AGENESIA DE INCISIVOS LATERALES SUPERIORES

In this section we review the literature on metamagnetic quantum criticality. Metamagnetic transitions at zero temperature have been studied theoretically under the same assumptions behind the Hertz-Millis action [51, 32, 52]. The assumption is that the electronic degrees of freedom can be integrated out and the critical behaviour can be described in terms of overdamped spin-fluctuations alone. The resulting theory is similar to Hertz-Millis theory for a z = 3 order parameter, but with a magnetic field which couples linearly to the order parameter S = 1 —V ÿ Ên,q ‰≠1(q, Ên) „ (q, Ên) „ (≠q, ≠Ên) ≠h⁄ dxd· „3(x, ·) + u ⁄ dxd· „4(x, ·) , (5.2.4) where ‰≠1(q, Ên) = r + q2+ | Ên| q . (5.2.5)

As explained in Section 5.1, it is believed that we do not need to worry about the presence of non-analytic terms due to the finite magnetic field.

IV

I

III

I

II

_II

0

r

_{/ u}

− r

_{/ u}

_h

T

r

T

〜√uh

T 〜(h/ u)

Figure 5.5: Regions in the phase diagram of a metamagnetic quantum critical end-point in the h-T plane at fixed r, taken from Ref. [52]. The regions are described in the main text.

[52] using a renormalisation group approach, from which the phase diagram and thermody- namic properties can be found. They find that just as for the ferromagnetic quantum critical point, the quartic and higher order terms are irrelevant in the renormalisation group sense. When the quintic and higher order terms are neglected, the action is symmetric under h æ ≠h and so the free energy is an even function of h. In the rest of this section we summarise the key aspects of their results.

The critical part of the free energy has two components. One component comes from the optimal field configuration ¯„ which is non-zero at any non-zero h, and the other component comes from the Gaussian fluctuations of the field about this value.

There are two main crossovers in the h-T plane as shown in Figure 5.5. One crossover is

R3 ≥ uh2, which is the difference between low-field and high-field behaviour, which depends on the form of ¯„. Here, R is the renormalised tuning parameter r which has acquired some temperature-dependence. In the high-field region, ¯„ ¥ --_-6h

u

- -

-1/3sign(h), and in the low-field

region, ¯„ ¥ h

regimes, which is determined by the properties of the Gaussian fluctuations. This is controlled by the curvature of the effective potential at ¯„. In the low-field region this leads to a crossover at R3/2 _{≥ T , which is the crossover between the Fermi-liquid and quantum critical regions for}

a single z = 3 quantum critical point. In the regime at temperatures above this, region III in Figure 5.5, there is an additional crossover associated with whether the correlation length is dominated by thermal contributions or not. This is the difference between the two distinct regions in the quantum critical regime for a single Hertz-Millis quantum critical point. In the high field regime this crossover is at T ≥ u1/2_{h. This leads to four distinct regions of the phase}

diagram in Figure 5.5. Regions I and II are in the non-linear regime, and regions III and IV are in the linear regime. Regions I and IV are in the Fermi liquid regime, and regions II and III are in the quantum critical regime. Region III is further subdivided depending on whether the correlation length is dominated by temperature or the parameter r.

In calculating the thermodynamic properties, h and r depend upon both the external magnetic field H and pressure p. Because of this, derivatives with respect to external field and pressure are related, as

ˆ ˆH = ˆr ˆH ˆ ˆr + ˆh ˆH ˆ ˆh, (5.2.6) and ˆ ˆp = ˆr ˆp ˆ ˆr + ˆh ˆp ˆ ˆh. (5.2.7)

The strategy of Zacharias and Garst [52] is to first examine the thermodynamic properties assuming that h(H, p) = H ≠ Hú(p) and treat r as a constant, and subsequently look at

corrections to this. Under this approximation, derivatives with respect to H and p are directly proportional to each other. The relation Fcr(H ≠ Hú(p), T ) = Fcr(Hú(p) ≠ H, T) implies

that the thermal expansion goes to zero at the critical field.

the temperature derivatives originate from R(T ) = r + uT4/3 _{in regions II and III. The h = 0}

component of the specific heat coefficient in region III is T(d≠3)/3 _{which is the contribution}

from Gaussian fluctuations, and is the same as for a ferromagnetic quantum critical point. While the thermal expansion is zero when h = 0 at zero temperature, corrections to the finite temperature thermal expansion arise for two reasons; the pressure-dependence of r, and the temperature dependence of the field h. Due to the pressure-dependence of r, the thermal expansion has an additive component proportional to ”– ≥ T(d+z≠2)/z_{in region III, which is the}

same for the quantum critical ferromagnetic transition. This effect is negligible compared to the contribution from the temperature-dependence of the field h. Since h does not acquire any temperature-dependent renormalisation from the spin-fluctuations, it has the standard Fermi liquid form of h(T ) = h + hTT2. In region III of the phase diagram, when the correlation

length is dominated by temperature, this correction becomes

”–≥ hT

u T

≠(d≠2)/3 _(5.2.8)

for temperatures T ∫ (r/u)3/(d+1). A physical consequence of this is that at finite tempera- tures, the field at which the thermal expansion is zero is shifted by a temperature-dependent amount

h_{|–=0 ≥ h}TT2. (5.2.9)

The contribution to the physical properties from r(p, H) is subleading in all regions.

There are two Grüneisen parameters which can be measured in this system. The usual Grüneisen parameter , and the magnetic analogue H, discussed in Section 2.2.1. When r

is approximated as a constant, these two quantities are directly proportional to each other. When the effect of r changing is taken into account, these parameters are no longer directly proportional to each other due to the relations between the derivatives in equations (5.2.6) and (5.2.7). Both and H are proportional to the sum of the same two terms, but the coefficients

Figure 5.6: The compressibility, thermal expansion, and specific heat coefficient in the various regions of the phase diagram, taken from Ref. [52]. The regions of the phase diagram are described in the main text.

of each term are different in both cases.

The Grüneisen parameter and its magnetic analogue change sign at the critical field, when

h(T ) = 0. The leading order behaviour of the Grüneisen parameter in region I of the phase

diagram is found to be h ≥ _hln1₍1

h), and in region III it is found to be

H ≥ h/

1

T10/32 in

three dimensions.

We now discuss a model of a metamagnetic quantum critical end-point and an antiferro- magnetic quantum critical point in close proximity in the phase diagram.

5.3 Quantum Critical Metamagnetism and Antiferro-