---
title: One-Dimensional Anderson Model
url: https://www.emergentmind.com/topics/anderson-model-in-one-dimension
type: topic
---

# One-Dimensional Anderson Model

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The Anderson model in one dimension is a random lattice or continuum Schrödinger problem in which hopping or kinetic energy competes with spatial disorder. In its standard discrete form it is the nearest-neighbor operator \( (H\psi)(n)=\psi(n+1)+\psi(n-1)+V_n\psi(n)\), equivalently a tight-binding fermion Hamiltonian with random onsite energies. It is a progenitor of many further models of disordered systems, and its one-dimensional version is distinguished by the fact that, in the thermodynamic limit, all states are localized for arbitrarily small disorder, even though finite-size scaling, spectral diagnostics, entanglement, and correlated or driven generalizations reveal a much richer internal structure [2507.06903] [1306.5712] [2110.00097].

## 1. Canonical formulation

In the discrete one-dimensional setting, the standard Anderson model appears in several equivalent notations. As a random Jacobi operator it is the \(W=1\) specialization of
\[
(H\psi)(n)=L_n\psi(n+1)+V_n\psi(n)+L_{n-1}\psi(n-1),
\]
with the classical Anderson choice \(L_n\equiv 1\), so that
\[
(H\psi)(n)=\psi(n+1)+\psi(n-1)+V_n\psi(n).
\]
In fermionic language one may write
\[
H=\sum_r \left(c_r^\dagger c_{r+1}+c_{r+1}^\dagger c_r\right)+\sum_r \varepsilon_r\,c_r^\dagger c_r
\]
or, with explicit hopping scale,
\[
\hat H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)+\sum_i(V_i-\mu)n_i.
\]
These are all nearest-neighbor, one-dimensional, noninteracting lattice realizations of the same basic disorder problem [2110.00097] [1408.5850] [1306.5712].

Representative formulations used in recent work are summarized below.

| Formulation | Operator | Disorder specification |
|---|---|---|
| Standard discrete chain | \((H\psi)(n)=\psi(n+1)+\psi(n-1)+V_n\psi(n)\) | i.i.d. onsite potential |
| Fermionic tight-binding chain | \(H=\sum_r(c_r^\dagger c_{r+1}+c_{r+1}^\dagger c_r)+\sum_r \varepsilon_r c_r^\dagger c_r\) | \(\varepsilon_r\in[-w/2,w/2]\) |
| Diagonal Anderson model in DCA/CTMT form | \(\hat H=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)+\sum_i(V_i-\mu)n_i\) | \(P(V_i)=\frac{1}{2W}\Theta(W-|V_i|)\) |
| Generalized tridiagonal model | random diagonal density \(h(x)\), random hopping density \(g(x)\) | arbitrary \(h\) and \(g\) |

The disorder law is not unique. Box-distributed onsite energies are common in numerical work, for example \(\varepsilon_r\in[-w/2,w/2]\) or \(V_i\in[-W,+W]\). Recent supersymmetric work also formulates a generalized one-dimensional tridiagonal model with diagonal density \(h(x)\) and nearest-neighbor hopping density \(g(x)\), so that the standard Anderson chain is recovered by taking deterministic hopping \(g(x)=\delta(x-1)\) and random onsite energies \(h(x)\) [1408.5850] [1306.5712] [2507.06903].

Boundary conditions and energy conventions are study-dependent. Open chains are used in entanglement calculations; periodic boundary conditions are used in large-system two-particle calculations; Dirichlet or Neumann restrictions are standard in finite-volume spectral theory. Half filling in the fermionic chain corresponds to \(E_F=0\) in the nearest-neighbor model [1408.5850] [1102.2109] [1602.08257].

## 2. Localization and thermodynamic scaling

A defining one-dimensional fact is that the thermodynamic critical disorder is
\[
W_c=0.
\]
Cluster typical medium theory reproduces this indirectly: at finite cluster size \(N_c=L_c\), the apparent critical disorder decreases with cluster length as
\[
W_c\sim \left(\frac{1}{L_c}\right)^{1/\nu},
\qquad
\xi=L_c\sim W_c^{-\nu},
\]
with fit
\[
1/\nu=0.46\pm 0.10,
\]
in basic agreement with previous numerical work reporting \(1/\nu=0.5\). In that framework the localization indicators are the typical density of states
\[
\rho_{typ}^c(K,\omega)=\exp\langle \ln \rho^c(K,\omega)\rangle
\]
and the hybridization rate \(\operatorname{Im}\Gamma(K,\omega)\), both of which vanish at the finite-cluster localization threshold [1306.5712].

