Exploring Anderson Localization in One Dimension

Anderson localization is a striking phenomenon in condensed matter physics where disorder causes electron wavefunctions to become spatially localized, halting diffusion even in the absence of interactions. In a perfectly periodic crystal, electrons propagate as Bloch waves that extend throughout the material, allowing current to flow easily. However, if random impurities disturb the lattice, quantum interference between multiple scattering paths can trap electrons, leading to an exponential decay of wavefunctions away from some center. Philip Anderson's pioneering 1958 work introduced this concept, demonstrating that in one dimension even infinitesimal disorder suffices to localize all electronic states. The localization length ξ characterizes the exponential decay: |ψ(n)| ∝ e^{-|n - n_0|/ξ}. Our calculator estimates ξ for a tight-binding chain with uncorrelated on-site disorder of strength W, relying on a perturbative formula valid for weak to moderate disorder. By entering W, the electron energy E, and the chain length L, users obtain the predicted localization length and the corresponding dimensionless conductance g ≈ e^{-L/ξ}, providing a practical glimpse into transport suppression in disordered media.

The one-dimensional Anderson Hamiltonian can be written as

$H=-t\sum n_{}^{}$

where t denotes the nearest-neighbor hopping amplitude, ε_n are random on-site energies drawn uniformly from [−W/2, W/2], and c_n†, c_n are creation and annihilation operators. For weak disorder, perturbation theory yields an approximate localization length given by

$\xi =\frac{24(4-E^2)}{W^2}$

This expression, derived under the assumption |E| < 2t, demonstrates that stronger disorder decreases ξ, while energies near the band edges (E ≈ ± 2t) also reduce ξ by shrinking the available density of states. Although more sophisticated techniques like transfer-matrix calculations provide exact results, the above formula captures essential trends and is widely cited in textbooks. Once ξ is known, the dimensionless conductance g of a finite sample of length L follows the scaling theory of localization:

$g\approx e^{-\frac{L}{\xi }}$

This exponential suppression explains why one-dimensional wires with substantial disorder behave as insulators despite containing mobile carriers. Importantly, the formula is dimensionless because g measures conductance in units of e²/h, the quantum of conductance. As L exceeds ξ, g drops precipitously, reflecting the low probability for an electron to traverse the sample.

The calculator's implementation is straightforward: the user-supplied W and E are inserted into the localization length formula, assuming t = 1. The result is expressed in units of the lattice spacing. Next, g is obtained from the above exponential relation using the provided L. To help interpret numbers, the tool prints both ξ and g. For very small W, the formula predicts large ξ, signifying nearly delocalized states. For W comparable to t, ξ becomes only a few lattice sites, and conductance plummets for moderate chain lengths. The table shows example values for E = 0:

W/t	ξ (sites)	L = 100 g
0.5	384	0.77
1.0	96	0.36
2.0	24	0.015

These numbers underscore how rapidly conductance decays as disorder increases. In experimental nanowires, disorder can arise from lattice defects, impurities, or surface roughness. Measuring transport as a function of sample length and comparing to localization theory helps infer the effective disorder strength and coherence length of carriers. Moreover, Anderson localization is not restricted to electrons: light, sound, and even matter waves can localize in disordered media. Photonic crystals with random index variations display suppressed light transmission, while ultracold atomic gases in speckle potentials localize matter waves, allowing simulations of Anderson's model with exquisite control.

Understanding localization also has implications for many-body physics. In interacting systems, localization competes with electron-electron scattering, leading to phenomena like many-body localization where an entire interacting system fails to reach thermal equilibrium. Although our calculator addresses the noninteracting case, the intuition it builds extends to these richer scenarios. For example, a small localization length suggests that localized orbitals are well separated, reducing overlap and hindering thermalization. Thus, estimates of ξ serve as starting points for analyzing interacting disordered systems.

Further refinements to the localization length formula account for energy dependence more accurately or incorporate different disorder distributions. For instance, Gaussian disorder or correlated potentials modify the numerical prefactor and may introduce mobility edges in higher dimensions. Nevertheless, the simple expression used here captures the essence of 1D localization and is commonly employed in introductory discussions.

Users can experiment with a range of energies and disorder strengths to observe how ξ and g respond. Trying E close to the band edges reveals that even weak disorder can localize states strongly. Conversely, setting E near zero highlights the quadratic dependence of ξ on W. The dimensionless conductance output shows how long chains become effectively insulating, which is essential knowledge when designing nanoscale conductors or interpreting measurements on disordered polymers, DNA strands, or semiconductor nanowires.

In conclusion, Anderson localization demonstrates how randomness can profoundly alter transport, turning metals into insulators. This calculator distills the key theoretical results for the 1D model into an accessible computational tool. By entering disorder strength, energy, and chain length, users gain quantitative insight into localization length and conductance, reinforcing core concepts of quantum interference, scaling theory, and the delicate balance between order and randomness in condensed matter systems.