Chapter 15
Magnetic ordering transitions of disordered systems

by T. Senthil and S. Sachdev

This chapter has been adapted from the Ph. D. thesis of T. Senthil, submitted to Yale University (1997), unpublished.

The last two chapters of this book move beyond the study of regular Hamiltonians which have the full translational symmetry of an underlying crystalline lattice, and consider the physically important case of disordered systems described by Hamiltonians with couplings which vary from point to point in space. By the standards of the regular systems already discussed, the quantum phase transitions of disordered systems are very poorly understood, and only a few well-established results are available: a large amount of theoretical effort has been expended towards unraveling the complicated phenomena that occur, and they remain active topics of current research. The aims of our discussion here are therefore rather limited- we highlight some important features which are qualitatively different from those of non-disordered systems, make general remarks about insights that can be drawn from our understanding of the finite T crossovers in Part 2, and discuss the properties of some simple solvable models.

In keeping with the general strategy of this book, we introduce some basic concepts by studying the effects of disorder on the magnetic ordering transitions of quantum Ising/rotor models studied in Part 2; we also make some remarks on the effects of disorder on the ordering transitions of Fermi liquids considered in Chapter 12. Models with much stronger disorder and frustrating interactions which have new phases not found in ordered systems will be considered in Chapter 16.

We begin by discussing a general stability criterion that must be satisfied by a quantum critical point in any disordered system: this leads to the requirement that the correlation length exponent n satisfy n ³ 2/d. Then we discuss the low energy spectrum of the phases away from the critical point: the presence of disorder introduces the so-called Griffiths-McCoy singularities. A first analysis of the random quantum Ising and rotor models is carried out by the field-theoretic methods of Chapter 8. Two solvable cases of the random quantum Ising model are considered next: models near the percolation transition and in d = 1. The latter are solved by a decimation renormalization group using a method developed by Fisher [23,24].