Disorder Localization

A disordered-system calculation averages over a specified probability distribution of Hamiltonians. One visually localized eigenstate or one irregular spectrum is not a localization diagnosis. Reliable conclusions compare eigenstate structure, spectral statistics, and system-size dependence within consistent energy and symmetry windows.

Helpful background. Tight-binding assembly constructs the lattice model; Anderson localization explains the physical phenomenon.

Disorder ensembles

For a one-dimensional example,

$$H=-t\sum_j(|j\rangle\langle j+1|+\text{h.c.}) +\sum_jV_j|j\rangle\langle j|,$$

with $V_j$ sampled independently and uniformly from $[-W/2,W/2]$. This distribution has variance $W^2/12$; another distribution with the same symbol $W$ may describe a different model.

Generate independent disorder realizations and retain their seeds or potentials. A seed repeated on parallel workers duplicates the sample. If the same realization is used across parameter values, that correlation can improve differences but must be preserved in uncertainty analysis.

Specify boundary conditions and the energy window. Averaging over the entire spectrum can mix different localization lengths and obscure mobility-edge behavior in models where it exists.

Participation ratios

For a normalized eigenstate on $L$ sites,

$${\rm IPR}=\sum_j|\psi_j|^4,\qquad {\rm PR}=1/{\rm IPR}.$$

A uniform state has IPR $1/L$, while a single-site state has IPR 1. A state spread approximately over $\ell$ sites has IPR of order $1/\ell$. On a finite sample, a localization length larger than $L$ can mimic an extended state.

Average both the distribution and suitable summaries. The typical IPR $\exp(\mathbb E[\log{\rm IPR}])$ differs from the arithmetic mean when rare events matter. Do not call their difference a numerical inconsistency.

In a many-body occupation basis, participation ratios depend strongly on basis choice and are not the same diagnostic as real-space single-particle localization.

An exactly summable localized profile supplies a more informative benchmark than a single occupied site. On the infinite unit-spacing lattice, take an amplitude envelope $\psi_j\propto e^{-|j|/\xi}$ and put $q=e^{-2/\xi}$. Normalization gives

$$|\psi_j|^2=\frac{1-q}{1+q}q^{|j|},\qquad \mathrm{IPR}=\frac{(1-q)(1+q^2)}{(1+q)^3}.$$

The second identity follows by summing the geometric series for $|\psi_j|^4$. At large $\xi$, $\mathrm{IPR}\sim1/(2\xi)$; at small $\xi$ it tends to one. Restrict and renormalize the profile on progressively larger boxes to measure finite-size bias explicitly. This relation is specific to the chosen envelope and definition of $\xi$, not a universal conversion between IPR and localization length for arbitrary eigenstates.

Adjacent-gap ratios

After resolving exact symmetries and sorting energies in a chosen window, define

$$\delta_n=E_{n+1}-E_n,\qquad r_n=\frac{\min(\delta_n,\delta_{n+1})} {\max(\delta_n,\delta_{n+1})}.$$

The dimensionless ratio avoids explicit spectral unfolding. Degenerate levels, mixed symmetry sectors, and disconnected components can drive ratios toward zero without implying localization. If both neighboring gaps vanish the ratio is undefined, not zero. Remove neither genuine degeneracies nor inconvenient levels silently; define the selection procedure and count any excluded undefined cases.

For independent Poisson spacings, the mean ratio is $2\log2-1\approx0.3863$. Random-matrix comparisons require choosing the symmetry class correctly, including antiunitary symmetries. A magnetic flux can change that class.

This Poisson reference can be tested without diagonalizing a disordered Hamiltonian: draw independent unit-mean exponential spacings. For $r=\min(a,b)/\max(a,b)$, symmetry between $a<b$ and $b<a$ gives density $p(r)=2/(1+r)^2$ on $0\leq r\leq1$. Integrating $rp(r)$ gives the stated mean. Generate fresh spacing pairs for independent test samples; ratios from overlapping triplets in a single spectrum are correlated and must not be assigned independent-sample error bars.

A controlled limiting case

At $t=0$ with a continuous onsite distribution, eigenstates are exactly site localized and IPR is one, except for probability-zero exact degeneracies. At $W=0$, translation eigenstates of a periodic chain have IPR $1/L$, but a generic eigensolver may return standing-wave combinations of degenerate momenta. Thus individual clean-limit IPRs can depend on the chosen basis inside degeneracies.

Fix a momentum basis or compare invariant subspaces when using the clean system as a benchmark. This illustrates why arbitrary eigenvector output should not be treated as a unique physical state.

Finite-size and error analysis

Increase $L$ at fixed model parameters and energy selection. Compare enough independent disorder samples that sampling errors are smaller than the size trend. If several eigenstates come from one realization, use realization-level resampling for uncertainty; those eigenstates are not fully independent disorder draws.

Report solver residuals separately. Near-degenerate eigenvectors can be sensitive even when energies converge, so participation quantities may require tighter solver control than density-of-states plots. For long localization lengths, complement exact diagonalization with transfer matrices.

Exercise

A normalized state has equal probability on exactly $m$ sites. Calculate IPR and explain what an $L$-independent value means.

Solution

Each occupied site has probability $1/m$, giving IPR $m/m^2=1/m$. If $m$ remains fixed as $L$ increases, the state occupies a finite number of sites. One finite-size value alone cannot establish that $m$ will remain fixed in the large-system limit.

References

• Anderson, P. W., “Absence of diffusion in certain random lattices,” Physical Review 109, 1492–1505 (1958), doi:10.1103/PhysRev.109.1492.
• Atas, Y. Y., Bogomolny, E., Giraud, O., and Roux, G., “Distribution of the ratio of consecutive level spacings in random matrix ensembles,” Physical Review Letters 110, 084101 (2013), arXiv:1212.5611.