Figure 1.3: Abrahams et al. [1] formulated the scaling theory of localization.
Renormalization group flows of the βfunction for the zero temperature conductance of a disordered system with dimensionalityd =1, 2 and 3. Here,d =2 is the lower critical dimension and there are two possible cases depending on the sign of β in the metallic limit: (i) Weak Localization (WL, β → 0− asg → ∞) and (ii) Weak Anti-Localization (WAL, β → 0+ as g → ∞). The fixed point at β = 0 defines the critical point corresponding to the Anderson transition. These are shown as red circles. Systems exhibiting WAL must undergo an Anderson transition at some critical disorder strength.
by Eq. (1.16). When the system is metallic, conductance is large, which means that when lng → ∞, the β function is a constant. When a sample is insulating, conductance is vanishing. Therefore, when lng → −∞, the β function must be a straight line with a positive slope.
The β function, which gives the flow as g as L is changed, is shown in Fig. 1.3.
Consider the flow of β as the system size is increased. If β > 0 (β < 0), the conductance must increase (decrease) flowing to a metallic (insulating) phase in the thermodynamic limit. The critical point for the transition occurs when the system is scale invariant, at β(lngc)= 0. Clearly, the fixed point of the βfunction is unstable.
One of the predictions of this theory is the strong dependence of localization on the dimensionality of the sample. As seen in Fig. 1.3, for dimensions, d < 2, there is no critical point for an Anderson transition. This means that even for infinitesimal disorder, all eigenfunctions of the system are localized. On the other hand, for d = 3, there must be a localization transition with distinctive metallic and insulating phases. The one-parameter scaling theory has been verified numerically [57,77].
Two dimensions is the marginal case for the one-parameter scaling theory. It is impossible to tell if there is a transition or not without further information. In the classical limit, βfunction vanishes in the metallic limit as expected from Eq. (1.16).
Therefore, the sign of the lowest order corrections to the in the metallic phase are important. These determine the nature of states in the disordered system. The two distinct cases corresponding to the sign of the βfunction in the metallic limit are
g→∞lim β(g)=
0−, : Weak Localization (WL), 0+, : Weak Anti-localization (WAL).
(1.17) The sign of the β function also determines whether there exists a critical point as shown in Fig. 1.3. In WL systems, β < 0 for all disorder strengths. Therefore the phase is localized for any finite disorder. On the other hand, WAL systems must cross β = 0 for some critical disorder strength. Therefore, generic WAL systems exhibit a localization transition even in two dimensions.
The phenomena of Weak Localization (WL) and Weak Anti-Localization (WAL) can be understood from the perturbative corrections by disorder. The sign of this correction to the conductance determines the sign of the βfunction. Intuitively, the correction is due to quantum interference effects on the return probability of the elec- tron scattering through the disordered medium. The return probability of an electron is obtained from the transmission probability,t(E,x,x)(defined in Section1.2). The
Symplectic (GSE) 3 7 4 218
36π3 64 9π
Table 1.1: Classification of the different Random matrix ensembles as first proposed by Wigner and Dyson [20,115]. The classification is based on the symmetries of the random matrix: Time reversal (T) and spin-rotation (S). We also list the parameterθ that uniquely identifies the ensembles, and the level spacing statistics (P(s)defined in Eq. (1.18).
non-zero contributions to the return probability comes from self-intersecting paths and their corresponding time-reversed partners. Clearly, the presence or absence of time-reversal symmetry becomes important. In the presence of time-reversal symmetry, the contributions from both these paths may either constructively or destructively interfere depending on the phase difference between the two paths.
A constructive interference corresponds to a higher return probability. This leads to localization and a negative correction to the conductance. This phenomenon is termed as Weak Localization. In contrast, destructive interference corresponds to lower return probability and leads to delocalization. This case corresponds to Weak Anti-Localization. In physical systems, the phase-difference between the two paths can be tuned using a magnetic field. When time-reversal symmetry is present, spinless systems are expected to be WL and systems with spin-orbit coupling are WAL. The spin-orbit coupling can be viewed as an effective magnetic field for the electron. This means that in order for a system to have delocalized states in two dimensions, the spin-rotation symmetry must be broken.
Symmetry plays an important role in determining the nature of the delocalized metallic states, as evidenced by the drastically different behaviors of disordered systems in two dimensions. The general classification of random matrices into distinct invariant ensembles on the basis of symmetry was done by Wigner and Dyson [20, 115]. The different ensembles of random matrices are based on the presence or absence of time-reversal (T) and spin-rotation (S) symmetry. The three ensembles are (i) Gaussian Unitary Ensemble (breaksT, Sis irrelevant), (ii) Gaus- sian Orthogonal Ensemble (respects bothT and S), and (iii) Gaussian Symplectic Ensemble(respects T, breaks S). Some properties of the ensembles are listed in Table1.1. The general properties of the eigenvalues of disordered Hamiltonians (in
the delocalized phase) can be understood from this classification. For example, the energy levels corresponding to delocalized states of a disordered Hamiltonian have
’level repulsion’. This is because it is impossible for two delocalized states to have the same energy, unless the degeneracy is protected by some symmetry. This level repulsion among nearly degenerate levels can be obtained from the random matrix ensemble corresponding to the same symmetries of the Hamiltonian.
The repulsion among the eigenvalues puts constraints on the form of the level- spacing statistics. As we discussed in Section 1.2, localized levels do not have any repulsion and therefore, must obey Poisson statistics. In comparison, extended states of disordered systems must repel. Wigner postulated that the level-spacing statistics of the random matrices can be approximated by
Pθ(s) = Aθsθexp
−Bθs2
, (1.18)
whereθ = 1, 2, 4 corresponding to Orthogonal, Unitary or Symplectic ensembles respectively. The constants A(θ) and B(θ) are determined from normalization of the distribution and mean spacing to 1. The exact values of these coefficients for the different ensembles are shown in Table 1.1. Comparing P(s) in Eqs. (1.13) and (1.18), we see that the level spacing distribution is drastically different. For the delocalized states, the level spacing distribution is vanishing, lims→0P(s) ∼ sθ.