Butterfly Effect and Holographic Mutual Information under External Field and Spatial Noncommutativity

We apply the transformation of mixing azimuthal and internal coordinate or mixing time and internal coordinate to a stack of N black M-branes to find the Melvin spacetime of a stack of N black D-branes with magnetic or electric flux in string theory, after the Kaluza-Klein reduction. We slightly extend previous formulas to investigate the external magnetic and electric effects on the butterfly effect and holographic mutual information. It shows that the Melvin fields do not modify the scrambling time and will enhance the mutual information. In addition, we also T-dualize and twist a stack of N black D-branes to find a Melvin Universe supported by the flux of the NSNS b-field, which describes a non-comutative spacetime. It also shows that the spatial noncommutativity does not modify the scrambling time and will enhance the mutual information. We also study the corrected mutual information in the backreaction geometry due to the shock wave in our three model spacetimes.


Introduction
Butterfly effects in the holographic geometry have been extensively studied in recent, which led to interesting new insights into quantum chaos and the behaviours of entanglement in nearthermal systems [1,2,3]. These investigations use eternal AdS black hole [4] which has two asymptotically AdS to describe the thermofield double state. After adding a small perturbation to one of the two CFTs at early time t w one then studies the effect on the structure of the state at t = 0. In this picture the perturbation can be modeled by a shock wave near the horizon of the black hole. In the dual holographic geometry the shock wave will travel across the horizon of the black hole and the mutual information of two space-like regions in each boundary given can be calculated by the Ryu-Takayanagi prescription for for entanglement entropy [5,6,7]. After an amount of time t * the mutual information between the two sides is disrupted and it shows the butterfly effect familiar from chaotic dynamics.
In the initial investigation [1] the dual black hole geometry in is the non-rotating BTZ black hole. The later analysis has generalized to AdS d [8], multiple shock waves [2], and localized shock waves [9,10]. The stringy corrections of the scrambling time were presented in [11]. The butterfly effect for black holes with rotation or charge had also been studied in [8,12,13]. That on black D p branes was investigated in [14]. In this paper we will studied the butterfly effects under external electric field, magnetic field or NSNS b-field, which are called as Melvin fields [15] In section 2 we first follow the method in [1] to derive the formula of scrambling time in the more general black branes background and apply it to our model spacetime. In section 3 we first apply the transformation of mixing azimuthal and internal coordinate [16,17] or mixing time and internal coordinate [18,19] to the 11D M-theory with a stack N black M5-branes [20,21] to find the spacetime of a stack of N black D4-branes with magnetic or electric flux in 10 D IIA string theory, after the Kaluza-Klein reduction 1 . Next, we follow [23] to begin with a N black D 3 branes and by applying the three chains of operation to find a model spacetime : 1. T-dualize along z to obtain a N black D 2 branes [24]. 2. Twist the compactification. 3. T-dualize along z to obtain Melvin Universe supported by the flux of the NSNS b-field. After applying the mapping of Seiberg and Witten [25] we then find a new Melvin Universe which describe the non-comutative spacetime 2 . We calculate the scrambling time in the above three spacetimes to see how the background field will affect it 3 .
In section 4 we calculate the mutual information in the above three spacetimes and determine the critial interval therein. In section 5 we follow the method in [1,8] to derive the formula of extremal surface in the more general black branes background and apply it to study the corrected mutual information in the backreaction geometry due to the shock wave in our three model spacetimes in section 6. Last section is used to summarize our results.

Shockwave Geometry
We first derive the formula of scrambling time in the following general black branes background in which the horizon locates at r = r H and f (r H ) = 0 while a(r H ) = 0, b(r H ) = 0. Contract to [1] above metric has two extra functions a(r) and b(r) which could help us to explicitly see how the external filed will affect the butterfly effect. The associated temperature is The line element can be expressed in the Kruskal coordinate . We now follow [1] to add a small null perturbation of asymptotic energy E M (M is the ADM mass of black branes) at time t w and radius r = Λ in the left asymptotic region. We denoteŨ ,Ṽ as coordinates to the left of the perturbation and U, V are those to the right. Then, the shell propagating on the constant U surface is described bỹ We can patch two side of geometry and the matching condition which relatesṼ to V is described byŨ In the case of linear order in small E and large t w the matching condition can be used to find the shiftṼ which is shown in figure 1.

Scrambling Time
Since the large value of t w will lead r → r H we can make a approximation f (r) ≈ f (r H )(r − r H ) + · · · and tortoise coordinate becomes In small E we can approxŨ w ≈ U [1] and To proceed we will express the shift value in terms of entropy. Using the black hole area law and thermodynamic first law we can find that and thus The scrambling time t * is defined to be the value of t w when α = 1 and E ≈ T . Then we finally find the formula It is interesting to see that while our general metric is different from that in [1,14] (in which a(r) = b(r) = 1) the formula of scrambling time t * has same form. Notice that, in the general metric the value of V d−1 (r H ), T, S BH may be different form that in the case of a(r) = b(r) = 1 and the scrambling time t * therein could be different therefore.

