Anderson Localization on Husimi Trees and its implications for Many-Body localization

Abstract

Motivated by the analogy between many-body localization (MBL) and single-particle Anderson localization on hierarchical graphs, we study localization on the Husimi tree, a generalization of the Bethe lattice with a finite density of local loops of arbitrary but finite length. The exact solution of the model provides a transparent and quantitative framework to systematically inspect the effect of loops on localization. Our analysis indicates that local loops enhance resonant processes, thereby reducing the critical disorder with increasing their number and size. At the same time, loops promote local hybridization, leading to an increase in the spatial extent of localized eigenstates. These effects reconcile key discrepancies between MBL phenomenology and its single-particle Anderson analog. These results show that local loops are a crucial structural ingredient for realistic single-particle analogies to many-body Hilbert spaces.

Figures (8)

• Figure 1: (a) Largest eigenvalue of the integral operator \ref{['eq:operator']}, obtained from the large-$p$ analytical expression \ref{['eq:lambda']} (blue line) and from direct diagonalization (red circles), for $k=1$, $p=6$, and $W = 103\,t \approx W_c$. (b) Corresponding eigenvectors $f_\beta(x)$ obtained from Arnoldi diagonalization for three values of $\beta$, described by the functional form \ref{['eq:f']}. The vertical dotted line marks $x = 2/W$, and the dashed lines indicate the expected power-laws with exponent $2\beta - 2$.
• Figure 2: Critical scaling of the correlation volume, $\ln \Lambda_c \propto -\langle \ln \mathrm{Im}\, G \rangle$ (a), and of the tail exponent $\beta$ (b) of the distribution of the imaginary parts $\hat{x}_i$ [Eq. \ref{['eq:ansatz']}], plotted versus the relative distance from the transition, $|W_c - W|/W_c$, for several pairs $(k,p)$ controlling the graph topology. Symbols correspond to results obtained by solving the cavity equations using population dynamics with large populations of size $\Omega = 2^{28} \simeq 2.7 \times 10^{8}$ (see App. \ref{['app:num']} for details). Dashed curves show the predictions of Eqs. \ref{['eq:Lambdac']} and \ref{['eq:beta']}, with coefficients $c_1$ and $c_2$ from Eq. \ref{['eq:lambda_exp']}.
• Figure 3: (a) Logarithm of the exponent, $\ln \lambda_{1/2}$, controlling the exponential evolution of the typical imaginary part of the cavity Green’s functions \ref{['eq:Ximaginary_self']} under iteration, plotted as a function of the disorder strength $W$. Each data set corresponds to a different loop topology $(k,p)$ at fixed connectivity $z=12$. The dashed lines represent the analytical prediction of Eq. \ref{['eq:lambda']}, while symbols show the population dynamics results with large populations of size $\Omega = 2^{28} \simeq 2.7 \times 10^{8}$ (see App. \ref{['app:num']} for details). (b) Ratio between the critical disorder of the Anderson model on the Husimi tree, $W_c(p,z)$, varying $k$ and $p$ at fixed $z$, and the effective diagonal disorder strength of an interacting disordered spin chain imbrie2016diagonalizationde2024absencebiroli_large-deviation_2024, $W_c^{\rm MBL}(n)= 2\sqrt{n}, h_c$, with $h_c \approx 9$ the estimated MBL transition biroli_large-deviation_2024, and $n=z$. Solid lines show the analytical predictions \ref{['eq:Wc']} for $p/z \le 1/2$ for $z=12$ and $z=36$, while dashed lines indicate their analytic continuation into the loop-rich regime relevant for MBL ($p/z > 1/2$, shaded area). Symbols (squares for $z=12$ and circles for $z=36$) denote the critical disorder estimated from the zero crossing of a linear fit to the population dynamics data for $\ln \lambda_{1/2}$.
• Figure 4: (a) IPR as a function of disorder strength $W$ for Husimi trees with fixed connectivity $z=12$ varying $k$ and $p$. The population dynamics results are obtained using the procedure described in App. \ref{['app:num_IPR']} with large populations of size $\Omega = 2^{28} \simeq 2.7 \times 10^{8}$. (b) Magnitude of the IPR jump at the critical threshold, $I_2^{(c)}$, as a function of the loop-to-connectivity ratio $p/z$ for different connectivities $z=12$ (squares) an $z=36$ (circles). Solid lines are fits to the toy model of Eq. \ref{['eq:toyIPR']}, using a single parameter $\gamma \simeq 5$ for both curves. The analytic continuation into the MBL-relevant regime (dashed lines, shaded region) shows that the jump can be suppressed to arbitrary small values for $p \lesssim z$.
• Figure 5: (a) Number of loops of length $\ell$ starting from a given node of a Husimi tree with $k=1$ and $p=11$, as given by Eq. \ref{['eq:loops']}. The dashed line shows the approximate exponential scaling ${\cal N}(\ell) \approx (k+1)p^{\ell}/2$. (b) Comparison of loop statistics for the Husimi tree and the hypercube with the same local connectivity $z=10$. Blue circles denote the number of loops of length $\ell$ starting from a given vertex of the hypercube formed by the $2^{10}$ bit strings $\{+1,-1\}^{10}$, with $\ell \ge 4$ and even, obtained via an exact counting algorithm. The dashed blue line corresponds to the asymptotic expression \ref{['eq:loopsHC']}, valid for large $n$ and $\ell \lesssim n$. Red squares denote the number of loops of length $\ell$ in a Husimi tree with $k=1$ and $p=5$.