JHEP01(2014)137 Springer August 2, 2013 January 7, 2014 : January 24, 2014 December 6, 2013 : : : , Received Revised 10.1007/JHEP01(2014)137 Accepted Published doi: a,b Published for SISSA by [email protected] , and Sandipan Kundu a . strongly coupled noncommutative gauge theory, both at zero and at 3 N Arnab Kundu 1307.2932 The Authors. a,b c Gauge-gravity correspondence, AdS-CFT Correspondence, Non-Commutative

In this article we investigate aspects of entanglement entropy and mutual in- , [email protected] Theory Group, Department of Physics,Austin, University TX of 78712, Texas, U.S.A. Texas Cosmology Center, University ofAustin, TX Texas, 78712, U.S.A. E-mail: [email protected] b a Open Access Article funded by SCOAP ArXiv ePrint: surfaces can only be defined at this cut-off andKeywords: they have distinctly peculiarGeometry, properties. Classical Theories of Gravity and subsequently compute theregions corresponding which entanglement do entropy. not We liesensible observe entirely results. that in for the In noncommutativeentanglement order plane, entropy, to the it make RT-prescription is sense yields essentialregions of to which the introduce lie divergence an entirely structure additional in of cut-off the the in noncommutative corresponding the plane, theory. the For corresponding minimal area Abstract: formation in a large- finite temperature. Using the gauge-gravity dualitytion, and we the Ryu-Takayanagi (RT) adopt prescrip- a scheme for defining spatial regions on such noncommutative geometries Willy Fischler, Holographic entanglement in agauge noncommutative theory JHEP01(2014)137 (1.1) 14 21 6 18 , ij iθ ], that quantum field theories can be defined 1

= 8  – 1 – j 6 ) , x 7 i 2 13 x  , x 4 1 x ( 7 F 18 7 11 17 21 ) = 5 3 4 , x 2 , x 1 x 1 ( 25 F is the noncommutativity tensor. 2.3.1 2.3.2 General case ij θ Hallmark of a noncommutative theory is the inherent non-locality associated with the 5.2 Mutual information 4.1 Entanglement entropy4.2 of an infinite Mutual rectangular information strip 5.1 Entanglement entropy 3.1 Commutative rectangular3.2 strip Non-commutative rectangular3.3 strip Mutual information 2.1 Holographic dual 2.2 Regime of2.3 validity Noncommutativity and entanglement entropy where description. One key motivation behind studying such noncommutative theories is that and physics. It was realized,on perhaps a first noncommutative in geometry [ wherea the conventional position Lagrangian coordinates description, do the not primaryassociative commute. ingredient but to Based not define on such necessarilyof a commutative theory such algebra. is an an algebra Perhaps is the the most commutation familiar relation example between the space-coordinates 1 Introduction Noncommutativity is an intriguing concept that has repeatedly appeared in mathematics 6 More general shapes: noncommutative7 cylinder Conclusions 4 Introducing finite temperature 5 More general shapes: commutative cylinder 3 Infinite rectangular strip Contents 1 Introduction 2 Noncommutative Yang-Mills JHEP01(2014)137 ]. is ρ [ A c ]. In (1.3) (1.2) A 6 , 5 = tr entangling A ] matrix the- . ρ 3 A : c sharp ∂γ A = ], which defines the A ρ ∂A : denotes the minimal area log , A A A entangling surface provided x ρ γ = z ] to analyze the issue in a large an ] and the IKKT [ tr [ = 4 respectively. Consequently, the 2 y −

c ) z A = ( g , A ) ) y S . Now we can define a reduced density and ( c A f γ A +1) A j d ( N ∂ The issue is now to carry out a computation.

∂z G i ⊗ H 1 4 ∂ – 2 – Area( ∂y A ij θ = quantum field theories. Thus it will be a fruitful H i 2 A e = S non-local quantum field theories, which may eventually ≡ tot ) simple x H )( by tracing over the degrees of freedom in simple A the AdS/CFT correspondence [ factorization is sensible. At this point we can appeal to Bohr’s f ? g c ( A + 1)-dimensional Newton’s constant, via ]. According to this proposal, entanglement entropy of a region d 8 ) are two arbitrary functions. ⊗ H z ( A g H is the ( = +1) ) and tot d y ( N ( H G f We will not attempt to perform a weakly coupled field theory computation. Instead, The astute reader will immediately notice potential subtleties in defining such a quan- Before proceeding further, let us briefly introduce the notion of entanglement entropy. In the current article we will attempt to investigate aspects of entanglement entropy We have elaborated more on this issue in the following sections. noncommutative gauge theory. By virtue of the gauge-gravity duality, we will need to = 4 super Yang-Mills theory, where we just need to replace every commutative product 1 where surface whose boundary coincides with the boundary of region perform a classical gravitygauge-gravity computation duality, in entanglement entropy a is geometricformula calculated background proposed using obtained in the [ in Ryu-Takayanagi (RT) [ given by we promote the classicalcorresponding algebraic operator equation and defining its such eigenvalues. a surface as awe statement will on make the a detour N entanglement entropy. tity in a noncommutative field theory.surface One obvious in issue a is how fuzzy totheory manifold, define a which is alsocorrespondence tied principle to and the declare issue whether that in we a can noncommutative define that divides theus entire denote manifold the intotal corresponding two Hilbert sub-systems sub-manifolds space by at factorizes: matrix a for given the instant sub-system Finally in we time. can use Let the von Neumann definition where In a given quantum field theory defined on a manifold, we imagine an entangling surface be relevant for a theory of quantum gravity. in a noncommutative fieldN theory. We willby the consider star-product the noncommutative version of the profound property of anytheories theory do appear of rather quantumories naturally, gravity. of have which been In the widely stringconceptual BFSS studied theory challenges [ in noncommutative even the for exercise literature. to understand On the the other hand, non-locality posits they come equipped with a natural infra-red (IR) ultraviolet (UV) connection which is a JHEP01(2014)137 3 A for a definition ]. 11 , and define spatial does not necessarily c ∂A ∂A 4 . It can be proved under general , Thus there is no straightforward Y B ∪ 2 is defined as , if exists, provides A B S ∂A We will find that mutual information is − B and 5 S A

