Universal terms for holographic entanglement entropy in noncommutative Yang--Mills theory

In this paper, we derive the universal (cut-off-independent) part of the holographic entanglement entropy in the noncommutative Yang-Mills theory and examine its properties in detail. The behavior of the holographic entanglement entropy as a function of a scale of the system changes drastically between large noncommutativity and small noncommutativity. The strong subadditivity inequality for the entanglement entropies in the noncommutative Yang-Mills theory is modified in large noncommutativity. The behavior of entropic $c$-function defined by means of the entanglement entropy also changes drastically between large noncommutativity and small noncommutativity. In addition, there is a transition for the entanglement entropy in the noncommutative Yang-Mills theory at finite temperature.


Introduction
The noncommutative gauge theories discussed in this paper is a gauge theories in which the product of any two fields is given by the Moyal-Weyl product [5,6] where θ µν is the deformation parameter. It has well known that these gauge theories naturally arise as low energy theories of D-branes in Neveu-Schwarz-Neveu-Schwarz (NS-NS) B-field background [1,2,3,4]. A remarkable phenomenon in these gauge theories is UV/IR mixing [5,6], where the ultraviolet (UV) and infrared (IR) degrees of freedom of the theory are mixed in a complicated way. Although it is very interesting to understand such quantum effects of the noncommutative gauge theory, hard to investigate them in the perturbative approach.
There exists a holographic description for the strongly coupled noncommutative gauge theories in the large N limit [7,8,9,10]. The holographic description of the noncommutative gauge theories is often useful to investigate how the noncommutativity (the deformation parameter) affect the quantum properties of the gauge theories.
For instance, the noncommutativity modifies the Wilson loop behavior [12,13,34] and glueball mass spectra [35]. The holographic duals of noncommutative gauge theories with flavor degrees of freedom have also been constructed by using probe techniques [11]. Employing the holographic description, we have been able to find the noncommutativity is also reflected in the flavor dynamics [36]. It should be emphasized that the noncommutativity can also modify phase diagram as for instance chiral symmetry breaking in the noncommutative gauge theory [37,38].
In this paper, we focus on quantum entanglement in noncommutative gauge theory.
Entanglement entropy is known as a measure for entanglement in quantum systems (see e.g. [14]). The entanglement entropy of a subsystem A is defined by the von Neumann entropy of the reduced density matrix ρ A of the system A, It is possible to compute the entanglement entropy by employing the holographic approach. Ryu and Takayanagi conjectured the holographic formula of entanglement entropy should be where A is the area of a minimal surface with a given boundary [15,16]. The proof of this formula is given by [17].
Quantum physics allows a superposition of states, causing a nonlocal correlation between subsystems far apart from each other. Entanglement is the distinctive concept of the quantum physics, including quantum field theories, and that is one of the important concepts to understand quantum aspects of the quantum physics.
It is known that entanglement entropy for quantum field theories with nonlocal interaction (nonlocal field theories) follows a volume law as long as the size of subsystem is smaller than a certain scale [22,23,24]. The noncommutative gauge theory is a kind of nonlocal field theories. It would be worth investigating how nonlocality of the noncommutative gauge theory affects the properties of the entanglement entropy.
The entanglement entropies in the noncommutative gauge theories has been studied based on the holographic approaches [18,19,20,21]. It have been pointed out that the divergence (cut-off dependence) part of the entanglement entropy in the large noncommutativity limit follows a volume law [18].
The holographic entanglement entropies for quantum field theories are often regularized for finite by introducing a cut-off parameter. Little attention, however, has been given to the cut-off independent part of the holographic entanglement entropies in the noncommutative gauge theories. In this paper, we try to derive the universal (cut-off-independent) part of the holographic entanglement entropy in the noncommutative Yang-Mills theory and discuss its properties. The properties of the holographic entanglement entropy in the noncommutative Yang-Mills theory should be discussed on the basis of universal (cut-off-independent) quantities.
The paper is organized as follows. In section 2, we introduce the holographic entanglement entropy conjectured by Ryu and Takayanagi and derive the universal (cut-off-independent) part of the holographic entanglement entropy in the noncommutative Yang-Mills theory. In section 3, we investigate the strong subadditivity for the holographic entanglement entropy in the noncommutative Yang-Mills theory. The properties of the mutual information written by the entanglement entropies is also discussed. In section 4, we investigate the properties of the entropic c-function in the noncommutative Yang-Mills theory. In section 5, we derive the universal part of the holographic entanglement entropy in the noncommutative Yang-Mills theory at finite temperature and discuss a kind of the transition based on the holographic entanglement entropy. Section 6 is devoted to conclusions and discussions.
2 Holographic entanglement entropy in noncommutative Yang-Mills theory We consider the dual description of the noncommutative Yang-Mills theory on a space- where R 2 θ is the noncommutative (Moyal) plane defined by a Moyal algebra [x 2 , x 3 ] = iθ. At Large N and strong 't Hooft coupling, a holographic description of the noncommutative Yang-Mills theory is given by where R 4 = 4πg s N l 4 s and a denotes the noncommutativity parameter with dimension of length.
