Fermion Fields in BTZ Black Hole Space-Time and Entanglement Entropy

We study the entanglement entropy of fermion fields in BTZ black hole space-time and calculate pre- factor of the leading and sub-leading terms and logarithmic divergence term of the entropy using the discretized model. The leading term is the standard Bekenstein-Hawking area law and sub-leading term corresponds to first quantum corrections in black hole entropy. We also investigate the corrections to entanglement entropy for massive fermion fields in BTZ space-time. The mass term does not affect the area law.


I. INTRODUCTION
The laws of black hole thermodynamics capture the essential features of macroscopic description of black holes in general theory of relativity. The surface gravity is constant over the horizon and it is related to the temperature of the black hole called as the Hawking temperature. In the context of black hole thermodynamics, the entropy is proportional to area of the event horizon. It is very interesting to explain the origin of area law either by direct state counting or entanglement entropy considerations. Quantum mechanically black holes emit Hawking radiation and an understanding of this process will lead to the resolution of information loss paradox. If one considers quantum fields in the vicinity of black holes, the area law of entropy receives quantum corrections known as logarithmic corrections [1]. The state counting process should also produce these corrections. The entanglement thermodynamics can also be used to study such corrections.
The entanglement entropy is the source of the quantum information and it measures the correlation between subsystems separated by the boundary. The entanglement entropy depends upon the geometry of the entangling surface (hyper surface (Σ)). The AdS/CFT correspondence provides the geometric way to calculate the entropy of the black hole called the holographic entanglement entropy [2]. Alternatively one can consider the matter fields in the background of black hole and calculate entanglement entropy.
The entanglement entropy is defined by the von Neumann entropy relation ( where ρ A is the reduced density matrix of the system A and is dominated by short range correlations across the entangling surface. These correlations give an area law (Entanglement entropy proportional to the area of entangling surface divided by the cutoff (ǫ)) and the sub leading term in the entanglement entropy contains useful cutoff independent information about the field theory.
The entanglement entropy approach was first used by Bombelli et. al. [3] and Srednicki [4] for scalar field in spherical systems. Then Peschel [5] developed a technique to calculate the entanglement entropy of fermions, where the reduced density matrix can be written in terms of correlator. The reduced density matrix in term of correlators is known for free fermions and obeys the wick theorem. This theorem fixes the (ρ = c exp(−H)), where H is a Hamiltonian, which is quadratic on the fields inside the region. In this paper we calculate the density matrix for fermions fields in BTZ space time and explicitly diagonalize the reduced density matrix in order to estimate the entanglement entropy.
We also consider the massive fermion fields in BTZ black hole spacetime and calculate the en-tanglement entropy. The corrections to entanglement entropy for massive fields in (2+1) dimension is related to the coefficient of logarithmic term in (3 + 1) dimensional massless theory [6]. The coefficient of the logarithmic contribution happens to contain the linear combination of central charge appearing in the trace anomaly in (3 + 1) dimensional theory [7,8]. We have calculated numerically the coefficients of the 1/µR term using the entanglement entropy approach [9,10].
The entropy is directly related to the coefficients of term 1/µR (IR expansion) and the coefficient of linear term µR (UV expansion) valid in the large (µR) limit. The area law contribution of entanglement entropy is not affected by this mass term and the universal quantities depend upon the basic properties of the system (spatial dimension) [11]. This paper is organized as follow. We study the fermion field propagating in BTZ black hole space time in section (2). The correlator method by which, we calculate the entanglement entropy of fermion field is described in section (3). The numerical computation of entanglement entropy of massless fermions and logarithmic correction of entropy are presented in the section (4). The logarithmic contribution of massive fermions is studied in section (5). Finally, we present our results and its physical implication of entropy for fermion fields in BTZ black hole space time .

II. FERMIONS IN BTZ BLACK HOLE SPACETIME
The BTZ black hole is a solution of (2+1) dimensional gravity with negative cosmological constant [12] and the metric is given by; where N 2 (r) and N φ (r) are lapse and shift functions; where −∞ < t < ∞ and 0 ≤ φ ≤ 2π. The solution is parametrized by the mass, M and the angular momentum J of the black hole and they obey the conditions, M > 0 and |J| < M l.
Defining (r 2 = l 2 (u 2 + M )) the proper distance from the horizon, ρ is given by, r 2 = r 2 + cosh 2 ρ+ r 2 − sinh 2 ρ, where r + and r − are outer and inner horizons of the black hole. The metric of BTZ black hole can be written in term of proper distance [13,14], The Dirac equation in the background of BTZ black hole is given by (see appendix), in order to absorb the term proportional toγ 1 in the Dirac equation. Finally the Dirac equation becomes, where σ 1 , σ 2 and σ 3 are Pauli matrices.
The wave function Ψ can be written in terms of two component spinors, 2π e ιθ(m−1/2) are the eigen-vector of the angular momentum operator [10]. Now we can express the the Hamiltonian of the system as the sum over azimuthal quantum number,m, The explicit form of H m can be obtained from (4) and is given by, whereγ 0 = ισ 3 ,γ 1 = σ 1 ,γ 2 = σ 2 , and σ 3 is multiplied by imaginary unit to change the Lorentzian to Euclidean signature.