+ – 3 – A S ) = A, B ( I ] for a local field theory with nearest neighbour interaction, ] and subsequently it is subtle to compare the correspond- 9 10 ]. It is, however, not immediately clear whether in a gauge theory the analogue 12 denotes the entanglement entropy of the region [ Y B S The above feature is more prominent in the so called mutual information, which is a Let us now mull over why entanglement entropy may be a potentially interesting ob- In the case of a noncommutative gauge theory defined on the boundary of the bulk We will discuss thisNote in that detail even in for section a 2.3. gauge theory, where theWe interactions will are discuss not this nearestWe in neighbour would detail type, like in such to a an thank later area Matthew section. Hastings For for earlier correspondences related on studies, this see issue. [ 2 3 4 5 be violated below a certain length scale, where the entanglement entropy also deviates from law may hold. Although there is no general proof of this statement. where considerations that mutual information is alwaysand/or bounded by the area ofof the boundary this of theorem canalways tolerate an some area-worth quantity, non-locality. when it is finite. This area-worth and finite behaviour can derived quantity that has some advantagesbetween over entanglement two entropy. disjoint, Mutual separated information sub-systems obeys a universal area-law [ since quantum entanglement occurs primarily acrosstheory the this common may boundary. not Inviolation a necessarily of non-local be the true. areait law We may will below not find a be in certain meaningful length-scale; several to examples however probe we that the will there theory further is below argue this a that scale. issues, can be antanglement interesting entropy probe reduces for tofor such noncommutative thermal theories. theories. entropy In Therefore which we thegauge-invariant observable. expect is large Furthermore, that still this temperature the exercise entanglement a limit, mayinherent entropy also en- gauge-invariant is non-locality shed concept also on light a quantum on entanglement. the role It of is an expected that entanglement entropy area surfaces have rather peculiar properties. servable for such aoperator theory. in such First, aing note results theory that with [ it an is ordinary theory. non-trivial Entanglement to entropy, define modulo a the above-mentioned gauge-invariant We will assume that,minimal since area the surface bulk withentanglement geometry the entropy. is classical Henceforth still boundary wefinite a will temperature. classical investigate one, this Perhapsscheme quantity, the most of both corresponding intriguingly, the at the prescription zero RT-formula— for — and a does as at subregion allow applied residing us entirely within to on our define the a noncommutative sensible plane entanglement. However, the corresponding minimal geometry, this raises another subtlety:have a as well-defined we have meaning arguedto in before, make a sense fuzzy of geometry.regions the as Perhaps boundary bounds the curve on simplestway as the generalization to an eigenvalues is operator, of construct this denoted an operator. by interpolating bulk minimal area surface for such a spatial region. JHEP01(2014)137 ]. 7 )[ (2.1) (2.2) (2.3) (2.4) (2.5) 2.1 , denotes the  represent the 2 5 2 5 t Ω Ω d d + and ) 1 u x 2 ( f du 2 . u is a momentum cut-off that 4 u b + 4 u
a 1 2 3 ] entanglement/disentanglement dx 1 + 14 , + 2 2 13 , where b ) = -plane. The radial coordinate is denoted u dx u 2 θ ( iθ . ) R → . u , ( u