We will use the generalized Ryu-Takayanagi formula for the ten-dimensional geometry with a varying dilaton. The holographic definition of entanglement entropy is given by where G (10) N = 8π 6 α 4 is the ten dimensional Newton's constant. The 5-dimensional Newton's constant G N is proportional to G N up to a volume factor G N /π 3 R 5 . Let us compute the entanglement entropy for an infinite strip specified by with L → ∞. Under this configuration, the entanglement entropy defined by (2.2) takes the form, where y (u) is the derivative of y with respect to u. We find the quantity u 3 y (u) y 2 (u) + 1/u 4 h(u) is a constant which does not depend on u. This quantity leads to where u * denotes an integral of motion and u = u * represents the point of closest approach of the extremal surface. Such surfaces have two branches, joined smoothly at u = u * and u * can be determined using the boundary conditions: The entanglement entropy given by (2.4) at the stable solution is given by where u Λ is a cutoff parameter. The (dimensionless) entanglement entropy functional defined by S A ≡ (πa 2 /N 2 L 2 ) S A can be rewritten as where t ≡ u * /u is a dimensionless variable. The ratio of the length l to the noncommutativity parameter a is also a function of the product of the noncommutativity parameter a and the integral of motion u * : In the deep infrared region au * 1, we can approximate the right hand side of (2.9) by 2 au * 1 0 dt t t 4 1 − t 6 , and we have where Γ denotes the Gamma function. The length l given by (2.10) is the same as that in the commutative (a = 0) version. Hereafter, we refer to the approximation in the deep infrared region as the commutative regime. The commutative theory (a = 0) and the noncommutative (a = 0) theory can be compared through the approximation in the deep infrared region.
The entanglement entropy functional (2.8) in the commutative regime can be divided into a universal(finite) part S U that is independent of the cutoff parameter u Λ and the divergence part S D that depends on the cutoff parameter u Λ : (2.11) In deriving this expression, we have utilized a formula , with λ > 1. Notice that the finite part S U is independent of the cutoff parameter u Λ and thus is universal quantity. Eliminating the parameter u * from (2.10) and (2.11), we find the relation between the universal part S U and the ratio l/a in the commutative regime: (2.13) Meanwhile, we find that the divergence part of the entanglement entropy S D in the commutative regime is proportional to the area L 2 : 2π (2.14) In the deep ultraviolet region au * 1, the ratio of the length l to the noncommutativity parameter a can be approximated as Hereafter, we refer to the approximation in the deep ultraviolet region as the deep noncommutative regime [18]. The entanglement entropy functional (2.8) in the deep noncommutative regime can be divided into the universal part S U and divergence part S D : 16) respectively. The relation between the finite part S U and the ratio l/a in the deep noncommutative regime is given by (2.17) We notice that the dependence of the finite part S U on the ratio l/a is quite different between the commutative regime and the deep noncommutative regime. Eliminating the noncommutativity parameter a from (2.15) and (2.16), we find the relation between the divergence part S D and the length l in the deep noncommutative regime: For u * ∼ u Λ , this expression exhibits that the divergence part of the entanglement entropy S D in the deep noncommutative regime is proportional to the volume L 2 l.
The difference in the l-dependence between (2.14) and (2.18) would be understood as the area/volume law transition [18].
The ratio l/a The length l has a minimum value l min ∼ 1.614 a at u * ∼ 0.7946/a in the noncommutative theory. The ratio l/a is proportional to the inverse of au * for l/a l min /a and is proportional to au * for l/a l min /a. The ratio l/a increases in case of au * → ∞ as well as in case of au * → 0. This behavior is reminiscent of the UV/IR relation [5].
In deriving the expressions above, we have utilized the formula with a constant k. The universal part S U given by (2.19a) is also a function of the dimensionless quantity au * . The variation with au * of S U is shown by Fig.1(b). The behavior of S U is also different between the commutative regime and the deep noncommutative regime. The universal part S U given by (2.19a) takes a minimum value at u * ∼ 0.7946/a. This is the same value at which the length l takes the minimum value l min in the noncommutative theory.
We can evaluate the dependence of the universal part S U on the ratio l/a numerically. The variation with l/a of the universal part S U is shown by Fig.2. There is major difference in the dependence of the universal part S U on the ratio l/a between in the commutative regime and in the deep noncommutative regime. The curve of S U is concave downward in the commutative regime (shown as a red line in Fig.2), while that is concave upward in the deep noncommutative regime (shown as a purple line in Fig.2). This concave upward curve suggests that the behavior of S U in the deep noncommutative regime becomes unphysical. In the noncommutative theory, the curve of S U branches into a concave downward curve and a concave upward curve at l/a = l min /a (shown as blue lines in Fig.2). Fig.2 shows that the blue concave downward curve (lower branch) and the blue concave upward curve (upper branch) asymptotically approach the red concave downward curve and the purple concave upward curve in the limit l/a → ∞, respectively.
We see that the universal part S U with identical the ratio l/a(> l min /a) actually have different value in the noncommutative theory. Therefore, the concave downward curve becomes presumably dominated for the curve of S U in the noncommutative theory. When the concave upward curve of S U is realized in the noncommutative theory, the derivative of the universal part S U with respect to the ratio l/a seem to be discontinuous at the point l/a = l min /a. This behavior could be interpreted as the area/volume law transition [18] from the viewpoint of the universal part S U .  Let us define the following quantity

