Anderson Localization Length Calculator

What this calculator estimates

Anderson localization is the idea that disorder can trap a quantum state instead of letting it spread freely through a material. In a one-dimensional tight-binding chain, even modest randomness in the on-site potential can suppress transport dramatically. This calculator gives a fast estimate of that effect by combining three inputs you can vary directly: the disorder strength W, the electron energy E, and the chain length L. From those values it reports a localization length ξ in lattice sites and a dimensionless conductance g for the chosen sample length.

That makes the page useful in a very specific way. Instead of asking a vague question such as whether a disordered chain is mobile or insulating, you can ask a sharper one: for this amount of disorder and this energy, how far does a state remain appreciably extended before the wave amplitude decays, and how much transmission should remain across a chain of length L? A quick estimate is often enough to compare parameter ranges, check a simulation setup, or build intuition before you move on to a more exact numerical method.

All inputs here are dimensionless in the standard convention where the hopping parameter is normalized to t = 1. That is why the energy field is limited to |E| ≤ 2: in the nearest-neighbor 1D tight-binding model, that is the clean-system band. The chain length is measured in lattice sites rather than meters, and the conductance output is a dimensionless transmission-style quantity rather than a laboratory conductance in siemens. Keeping those conventions in mind is the main step that makes the results easy to interpret correctly.

How to choose the inputs

Disorder strength W describes how strong the random on-site potential is compared with the hopping scale. Larger W means stronger scattering, shorter localization length, and more rapid decay of transport. Because the formula contains W² in the denominator, disorder matters a lot: doubling W cuts the estimated localization length by a factor of four. If you are exploring a model rather than fitting a measured system, it is often helpful to test a low-disorder case, a medium case, and a high-disorder case to see how abruptly the behavior changes.

Energy E sets where you are within the tight-binding band. In this approximation, states near the band center tend to have longer localization lengths than states close to the band edges. That is why the factor 4 - E² appears in the numerator. When E = 0, that factor is largest; as E approaches ±2, it shrinks. The calculator accepts any value in the allowed interval, but you should remember that weak-disorder formulas become less trustworthy very close to the band edges, where the physics is more delicate than a single compact estimate can capture.

Chain length L is simply the number of sites the wave must cross. The localization length tells you how quickly amplitude decays, while the sample length tells you how much distance the state must survive. A short chain can still transmit reasonably well even when the disorder is noticeable, while a long chain with the same parameters can look effectively insulating because the exponential decay has more room to act. In practice, the most useful question is not whether ξ is big or small by itself, but whether it is big or small relative to L.

If you are new to the model, an easy workflow is to hold two inputs fixed and vary the third one intentionally. First change only W and watch the disorder-scaling table in the result panel. Then change E to compare the band center with an off-center state. Finally change L to see how the same microscopic disorder produces very different transport behavior in short and long chains. That kind of one-variable-at-a-time testing is more informative than entering a single set of numbers, reading the answer once, and moving on.

Formula used on this page

The calculator uses the weak-disorder 1D Anderson-model estimate already built into the page script. In the normalized convention t = 1, the localization length is approximated by:

Once ξ is known, the page estimates a dimensionless conductance using an exponential decay with chain length:

The physical meaning is straightforward. The factor 24(4 - E²) says the clean band structure and the chosen energy influence how tolerant the state is to disorder. The division by W² says stronger disorder very quickly suppresses the distance over which a wave remains coherent. Finally, the exponential for g says transmission falls off rapidly once the chain length exceeds the localization length. If L is much smaller than ξ, the conductance estimate stays relatively close to 1; if L is several times larger than ξ, the conductance becomes very small.

More abstractly, this page still follows the same structure as many scientific calculators: inputs are assembled into a function, and the output is then compared across scenarios. The generic function form below is preserved because it helps frame the calculator mathematically even though the specific Anderson-model formula above is the one that actually drives the result here.

Likewise, when you compare several contributions or scenario adjustments, a weighted sum can be a helpful way to think about how different influences accumulate. The next MathML block is not the transport equation used by the calculator, but it is a useful reminder that many modeling tools are built by combining a small number of meaningful inputs in a transparent way.

Worked example

Suppose you keep the default example values: W = 1.2, E = 0, and L = 200. Start with the localization length. Because E = 0, the energy factor becomes 4 - 0² = 4. Multiplying by 24 gives 96. The disorder term is W² = 1.44. Dividing those numbers gives ξ ≈ 66.667 sites. That tells you the wave remains appreciably extended over a scale of a few dozen sites, not hundreds.