III. A MODEL OF ENTANGLEMENT ENTROPY FOR FERMION FIELDS
The entropy of the black holes may be understood by the degree of freedom encoded by area law of black hole thermodynamics. In order to have a model of entanglement entropy, we consider the black hole spacetime divided into two parts one inside and the other outside the horizon. The appropriate formalism to describe entanglement entropy is the density matrix,defined in the term of local operator O in the region V of space and it is given by; The reduced density matrix of the system can be written in the exponential form [15,16]; where H is the hermitian matrix to be identified with the Hamiltonian of the system and c is the normalization constant.
The Hamiltonian of the system with fermions can be written as [16], This is the analogous to the discrete Hamiltonian of the bosonic system, which is quadratic in the fields and momentum operators. Using this Hamiltonian the form of reduced density matrix (7) is given by, We can diagonalize the exponent of density matrix using the relation, U HU † , U being a unitary operator. The relation between ρ V and H can be rewritten as, In case of fermions the density matrix is an exponential of a quadratic form in the creation and annihilation operator. The local creation and annihilation operator ψ i ψ † j satisfy the anti commutation relations. The two point correlators are given as; The hermitian matrix C ij are the correlator that determine the density matrix of the region V of space. Then we have,using (7) as, The entropy is given by the sum over different angular momentum quantum numbers, The eigen values of C lies between 0 and 1, except in case where the global state is pure, then the eigenvalue values of the C are 0 or 1's.
For the general quadratic case the discretized Hamiltonian can written as , The discrete Hamiltonian in these variable for a N lattice takes the form where i, j are the discrete variable corresponding to the radial distance ρ and M ij for fixed i, j is the (2 × 2) matrix.
These are the matrix elements of massive fields with angular momentum J. If J = 0, these elements reduce to non rotating BTZ black hole.
Since, the fermion correlator appearing in the entanglement entropy formula is related to the M ij , the matrix appearing in the Hamiltonian, for general quadratic case the entropy of the system is given by the relation, where λ m are the eigenvalue values of the matrixM ij

IV. NUMERICAL COMPUTATION OF ENTANGLEMENT ENTROPY
The total entropy of the discretized system is given by the sum all modes "m" of the angular momentum, The factor ( 1 2 ) appears due to fermion doubling in the radial direction as we have decomposed fermion field into two component Weyl spinors. We consider the system discretized in radial direction with N=200 lattice points and "a" is characteristic cut-off scale of the system and it is identified as Planck length in fundamental theories of gravity. The partition size is specified by the integer n B .
To compute the entanglement entropy of the massless Dirac field in the background of BTZ black hole, we fit the data in the following form, We extend our results to calculate the logarithmic contribution of entropy for fermion field. We fit the data to calculate the logarithmic corrections to entropy; The numerical value of these coefficients is obtained by fitting procedure and is given as; a=0.304, b=-0.315, c=-0.327. The coefficient of logarithmic parameter -0.315 is consistent with CFT results with the -1/3 [15].
The graph is shown in figure (1) We have also studied the dependence of the entropy on the angular momentum as shown in the figure (2). The entropy is weakly dependent on angular momentum.

V. ENTANGLEMENT ENTROPY IN FREE MASSIVE THEORY
In this section, we study the behavior of the universal term contributing to the entanglement entropy for the fermion field propagating in BTZ space time.
The universal term in the entanglement entropy in (2+1) dimensions massive theory can be obtained via dimensional reduction from (3+1) dimensions massless theory [16,17]. The one  The Fourier decomposition of the field modes in the compactification direction, one obtain the (2+1) dimension massive theory [18,19]; where M is the mass of the free fields, and acts as infrared regulator. The contribution of entanglement entropy of two dimensional fields are given by the relation [10], The series expansion of S(µ) with mass is given as [10], We calculate the entanglement entropy of the fermion field for different masses in the range (.05 < µ < .5), we fix the the cutoff (ǫ = 1) and µ ρ > 1. The coefficient c 1 and c −1 are expanded as, [10] , The value of c 0 (µ), c 1 (µ) and c −1 (µ) for different masses are tabulated in table (I),

VI. RESULTS AND CONCLUSION
In this paper, we have studied the entanglement entropy of the fermion field propagating in the background of BTZ black hole numerically. We have calculated the leading and sub-leading correction of entropy for massless fermion field. This sub leading correction gives the first quantum-correction of entropy, which is logarithmic correction. We have calculated the coefficient of logarithmic using fitting procedure. The coefficient of logarithmic correction term seems to agree with the conformal field theory results (−1/3) for fermion [15]. We have also showed that, the entropy of the fermion fields depends weakly upon the angular momentum (J = 0.1, 0.2 and 0.9).
The logarithmic contribution of entanglement entropy for a massive fermion field is obtained by the dimensional reduction, where the entanglement entropy for the fermion field can be calculated as an integral over masses. We have studied the entanglement entropy for massive fermion numerically, and calculated different coefficients c 1 and c −1 , which are consistent with the analytic results (27).
One may also compute the entanglement entropy for fermion field numerically in the context of non-vacuum states (first excited state and mixed state) using the correlator method and study the correction of area relation.