Rigorous single-scale localization theory for quasi-one-dimensional random operators reduces, in the \(W=1\) scalar case, to the standard one-dimensional Anderson model. Under assumptions allowing singular distributions, the spectrum is almost surely pure point, and every eigenfunction satisfies exponential decay at a rate at least the Lyapunov exponent \(\gamma_1(E)\). The same framework yields exponential decay of eigenfunction correlators on compact energy windows, which implies exponential dynamical localization and exponential decay of the Fermi projection [2110.00097].

The transfer-matrix viewpoint remains central across formulations. For the standard scalar model the transfer matrix is
\[
T_n(E)=
\begin{pmatrix}
E-V_n & -1\\
1 & 0
\end{pmatrix},
\]
or, for random Jacobi hopping,
\[
T_n(E)=
\begin{pmatrix}
L_n^{-1}(E-V_n) & -L_n^{-1}L_{n-1}\\
1 & 0
\end{pmatrix}.
\]
Large-deviation control of products of these matrices is the mechanism behind positivity of the Lyapunov exponent and the corresponding exponential decay bounds [2110.00097].

## 3. Finite-volume spectral structure

For finite one-dimensional boxes, the spectrum is simple and quantitatively nondegenerate. For the discrete operator
\[
(Hu)_n=u_{n-1}+\varepsilon_n u_n+u_{n+1},
\qquad |\varepsilon_n|\le W<\infty,
\]
with Dirichlet or Neumann boundary conditions, distinct eigenvalues satisfy
\[
|E_i-E_j|\ge \frac{\pi(\eta(W)^{-1}-1)}{\eta(W)^{-N}-1},
\]
hence
\[
|E_i-E_j|\ge Ce^{-bN}.
\]
The proof is one-dimensional in a strong sense: it uses the recursion
\[
u_{n+1}(E)=(E-\varepsilon_n)u_n(E)-u_{n-1}(E),
\]
a Prüfer-angle representation, and strict monotonicity of the terminal angle in energy [1102.2109].

A complementary spectral development is the supersymmetric reduction of the disorder-averaged density of states to a scalar integral equation. For the generalized tridiagonal model, the dominant supersymmetric transfer eigenfunction \(\Psi_*(r,E)\) satisfies
\[
\Psi_*(r)= -\sqrt{\tilde h(r)}\,e^{iEr/2}
\int_0^\infty dr'\,\partial_{r'}
\Big[\sqrt{\tilde h(r')}\,e^{iEr'/2}\Psi_*(r')\Big]
\int dx\,g(x)\,J_0\!\big(2x\sqrt{rr'}\big),
\]
and the density of states is then
\[
\rho(E)=\frac1\pi \Re\int_0^\infty dr\,\Psi_*^2(r,E).
\]
For deterministic hopping \(g(x)=\delta(x-1)\), Thouless’ formula gives the localization exponent directly from \(\rho(E)\):
\[
\lambda(E)=\int dE'\,\rho(E')\log|E-E'|.
\]
This provides analytic access to spectral and localization data for arbitrary onsite disorder distributions [2507.06903].

Another finite-volume perspective uses the principal eigenvalue \(\lambda_{n,V}\) and the landscape function \(L_{n,V}=(-\Delta_{\Lambda_n}+V)^{-1}1\) on the box \(\Lambda_n=[-n,n]\cap\mathbb Z\). For nonnegative i.i.d. potential satisfying either \(0<F(0)<1\) or
\[
F(t)=c\,t^\eta(1+o(1))\qquad \text{as }t\downarrow0,
\]
the one-dimensional result is
\[
\lim_{n\to\infty}\lambda_{n,V}\,\|L_{n,V}\|_\infty=\frac{\pi^2}{8}
\qquad \mathbb P\text{-a.s.}
\]
The asymptotics are governed by the longest interval where the potential is zero or very small, with \(\lambda_{n,V}\sim \mu_1 y_n^{-2}\) and \(\|L_{n,V}\|_\infty\sim y_n^2/2\) [2203.01059].

## 4. Entanglement, local structure, and conserved charges

For the free-fermion ground state of the canonical one-dimensional Anderson chain at half filling, bipartite entanglement is computed from the subsystem correlation matrix
\[
C_{mn}=\langle c_m^\dagger c_n\rangle,
\]
with entropy
\[
S=-\sum_{l=1}^{N_A}\left[\zeta_l\ln\zeta_l+(1-\zeta_l)\ln(1-\zeta_l)\right].
\]
In the clean chain the entropy grows logarithmically with system size. For any nonzero box disorder, the entropy first follows the clean trend, then departs near the mean free path \(\ell\), and saturates around
\[
L\approx 3\ell.
\]
At \(E_F=0\), perturbation theory gives
\[
\ell \approx \frac{48}{w^2}.
\]
In one dimension \(\xi\sim \ell\), but the higher-dimensional comparison in the same work leads to the interpretation that the entanglement crossover is controlled by the mean free path rather than the localization length [1408.5850].

