Anderson Localization Length Calculator

Introduction to Anderson localization length in a 1D disordered chain

Anderson localization in a one-dimensional tight-binding chain means that random on-site disorder can prevent a quantum state from remaining extended over long distances. Instead of spreading freely across the lattice, the wave amplitude decays away from a typical center, and the rate of that decay is summarized by the localization length ξ. This calculator turns that idea into a quick numerical estimate. You enter the disorder strength W, the energy E, and the chain length L, and the page reports both the estimated localization length and a simple finite-chain conductance proxy g.

Anderson localization is often introduced with broad statements such as disorder suppresses transport, but the useful question is more specific: how far does a state remain effectively extended for the chosen parameters, and what does that imply for a chain of finite length? The answer depends strongly on scale. A state with a localization length of 100 sites may look almost mobile in a chain of 20 sites but effectively insulating in a chain of 500 sites. That is why this page pairs ξ with g rather than stopping at a single abstract number.

The conventions matter. This calculator assumes the standard dimensionless 1D Anderson-model normalization where the nearest-neighbor hopping is set to t = 1. Under that convention, the clean tight-binding band runs from E = -2 to E = 2, so the energy input is restricted to |E| ≤ 2. The chain length is measured in lattice sites, not meters, and the conductance value is a dimensionless transmission-style estimate rather than a laboratory conductance in siemens. Those choices make the formulas compact and let you compare trends without carrying extra unit conversions.

Used that way, the calculator is best thought of as a fast physics sanity check. It is helpful for classroom work, parameter scanning, rough comparisons between disorder strengths, and checking whether a numerical simulation is being launched in a regime that is expected to be strongly localized or only moderately suppressed. It is not trying to replace transfer-matrix calculations, exact diagonalization, or a full transport solver with contacts and dephasing. Its value is speed, clarity, and physical intuition.

How to use the Anderson localization length calculator

Using this Anderson localization tool is easiest when you treat the three inputs as separate levers that control different parts of the physics. The disorder strength W governs how strongly the random on-site potential scatters the wave. The energy E places the state within the tight-binding band. The chain length L tells the model how much distance the state must survive before reaching the far side of the sample. Because the result depends on the comparison between ξ and L, the most informative runs are the ones where you intentionally vary one input at a time.

Disorder strength W has the strongest leverage in this formula because it enters as W² in the denominator. If you double W, the estimated localization length falls by a factor of four. That means a modest increase in disorder can make the same chain look dramatically more insulating. In practical use, try one low-disorder value, one intermediate value, and one higher-disorder value to see whether your conclusion is stable or extremely sensitive.

Energy E controls where you sit inside the clean-system band. In this approximation, states near the band center tend to localize less strongly than states near the band edges. That is why the factor 4 - E² appears in the localization-length formula. When E = 0, the factor is largest; as E moves toward ±2, it shrinks and the estimated localization length becomes shorter. The page accepts any energy inside the band, but the numerical estimate should be interpreted more cautiously near those edges.

Chain length L does not change the localization length itself; instead, it changes how much the finite sample amplifies the consequence of localization. A chain much shorter than ξ can still have a conductance estimate close to 1, while a chain several localization lengths long will give a very small g. That is why a result becomes most meaningful when you read the ratio qualitatively: is the chosen sample shorter than the decay length, roughly comparable to it, or far longer than it?

1. Enter W, E, and L in the form below.
2. Press Compute localization to generate the baseline estimate.
3. Read the reported ξ first, then compare it mentally with L.
4. Use the built-in scenario table to see how a 20% decrease or increase in disorder changes the outcome at fixed E and L.

If you are new to the model, a good habit is to keep two inputs fixed and change only the third. First vary disorder to build intuition for the strong 1/W² scaling. Then move the energy away from the band center and watch the localization length shorten. Finally hold the microscopic parameters fixed and increase the chain length to see how finite-size transport collapses even when the underlying state is not localized on the scale of only a few sites.

Formulas for the weak-disorder 1D Anderson estimate

The Anderson-model formulas used on this page are the actual formulas implemented by the page script. In the normalized convention t = 1, the localization length is estimated by the weak-disorder expression

That expression already tells you most of the physics. The numerator shows that localization is weakest near the band center and stronger near the band edges. The denominator shows that disorder does not merely reduce ξ linearly; it suppresses it quadratically. This is why a disorder sweep is so revealing. Even a 10% or 20% adjustment in W can noticeably change the transport picture over a fixed chain length.

After computing ξ, the page estimates a dimensionless conductance for a finite chain through an exponential attenuation law,

This second formula converts a localization length into an easily compared finite-length quantity. If the chain is short compared with ξ, then L/ξ is small and g stays closer to 1. If the chain is several localization lengths long, then L/ξ is large and the conductance estimate falls exponentially. Because g is dimensionless here, it should be read as a compact transport indicator rather than as a full experimental conductance prediction.