Next compare that scale with the sample length. The chain is 200 sites long, so L/ξ ≈ 3. Plugging that into the conductance expression gives g = e^-3 ≈ 0.0498. In plain language, the state is strongly attenuated across the full chain. It is not that the disorder is astronomically large; rather, the chain is long enough that repeated scattering compounds into a very small transmission estimate. This is exactly the sort of result that helps when you are deciding whether a parameter set belongs in the localized regime for a simulation or a classroom example.

The scenario table below illustrates how sensitive the estimate is to disorder alone. Reducing W by 20% raises ξ from about 66.7 sites to about 104.2 sites and lifts the conductance to roughly 0.1466. Increasing W by 20% pushes ξ down to about 46.3 sites and the conductance down to roughly 0.0133. That spread is the practical lesson of the calculator: localization in one dimension reacts strongly to disorder, so apparently small changes in W can produce order-of-magnitude differences in transport over the same length.

How to read the result panel

When you press the button, the first line in the result panel is the main estimate for your exact inputs. Read the localization length first. If ξ is much larger than L, your chosen chain is short compared with the decay scale, so transport over that finite sample should be relatively less suppressed. If ξ is comparable to L, you are in a crossover region where finite-size effects matter and small parameter changes can visibly alter the conductance. If ξ is much smaller than L, the chain length dominates and the exponential decay drives the conductance toward zero very quickly.

The conductance output is then a compact way to summarize that comparison numerically. Values near 1 correspond to weak suppression across the chosen sample, while values that are tiny indicate strong localization over the distance you asked about. It is worth stressing that the conductance here is a model output, not a substitute for a full transport calculation with contacts, temperature, dephasing, or multi-channel effects. Still, as a quick indicator, it is highly useful: it converts the localization length into a finite-length prediction that is easier to compare from case to case.

The built-in scenario table is especially helpful if your main uncertainty is the disorder amplitude. It automatically evaluates W × 0.8, the baseline, and W × 1.2 at fixed E and L. That lets you judge sensitivity without opening a spreadsheet. If a 20% shift in disorder barely changes the answer, your conclusion is robust. If the answer swings sharply, you know the result depends on precise knowledge of W and you should be cautious about over-interpreting a single run. The copy button then gives you a quick plain-text snapshot you can paste into notes, a lab log, or a message to a colleague.

Assumptions and limitations

This page is intentionally a quick estimator, not a full transport solver. The formula assumes a one-dimensional tight-binding chain with weak disorder and the usual normalization t = 1. It does not include interactions, temperature effects, dephasing, multi-band structure, correlated disorder, or higher-dimensional physics. The result is therefore best used as a compact guide to scale and trend. It answers the question, 'What does the standard weak-disorder picture suggest?' rather than, 'What would every real device or numerical experiment do under all conditions?'

There are also two specific edge cases to watch. First, near the band edges where |E| approaches 2, the factor 4 - E² becomes small, so the calculator predicts a short localization length. That direction is sensible, but the approximation is also more fragile there, so treat the exact number cautiously. Second, for very strong disorder the weak-disorder derivation is no longer the right asymptotic tool even if the interface still returns a finite value. In those regimes, the calculator is still useful for intuition, but detailed conclusions should come from direct numerical methods or a theory adapted to the stronger-scattering limit.

A good sanity check is to ask whether the trend matches physics. Increasing W should reduce ξ. Moving E away from zero toward the band edges should also reduce ξ in this model. Increasing L while holding ξ fixed should lower g. If your result moves in the opposite direction, the most likely cause is an input mistake rather than a subtle transport phenomenon. That is why the cleanest way to use the tool is to run a baseline case, then alter one variable at a time and confirm that each response makes qualitative sense before you rely on the exact digits.

Optional mini-game for intuition

If you want a fast mental model for the same comparison the calculator performs, try the mini-game below. Each incoming wave packet carries its own W, E, and L. Your task is to route it left to Transmit when the packet's localization length is long enough for that chain, or right to Localize when the chain is too long relative to ξ. Early rounds show a clear ξ/L bar so the rule becomes obvious. Later rounds add band-edge fog and higher-disorder storms so you start recognizing the pattern from the variables themselves.

The game does not change the calculator's math, but it reinforces the core lesson in a way that is hard to forget: long localization length compared with sample size supports transmission, while strong disorder and unfavorable energy choices shrink ξ and trap the wave. It is optional, quick to replay, and meant to make the formal equations feel intuitive rather than abstract.