Resultado del Diagnostico e Interpretación y Análisis de la Información

Figure 6.1 (left) shows the speed of sound squared calculated from the dispersion relation (6.92), i.e.,

c2_s = βˆ1

2

Figure 6.1: Top: Speed of sound squared calculated from the dispersion relation (6.92) versus the dimensionless control parameter αp1

8πT. The red curve corresponds

to the symmetric phase where χ = 0. The purple curve corresponds to the first symmetry-broken phase withχ, _{0. Bottom: Percent di}fference between the speed of sound squared calculated in this section and that shown in figure 5.4.

versus the dimensionless control parameter αp1

8πT, where T is given by equation

(5.59). Figure 6.1 (bottom) shows the percentage difference between the results in figure 5.4 (bottom) and figure 6.1 (left). The difference is everywhere less than 0.007 %. This is a non-trivial check that our analysis of the background fluctuations in this chapter is correct.

The bulk-to-shear viscosity ratio can be related to ˆβ2 by comparing (6.92) and

(6.88). We find

ζ

η = βˆ2−1. (6.108)

Figure 6.2 shows the results for the bulk-to-shear viscosity. In the top figure, the red curve corresponds to the symmetric phase, and the black dashed curve is the correction to the conformal value ζ/η = 0 given by equation (6.90). There is an excellent agreement at small values of p1, which is another check that our analyses

are consistent and correct. In the bottom figure, the red curve corresponds to the symmetric phase, and the purple curve corresponds to the symmetry-broken phase. The green dashed line corresponds to the critical value of αp1

8πT ≈ 0.0771. The bulk

viscosity in the symmetry-broken phase appears to diverge at the transition. By pushing the calculation closer and closer to the transition, we can confirm that this is indeed the case.

As a final remark in this chapter, we will verify the bulk viscosity bound pro- posed in [6], which conjectures that

ζ η ≥2 1 p −c 2 s ! , (6.109)

where p is the spatial dimension of the field theory. Figure 6.3 shows the bulk-to- shear viscosity ratio as a function of 1/2₋ c2

s. The red curve corresponds to the

Figure 6.2: Top: Speed of sound squared calculated from the dispersion relation (6.92) versus the dimensionless control parameter αp1

8πT. The red curve corresponds

to the symmetric phase where χ = 0. The purple curve corresponds to the first symmetry-broken phase withχ, _{0. Bottom: Percent di}fference between the speed of sound squared calculated in this section and that shown in figure 5.4.

Figure 6.3: Bulk-to-shear viscosity ratio as a function of 1/2₋c2

s. The red curve

corresponds to the symmetric phase, and the purple curve corresponds to the first symmetry-broken phase. The dashed blue line indicates the lower limit of the bound

ζ/η _≥ 2(1/2 ₋ c2

s). The dashed green line corresponds to the high-temperature

approximation.

phase. The dashed blue line indicates the lower limit of the boundζ/η_≥ 2(1/2₋c2_s). We find that the bound is satisfied in both the symmetric and symmetry-broken phases. In fact, the bound is satisfied trivially in the symmetry broken phase since

c2

s > 1/2 (see figure 6.1). We also see once again the divergent behaviour ofζ/ηin

the symmetry-broken phase.

In this chapter we calculated the dispersion relation of small fluctuations of the background fields and extracted the hydrodynamics of the dual field theory. We found that the bulk viscosity in the symmetry-broken phase diverges at the phase transition. We verified the correctness of our results by comparing them against those of the near-conformal limit. Next we will do an in-depth study of the critical behaviour of the symmetry-broken phase near the phase transition.

Chapter 7

Critical behaviour in hairy AdS

₄

The observation of a second-order phase transition beckons a study of the critical behaviour of this system. In this chapter we will define a set of critical exponents by relating our parameters to those of models of ferromagnets. We will explicitly calculate all of the critical exponents, not all of which will be found to be of mean- field type. We find that some of the scaling relations that arise from the static scaling hypothesis are violated. We will also find that the symmetry broken phases are perturbatively unstable.

7.1

Criticality in ferromagnets and hairy AdS

To make our analysis as transparent as possible, it is convenient to cast the crit- ical behaviour in terms of the language of ferromagnets. We define the reduced temperature t to be

t = T

Tc −

So that the transition occurs at t= 0. In ferromagnetism, we also denote the external magnetic field by_H. The Gibbs free energy density is given by

W(t,_H)=ǫ ₋sT _{− MH}

= Ω_o(t,_H)₋Ω_d(t,_H),

(7.2)

where ǫ is the energy density, s is the entropy density, and Ωo and Ωd are the

Helmholtz free energies in the ordered phase and disordered phase respectively. As we traverse the critical temperature, there is a spontaneous magnetization in the system given by M =₋ ∂W ∂_H ! t . (7.3)

The two-point correlation function of the magnetization is defined as

G (r)= _hM(r)_M(0)_i (7.4)

The critial exponents1_{α, β, γ, δ, ν, η_}are defined [43] as follows:

c_{H ∼ |}t_|−α (7.5) M ∼ |t_|β (7.6) χT ∼ |t|−γ (7.7) M(t= 0)_{∼ |H|}1δ _(7.8) G (r)_∼          e−|rξ| _{, t, 0} |r_|−p+2−η, t =0 , where ξ_{∼ |}t_|−ν, (7.9) 1_{Do not confuse the critical exponentαwith the parameterαin (5.51). Likewise with the ex-}

ponentηand the shear viscosityη. Their use should be clear from the context in which they are used.

where c_H is the specific heat,χT is the isothermal susceptibility, andξis the corre-

lation length. Under the scaling hypothesis [43], we have the following relations

α+2β+γ= 2

γ =β(δ₋1)=ν(2₋η) 2₋α= νp,

(7.10)

where p is the spatial dimension of the system.

In order to employ this language we need to relate the quantities that we have been working with in our Exotic AdS4 model to those given so far in this section.

In ferromagnetism, the order parameter is _M; that is, it is zero below the transi- tion, and non-zero above the transition. The quantity in our model that fits this description isχ4(see figure 5.1). So we identify that

M ⇔χ4. (7.11)

The external control parameter in our model isαp1, so we identify it with the ex-

ternal magnetic field,

H ⇔αp1, (7.12)

whereαhere is as in (5.51). Having established the language in which to define the critical exponents, we will now calculate them.