Casimir Energy in Non-relativistic Backgrounds: Numerical Approach

Subject Areas : International Journal of Mathematical Modelling & Computations

Mozhgan Belyad ¹ (Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, P.O. Box 14676-86831, Iran)
Mohammad Reza Tanhayi Tanhayi ² (Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, P.O. Box 14676-86831, Iran)

Received: 2023-03-01 Accepted : 2023-06-17 Published : 2023-06-01

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Casimir Energy in Non-relativistic Backgrounds: Numerical Approach

M. Belyad

Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran, P.O. Box 14676-86831, Iran

In this paper we use numerical methods to investigate the Casimir effect for a scalar field in a specific boundary condition. In order to calculate the energy-momentum tensor, the holographic method is used, and, the background is Schrdinger-type metric which is close to the classical metric. We also compute the holographic entanglement entropy, and, for two steps the mutual information is also studied. By numerical analysis, we argue that the mutual information is always positive. Furthermore, for three entangling regions, we show that the corresponding tripartite information becomes negative.

Key words: Holographic method, Casimir Energy, Entanglement Entropy, n-Partite Information, Numerical Analysis

Contents

1- Introduction

2- Review

2.1- Lifshitz Metric

2.2- Schrdinger Metric

3- Energy-Momentum Tensor

4- Holographic Energy-Momentum Tensor

5- Casimir Energy

6- Holographic n-Partite Information

7- Conclusion

 

 

1- Introduction

One of the manifestations of macroscopic zero-point energy in the Quantum Field Theory is the Casimir effect, which expresses the non-trivial properties of the vacuum state. In simple experimental terms, the attractive force between two parallel conductive plates that are electrically neutral and located in a vacuum is called the Casimir effect, which is caused by the presence of the vacuum. Theoretically, the Casimir effect can be considered as a result of the zero-point fluctuation spectrum in the presence and absence of these plates. It is noteworthy that the zero-point energy of any relativistic field is obtained under boundary conditions. In this view, the virtual particle-antiparticle pairs have the ability to create and annihilate in vacuum [1,2,3,4,5].

Basically the Casimir effect is a completely quantum effect and its computation needs to consider the quantum fields with some specific boundary conditions. Usually there are no analytic solution and in most cases numerical methods are needed. In this paper we study a scalar field and use numerical method to calculate Casimir effect. We use Anti-de Sitter space/Conformal Field Theory correspondence (AdS/CFT) to calculate the vacuum energy of a given system in the non-relativistic background.

In order to expand the initial studies of Casimir energy in reference [8], we can used the AdS/CFT correspondence, which expresses a relation between the quantum physics of strongly correlated    many-body systems and classical gravitational dynamics of a higher dimension. According to this correspondence, the asymptotic metric determines the expectation value of all the individual components of the energy-momentum tensor and the corresponding energy is calculated from relation  [6,7,8].

Nowadays, more general classifications of metric than metrics with asymptotic AdS boundaries are used in investigating some feature of condense matter systems. For example, in recent AdS/CFT applications, a hypersurface violation of the dual quantum field Theory is shown by a metric that transforms covariantly under dilatation. We use this tool to extend in the context of Schrdinger holography and lifshitz spacetime [9,10,11].

2- Review

2-1. Lifshitz Metric

Holographic dual of physical systems in critical points with different space-time scales is given by Lifshitz metric. The time and space are scaled as ; where z is the dynamical critical component. The Lifshitz metric is:

Where L is the geometric radius. Due to an anisotropy between space and time, the above metric cannot be an ordinary solution for the Einstein equation; To break this anisotropy, some kinds of material fields are needed [8,27,46]. In general, the metric corresponding to the hyperscaling violating geometries is as follows:

(1)

Where  is the hyperscaling violating factor,  is a dynamical scale. The appeared parameters are included the two conditions by considering null energy [26,16] and scale transmation:

(2)

Because metric (2) is not invariant, the above scale transmation has been applied.