, where noncommutativity parameter is non- 4 ] = is the string tension. Finally, 6 ) , h h ] 3 u 2 0 u gauge theories. Perhaps more importantly, we – 4 – 4 2 ( 6 u 1+1 α , x , a  s h 2 2 R 5 g N 2 s + H x u g R [ ) is also characterized by two functions, denoted by 2 1 u × 2 θ  = = dx 2.2 R 2 − , where 01 2Φ u represents the 2 0 } + plane is defined by a Moyal algebra 3 denotes the string coupling which is related to the radius of 2 2 θ Nα ) = 1 s , x s dt g u R 2 , e ) ( ,C x ) u πg f ) { ( u u ( f ( 2 h = 4 h 4 u 3 4 plane. u u − 2 R 2 θ 4  a , s R 2 2 R g 4 R R via ) respectively. These functions are explicitly given by = = = u ( 2 u h 23 ds B denotes the radius of curvature of the background geometry, 0123 R F ; the ultraviolet (boundary) is located at Note that, the background in ( This article is divided in the following sections: we begin with a brief review of the -directions, whereas ) and Note that the simplest way to understand why this corresponds to a noncommutative gauge theory is to u 1 6 , u 1 ( f consider an open string in the corresponding background, which yields the commutation relation in ( by is taken to bethe large. geometry Also, metric on an unit 5-sphere. Here R At large N andgiven strong which, ’t in Hooft the string coupling, frame, a reads holographic [ description of this theory can be In this section, we will considerYang-Mills a theory four-dimensional on maximally supersymmetric a SU(N) spacetime zero super only in the 2 Noncommutative Yang-Mills 2.1 Holographic dual A mild form of non-locality can be realized by considering noncommutative gauge theories. we then discuss onenoncommutative geometry. possible We then way discuss toinformation features for define of a entanglement “rectangular sub-regions entropy strip”in of and geometry mutual both section various at 3 possible vanishing and and shapesmore at 4 general finite on respectively. shapes: temperature a a Afterin commutative this, section cylinder we 6. in discuss section Finally some 5 we and features conclude a in of noncommutative section entanglement cylinder 7. for transition as observed inwill generic also large show that mutualnoncommutative theory information with is the the corresponding right quantity commutative to one. compare thenoncommutative results gauge theory of and the its holographic dual geometry. Within the same section an area-law. Furthermore, it undergoes the familiar [ JHEP01(2014)137 = a (2.6) (2.7) (2.8) (2.9) — as can be 2 θ ) bears is larger u T a ( 2 θ h or T 2 θ R or 2 θ R can also be thought denotes the location b . The strict limit of 2 u H Z u along / 2 θ l . The parameter R s ∼ = πg 2 θ T -background. The non-trivial B- = 4 5 S λ × . . , 2 2 / = 0, the infrared limit of the geometry is / λ . 1 4 1 4 2 ) / ) / H b √ 1 a 1 a u b

b λ λ au u u  ( ( – 5 – is the string length. This leads to the condition ) can be simply obtained by a chain of T-duality b = s ∼ l  α au 2.2  a l a ) — of a coordinate distance -directions as a 2-torus , where 2.2 } 0 and thus the geometry degenerates. Hence we need to 2 s ) — does not degenerate. To sharpen this constraint, let us 3 /l is related to the noncommutativity parameter through: , x 2.2 → 2 0. In this limit, we recover an AdS-space. On the other hand a ) x = 1 { u 0 → ( h /α u , 1 ∼ is the ’t Hooft coupling defined as ) represents the existence of a black hole in the geometry and u λ . ( is the short distance cutoff then → ∞ f  ∞ b u < b can also be viewed as the degeneration of this 2-torus. u , where : s θ l Before proceeding further, a few comments are in order: first, note that when there is Also note that the background in ( √ 4 → ∞ / 1 b Later we will observestructure of that entanglement this entropy. “cut-off” does play an important role in the divergence Finally, as explained before, wemeasured also by need the to metric impose inintroduce the ( constraint the that following the dimensionless “cut-off” than Therefore, if which is trivially satisfied for largeas ’t the Hooft momentum coupling. cut-off. The UV Anotheras cut-off constraint measured comes by from the the fact metric that in proper ( distance — 2.2 Regime of validity We can trust the supergravity solutionsmall only compare when to the scalar curvature of the background is transformations on the familiarfield and AdS-Schwarzschild the dilaton are generatedHence as a we consequence can of this view chainu of the duality transformations. no black hole present inobtained the by background, sending i.e. if we send impose the signature that the dualof gauge theory the is event noncommutative. horizon Here λ and thought of as theparameter “renormalized” that noncommutativity will at enter in strong every coupling, holographic since computation. this is the The function JHEP01(2014)137 ) ∈ 3 , x 2 ∂B (2.19) (2.15) (2.16) (2.17) (2.18) (2.10) (2.11) (2.12) (2.13) (2.14) should , , x ). Now, 1 2 F ˆ x x ∂A ( , 1 F x ( ˆ F as operators with 3 x and 2 to be the noncommutative . x , 3 . x } } 0 0 ) . F, 3 ˆ i x ≤ ≥ and = . 2 , ) ) 2 2 x i 3 3 F | ˆ x x . ) F , 2 | , x , x , ∂γ 1 } } 2 2 respectively F x 0 0 , x ( 1 , x , x B ˆ basis: = ≤ ≥ 1 1 x F iθ . ( x x i i ) = 0 ) ( ( F F 2 3 F F F

x | ] = and ) = F F | γ ) 3 , x i | i | 3 = | | 3 2 ˆ – 6 – x ˆ x A (4+1) N F F i ˆ ) ) x . Let us take , , 2 3 3 , 2 θ 2 , x G 2 {| {| 2 x 1 4 x ˆ T | x [ˆ Area( ˆ , x , x x x ) , = = 2 2 , ( ⊗ 2 1 1 1 ˆ x , x A ) B F , x , x x 1 , (ˆ ) = 2 ( 1 1 1 ˆ R F ˆ x x F x F ( ( ( ( , x { { or ) form a complete basis 1 ˆ → . In the boundary theory, we should look at F 3 θ 2 RT 2 θ x ˆ that lies entirely in the commutative submanifold: = = x ( S F R , R 2 A F B ⊗ ˆ ⊗ x 1 ∂B , , 1 R 1 , this division is unique. x R or ) = F ( ⊂ 3 ˆ F ∂A , x com 2 M , x 1 x commute and we can work in the ( 2 F x , where Now, we can ask the following general question: if we choose a function com and ˆ ˆ This is the simplest case.have a In boundary this case, theM answer is straight forwardF since the sub-regions what doescommutative background? it correspond to2.3.1 when the boundary theory is defined on a non- For a given function on the boundary and use the bulk geometry to calculate the RT-entropy Now the system can be divided in two sub-regions in a unique way: The eigenstates of In the noncommutative case, followingbe the promoted correspondence to principle an the function operator In the commutative case, an entangling surface can be defined as an algebraic equation which defines two sub-regions denoted by Let us begin within an a elementary noncommutative discussion geometry.spacetime of of We the defining will form regions consider using adirections. (3 entangling In + analogy surfaces 1)-dimensional with quantum noncommutative mechanics, we should treat 2.3 Noncommutativity and entanglement entropy JHEP01(2014)137 A (3.1) (3.2) (3.3) (2.20) (2.21) (2.22) (2.23) ) following F and the area ( 3 RT , x S 2 x , , up to a volume factor.  . (5) N 2 L A ) . G , u (10) N 4 ( 2 } } L 1 G f 0 0 = 4 − u  ≤ ≥ . (8) ind ) ) ∈ } } ) is the entanglement entropy between + 2 2 0 0 G 3 2 F 0 ( , x , x q ≤ ≥ 1 1 , x X 2 x x RT 2Φ F F ( ( s S −