With nearest-neighbor repulsive interactions added to the canonical onsite-disordered chain,
\[
H=\sum_{j=1}^{L}\epsilon_j c_j^\dagger c_j
-t\sum_{j=1}^{L-1}c_j^\dagger c_{j+1}+h.c.
+U\sum_{j=1}^{L-1}\left(c_j^\dagger c_j-\frac12\right)\left(c_{j+1}^\dagger c_{j+1}-\frac12\right),
\]
the entanglement-entropy distribution changes qualitatively from the random-bond random-singlet case. DMRG for \(L=700\) and \(100\) disorder realizations shows a broad skewed low-entanglement peak and one narrower peak near \(\ln 2\), but no peaks at higher integer multiples of \(\ln 2\). The proposed explanation is that onsite disorder breaks the symmetry across the cut, so singlet-like contributions are broadened below \(\ln 2\) rather than being sharply quantized [1710.05322].

At the operator level, the localized phase admits explicit conserved charges. For the disordered hopping Hamiltonian
\[
H(y)=\sum_i \epsilon_i n_i-y\sum_{ij} t_{ij}c_i^\dagger c_j,
\]
one can construct
\[
Q_i=P_{i0}+yP_{i1}+y^2P_{i2}+\dots,
\qquad P_{i0}=n_i,
\]
with quadratic corrections
\[
P_{im}=\sum_{jk}\eta^{(m)}_{jk}(i)c_k^\dagger c_j.
\]
The coefficients satisfy a recursion determined by \([Q_i,H]=0\). In the standard gauge \(R_a^{(m)}(i)=0\), the Hamiltonian itself may be written as
\[
H=\sum_i \epsilon_i Q_i.
\]
For nearest-neighbor Anderson hopping the series is generically infinite; for the Type-1 infinite-range model \(t_{ij}=\gamma_i\gamma_j\), it truncates at linear order. The same construction converges in the localized phase of the Aubry–Andre model and fails in its extended phase, making convergence of the charge expansion a localization diagnostic [1503.07019].

## 5. Interactions and multiparticle extensions

The two-interacting-particle problem on the one-dimensional Anderson chain has become a benchmark for interaction-enhanced localization length. For two particles with onsite disorder \(V(x)\in[-W/2,W/2]\), periodic boundary conditions for eigenfunctions, and finite-range interaction
\[
\hat U = U P,
\qquad
P=\sum_{|x_1-x_2|<U_R}|x_1,x_2\rangle\langle x_1,x_2|,
\]
large-scale Green function Arnoldi calculations reach \(N=5000\) sites, corresponding to a Hilbert-space dimension \(\approx 10^7\). The one-particle band-center localization length is taken as
\[
L_1=\frac{105}{W^2},
\]
and one-parameter scaling is found to be reliable only when \(N\) is already significantly larger than \(L_1\). In the optimal interaction window, the two-particle localization length obeys
\[
L_2\sim L_1^2,
\]
more precisely fitting forms such as
\[
L_2=aL_1+bL_1^2,
\]
and, for \(U=2\), \(E=0\),
\[
L_2=0.569\,L_1+0.047\,L_1^2.
\]
The same study shows that the interacting eigenstates are sparse in the basis of noninteracting product eigenfunctions and are organized by energy resonance together with approximate momentum conservation [1602.08257].

For finitely many interacting particles and singular disorder, one-dimensional Bernoulli models admit Wegner bounds strong enough for multiscale analysis. On the lattice, the \(N\)-particle Hamiltonian
\[
\mathbf H_h^{(N)}(\omega)= -\boldsymbol\Delta+\sum_{j=1}^N V(x_j,\omega)+h\,\mathbf U
\]
obeys one-volume and two-volume Wegner estimates in the weak-interaction regime \(|h|<h^*\), which imply spectral exponential localization and strong dynamical localization for the one-dimensional multi-particle Bernoulli–Anderson model [1612.08303]. In the continuum, the analogous one-dimensional \(N\)-body alloy-type operator
\[
\mathbf H^{(N)}(\omega)=-\boldsymbol\Delta+\mathbf U+\mathbf V
\]
with Bernoulli-type single-site amplitudes also satisfies one- and two-volume Wegner bounds for sufficiently weak interaction, again yielding Anderson localization via multiscale analysis [1701.05050].