Scrambling Time in Electric Field Deformed Geometry
We now apply the transformation of mixing time and internal coordinate to the 11D M-theory with a stack N black M5 branes to find the spacetime of a stack of N black D4 branes with electric flux after the Kaluza-Klein reduction. The full N black M5-branes solution is given by where H is the harmonic function defined by in which G 11 is the D-dimensional Newton's constant and T 5 is the M 5 brane tension. The function f (r) specified by the horizon at r H is Now we transform the time t by mixing it with the compactified coordinate x 5 in the following substituting Using above substitution the line element (3.1) becomes Using the relation between the 11D M-theory metric and string frame metric, dilaton field and 1-form potential the 10D IIA background is described by In this decomposition into ten-dimensional fields which do not depend on the x 5 , the tendimensional Lagrangian density becomes and from (3.10) we see that the parameter E represents the magnitude of the external electric field. In the case of E = 0 the above spacetime becomes the well-known geometry of a stack of N black D4-branes. Thus, the background describes the spacetime of a stack of N black D4-branes with electric flux.
To calculate the thermal quantities of the black branes system we have to change the metric in string frame into the Einstein frame by the relation Thus the line element in Einstein frame is In the "near horizon'' limit we can approximate H ∼ R 3 r 3 and metric become Use above metric we can calculate the black brane temperature where T 0 is that without electirc field and scrambling time in which (t * ) 0 is the scrambling time without electric field. The trivial property is the consequence of E 2 f (r H ) = 0.

Scrambling Time in Magnetic Field Deformed Geometry
We now apply the transformation of mixing azimuthal and internal coordinate to the 11D Mtheory with a stack N black M5-branes to find the spacetime of a stack of N black D4 branes with magnetic flux. Using the full N black M5-branes metric described in (3.1) we can transform the angle ϕ 1 by mixing it with the compactified coordinate x 5 in the following substituting Using the above substitution the line element (3.1) becomes Using the relation between the 11D M-theory metric and string frame metric, dilaton field and 1-form potential, the 10D IIA background is then described by In this decomposition into ten-dimensional fields which do not depend on the x 5 , the tendimensional Lagrangian density will be described by (3.10) and the parameter B is the magnetic field defined by In the case of B = 0 the above spacetime becomes the wellknown geometry of a stack of N black D4-branes. Thus, the background describes the spacetime of a stack of N black D4-branes with magnetic flux.
The line element in Einstein frame is In the "near horizon'' the metric become Use above metric we can calculate the black brane temperature where T 0 is that without magnetic field and scrambling time in which (t * ) 0 is the scrambling time without magnetic field. Therefore, likes as the Malvin electic field the Malvin magnetic field also does not modify the scrambling time.

Scrambling Time in Noncommutative Geometry
To begin with, we quote the formula of T-duality [24]. After the T-duality on z coordinate the metric, dilaton field and B field become: Now consider the metric of a stack of N black D 3 branes which has zero dilation field and f (r) = 1 − r 4 h /r 4 . T-dualize along z gives a stack of N black D 2 branes which is described by After performing the twist by replacing and then T-dualize alongz will give a stack of N black D 3 branes described by in which b zφ is the flux of the NSNS b-field. After applying the mapping of Seiberg and Witten [25] (G + θ) µν = [(g + b) µν ] −1 (3.33) the metric G and non-commutativity parameter are which describes the black branes geometry with non-commutativity θ zφ = b [23]. Note that in Cartesian coordinates the non-commutativity parameter is non-constant. The line element in Einstein frame is in which the "'near horizon' lime H → R 4 r 4 is used. Use above metric we can calculate the black brane temperature where T 0 is that without NSNS B-field and scrambling time in which (t * ) 0 is the scrambling time without NSNS B-field. The trivial property is a consequence of the invariance of entropy under T duality [26].

Mutual Information in Melvin Field Deformed Geometry
In this section we will compute the mutual information of a region A on the left asymptotic boundary and its partner B on the right asymptotic boundary. Note that A = B when the left and right boundaries are identified. For simplicity we consider the mutual information (1.1) between two strips contained in the left and right side of the geometry respectively. Therefore we can use the Ryu-Takayanagi prescription [5,6,7] to calculate the entangle entropy (EE) between each strip and the rest of the system (S(A) and S(B)), and the EE of their union (S (A ∪ B)).
To consider a strip we can pick up an interval L in a coordinate y with − L 2 < y < L 2 (4.1) Since that in the black brane system there is a function f (r) which is zero on horizon r = r H we can let y = y(r) in the Ryu-Takayanagi prescription and, after integrating other coordinates besides r, the action can be described by in which r min is the turning point. This is the general action which includes those in [8,14] and mode spacetimes studied in this paper. A minimal bulk surface is found by extremzing the area functional, which leads to in which C min ≡ C(r min ) and D min ≡ D(r min ). Note that the interval L can be calculated by In this way the minimum surface becomes where the second term is S(A ∪ B) that from area of surface which passes through the horizon to connect to the other side [1]. We now want to find the critical interval L c in which the mutual information becomes zero. To proceed, we first rewrite the mutual information function as Notice that above result is an exact relation. Since the mutual information becomes zero when r min ≈ r H above relation leads to Above formula gives same scaling relation between L c and R as that in [8]. We now apply these formulas to the Melvin field deformed geometry.