There are two especially useful interpretations to keep in mind. First, the formula can produce very large ξ when disorder is tiny and the energy is near the band center; in that case the page will return a conductance estimate close to 1 for moderate sample lengths, which is exactly what you would expect from weak scattering. Second, the conductance estimate can become extremely small once the sample length exceeds the localization length by a few multiples. That rapid drop is the signature feature the calculator is meant to highlight.

Worked example: W = 1.2, E = 0, and L = 200 sites

This Anderson localization worked example uses the page's default values so you can verify the arithmetic directly from the formulas above. Start with W = 1.2, E = 0, and L = 200. Because the energy is at the band center, the factor in parentheses becomes 4 - 0² = 4. Multiplying by 24 gives a numerator of 96. The disorder denominator is W² = 1.44. Dividing gives

ξ ≈ 96 / 1.44 = 66.667 sites.

Now compare that scale with the sample length. The chain is 200 sites long, so L/ξ ≈ 200 / 66.667 ≈ 3. Substituting into the conductance expression gives g = e^-3 ≈ 0.0498. In plain language, the state is not confined to only a handful of sites, but the chain is still long enough that the cumulative exponential decay leaves only a small transmission-style conductance estimate across the whole sample.

The same arithmetic also makes the disorder sensitivity intuitive. Lowering W by 20% raises ξ substantially because of the squared dependence, while raising W by 20% pushes ξ down sharply. The worked table below uses the same fixed E and L and changes only disorder so you can see how quickly the transport estimate responds.

The important lesson is not the last decimal place. The important lesson is the structure of the response: in this weak-disorder estimate, transport over a fixed chain can swing by a large factor when W changes by what might look like a modest amount. That is exactly why the scenario table in the calculator is useful for fast comparisons.

How to read Anderson localization results and the disorder-scaling table

The Anderson localization result panel is easiest to read in two passes. First, look at the reported localization length ξ. That number tells you the characteristic scale, in lattice sites, over which the wave remains appreciably extended before strong decay sets in. If ξ is much larger than your chosen L, the chain is short compared with the localization scale. If ξ is comparable to L, you are in a crossover regime where finite-size effects matter. If ξ is much smaller than L, the finite sample is deep in the exponentially suppressed regime.

Second, read the reported dimensionless conductance g as a compact summary of that comparison. Values near 1 correspond to weak attenuation across the chosen chain, while values near 0 indicate strong suppression over that distance. Because the formula is exponential, the number can change from moderate to tiny very quickly once L exceeds ξ by a few multiples. That is not a bug in the page; it is the main physical point the calculator is illustrating.

The automatic scenario table keeps E and L fixed and evaluates three disorder cases: W × 0.8, the baseline, and W × 1.2. This makes it easy to judge sensitivity without leaving the page. If all three rows tell the same qualitative story, your conclusion is robust. If the rows differ strongly, your transport picture depends sensitively on disorder amplitude and should be interpreted with more caution. The Copy Summary button then lets you save that text snapshot for notes, a research log, or a discussion with a collaborator.

Assumptions and limitations of the weak-disorder 1D Anderson estimate

This Anderson localization calculator is deliberately a compact estimator, not a full transport package. The formulas assume a one-dimensional nearest-neighbor tight-binding chain, dimensionless units with t = 1, and a weak-disorder regime in which the simple analytic expression for ξ is a useful guide. The page does not include electron-electron interactions, temperature dependence, dephasing, contacts, multiple channels, correlated disorder, magnetic fields, or higher-dimensional effects. As a result, the output should be read as a scale-setting estimate rather than as a universal prediction for every real material or device.

The Anderson-model approximation used here also becomes more delicate near the band edges. As |E| approaches 2, the factor 4 - E² becomes small, so the calculator predicts shorter localization lengths. That trend is physically sensible, but the exact numerical value is less reliable there than it is near the middle of the band. The same caution applies when disorder becomes strong enough that a weak-disorder derivation is no longer the right asymptotic description. The calculator will still return a finite number, but that number should be treated as qualitative guidance, not as a precision result.

A practical self-check is to verify the direction of change. Increasing W should reduce ξ. Moving E away from zero toward the band edges should also reduce ξ in this model. Increasing L while keeping ξ fixed should lower g. If your run behaves opposite to those expectations, the likeliest explanation is an input mistake or a misunderstanding of the dimensionless conventions. For serious edge-case work, direct numerical methods remain the safer next step.

Optional mini-game for Anderson localization intuition

This Anderson localization mini-game turns the same comparison into a quick visual reflex test. Each falling packet has its own W, E, and L, and your task is to route it left to Transmit when the localization length is long enough for that chain or right to Localize when the chain is too long compared with ξ. Early rounds make the decision easier by showing a direct ξ/L bar. Later rounds hide part of that information and push you to recognize the pattern from disorder and energy alone.

The game is optional, but it reinforces the same lesson as the calculator: stronger disorder and less favorable energies shorten localization length, and long chains magnify the effect through exponential decay. It does not change any calculator result, yet it can make the formulas easier to remember because you start associating the numbers with a transport outcome rather than reading them as abstract symbols only.