mathrm{\Gamma }=0}=\frac{1}{2|i-j|}$ (red straight lines in log-log scale). The fields $h_i$ are chosen to be uniformly distributed between 0 and 1. For initial distributions with exponents $\alpha <0.5$, the exponent changes and flows to $\alpha =0$ as $\mathrm{\Gamma }$ increases. For exponents $\alpha >0.5$, the long-distance tails are not altered by the flow.

Figure 5

Final evolution $\left(\mathrm{\Gamma }\to \infty \right)$ of the number operator initialized in the middle of a 45-site chain, at site number 23, ${\mathfrak{n}}_{23}(\mathrm{\Gamma }=0)=c_{23}^{†}{c}_{23}^{}$, for some representative exponents. At $\mathrm{\Gamma }=0$, all ${\mathfrak{J}}_{23}^i$ are zero, and only ${\mathfrak{h}}_{23}$ is equal to one. The asymptotic values are obtained by measuring the final values of ${\left({\mathfrak{J}}_{23}^i\right)}^2$ [see Eq. (19)]. The tilde indicates the set of variables related to decomposition of the operator flow in terms of an instantaneous basis [Eq. (18)]. The results are averaged over 20 disorder realizations.

Figure 6

Schematic of the steps in the strong-bond RG method. The first part consists of finding the bond $\left(i,j\right)$ with the maximum $r_{ij}$. Using an appropriate unitary, the hopping on the bond is transformed to zero. Hoppings connecting to the bond $\left({\tilde{J}}_i^k,{\tilde{J}}_j^k\right)$ and fields on its sites $\left({\tilde{h}}_i,{\tilde{h}}_j\right)$ are renormalized. This procedure is iterated until all bonds are set to zero. The strong disorder allows us to make a crucial simplification: Once a bond is set to zero, we neglect its regeneration in subsequent steps. This produces a negligible error if the generated ${\tilde{r}}_{ik}$ and ${\tilde{r}}_{jk}$ are smaller than the removed $r_{ij}$. After $\mathcal{O}\left(N^2\right)$ steps, where $N$ is the system size, the Hamiltonian is diagonal.

Figure 7

The representation strong-bond RG procedure in the $x\text{−}J$ space. Each point represents a bond, and its distance from the origin is $r_{ij}$. The strong-bond RG rotates the bonds within a large-$r$ shell. In the first step, the bonds with largest $r_i^j$ are rotated to the $x_i^j$ axis. Next, the bonds connected to the decimated bond undergo a renormalization via Eqs. (24), (25), and (26). We perform one approximation: once eliminated, a bond is not allowed to assume finite values again, and these points which lie on the $x$ axis beyond the $r$ cutoff move horizontally only.

Figure 8

Marginal distribution $P_{\mathrm{\Gamma }}\left(logJ\right)$ in the log scale for different RG steps $\mathrm{\Gamma }$. From (a)–(c), we plot the evolution of the marginal distribution for exponents $\alpha =0.7$, 1.0 and 2.0. Different colors represent different RG steps; $\mathrm{\Gamma }=1$, 1000, 2000, 3000 are represented by blue, red, green, magenta, and black, respectively. As seen from Eq. (34), there are two distinct regimes in the probability distribution. The crossover scale is given by $J_c$. Below this, $J<J_c,P_{\mathrm{\Gamma }}\left(logJ\right)\sim logJ$ and above the scale, for $J>J_c,P_{\mathrm{\Gamma }}\left(logJ\right)\sim -\frac{1}{\alpha }logJ$. We note that as the bonds are decimated, the behavior of the distribution below and above the crossover remains unchanged. The system has size $N=100$ and we average over 20 disorder realizations.

Figure 9

Decimated $r=r_{\text{max}}$ as function of the RG step in a given disorder realization, for distinct exponents, (a) $\alpha =0.1$, (b) $\alpha =0.7$, (c) $\alpha =1.0$, and (d) $\alpha =2.0$. The behavior of the slopes of the peaks in the curves differs significantly, as in (a) $r$ increases in several consecutive RG steps, while in (b)–(d) a bond that is generated with $r>r_{\text{max}}$ is immediately removed. Notice additionally that, in contrast with (a), in (b)–(d) the decimated $r_{\text{max}}$ decreases consistently during the RG flow.

