Anderson localization: a density matrix approach

TL;DR

This work introduces a density-matrix based framework for Anderson localization by leveraging the modular density matrix (MDM) and its slowest-decaying mode to extract the localization length, providing a bridge to the transfer matrix method. The authors validate the approach against the transfer matrix method in the 3D Anderson model and the 2D spin-orbit-coupled model, and extend it to multiorbital systems, demonstrating consistent mobility-edge behavior. They then generalize to interacting systems with a many-body subtraction density matrix (SDM) and apply it to a 1D spinless model and the 2D Anderson-Hubbard model, revealing interaction-driven delocalization and a correlated metallic phase at finite disorder. The results establish MDM as a versatile, efficient framework for studying localization and its interplay with electronic correlations, with promising implications for integrating into DMET/DMFT and tensor-network approaches for realistic materials.

Abstract

Anderson localization is a quantum phenomenon in which disorder localizes electronic wavefunctions. In this work, we propose a new approach to study Anderson localization based on the density matrix formalism. Drawing an analogy to the standard transfer matrix method, we extract the localization length from the modular density matrix in quasi-one-dimensional systems. This approach successfully captures the metal-insulator transition in the three-dimensional Anderson model and in the two-dimensional Anderson model with spin-orbit coupling. It can be also readily extended to multiorbital systems. We further generalize the formalism to interacting systems, showing that the one-dimensional spinless attractive model exhibits the expected metallic phase, consistent with previous studies. More importantly, we demonstrate the existence of a two-dimensional metallic phase in the presence of Hubbard interactions and disorder. This method offers a new perspective on Anderson localization and its interplay with interactions.

Figures (16)

• Figure 1: Schematic diagrams of wave function localization and finite-size scaling in disordered systems. (a), (b) schematically show the real-space distribution of the extended and localized wave function respectively. The extended state exhibits undamped oscillations with finite mean free path $l$, whereas the localized state decays exponentially as $e^{-x/\xi}$. (c) shows the quasi-1D system of width $M$ and length $L$, $\alpha$ and $\beta$ are the site indices in the slice $i$ and $j$, respectively. $x$ represents the distance between slice $i$ and $j$. (d), (e) schematically show the finite-size scaling law of the dimensionaless ratio $\xi_M/M$, which increases with $M$ for the extended states and decreases with $M$ for the localized states.
• Figure 2: Computational workflow for extracting the localization length from the MDM in one-dimensional non-interacting systems. (a) schematically shows different disorder samples with hopping energy $t$ and onsite random potential $\epsilon_i$. For each disorder sample, the single-particle eigenstate with the energy level $E_n$ nearest to the Fermi energy $E_F$ is chosen to calculate $\rho^m(x)$, as illustrated in (b). (c) shows $\gamma(x)$ as a function of $x$ for the systems with disorder strength $W=2t$, which exhibits exact exponential decay with a localization length $\xi=6.08$. (c) is calculated by averaging 500 disorder realizations of length $L=120$ under periodic boundary condition at $E_F=0$.
• Figure 3: Comparison of localization length scaling between MDM and TMM. (a)-(b) are finite-size scaling of $\xi_M/M$ obtained from MDM for the 3D Anderson model at $E_F=0$ as a function of $W$ (a) and at a fixed disorder strength $W=6t$ as a function of $E_F$ (b). (e)-(f) are the corresponding results from the TMM. Black dashed lines mark the metal-insulator transition points. (c), (g) are the phase diagrams of the 3D Anderson model and the spinful 2D Anderson model with SOC from both methods, respectively. (d), (h) are finite-size scaling of $\xi_M/M$ for the spinful 2D Anderson model with SOC at $E_F=0$ from MDM and TMM, respectively. Each $\xi_M/M$ data point from the MDM is obtained by averaging over 3000 disorder realizations of size $M^{d-1}\times L$ with $L=300$ for 3D Anderson model and $L=500$ for 2D Anderson model with SOC ($d$ is the dimension of the considered model), while in TMM each data point is obtained from quasi-1D systems of length $L=3\times10^6$ by averaging over three disorder realizations. In (c) and (g), the black dot lines indicate the energy spectral boundaries, determined as stable values of the largest eigenenergy averaged over 50 disorder realizations of size $L^d$ with $L=100$.
• Figure 4: Comparison of results from MDM and TMDCA for the two-orbital 3D Anderson model at a fix disorder strength of $W=5t$. (a) Finite-size scaling of $\xi_M/M$ obtained from MDM, with system size $M^{d-1}\times L\times2$ ($L=300$) averaged over 3000 disorder realizations. (b) TDOS obtained from TMDCA with cluster sizes $N_c=216$ (black) and $N_c=64$ (red). Model parameters are $t^{11}=t^{22}=t$, $t^{12}=t^{21}=0.3t$ and $W_A=-W_B=5t$. Vertical dashed lines in both panels mark the phase boundary between localized and extended states.
• Figure 5: Computational workflow for extracting the localization length using the many-body extension of MDM approach in spinless 1D interacting systems. (a) schematically shows different disorder samples with hopping energy $t$, onsite random potential $\epsilon_i$ and nearest-neighbor electron-electron interaction $V$. For each disorder sample, $\left|{GS_N}\right>$ and $\left|{GS_{N-1}}\right>$ are obtained by performing DMRG calculation, as illustrated in (b). Then the SDM $\rho^{sub}_{(i);(i+x)}$ can be obtained from the two states according to Eq.\ref{['eq:SDM_definition']}. $A_N^i$ ($A_{N-1}^i$) represents the matrix product state of $\left|{GS_N}\right>$ ($\left|{GS_{N-1}}\right>$) at site $i$. (c) shows $\gamma(x)$ as a function of $x$ for the systems with $W=2t$ and $V=t$, which exhibits clear exponential decay with a localization length $\xi=4.08$. (c) is calculated by averaging 500 disorder realizations of length $L=120$ under open boundary condition at half-filling. In the site averaging for each sample, the 20 sites nearest to each boundary are excluded to minimize open-boundary effects.