Mutual Information and Critical Interval in Electric Field Deformed geometry
For the electric field deformed geometry the action becomes Using above formulation we present figure 2 to show the mutual information in Melvin electric field spactime.  Use the approximation relation the increasing property of the function L c (E) can be easily read.

Mutual Information and Critical Interval in Magnetic Field Deformed geometry
For the magnetic field deformed geometry the action becomes Using above formulation we present figure 4 to show the mutual information in Melvin magnetic field spactime. The associated critical interval L c (B) calculated from above formula is plotted in figure 5. Use the approximation relation the decreasing property of the function L c (B) can be easily read.

Mutual Information and Critical Interval in Noncommutative Geometry
For the NSNS b-field deformed geometry the action becomes y (r) 2 (4.14) Using above formulation we present figure 6 to show the mutual information in Melvin NSNS b-field spactime.  Use the approximation relation the increasing property of the function L c (b) can be easily read.
Above results show that the Melvin electric field, Melvin magnetic field and NSNS b-field will increase the mutual information.

Extremal Surface in Shockwave Melvin Field Geometry
We now consider the mutual information in the backreaction geometry due to the present of shock wave.

Extremal Surfaces
The area of minimal surface is reduced to two-dimensional problem and is dercribed by r(t) [8]. It can be studied from the general form Regarding A as a particle action we see that the t-translation symmetry in there gives a conserved quantity where r 0 is define as the radial position whereṙ = 0. Since that r 0 is behind the horizon and thus f (r 0 ) is negative as shown in figure 8. Above relation implies that dt = dr a(r)b(r)f (r) 2 1 + a(r)f (r) Therefore the entanglement entropy can be evaluated from the extremal surface [5,6,7] S A∪B (r 0 ) = A 4 = 1 4 dr a(r) b(r) γ 2 + a(r)f (r) (5.4) In order to examine how the EE depends on α we have to find the relation between α and r 0 . In below, for self-consistent, we will slightly generalize [8] to derive this relation in our model spacetimes.

Surface Location
For the segment I, the minimal surface stretches from the boundary at (u, v) = (1, 1) to (u; v) = (u 1 ; 0). Using the definition of u we have Since that r = ∞ at left boundary and r = r H at right boundary (u 1 , 0) as shown in figure 5.
For the segment II, the u 2 can be evaluated in a same way and Since that r = r 0 at right boundary (u 2 , v 2 ) as shown in figure 8. We now need to find the formula about v 2 . In this case we can define a reference point (ū,v) which is in the interior of the black brane and at some radial coordinate r =r < r H . Then Note that if we choose the point located at r * (r) = 0 thenūv = 1 [8].
Since that r = r 0 at left boundary (u 2 , v 2 ) and r = r H at right boundary (0, α 2 ) as shown in figure 8.
Combine above three relations we can find that Using above relation the entanglement entropy can be expressed as function α. Since it is divergent we can renormalized it by subtracting its value at α = 0. Then In next section we will use above formulas to numerically study the mutual information in three kinds of shockwave Melvin field geometry.
6 Mutual Information in Shockwave Melvin Field Geometry

Mutual Information in Shockwave Electric Field Deformed Geometry
For the Melvin electric field deformed geometry the minimal surface is described by The relation between α and r 0 is shown in figure 9.

Mutual Information in Shockwave Magnetic Field Deformed Geometry
For the Melvin magnetic field deformed geometry the minimal surface is described by in which we adopt the approximation for the case of small B field. The relation between α and B, and renormalized entanglement entropy −S R A∪B as function of Malvin magnetic field B are shown in the figures 11 and 12.

Mutual Information in Shockwave Noncommutative Geometry
For the NSNS b-field deformed geometry the minimal surface is described by   All above figures show that Melvin fields will enhance the mutual information in the backreaction geometry due to the shock wave.

Conclusion
In this paper we extend previous method [22,23] to derive three kinds of Melvin geometry. First and second geometries describe the spacetime of a stack of N black D4-branes with magnetic or electric flux in 10 D IIA string theory. The third geometry describes a Melvin Universe supported by the flux of the NSNS b-field, which relates to the non-comutative spacetime. We then follow the method in [1,8] to derive the new formula of scrambling time and holographic mutual information in more general spaceimes and use the new formulas to investigate the scrambling time and holographic mutual information in the three model spacetimes. We first find that while the three kinds of external field do no modify the scrambling time they can enhance the mutual information. We also study the mutual information corrected by the backreaction geometry due to the shock wave. It also shows that the three kinds of external field will enhance the mutual information. It will be interesting to see that whether these properties could be found in any simple field theory models. We are now searching the toy mode and investigating the properties from the field theoretic side.