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Inspired by the holographic entanglement entropy, for geometries with nonzero abelian charges, we define a quantity which is sensitive to the background charges. One observes that there is a critical charge below the system that is mainly described by the metric, and the effects of the background charges are just via metric's components. For charges above the critical one, the background gauge field plays an essential role. This, in turn, might be used to define an order parameter to probe phases of a system with fractionalized charges.

In application of AdS/CFT correspondence [

We note, however, that this is not the only way to construct a gravity model whose dual theory is a system at finite density. Indeed, finite density holographic duals may be obtained by two, rather distinctive, ways. Actually, the asymptotic electric flux—to be identified with the chemical potential at the boundary theory—may be supported by either nonzero charges from behind an event horizon or charged matter in the bulk geometry. If we are interested in a phase with unbroken

Of course one can distinguish between these two cases due to the fact that in the first case (fractionalized phase), the charge density is of order

Since the charge density of a system may be originated from both behind an even horizon and a charged matter, it could be in different phases depending on the origin of the asymptotic flux. To classify possible phases, an order parameter has been introduced in [

To proceed, let us consider a

A generic solution of the equations of motion of the above action could be a charged black hole (brane) with nontrivial dilaton profile. We may assume the background solution to be an asymptotically locally

The gravity description may be used to extract certain information about the dual field theory. In particular, one may study certain nonlocal observables. Prototype examples include holographic entanglement entropy [

We note, however, that since typically we are interested in backgrounds with electric field, it is not appropriate to work with fixed time as one does for the holographic entanglement entropy. In other words, it would be more natural to consider the geometric entropy [

Let us change the periodicity of

From gravity point of view, it is essentially similar to the entanglement entropy, where one should minimize a codimension two hypersurface in the bulk. However, in the present case-one considers a hypersurface with a spatial direction fixed. Indeed, to compute the geometric entropy, one usually utilizes a double Wick rotation to promote a spatial direction to time direction. Of course as far as the computations in the gravity side are concerned, it is not necessary to do that.

Now consider a codimension two hypersurface in the bulk parametrized by coordinates

For sufficiently small charges, one may expand the square root which for

For arbitrary charges, following the general idea of AdS/CFT correspondence, it is then natural to minimize

The paper is organized as follows. In the next section, we will consider charge black branes with one

In this section, in order to explore a possible information encoded in the expression defined by (

Let us consider a

This geometry is supposed to provide a gravitational description for a

Let us consider the following strip as a subsystem in the dual

Now, the aim is to minimize

Actually, as we will see when we increase the background charges the effects of gauge field become important leading to a new scale in the theory which could take over the role of the horizon. More precisely, as it is evident from (

Let us assume

where

If one drops the factor of

For the RN background given in (

From expression (

This behavior can be demonstrated by solving the integral (

From (

It should also be noticed that since the space time has a horizon, one could always imagine the case where the function

The disconnected solution is given by setting

One observes that for sufficiently small

Therefore, as far as the qualitative behavior of

To study the effects of the background gauge field, one may increase the background charge so that

Indeed looking at (

Moreover, since in the present case we do not have the disconnected solution, it does not make sense to compute the difference

From our numerical results, one observes that as long as we are in the range of

An interesting observation we have made is as follows. Although there is a maximum width (or correspondingly a maximum turning point) over which there is no closed hypersurface which minimizes

In this section, we extend our study to a charged black hole in a global AdS geometry. The action is still given by (

Holographic geometric entropy in this background has also been studied [

Therefore, we will consider a subsystem in the form of

Alternatively, for

Now the aim is to minimize

Although it is not explicitly clear from the above equation, there is still a special point at

To proceed, let us first consider

In order to calculate

On the other hand, as one increases the background charge so that

It is worth to mention that for

On the other hand, using the expression (

In order to explore the new feature, let us consider the situation where

In this paper, we have introduced a quantity which is sensitive to the background fractionalized charge not only due to its effects in the components of the metric, but also directly from the gauge field. To explore its properties, we have explicitly computed the quantity for the RN black brane and a black hole in an asymptotically AdS geometry.

For sufficiently small charges, the metric plays the essential roles; while as one increases the charge, one would expect to see the effects of the gauge field. Indeed, following our definition in the quantity (

On the other hand, in the opposite limit when

Note also that for the special value of

Probably the most interesting, but rather difficult, aspect of our study, is to find an interpretation for the quantity defined by (

The authors would like to thank A. Davodi, M. M. Mohammadi Mozaffar, A. Mollabashi, A. E. Mosaffa, M. R. Tanhayi, and A. Vahedi for the useful discussions. They would also like to thank D. Tong for the comments on the draft of the paper.