Strong subadditivity and Mutual information
The quantity D = πa 2 N 2 L 2 D becomes positive when the strong subadditivity inequality is satisfied. The variation of d with x/a is shown by Fig.4.
x a  Let us define the following quantity The variation of the quantity I = πa 2 N 2 L 2 I with x/a is shown by Fig.6.

Entropic c-function
It is well known that there exists a so called c-function that is a positive real function and is monotonically decreasing under the renormalization group (RG) flow. The c-function can be defined by means of the entanglement entropy, and that is called entropic c-function [27,28,29]. For the infinite strip subsystem with the length l, the entropic c-function denoted by C can be rewritten as [30,26,18] Note that this quantity does not depend on the cutoff parameter u Λ . The variation of C(l) = πa 2 N 2 L 2 C(l) with l/a is shown by Fig.7. Since the c-function denoted by C(l) measures the number of degrees of freedom, it is expected to satisfy the inequality C(l UV ) ≥ C(l IR ) (for l UV ≤ l IR ) and the derivative of C with respect to l is expected to be negative. As we expected, the entropic c-function in the commutative regime (shown as a red line in Fig.7) satisfies such properties.
The entropic c-function in the deep noncommutative regime (shown as a purple line in Fig.7), however, does not follow the expected behavior. It diverges as l approaches to ∞. The entropic c-function in the noncommutative theory (shown as a blue line in Fig.7) under the condition of u * < 0.7946/a also satisfies the inequality C(l UV ) ≥ C(l IR ) (for l UV ≤ l IR ) and the derivative of C with respect to l is negative.
The behavior of the entropic c-function in the commutative regime under the condition of l → ∞ is similar to that in the deep noncommutative regime under the condition of l → 0. This phenomenon is also reminiscent of the UV/IR relation [5].

Finite temperature
In this section, we consider the holographic entanglement entropy in the noncommutative Yang-Mills theory at finite temperature. A holographic description of the noncommutative Yang-Mills theory at finite temperature is given by where u T is a parameter with dimension of mass. The corresponding temperature T of the background can be obtained to be T = u T /π.
We compute the entanglement entropy for an infinite strip specified by (2.3) with L → ∞. Under this configuration, the entanglement entropy denoted by S AT at the stable solution is modified to include the parameter u T : .
The length l is also modified to include the parameter u T : .