3 F F , then in the boundary theory the function i | i | 3 | | , x σe – 7 – x F F ) ) 8 2 2 duu  d {| {| l 2 is proportional to , x , x Z , Z = = 1 1 and ). l 2 3 x x (5) N A A ( ( B 2 2 s R − ( { { ) gives the geometric entropy between spatial regions (10) N x g G 2 1  defines the quantum analogue of an entangling surface. It S L F G = = ( ˆ ∈ 4 F 1 = A B ) = RT x = S F A parametrizes the worldvolume of the minimal surface and the ( A ≡ S σ RT X S , 4 0 : α ) is written in the string frame, hence we will use the generalized RT- B 6 π 2.2 and . The extremal surface is translationally invariant along = 8 -plane. In the bulk theory however, we still can calculate A 2 θ ) does not have a well-defined meaning since we cannot draw a sharp curve in ) in the bulk theory, there is a unique decomposition in the boundary theory: is a function of both 3 3 R (10) N , x , x F → ∞ 2 2 G : L , x , x B Although an entangling surface cannot be drawn in the general case, for any function 1 1 = 4 SYM case. x x ( ( The above expressionSchwarzschild background. coincides Hence, with theN entanglement the entropy is corresponding the same expression as that in for a the pure AdS- with of the extremal surface (in the Einstein frame) is simply given by where 5-dimensional Newton’s constant 3.1 Commutative rectangularLet strip us choose the explicit computations.background In in particular, ( we willformula take for the the “infinite 10-dimensional geometry strip” with geometry. a The varying dilation subsystems 3 Infinite rectangular strip In this section we will investigate one particular example for which we can easily perform F Thus in this case, the operator is therefore interesting to investigate whether When, F the fuzzy the usual bulk prescription. and 2.3.2 General case Therefore, the RT-entropy JHEP01(2014)137 - ) 3 b 1 = x u , x ( 1 (3.6) (3.7) (3.8) (3.9) (3.4) (3.5) h ,X x c and p u l } = 3 ∼ ) u , x 2 3.5 x { should be 3 x . or  . . 2 2 L ) x ) , ). Area of this surface (in the u , u ( ) u ( 2 L 1 joined smoothly at ( ) ( h u h ) is anisotropic in 4 u ( 7 − ( X  u h  h 6 c 6 2.2  u u ∈ +  6 c 6 , udu 6 c 6 2 , we can write u u 3 0 − u u b l du 2 3 c 3 c u 1 , x X − u ± represents the point of closest approach of − u 1  1 s 1 c

 3 r u  ) = b r – 8 – b , x c u r = 5 u u 5 duu  ( u Z u u ) is still AdS and hence the physical distance is the l 2 b X Z 3 , c u ± u l 3 2.2 2 R 2 s Z 2 = 2 s g R − L g 2 =  2 L l du 2 ∈ dX = = 2 x A A ≡ . However, the relevant metric one should use to compute physical b X u can be determined using the boundary conditions: = c u u ]. 11 . The corresponding extremal surface is translationally invariant along ) and is an integral of motion and c → ∞ u → ∞ L Now we will go ahead and obtain the equation of motion using action ( 0 ) = 1. Let us now compute the entanglement entropy for an infinite strip specified by For convenience, we will refer to these as the U-shaped profiles. We also note our results here are 7 u ( ,X This area is divergent and using the UV-cutoff consistent with [ and finally the area functional is which leads to where, the extremal surface. Such0 surfaces have two branches, string metric for the backgroundsame in as ( the coordinate distance. One crucial comment is indirections. order: Naively, the it bulk looks metric likeat in the a ( given physical cut-off distance along distances is not the bulk metric, but the open-string metric. One can check that the open- with and the profile ofEinstein the frame) surface is given in by the bulk is given by We will begin withf the case when there is no black hole in the geometry, i.e. by setting 3.2 Non-commutative rectangular strip JHEP01(2014)137 b ∼ u l a . It and c , = Λ. In 8 has 8 θ (3.11) (3.12) (3.13) (3.10) . . θ u . Thus 0 0 b ∼ l < l u ∼ 2 a a a < c c . In fact for u ∼ a u 6 2 . x 1 c u ∼ a , which does not pen- 0 b l 10 u 6, U-shaped solution does . ≥ = 1 , l u < .  l ) a b b u 8 2 2 ). But before we proceed, a few au a ), we can conclude that the 3.9 . ln( . For ∼ . l 4 ) 3 b a 1 b 3.12 u 2 finite 3 au 6 2 s A + ; however the solution with g lR , where, l ln( c 4 b 2 +

