model, we first perform general scaling analysis for a quantum multicritical point. We adapt the arguments of Sections 2.6.1 and 1.5.1.
We begin by generalising equation (2.6.1) for a quantum multicritical point between two different types of order, each associated with a different dynamical exponent, z< and z>. To
tune to a quantum multicritical point we require two non-thermal tuning parameters r< and
r>, which tune to the quantum multicritical point at r> = r< = 0. There are now two
correlation lengths in real space, corresponding to the fluctuations of each order parameter. These lead to two correlation lengths in the imaginary time dimension. In addition, there are two temperature scaling fields T< and T> as in Section 2.6.1. The free energy splits into the
sum of the parts associated with each dynamical exponent, as
F (r<, r>, T) = b≠(d+z> >)f> 1 r<b1/‹> <, r>b1/‹> >,T<bz><,T>bz>> 2 + b≠(d+z<) < f< 1 r<b1/‹< <, r>b1/‹< >,T<bz<<,T>bz<> 2 . (3.2.1)
For generality we have allowed the correlation length exponents ‹< and ‹> to be different for
with the order with the dynamical exponent z>.
Because this energy depends on so many variables, it becomes difficult to analyse without knowing the scaling functions themselves. This model is even more complicated than the free energy in Section 2.6.1, and we are unable to make physical predictions about coupled multiple dynamic scaling.
We can make physical predictions in the special case of decoupled multiple dynamic scaling, where f> becomes independent of T< and r<, and vice-versa. We find that up to logarithmic
corrections, this method correctly predicts the leading order physical properties and regions of the phase diagram. However, it does not capture the effects of multicriticality which we find in the full Hertz-Millis model of a quantum multicritical point. Specifically, it does not predict the boundaries of the ordered phases or the temperature-dependence of the correlation length.
The free energy in the case of decoupled multiple dynamic scaling is
F (r<, r>, T) = b≠(d+z> >)f> 1 r>b1/‹> >,T>bz>> 2 + b≠(d+z<) < f< 1 r<b1/‹< <,T<bz<< 2 , (3.2.2)
which is the sum of the free energies in two completely independent quantum critical points, of the type analysed in Section 2.2.1. In this case, the specific heat and thermal expansion are just the sum of the specific heat and thermal expansions for two independent quantum critical points.
The free energy above implies that there are now four possible regions of the phase diagram in the disordered phases, separated by the two crossovers T< ≥ r‹<<z< and T> ≥ r‹>>z>. This is
in contrast to the model in Section 2.2.1 where there were only three regions in the disordered phase.
The crossovers Ti ≥ ri‹izi are interpreted as the crossovers when the modes associated with
zichange from the quantum critical region to becoming Fermi liquid-like. We label the regions
of the phase diagram (a) through (d), and discuss the physical properties in each region. In region (a), defined by T> ∫ r‹>>z> and T< ∫ r<‹<z<, both types of mode are quantum
critical. The specific heat and thermal expansion are given by
C ≥ c>Td/z>+ c<Td/z<, (3.2.3)
and
–≥ a>T[d≠(1/‹>)]/z>+ a<T[d≠(1/‹<)]/z< (3.2.4)
where the a> and a< constants are not necessarily of the same sign, as the derivatives drdp<
and dr>
dp may independently be positive or negative. These constants also contain the kinetic
coefficients, in the same way as in Section 2.6.1. The specific heat is dominated by the higher dynamical exponent, whereas the thermal expansion depends on both ‹> and ‹< as well as
the dynamical exponents. In situations where ‹> = ‹<, the higher dynamical exponent also
dominates the thermal expansion.
In the remaining three regions of the phase diagram, there are two components contributing towards both the thermal expansion and specific heat, and neither contribution can be argued to vanish.
In region (b), defined by T> ∫ r‹>>z> and T< π r<‹<z<, the modes associated with z> are
quantum critical and the modes associated with z< are Fermi liquid-like. The specific heat
and thermal expansion are given by
C ≥ c>Td/z>+ cÕ<|r<|‹<(d≠z<)T, (3.2.5)
and
–≥ a>T[d≠(1/‹>)]/z>+ a<Õ |r<|‹<(d≠z<)T. (3.2.6)
Fermi liquid-like and the modes associated with z< are quantum critical. The specific heat
and thermal expansion are given by
C≥ cÕ>|r>|‹>(d≠z>)T + c<Td/z<, (3.2.7)
and
–≥ aÕ>|r>|‹>(d≠z>)T + a<T[d≠(1/‹<)]/z<. (3.2.8)
In region (d), defined by T> π r‹>>z> and T< π r‹<<z<, both types of mode are Fermi-liquid
like. The specific heat and thermal expansion are given by
C ≥ cÕ>|r>|‹>(d≠z>)T + cÕ<|r<|‹<(d≠z<)T, (3.2.9)
and
–≥ aÕ>|r>|‹>(d≠z>)T + aÕ<|r<|‹<(d≠z<)T. (3.2.10)
We again note that these equations only hold if the modes decouple. When we reviewed the scaling analysis of classical multicriticality in Section 1.5.1, we were able to identify a crossover exponent which characterised the boundaries of the ordered phases. This is not possible to find for a quantum multicritical point via scaling, as in Hertz-Millis theory it is the temperature- dependence of the renormalised tuning parameter which governs the ordered phase boundary. As explored in Chapter 2, this is given by the dangerously irrelevant interactions.
The aim of the rest of this chapter is to use a specific model and perform a more thorough renormalisation group treatment in order to find the phase diagrams, and to determine whether these equations hold. As in the case of a single quantum critical point, we shall find that they do give the leading order contributions to specific heat and thermal expansion in each regime,
up to logarithmic corrections. However, the dangerously irrelevant interactions shape the phase diagram and give the renormalised tuning parameters their temperature-dependence.