Link to Experiments and Outlook

We now look back to the experiments discussed in Section 3.6.2 and compare our results to them. In Section 3.5 we found that in the presence of quantum critical fluctuations of both ferromagnetism and antiferromagnetism, ferromagnetism dominates thermodynamic proper- ties. These results are consistent with the experimental data on NbFe2 and Ta(Fe1≠xVx)2

where the specific heat obeys C ≥ T ln11 T

2

usually associated with three dimensional ferro- magnetism. To our knowledge no thermal expansion measurements have been performed on these materials, but we would expect to see the thermal expansion vary as – ≥ T1/3 _{and a}

Grüneisen parameter obeying ≥ 1/1T2/3ln1_T122, as would be expected of a ferromagnet in

three dimensions.

Another prediction of this theory is the shape of the boundaries on the phase diagram, which are predicted in Sections 3.4.4 and 3.4.7. While the phase diagrams of both NbFe2 and

Ta(Fe1≠xVx)2 are known [4, 8], for both materials there are not enough data points on the

boundary of the antiferromagnetic phase to determine the power law and test the prediction of the theory.

The theory we have developed appears to be consistent with experimental data on NbFe2

and Ta(Fe1≠xVx)2, but questions still remain. In the analysis presented here ferromagnetism

seems to dominate everything, so the experimental evidence that resistivity in NbFe2 and

Ta(Fe1≠xVx)2 is dominated by antiferromagnetism is somewhat of a surprise. In Chapter 4 we

solve the Boltzmann transport equation to find the resistivity of the model described in this chapter and show that the power-law usually associated with the antiferromagnetic QCP does dominate. We are able to explain why this is the case, and why this power law is actually stabilised in the presence of ferromagnetic fluctuations.

The results we have derived rely upon the antiferromagnetic ordering wavevector Q being sufficiently large. There has been some discussion in the literature as to whether the anti- ferromagnetic wavevector in Nb1≠yFe2+y is always finite or whether it smoothly evolves from

zero with doping [7]. In Chapter 4 we show that the observed T3/2 _{power law relies upon the}

presence of strong scattering at a finite Q.

Another issue that we wish to address is the problems with the model itself. As explained in Section 2.5, Hertz-Millis theory is hindered by the fact that integrating out the electronic modes to construct an effective theory in terms of spin-fluctuations alone is not a safe thing to do, as non-analytic terms arise in the effective action when a more careful analysis is done. Despite this, Hertz-Millis theory correctly predicts power laws in many systems [1]. Since the model we derive in equation (3.3.13) also ignores these non-analytic terms, it is also plagued by the same illnesses. We must then question to what extent we trust the predictions of this model.

For the same reason that no true ferromagnetic QCPs exist, we would expect that a true quantum multicritical point between ferro- and antiferromagnetism could never exist. At some low temperature the ferromagnetic QCP would be driven first order by the non-analytic terms

which should exist in the model if the electrons are integrated out more carefully, as in Section 2.5. However, above a typical energy scale induced by these non-analyticities we would expect the thermodynamic properties of the material to be dominated by quantum critical fluctuations of both ferromagnetic and antiferromagnetic order. Above this energy scale, we expect our predictions to hold.

Even though the experimental links suggest our model is to be trusted, in Chapter 5 we analyse a quantum multicritical model where the non-analytic terms do not arise. Specifically, we analyse a metamagnetic quantum critical end-point described by the dynamical exponent

z = 3 interacting with a z = 2 antiferromagnetic QCP. Metamagnetic transitions occur at

finite magnetic fields, and these fields wipe out the non-analyticities associated with the fer- romagnetic QCP. However, the magnetic field adds its own complications so we devote all of Chapter 5 to an analysis of this model.

Chapter 4

TRANSPORT NEAR A QUANTUM

MULTICRITICAL POINT

In the previous chapter we demonstrated that at a quantum multicritical point, ferromagnetism seems to dominate the thermodynamic quantities. Our prediction that the ferromagnetic contribution dominates the specific heat agrees with experiments on NbFe2, Ta(Fe1≠xVx)2 and

YbRh2Si2. In YbRh2Si2 our prediction that the Grüneisen parameter is dominated by the

ferromagnetic term is also seen.

One quantity which cannot be obtained directly from the renormalisation group approach of the previous chapter is the resistivity, as it is not a simple derivative of the free energy. In both NbFe2 and Ta(Fe1≠xVx)2 this is seen to obey a fl = fl0 + cT3/2 power law, which would

usually be associated with antiferromagnetic quantum criticality in a disordered system. The resistivity near quantum critical points is well-understood. In a material near a fer- romagnetic quantum critical point the resistivity obeys a T5/3 _{power law [44]. Near an anti-}

ferromagnetic quantum critical point, in perfectly clean systems a T2 _{resistivity is expected}

whereas in disordered systems a T3/2 _{power law is expected at low temperatures. However,}

in these disordered systems there is a large crossover region with an almost-linear power law [45].

If one were to naively suppose that the ferromagnetic scattering process provides a T5/3

power law, then we could conclude that the antiferromagnetism would dominate, as is experi- mentally observed. We shall show in this chapter that this justification is flawed. We perform numerical calculations to investigate the resistivity near a quantum multicritical point in three dimensions. We shall show that the ferromagnetic fluctuations actually stabilise the T3/2_power

law, and even in clean systems we expect this power law to hold.

We first discuss the methods involved in calculating the resistivity, and how they have been applied to quantum criticality. Section 4.1 is a background section where we review the Boltzmann equation, and how it can be used to find the resistivity. In Section 4.2 we review the literature documenting the application of the Boltzmann equation to the ferromagnetic and antiferromagnetic quantum critical points. We also write the equations in a form which we find convenient to numerically study in subsequent sections. In Section 4.3 we discuss original research on how the Boltzmann equation can be used to find the resistivity near a quantum multicritical point. We discuss the results in Section 4.3.2 and conclude by referring back to the experimental data in Section 4.5.

4.1 The Boltzmann Equation