(5.3)
We can find that in the large l limit, the main contribution of (the finite part of) the integrals (5.2) and (5.3) coming from the region near u ∼ u * ∼ u T leads to the relation, The (dimensionless) entanglement entropy functional S AT = (πa 2 /N 2 L 2 ) S AT can be rewritten as where t ≡ u * /u and τ is a dimensionless parameter defined by τ ≡ au T . The ratio of the length l to the noncommutativity parameter a is also modified to include the parameter τ : and τ : The entanglement entropy functional given by (5.5) can be divided into the universal part S U T and divergence part S DT : (5.8b) Although the universal part S U T depends on the parameter τ , related to the temperature T , the divergence part S DT does not depend on the parameter τ . This fact indicates that the property of the universal part of the entanglement entropy is modified at finite temperature, while the property of the divergent part of the entanglement entropy is not modified at all.
We can also evaluate the dependence of the universal part of the entanglement entropy functional S U T given by (5.8a) on the length l given by (5.6) numerically. The variation with l/a of the universal part S U T is shown by Fig.8. Notice that the minimum length l min exists even at finite temperature. It can be found that the value of l min increases with increasing temperature τ . The change in the minimum length l min with temperature τ is however slight. There are no significant changes in the l-dependence of the entanglement entropy in the region of au * > τ (u * > u T ). In contrast, Fig.8(b) shows that the relationship expressed in (5.7) is approximately satisfied between S U T and l/a in the large l limit (in the region of Generally, there is another surface that satisfy the boundary condition (2.6) because of the holographic dual of the field theory at finite temperature involves black hole horizons [33]. The surface is parametrized as We call the surface that is parameterized as (5.9) piecewise smooth surface, to distinguish it from the smooth minimal surface. Let us compute the area of the piecewise smooth surface denoted by A and examine the behavior of two areas A and A as a function of the ratio l/a. The induced line elements for different segments are The entanglement entropy denoted by S AT is then given by: The (dimensionless) entanglement entropy functional S AT = (πa 2 /N 2 L 2 ) S AT can also be divided into the universal part S U T and divergence part S DT : (5.12b) The divergence part S DT is the same as S DT , and thus S DT does not depend on the parameter τ . The variation with l/a of the universal part S U T and S U T is shown by We would like to focus on the behavior in the curves of the entanglement entropy functional S U T and S U T near au * ∼ τ (u * ∼ u T ). As shown in Fig.9, the value of the entanglement entropy functional S U T also increases with the increase of l/a. As is clear from the fact that the slope of the curve of S U T is almost flat, however, the increase rate of S U T with respect to l/a is smaller than that of S U T . Therefore, the curves of the entanglement entropy S U T and S U T cross at l = l crit > l min . This fact exhibits that the entanglement entropy is governed by the configuration of the piecewise smooth surface for l > l crit . In other words, there is a transition for the entanglement entropy in the noncommutative theory. It has been shown that such transitions do not occur in corresponding (four-dimensional) commutative theory [33].
The variation with l/a of the difference S U T − S U T is shown by Fig.10.

Conclusions and discussions
In this paper, we have examined the properties of the holographic entanglement entropy in the holographic dual of the noncommutative Yang-Mills theory. The finite part of the holographic entanglement entropy in the noncommutative Yang-Mills theory can be derived without cutoff dependence, and thus is universal. Although the divergence part of the holographic entanglement entropy in the commutative regime (au * → 0 limit) follows the area law, that in the deep noncommutative regime (au * → ∞ limit) follows the volume law. This area/volume law transition [18] could be understood as a feature of nonlocal field theories.
The universal part of the holographic entanglement entropy as a function of length l in the noncommutative theory exhibits a peculiar behavior. There exists a minimum length l min in noncommutative theory, and the curve of the entanglement entropy branches at points of the minimum length l min . This behavior seems to be a remarkable feature that somehow reflects the area/volume law transition. On the other hand, the behavior of the entropic c-function in the IR limit of the commutative regime is similar to that in the UV limit of the deep noncommutative regime, if we interpret l → ∞ as infrared limit and l → 0 as ultraviolet limit. This phenomenon seems to be a kind of the UV/IR relation. It would be interesting to discuss such arguments with the entropic c-theorems in four dimensions (a-theorem) [31,32].
There exist a minimum length l min that gives a branch point of the curve of the entanglement entropy even in the noncommutative theory at finite temperature. On the other hand, the effect of temperature on the behavior of the entanglement entropy curve becomes remarkable for large l/a (u * ∼ u T ). It is also notice that there is a transition from the configuration of the smooth surface to the piecewise smooth surface for the entanglement entropy in the noncommutative theory. The anisotropy of the