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Focusing on the connection between the Landau theory of second-order phase transitions and the holographic approach to critical phenomena, we study diverse field theories in an anti de Sitter black hole background. Through simple analytical approximations, solutions to the equations of motion can be obtained in closed form which give rather good approximations of the results obtained using more involved numerical methods. The agreement we find stems from rather elementary considerations on perturbation of Schrödinger equations.

Much activity has been centered in the last few years on the application of the gauge/gravity duality, originally emerged from string theory [

In the holographic approach to the study of phase transitions one starts, on the gravity side, with a field theory in asymptotically anti de Sitter (AdS) space-time with temperature arising from a black hole metric, either introduced as a background or resulting from back reaction of matter on the geometry. Then, using the AdS/CFT correspondence, one can study the behavior of the dual field theory defined on the boundary, identify order parameters, and analyze the phase structure of the dual system.

Several models with scalar and gauge fields in a bulk which corresponds asymptotically to an AdS black hole metric have been studied [

The main point in this AdS/CFT based calculation is that the asymptotics of the solution in the bulk encodes the behavior of the QFT at finite temperature defined on the border. One finds in general a typical scenario of second-order phase transition. The critical exponents can be computed with a rather good precision and coincide with those obtained within the mean field approximation in a great variety of models.

It is the purpose of this paper to get an analytical insight complementing the numerical results, with a focus on the connection between the holographic approach and mean field theory for the calculation of critical exponents. To this end, it will be important to first stress some connections, already signaled in [

It should be noted that our calculations assume that the backreaction of dynamical fields is negligible (probe approximation) which is valid when the gauge coupling constant is large. This approximation is useful to study the behavior near the phase transition which is precisely the domain we will analyze, comparing our analytical results with those obtained numerically. In fact, the holographic results we have previously obtained solving numerically the Einstein-Yang-Mills-Higgs equations of motion both considering the probe approximation [

With this in mind and using an analytic approach proposed in [

In the Landau approach to second-order phase transitions one considers an order parameter

Let us give a brief description of gauge/gravity approach to phase transitions to connect it to the Landau theory. On the gravity side one considers a classical field theory in a Schwarzschild-AdS black hole background (or one in which backreaction of fields on an asymptotically AdS space leads to a black hole solution). The choice of such a geometry is dictated by the fact that the warped AdS geometry prevents massive charged particles to be repelled to the boundary by a charged horizon and as a result a condensate floating over the horizon can be formed. To find such condensate one should look for nontrivial static solutions for the fields outside the black hole by imposing appropriate boundary conditions. The behavior of fields at infinity then allows to determine the dependence of the order parameter on temperature, identified with the Hawking temperature of the black hole.

Basically, the free energy

Dynamics of the system is governed by the action

In order to look for simple classical solutions for this model one can propose the following ansatz:

It will be convenient for the analysis that follows to change variables according to

Conditions (

In view of (

In the limit

Now, (

To go further in the analysis without resorting to a numerical analysis, we will consider expansions of the fields in the bulk near

For the solution near the horizon (

As announced, imposing the conditions of continuity and smoothness at an intermediate point

We will first analyze the boundary condition

One can see from (27) that for

Our results for the critical temperature, as inferred from (

We now consider the case of an

At the horizon (

At the

As in the scalar field case, we will match both solutions at an intermediate point which we again choose as

Substituting these values in (

This equation implies that

Finally, introducing the temperature

We will consider here a system with dynamics governed by the action (

In terms of the variable

The boundary conditions for system (

For the solution of system (

As before, we impose matching conditions at

The equation for

The minimum value that

Using (

The phase diagram of

Note that

A plot of the order parameter

The results are qualitatively in agreement with those described in [

We have analyzed a number of models which have been proposed to study phase transition through the AdS/CFT correspondence. The common feature of all three models we discussed was that the space time bulk geometry was an Anti de Sitter black hole. Although the dynamical field content was very different—a charged scalar coupled to an electric potential, the same model in an external magnetic field, and a pure non-abelian gauge theory—the emerging scenarios are very similar and always include a second-order phase transition with mean-field critical exponents.

On general grounds, we were able to explain why the highly symmetric ansätze generally used produce the critical behaviors seen in mean field theory or the Landau approach. Founded on basic principles as the connection between the equations of motion and the Schrödinger equation, we clarify the similarity between several relevant quantities along a variety of models. In particular we showed that resorting to simple matching conditions, we obtain closed form solutions that significantly agree with the results obtained by numerically solving the exact set of equations of motion. This uncovers the important role played by analyticity to explain the universal behavior of certain physical constants.

The method seems to work very well near the critical temperature, though it deviates from the numerical results as we approach

Alternative analytic calculations have been recently presented in [

Although the matching method works very well near the critical temperature, it deviates from the numerical results as

F. A. Schaposnik is associated with CICBA.