4 u 3 b L ) and ( u – 9 –  1 div 2 = l < l 2 a A a 3.11 4 3 + deg = 2 s is not a sensible regime, although the entanglement R b g A 2 c A u 2 L 2 2 a = l < l = for U-shaped profiles. The curve has a minimum at plays the role of a momentum cut-off. Thus for a momentum 2 c c b l div u ) plays the role of a renormalized noncommutativity parameter u 2 A a with l l a 5 10 25 20 15 , the uncertainty in the transverse direction becomes ∆ 3 can be calculated from the equation ( 6, two U-shaped solutions exist for each x . 1 > finite l A a . Variation of along b , two U-shaped solutions exist for each u 0 There is another solution to the extremal surface equation: ∼ 6. For 3 . is well-known that ina a momentum noncommutative cut-off theory Λ, with theour noncommutativity transverse case, parameter (to the the parameter momentum) ( and direction the stretches bulk to radial cut-off p is the minimum area solution only when Let us now commententropy on for why this particular case is still a perfectly well-defined quantity even for etrate the bulk at all. For this class of solutions, we get the following divergence structure Comparing the leading behaviours in ( whereas comments are in order.l > The l U-shapedsmaller profile area. exists only for For the case in hand we get Figure 1 1 not exist. JHEP01(2014)137 . 1 α , it  (3.14) (3.15) a b 0 limit. → ∞ u → l/a a . Evidently b u ) at the boundary. and for ∞ ( l/a ). Using this additional ) as Let us now focus on the c Yang-Mills case, the finite → O 2.9 N 0 8 l > l . X in ( . α , diverges at the boundary, result c l  0 α l a log X  3 2 s a R quadratic divergence in 2 2 s henceforth.  g L 2 2 b a L u + 

2 b solutions are also rather peculiar. First of all, = 2 u b π 2 0 limit and then perform the RT-computation. familiar – 10 – u 2 N u 0 limit does not reproduce the known result for α 3 = + → 2 s ). Naively, it seems that the entanglement entropy R → g u solution, indicates that it is only sensible to consider 2 a 2 a div b L 3.11 S u and hence we always have U-shaped solutions. = 0 = = l , as opposed to the quadratic divergence in the ordinary b u S div u  A c l 5966. Note that, in the ordinary large . 0 , this particular kind of surface obeys b . Here we should also note that we are interested in the regime ), we can first perform an RT-computation and then take the → c u ) 2.2 → l/a l > l ( ) is a monotonically increasing finite-valued function of u s l/a . Hence existence of ( b s 0 as u 2 a We can schematically write the entanglement entropy ( We will now argue that there is one way to reconcile with the above expectation. Let Thus we will discard the solutions From the bulk point of view, the → In other words, we consider length scales that are greater than 8 ∼ 0 c Where approaches part of the entanglementasymptotes entropy to approaches infinity. This zero is as clearly the not length the case of here. the This rectangular again strip makes the comparison part of the entanglement entropy as well, which we will discuss now. Note however, that takingordinary a Yang-Mills naive theory. Atbackground in this ( level, we canAlternatively, we make can a curious alsoThese observation: take two processes the given do the not yield the same result. This observation holds true for the finite which can be re-interpretedthis as comes at having the a cost of having to introduce an additional scheme-dependent quantity of freedom inside oneshould Moyal cell, not which worsen implies the divergence that, in at a the quantum very field least,us theory. noncommutativity recall that we“cut-off” introduced we can a rewrite dimensionless the “cut-off” divergent piece as similar behaviour. divergence structure obtained inhas ( a quartic divergenceYang-Mills case. in This is evidently counter-intuitive. We cannot fit more than one degree note that the corresponding RT-surface doesunlike not the probe other the familiar bulk well-defined geometry RT-surfaces,X which at obey all. the Moreover, boundary conditionIt that is important toin note an that apparent such volume solutions, dependencea for for divergent which entanglement mutual entropy. information. This subsequently In results a in later section we will observe a stronger presence of is still al well-defined quantity, it doeslength scales not make sensewhere one to can probe check that below the length scale for the noncommutative theory at this energy scale, though formally entanglement entropy JHEP01(2014)137 - 2 x (3.16) has the and thus } l 1 along the t, x { a { x 6 . . . ) 2 . The corresponding mutual SO(2), where l/a l/a 3 ( × s l 5 1) l∂ , ) with  2 2 l/a separated by a distance a L ( l SO(1 s  ] and hence measures the number of degrees 2 π

