Conductance distribution in the magnetic field

ABSTRACT

Using a modification of the Shapiro scaling approach, we derive the distribution of conductance in the magnetic field applicable in the vicinity of the Anderson transition. This distribution is described by the same equations as in the absence of a field. Variation of the magnetic field does not lead to any qualitative effects in the conductance distribution and only changes its quantitative characteristics, moving a position of the system in the three-parameter space. In contrast to the original Shapiro approach, the evolution equation for quasi-1D systems is established from the generalised DMPK equation, and not by a simple analogy with one-dimensional systems; as a result, the whole approach became more rigorous and accurate.

KEYWORDS:

• Disordered systems
• localised states
• phase transitions
• quantum critical phenomena
• electrical conduction

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. We shall return later to validity of this statement. For the moment it suffices to note that this statement is rigorous in the framework of the orthodox scaling of the paper [16].

2. For definiteness, we keep in mind the case of the external magnetic field, while the following analysis is equally applicable for the case of the magnetic impurities.

3. In this case, it does not contain the parameter β, distinguishing the orthogonal and unitary ensembles. It agrees with the previously made conclusion that a strictly 1D system does not ‘feel’ the magnetic field.

4. A somewhat less general form of the equation was derived previously by Muttalib et al. [35–37] and was used in [38,39] for the description of the conductance distribution.

5. The analysis of the DMPK equation shows [30] that for large number of channels the structure of its solution does not depend on N. Formally it is obtained in the limit , , , when the DMPK equation reproduces the diagrammatic results.

6. In the higher orders in ε, one should take into account the difference of from the corresponding Wigner–Dyson values.

7. In fact, such terms should be cancelled, since there are no grounds for the term in the square bracket of (39(39) ).

8. The form of the matrix (A6(A6) ) is chosen from the analogy with a point scatterer, allowing to accept a zero value for the mean of ε [5].