question is to ask what e↵ect disorder has (if any) on the nature of the quantum phase transition itself and the nature of the phases either side of it. It could be that disorder destabilises one phase in favour of another, or that the phase boundary stays the same but critical exponents are modified. In the case of Griffiths phases, disorder causes entirely new ‘mixed’ phases to appear in between the phases of the clean system. Our first port of call in determining the relevance of disorder is known as the Harris criterion - the following discussion is based upon Refs [76] and [77].
Harris Criterion
The Harris criterion states that disorder is a relevant perturbation to a renormalisation group fixed point in the clean system if ⌫d < 2 where ⌫ is the correlation length exponent and d is the spatial dimension. In the case of quenched disorder this criterion is equally valid for quantum and classical phase transitions due to the perfect correlation of disorder in time (i.e.dis not replaced with d+dz).
Critical points with quenched disorder can be classified by the behaviour of the average disorder strength [78]. For systems which fulfil the Harris criterion (i.e. for which ⌫d > 2), the disorder renormalises to zero on large length scales and the critical behaviour is that of the clean system in the absence of disorder. If the Harris criterion is violated (⌫d <2), there are two possibilities:
i) The average disorder strength remains finite at all length scales and the transition is controlled by a finite-disorder critical point, where the scaling is of conventional power-law form but the critical exponents have been modified from the clean system values.
ii) The average disorder strength increases under renormalisation, becoming more and more relevant at longer and longer length scales. The transition is there- fore controlled by an exotic infinite-randomness fixed point with unconventional scaling relations.
The Harris criterion and the classifications arising from it deal solely with the behaviour of the average disorder strength. This is not sufficient for a full description of disordered systems - randomness which leads to fluctuations on finite length scales can give rise to new physics not captured by the behaviour of the average disorder strength.
The presence of these relevant fluctuations on finite length scales lead to situations where disorder can cause certain rare regions of the system to be locally in one phase even when (based on the average disorder strength) the bulk system is predicted to be in another. These rare regions lead to nonanalyticities in the free energy known as Griffiths singularities in the vicinity of the phase transition and they can dramatically change the behaviour of the system.
Transitions in the presence of rare regions
The presence of rare regions can lead to three di↵erent situations, governed by the relation between the e↵ective dimensionality of the rare regions dRR
(which includes imaginary time) and the lower critical dimension of the ordering transition dc .
i) dRR < dc : When the dimension of the rare region is lower than the
lower critical dimension of the ordering transition, the rare regions cannot order independently of the rest of the system. Their contribution is at most power-law in the volume of the system, whereas the density of the rare regions decreases exponentially. The fixed point is a conventional finite-disorder type with power- law correlations. The rare regions thus lead to exponentially small corrections to the thermodynamic properties.
ii) dRR = dc : When the rare regions are precisely at the lower critical
dimension, they still cannot order independently of the bulk, however their contribution to the thermodynamics is exponential in the system volume. The
Figure 1.9 An illustration of an Ising spin system with di↵erent regions of the system locally in di↵erent phases. The black regions are paramagnetic and exhibit no long-range order, while the red and blue regions refer to patches of locally ordered ferromagnetism.
e↵ects of the rare regions can therefore overpower the low density, leading to Griffiths singularities in the free energy and non-power-law scaling. The renormalisation group fixed point associated with this type of transition is the so-called ‘infinite randomness’ type, where the e↵ects of disorder increase as the short-wavelength modes are integrated out and the disorder strength diverges.
iii) dRR > dc : In this case, the rare regionsare able to order independently
of the bulk system. Their contribution to the thermodynamics is such that not only does it change the behaviour, but in fact it destroys the sharp transition entirely, smearing it out over a wide region of phase space and leading to the formation of a new ‘mixed’ phase where rare ordered regions exist within an otherwise disordered system.
The case in which the transition is smeared out leads to a region known as a Griffiths region, or Griffiths phase. In this region, exponentially rare pockets of one phase exist within a bulk system which is in another phase. It is this type of phase that will be the main focus of Chapters 3, 4 and 5.