→ 15 2 4 N – 11 – 1) , ) = l/a is a concave one. ( . Variation of S 2 l has the SO(2) symmetry. Thus it is non-trivial that the 3 ) is independent of the length of the interval l∂ } 3 3.11 , x ) = l 2 Figure 2 ( x { C 2 0. It is straightforward to check that the above inequality is satisfied 0 is still satisfied, since there is no “effective” Lorentz symmetry in ≤ L ≤ ) a l  ) { ( l H ( s C l 0.45 0.55 0.50 C l l∂ l∂ -plane to protect it. } 2 ) since the curve in figure 1) symmetry and , t, x ) can be evaluated numerically, which is shown in figure In general the above result follows from three criteria: (i) Lorentz invariance, (ii) Before proceeding further in discussing aspects of mutual information, let us ponder 3.16 { l/a ( 3.3 Mutual information Let us now moveconsider on two to “rectangular strips” discussdirection. each mutual of information. width For a To define visual this rendition quantity of we the need set-up, to see figure unitarity and (iii) strongfull subadditivity of Lorentz entanglement invariance entropy.SO(1 is In our broken, construction, SO(3 the inequality the In an (1 +monotonically 1)-dim decreasing CFT, under an this RG-flow [ definesof freedom: a central chargeby function ( that can be shown to be that the divergent pieceone in can ( define a finite quantity derived from the entanglement entropy as follows between the commutative and thethis ordinary subtlety Yang-Mills is theories absent subtle. ins We the will mutual show information. that Meanwhile, the functional behaviour of over a key aspect of entanglement entropy in this case. It is straightforward to observe JHEP01(2014)137 , c 732 . l < l (3.17) (3.18) (3.19) (3.20) and/or β , x ≤ (b)), which is not 4 , x/l  x . Y 0 (see figure a + , l . The corresponding “phase dia- B , 2 → a ∪  a A x,a s S  l − − approaches the commutative result 0   B β is defined as a x S 

B . Clearly, to compute the entanglement entropy of + SYM s u NCYM ) ) A – 12 – − ≡ S and r  A, B A l a A, B ( ( ) = I  I and that for the computation of entanglement entropy of s  2 2 (a). Note that 3 x A, B ( 4 ≡ I   2 2 X a L ]. Here we will observe a similar physics. In this case mutual  14 9 2 π , x/l > β , 2 N , as we have already mentioned. It can be checked easily that for c = 0 , we have two candidates for the corresponding minimal area surface. ) = B , we have two choices for the minimal area surface. x, l > l B ∪ . A couple of comments are in order: first, we note that mutual information A, B ∪ A ( denotes the entanglement entropy of the region I A x/a depends on the non-commutative parameter . A diagrammatic representation of the two rectangular strips which are used to analyze Y β S To sharpen the latter statement, let us define the following quantity It is demonstrated in figure We consider 9 mutual information is divergent. gram” has been shown in figure for large again picks out the finite partthe of results entanglement entropy. for Second, ordinary it Yang-Millsthe seems theory case possible by to for recover setting entanglement entropy. where the region This gives rise to andiscussed interesting “phase in transition” details for mutual ininformation information, [ is which given has by, been the region information between the sub-systems where Figure 3 the mutual information. Here JHEP01(2014)137 x 0 we → 3.0 a 2. Dashed . all possible 2.8 = 1 a x a 2.6 0.6 2.4 , indicating an increase of (b) 2.2 a/x 0.5 2.0 L 2 L 2 N 1.8 H 2 limit) with l L l 0.4 0.8 0.6 0.4 0.2 A,B H = 1 and red solid line is for I Π a 2

– 13 – . Since mutual information encodes 0.3 5 0.2 6= 0. After Euclidean continuation and periodically identifying H a u , (a) x = 4. Blue solid line is for p l implies that noncommutativity introduces more correlations between l 5 for ) with SYM 1.20 1.15 1.10 1.05 1.00 NCYM I I x 2.2 . Variation of mutual information (at large . Panel (a): 2-dimensional parameter space for the (3+1)-dimensional NCYM boundary ) with A, B ( I 4 Introducing finite temperature We will now discuss the physicsgeometry at finite in temperature. ( To this end, we will now consider the where “NCYM” stands for non-commutative Yang-Mills andMills. “SYM” stands which for super we Yang- havecorrelations, plotted figure in figure two sub-systems of the full system. Figure 5 correlations as we increaserecover the the noncommutativity known result of for the pure theory. SYM theory. Also note that in the theory. The mutual informationalof is non-zero only inblack the line blue shows shaded the region. corresponding Panel mutual (b): information for variation the commutative case. Figure 4 JHEP01(2014)137 (4.1) (4.2) (4.3) (4.4)

The one that goes , and we are left with only

10

岻 岻 張 ). D ⻑ → ∞ aT ( approaches the constant solution 0 2 HORIZON 岫憲 噺 憲 BOUNDARY 岫憲 噺 憲 U ,L ). Just like the zero temperature  ) solutions. l > l T. 6 2 2 L D coincides with , . πa 2 2 L ) U 2 0, solution − U . = 2  ) when ( → 2

H ∈ H a 態 怠 π

u u ,U 3 経 戟 戟 2 ≤ A 1 a – 14 – ) , x U = 2 1 1 T U ≈ ( ) A , x ) and one constant ( aT  P ( l 2 0 l ,

l 2 − 鶏  ∈ 2 x ), one parallel ( ≡ , there are four solutions (see figure 2 l X ,U 1 U ), U-shaped solutions do not exist. It can be shown that at large temperature aT ( . All possible extremal surfaces for an infinite rectangular strip. There are four solutions: 0 1)