Figure 10

Fraction of decimations $\left(f\right)$ that does not lower the energy scale $r$ in the strong-bond RG scheme. There is a transition at $\alpha =0.5$, indicating the failure of the strong-bond RG for $\alpha <0.5$. The strong-bond RG has vanishing fraction of decimations in the thermodynamic limit for $\alpha >0.5$.

Figure 11

Level-spacing comparison for eigenvalues obtained through strong-bond RG (blue circles) and exact diagonalization (red squares), normalized by the mean level spacing value. (a)–(d) correspond to exponents $\alpha =0.9$, 1, 1.2, and 2, respectively. For comparison, we also plot the analytical expressions for Poisson (green) and Wigner-Dyson (magenta) statistics. The system size is $N=400$ sites, averaged over 500 disorder realizations. The $G_i^j=J_i^j{\left|i-j\right|}^{\alpha }$ and $h_i$ variables follow Gaussian distributions, with unit standard derivation.

Figure 12

Distribution of nondecimated distance-independent bond couplings $G=J_i^j{\left|i-j\right|}^{\alpha },\tilde{P}\left(G\right)$, as the RG flows, at RG steps $N_{\mathrm{steps}}=1$ (blue), 100 (red), 1000 (green), 2000 (magenta), and 3000 (black). The number of sites is $N=100$, and the total number of steps to diagonalize the Hamiltonian is $N_{\mathrm{steps}}=4950$. The exponents shown are (a) $\alpha =0.7$, (b) $\alpha =1.0$, (c) $\alpha =2$. In all cases, the initial distribution of $G$ is uniform, from $-1$ to 1 (blue curve). At later RG steps, the $G$ distribution becomes Gaussian.

Figure 13

(a) Density plot of the distribution $x$ and $logJ$ at $\mathrm{\Gamma }/4=6$. In red, the curve $x^2\mathrm{\Gamma }=-log\left(J\right)$, showing the maximum intensity for small $logJ$. (b) Cuts of $logJ=-5$ (blue), $-4$ (purple), $-3$ (yellow), $-2.5$ (green), $-2.2$ (blue) at time $\frac{\mathrm{\Gamma }}{4}=\frac{10}{3}$. The two distinct peaks collapse at large values of $logJ$.

Figure 14

Initial probability distribution $P\left(y=logJ\right)$ of couplings connected to an arbitrary test site, in log scale. The distributions have been shifted horizontally so that the maximum of all the curves is located at $y=0$. For $y<0$, the uniform part of the distribution $P\left(J\right)$, corresponding to $logP\left(y\right)\sim -y$, is independent of $\alpha$. For $y>0$, the angular coefficient, expected to result in $-\frac{1}{\alpha }$, gives $-2.07$ for $\alpha =\frac{1}{2}$ (red line and points), $-1.01$ for $\alpha =1$ (purple line and points), and $-0.51$ for $\alpha =2$ (black line and points). The blue points correspond to $\alpha =0$, where the saddle-point approximation fails.

Figure 15

Comparison of the single-particle spectrum obtained from exact diagonalization with the one obtained from the strong-bond RG technique. (a)–(d) In ascending order of exponents, $\alpha =0,0.7$, 2.0, 5.0. In all cases, both spectra look reasonably similar. A careful inspection of the level-spacing statistics, however, reveals that the eigenvalues obtained in case (a), $\alpha =0$, does not experience repulsion, like a delocalized phase should (see main text for further details about the level spacing).

Figure 16

Finite-size scaling of the IPR for different system sizes $N=100$ (blue), $N=200$ (red), and $N=400$ (green). The finite-size dependence and scaling collapse are shown in (a) and (c) for the proposed RG procedure, and (b) and (d) for exact diagonalization, respectively. The data are taken for the critical point $\alpha =1$. The initial distributions of $G_i^j$ and $h_i$ are Gaussian. The fractal dimensions are found from the scaling $log\text{IPR}\to log\text{IPR}+D_{2}logN$. We find $D_2=0.5$ for the RG case (a), and $D_2=0.6$ for the case of exact diagonalization (b).