Transfer Matrices for Lattice Systems

Lattice transfer matrices propagate stationary wavefunction data from one slice to the next. Long products reveal exponential growth and decay rates used to infer localization lengths. Those rates must be extracted with repeated orthogonalization; direct multiplication eventually retains only the most rapidly growing direction.

Required background. Tight-binding models fixes the Hamiltonian, and disorder and localization distinguishes wavefunction localization from a finite-sample transmission measurement.

The nearest-neighbor recurrence

For real constant hopping $t>0$ and onsite potential $V_n$, use

$$-t(\psi_{n+1}+\psi_{n-1})+V_n\psi_n=E\psi_n.$$

Then

$$\begin{pmatrix}\psi_{n+1}\\\psi_n\end{pmatrix} =T_n\begin{pmatrix}\psi_n\\\psi_{n-1}\end{pmatrix}, \qquad T_n= \begin{pmatrix} (V_n-E)/t&-1\\ 1&0 \end{pmatrix}.$$

Here $\det T_n=1$. Here $k$ denotes the dimensionless phase advance per lattice site. In a uniform chain $E=V-2t\cos k$, its eigenvalues are $e^{\pm ik}$ in the band. Outside the band, they are reciprocal real numbers, giving evanescence even without disorder.

For a strip with $M$ orbitals per slice and interslice hopping $-tI$, replace the top-left entry by $(H_n-EI)/t$ and the scalar off-diagonal entries by $\mp I$ as displayed. More general interslice hopping requires solving with that coupling matrix; singular coupling may invalidate this simple transfer form.

QR accumulation of growth rates

Choose an orthonormal matrix $Q_0$. Propagate through a modest block of slices, then factor

$$T_{n+b-1}\cdots T_n Q=Q_{\rm new}R.$$

Accumulate $\log|R_{jj}|$ and repeat. Over total physical length $\mathcal L$, the accumulated averages approach Lyapunov exponents $\gamma_j$. QR prevents all propagated columns from collapsing onto the largest-growth direction. Normalize diagonal phases consistently and retain logarithms instead of enormous product magnitudes.

For a conservative reciprocal setup with the appropriate current-preserving transfer structure, positive and negative exponents pair. In the scalar determinant-one case their sum should approach zero; loss of that check diagnoses arithmetic or accumulation errors.

Localization length and a clean benchmark

In a quasi-one-dimensional disordered strip, the smallest positive exponent gives the longest characteristic localization length $\xi=1/\gamma_{\min}$, in the declared physical length units. In a localized single-channel asymptotic regime, typical transmission behaves as $\log T\sim-2\mathcal L/\xi$; the factor two reflects probability versus amplitude decay.

For a clean chain with $V=0$ and $E=3t$,

$$\gamma a=\operatorname{arcosh}\frac{|E|}{2t} =\operatorname{arcosh}(1.5)\approx0.962424,$$

where $a$ is the lattice spacing. This is an evanescent gap solution, not Anderson localization. Inside the clean band the asymptotic growth exponents vanish, although finite products can have oscillatory transients.

Use both tests before adding random onsite energies.

Length, orthogonalization, and disorder statistics

Vary the QR block length. Too infrequent orthogonalization loses subdominant directions; excessively frequent QR can waste work but should not change converged exponents. Discard initial alignment transients and estimate uncertainty from sufficiently long blocks or independent disorder realizations.

Then vary total length and transverse strip width separately. A long strip at fixed width estimates a quasi-one-dimensional quantity; determining a higher-dimensional transition requires controlled finite-width scaling, including corrections and uncertainty. A crossing that drifts with strip width may disappear in the large-width limit.

Specify disorder distribution, correlations, seed/stream policy, boundary conditions, energy, width, and all length units. Average logarithmic quantities when studying typical transmission; arithmetic means can be dominated by rare high-transmission samples.

Exercises

What error results from interpreting an inverse length measured per lattice slice as an inverse physical length?

Solution

If the accumulated exponent per slice is $\widetilde\gamma$, then $\gamma=\widetilde\gamma/a$ and $\xi=a/\widetilde\gamma$. Omitting the factor $a$ changes the units and gives an incorrect physical localization length. For blocks of several slices, divide by the full propagated length, not by the number of QR factorizations.

References

• MacKinnon, Angus, and Bernhard Kramer, “The scaling theory of electrons in disordered solids: Additional numerical results,” Zeitschrift für Physik B 53, 1–13 (1983), doi:10.1007/BF01578242 — transfer-matrix localization lengths and finite-width scaling.