 l < l Presence of the second U-shaped solution probably is the bulk reflection of the boundary UV/IR 10 . When the noncommutativity is turned off completely, aT ( connection. It is interestingD to note that inone the U-shaped limit solution. case, there are twodeeper into U-shaped the solutions bulk ( has smaller area For let us discuss the nature offor extremal a surfaces given for this length particular geometric shape. Typically, Here we will confinebeen ourselves discussing to so far. discuss the “rectangular4.1 strip” geometry that Entanglement we entropy have ofBefore we an go infinite ahead rectangular andrectangular strip calculate strip the specified entanglement by entropy of a non-commutative infinite two U-shaped ( the time-direction, the corresponding temperature of the background can be obtained to be Figure 6 JHEP01(2014)137 . c T l . 2 c l (4.6) πa is the ) it is 2 b c u l ) and ∼ ∼ ) aT ( T 0 l aT T ( ) grows linearly 0 a l aT ( = 0 c ). At sufficiently high l l l and uncertainty along 3.0 ) which has larger area aT ( T P 0 is an effective cut-off that l c ∼ l i . In this regime, there seems  p c 7 , l 2.5 ) . At zero temperature, b P ( u A only at sufficiently high temperature . Dashed black line corresponds to some ). As before, there is also a constant , c ) (4.5) ) 2 aT 2.0 aT P . Hence one perhaps can argue that this U ( ( > l ( 0 is a fundamental length scale that comes T ) 2 A a , a

≤ A ) aT 6 . 1 ( l < l ) ∼ 1 0 does not depend on temperature. As we argued 1 – 15 – U l 1.5 3 ( b U , ( 2 ∼ u A 2 x A . At low temperature ). . 0 a b c < l ; this should be investigated more carefully in future. . However, at high temperature, l 1 2 u b 0 ) l aT u ( ∼ D 0 1.0 ( l  c l A c ∼ l l ) (solid blue curve) with c aT ( 0.5 0 ) which probably indicates that the space-time becomes fuzzier at l l < l , that does not go inside the bulk at all. This solution becomes b u 4.4 , average momentum along any direction L ) can be larger than T = T a H 5 aT 0 u . However, this phase transition can be an artifact of having temperature 15 10 { ( 0 P l ): to . Variation of D ); this feature is schematically represented in figure 1 b π u 4 U From the above discussion, it is clear that we have two length scales: & close to the momentum cut off for a particular momentum cut-off T c T At temperature noncommutative directions become ∆ phase transition is physicallynot irrelevant sensible because to at probe very below high temperature ( relevant cut-off scale because with temperature ( finite temperature. It is( possible to have to be a secondfrom order phase transition of entanglement entropy at before, presence of this solution indicates that it doesAt not zero make temperature, sense tofrom we probe the saw noncommutative below nature that ofappears the space-time. only Whereas, after we introduce a momentum cut-off solution( important only when where, we will show later that Similar to the commutative case, there exists a parallel solution ( and hence it can only be important when Figure 7 l temperature JHEP01(2014)137 (4.7) (4.8) (4.9) (4.10) c u a , 2.0  ) α . ln(  ,
4 4 H 4 u  u T = 1 respectively. 4 4 H 4 du u u − a a 4 1.5 h 4 1 u u − du ), there are two U-shaped solutions; 4 π 4 a 1 H 3 b   u 6 c 6 4 u 1 + u u au 3 a   2 ( s 1 + 5 and 6 c 6 . This feature is pictorially represented in 0 . g c u u − lR l √ + 2 1 1 + u

2 au − L = 0  √ a 1 a 3 c 2 = r – 16 – 2 h  u α 1.0 u 2 b c ∞ r u deg u 5 Z u A 3 corresponds to smaller area. For increasing values of above some   c ∞ R 2 c u 2 l 2 s 2 Z a g L L au for u-shaped profiles at finite temperature. The black curve is for =  2 c 2 l 2 π u = 2 0.5 N A = with l div S with a volume-worth of “area” l l a 2.5 2.0 4.0 3.5 3.0 . Variation of monotonically increases and so does . 0 8 l Similar to the zero temperature case, the constant solution(D) becomes important at Now, at finite temperature the divergence structure is slightly different: Now we will compute the area of the physically relevant U-shaped solutions. Proceed- , H cut-off dependence in ordinary quantum field theories. smaller value of which seems to receivethat an this additional is cut-off rather dependent unique, term since at usually finite finite temperature. temperature Note does not introduce additional As we mentioned earlier, forthe any one with smallerau value of figure and ing as before, at finite temperature we obtain Figure 8 zero temperature, blue and red lines are for JHEP01(2014)137 the , (red), (4.14) (4.11) (4.13) (4.12) T lT 7 β . π 0 ≤ { a = , T a  ), we get 4.0 0 l . For large , x/l , T a l a 9 (blue), .   5 (and . b π 0 S u c 2 2 , T a  = a x 2 2 a L l > l 3.5 ∼ a + T a . , exactly like the commutative l )  l a b 2 2 π  au 2 N . ln( + − S 3    ) 4 α = 0 (black), 3.0 T VT 4 2 3 b , T a a ln(