## 6. Correlated disorder, extended hopping, and driven analogues

Several one-dimensional generalizations modify the localization mechanism without restoring an ordinary metallic phase. A recent example is the Anderson chain with stealthy disorder, defined in Fourier space by
\[
S(q)=\frac{|w_q|^2}{W^2},
\qquad
S(q)=\Theta(|q|-k_0),
\qquad
\chi=\frac{k_0}{2\pi}.
\]
For the clean dispersion \(\epsilon_k=-2\cos k\), weak-disorder perturbation gives
\[
\frac{1}{\xi(k)}=\frac{W^2}{8\sin^2 k}\,S(2k).
\]
Thus first-order backscattering is suppressed when \(2k<k_0\). Higher-order processes then dominate, producing regimes \(\xi\sim W^{-4}\) and \(\xi\sim W^{-6}\). For fixed finite \(L\), fixed small \(W\), and suitable \(\chi\), the localization length can exceed the system size, a regime described as effective delocalization [2509.13502].

Adding next-nearest-neighbor hopping changes weak-disorder anomalies while preserving localization. In the \(t\)-\(t'\) chain
\[
t'(\Psi_{n+2}+\Psi_{n-2})+t(\Psi_{n+1}+\Psi_{n-1})+\epsilon U_n\Psi_n=E\Psi_n,
\]
the standard Kappus–Wegner anomaly survives as a wavelength resonance but shifts in energy from \(E=0\) to
\[
E=-2t'.
\]
Additional nonuniversal anomalies appear at wave vectors determined by
\[
x^3=3px-2q,
\qquad
x=2\cos k,
\qquad
p=\frac23,
\qquad
q=\frac h2,
\]
showing that commensurability effects in one dimension persist beyond the nearest-neighbor chain [1311.1689].

Sparse off-diagonal disorder leads to a more radical departure from the canonical i.i.d. onsite model. For the exponentially sparse Jacobi matrix with weakened hoppings \(p_n^\omega\in\{1,p\}\), \(p\in(0,1)\), one finds a one-dimensional spectral transition: singular continuous spectrum near the band center and dense pure point spectrum near the band edges. The transition interval is controlled by
\[
\lambda^\pm=\pm 2\sqrt{1-\frac{v^2}{v_c^2}},
\qquad
v=\frac{1-p^2}{p},
\qquad
v_c=2\sqrt{\beta-1}.
\]
This model is “Anderson-like” but not equivalent to the standard i.i.d. onsite-disordered chain [1106.4852].

Time-dependent and unitary analogues preserve one-dimensional localization in different forms. For quasiperiodic modulation of onsite disorder with finitely many incommensurate frequencies, the effective channel number scales as
\[
\Delta\sim \frac{\mu W}{\omega},
\]
and the localization length can be strongly enhanced in the multi-channel regime, yet remains finite for any finite number of driving frequencies [1602.02476]. In a distinct Floquet map with maximal phase disorder \(W=2\pi\), all quasienergy eigenstates are exponentially localized with the same localization length,
\[
\frac{1}{L_{\mathrm{loc}}}=\left|\ln\left(|\sin\theta|\right)\right|,
\]
providing an exact instance of uniform Anderson localization in one dimension [2204.02717].

A further non-ergodic continuum variant replaces stationary disorder by a decaying random envelope,
\[
H_{\omega,\lambda}=-\frac{d^2}{dx^2}+\lambda\sum_{n\in\mathbb Z} a_n\omega_n u(x-n),
\qquad
a_n n^\alpha\to 1.
\]
For
\[
0<\alpha<\frac12,
\]
one has dynamical localization on compact positive-energy intervals, but with stretched-exponential rather than exponential spatial decay:
\[
\mathbb E\bigl[Q_{\omega,\lambda}(x,y;I)\bigr]\le C e^{-c|x|^{1-2\alpha}},
\qquad
\|\chi_x\psi_{\omega,E}\|\asymp e^{-c|x|^{1-2\alpha}}.
\]
This matches the cumulative strength of the decaying randomness \(\sum_{j\le n}j^{-2\alpha}\asymp n^{1-2\alpha}\) [2001.02197].

In this enlarged one-dimensional landscape, the canonical Anderson chain remains the reference problem: nearest-neighbor hopping plus onsite disorder, pure-point localization for arbitrarily small disorder, and a transfer-matrix/Lyapunov framework that continues to organize later developments. What changes across extensions is not the central role of disorder, but the scale at which backscattering, resonance, conserved structure, or entanglement reconfigure the localized phase.

Source: https://www.emergentmind.com/topics/anderson-model-in-one-dimension