2-2. Schrdinger-type Metric

The non-relativistic Schrdinger-type metric with Galilean scaling and for 2+d dimensional theories is as:

(3)

Under the special conformal transformation, we will have:

(4)

3- Energy-Momentum Tensor

The Ricci scalar curvature for the metric (4) is obtained as follows:

(5)

The Ricci scalar curvature is constant for . And the quasilocal energy-momentum tensor with a boundary metric of a given gravity theory  is as follows:

(6)

Where S is action. The energy-momentum tensor is divergence at the boundary. To eliminate this divergency, a boundary term (that does not affect the bulk equations of motion) can be added to the action. In other words, using the AdS/CFT correspondence, (7) is as the expectation value of the stress tensor in the CFT:

(7)

The zero-zero component of the energy tensor is calculated as follows:

(8)

Therefore, the energy between two infinite planes at  for d=3 with will be as follows:

(9)

4- Holographic Energy-Momentum Tensor

The energy-momentum tensor gives us information about the number of degrees of freedom and conformal anomalies. Considering the high importance of the energy-momentum tensor and its expectation value, here we briefly review the method of calculating the Brown-York stress tensor. The induced metric with a radial cutoff at the hypersurface  from the metric (4) and the extrinsic curvature is given by:

(10)

Where  is the unit normal vector of timelike surface. And the quasilocal stress tensor is expressed by:

(11)

According to the AdS/CFT correspondence and the limit of , we will have:

(12)

Which in the hypersurface , the induced metric will be as follows:

(13)

Now, by using the above equation, the expectation value of the zero-zero component of the stress tensor is obtained by:

(14)

5- Casimir Energy

Now let us consider the zero-point energy in a box of size a at the boundary  which is located between  and . Putting these plates will affect the modes inside and outside the plate and a 1+1-dimensional box, one obtains:

(15)

Where as usual we have define . The above integral is clearly infinite. However, making use of the damping function as  where  is defined as a natural size leading to a high frequency cutoff,  which leads to:

(16)

Now as usual in computing the Casimir energy one can obtain the renormalized Casimir energy in this setup by subtracting the above integral with same value which is calculated without the boundary, namely one has:

(17)

CFT at the boundary causes the creation of this negative energy density, which can be considered the same as Casimir energy. Therefore, the Casimir force can be obtained as follows:

(18)

For this one dimensional box, the null energy consideration implies that z=1 and . From these expressions, we observe that the hyperscaling violation has the effect of modifying the Casmir energy by shifting its value.

6- Holographic n-Partite Information

In this section, we want to obtain mutual information and tripartite information. Consider a strip as an entangling region located at  in the given background:

(19)

Holographic entanglement entropy is given as follows:

(20)

For  and :

(21)

Mutual information is a subset of n-partite information with  in quantum information literature. As we know, entanglement entropy measures the quantum correlation between two subsystems A and its complement. Therefore, mutual information is used to determine the amount of entanglement or shared information between two subsystems. Consider a system of two disjoint parts  and . Mutual information is given by:

(22)

Where , the union of two entangling regions, is the entanglement entropy [59]. For two operators as  and  in the regions  and  respectively, mutual information sets an upper bound in both quantum and classical correlations between them is as follows [57]:

(23)

Therefore, mutual information expresses not only the total correlation between two subsystems, rather from its definition, it can be seen that it is UV-cutoff independent and free of divergences.

There is a phase transition [58] that be raised from the union of two entangling regions and affects the calculation of mutual information. Here, the presence of a critical distance causes the two subsystems  and  to be completely separated; in other words, the mutual information vanishes as the distance between two subsystems increases, which is called disentangling transition. Within the holographic point of view, this phase transition has a simple explanation [59]. The minimal area surfaces for union entanglement entropy exists in two ways shown in Figure 2 for two strips with length l and distance h from each other, one of which is used according to the separation between two subsystems.

(24)

The corresponding minimal configurations can change from one shape to another, depending on the value of , which leads to the definition of a critical ratio as follows:

Figure 2. Showing two different structures of calculation . According to the value of the parameters, the minimum state can be changed from one to another.

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