u T a 4 a x N 2
2 2 π 4  a – 17 – π for T 4 1 + a it is given by − S 4 l/a  S ∼ 3 π  2.5 + b 1 + , T a u 2 ) with l 2 a , a , the leading finite part of the entanglement entropy follows +  T does not receive strong contribution coming from the temper- 2 lT = S c l α 2 c 2 (green). 2 b l/a, T a ( u 2 . S 2.0   π 1 2 l < l 2 a L   = 2 2 L  a L T T a 2 π a , x/l > β  2 , N 2 a π  2 1 4 3 2 { N = 0 H ], and it is given by ) = is the finite part, which is pictorially represented in figure S . Variation of = 16 (brown), S S 1 π A, B ( = I for the configuration shown in figure which is independent of the noncommutativity parameter 4.2 Mutual information Once again mutual information can be easily obtained from the entanglement entropy and finite part of thecase. entanglement entropy That becomes means linear ata in large volume law. At highthe temperatures, entanglement entropy the is most expected dominantsurface to contribution [ to come the from finite the part near of horizon part of the extremal ature. As argued before, we will discard such solutions. For where for It is noteworthy that this Figure 9 T a JHEP01(2014)137 a 10 (5.1) , mutual c 0. Mutual In figure l < l → 11 x . a T 3.0 plane and length . Dashed black line 2 and/or 2 . π x 0 x − = 1 2.8 x T in r 2.6 and temperature a . = 1 for different temperatures. Blue solid 2 2 x a 2.4 − mutual information is again a well-defined 2 r

c l q and red solid line is for = 4 and – 18 – . One can again check that for ± b l ] disentangling transition and the corresponding 2.2 15 u π . = 0 14 . As expected, we observe that a non-zero value of for 1  = x x : T 11 T = 0. 2.0 and T → ∞ c ) with L L 2 L 2 A, B ( N 1.8 l, x > l I H 2 l L 0.2 1.2 1.0 0.8 0.6 0.4 A,B H , brown line is for I 1 . π Π 0 2 = and in the limit . Variation of depends on the noncommutative parameter 3 T x T β We have assumed that along 11 L information is divergent. Rather, we will focus onconsider some the qualitative background features at henceforth. vanishing temperature. For these purposes,5.1 we will Entanglement entropy Let us begin by considering a cylinder: a circle with radius So far weThe have primary studied reason the forto simplest this consider geometry, is technical namely more simplicity.computations the general become However, “infinite shapes involved to in rectangular gain as such intuition strip”. cases we one and will we needs will see not below. attempt a thorough However, analysis. many of the explicit information undergoes the expected“phase diagram” [ is shown inresults figure in a larger region in the phase space where5 mutual information is non-zero. More general shapes: commutative cylinder where we have pictorially shownishing how temperature, mutual above information thefinite behaves. length-scale quantity As which in yields the the the ordinary case Yang-Mills result of in van- the limit Figure 10 line is for shows the mutual information at JHEP01(2014)137 (5.2) (5.3) (5.4) (5.5) (5.6) . 2  ). The minimal ) , . u, x ... 2.19 2 = 0 ( ∂u  +  ) ∂F , 2 ) ) x  ) u, x 4 ( ∂u − u, x u ( ∂u u, x ua 2 ( r ∂F 1 + ∂F √ F 2  log( 2 0 5 4  2 x + u L ) u r 2 2  u x + 2 u, x 2 ∂ − ( ∂x ∂u ∓  2 Near the boundary, the extremal surface ) 2 r + ∂F x 12 p   u, x −

). ) ± ( ∂x 2 2 r – 19 – + 1) ∂F -plane using a polar coordinate is inconvenient, since there u, x ) = p u, x ( } ∂x 4 2 (  2 u ± 1 4 , x x ∂F 1 a u, x 0 x ( + 1) 1 ) = { 1 L 4 x  u 1 + ( u, x 4 ≡ ( a ∂ s ) ∂x F
). u u, x 1 + ( ( ) = ( + 1 h 2 ududx s F 4 u Z = 4 u, x ( a 0 L 1 2 0. The area functional is given by: s 3 L x g u R → ) = ). , x ) representing a circle in the . 2-dimensional parameter space for the (3+1)-dimensional boundary theory. Mutual A b 1 2 u 2.2 ( = 1 a x F For our purposes, it is particularly convenient to consider Cartesian coordinates. In the bulk geometry 12 given in ( is a non-trivial warp factor where The solution near the boundary is given by: Corresponding equation of motion is given by area surface can be parametrizedcan be by written in the following form: with informational is non-zero onlybelow in the the shaded black region. curvecase for Mutual ( the information commutative is case non-zero and for the below region the red curveNote for that the the noncommutative above curve falls under the general category discussed in ( Figure 11 JHEP01(2014)137 ) 12 (5.7) (5.8) = 5 and -direction 13 1 ua x 1 . x  ) 2 α ln( 2 r a . 3 c = 0 )) + ) 1 α ua ln( 2 2 u log( c 2 + r 2 1 c

( − 2 2 2 x r – 20 – ) 1 2 1 2 α 2 1 x - - + 2 b − 2 ru r 2 ( α r  1 − L π 2 2 2 - 2 = 2 at the boundary (black curve). The red curve is at x N ra = div ) . = 2. α A ( ln 2 ua S - Lr 4 T 2 . Behavior of the extremal surface for the commutative cylinder near the boundary. a 2 N ) numerically. Here we will not attempt so, instead let us focus on the divergence ∼ To obtain the corresponding entanglement entropy, we need to solve the equation 5.4 At finite temperature, similar to the infinite strip case, there will be an additional cut-off dependent 13 As we have argued before, the corresponding minimal areaterm surface should have less area in ( structure of the entanglement entropy which cannear be the deduced boundary. from the The leading divergent order solution part of the entanglement entropy is given by The curve above clearly demonstratesas that we the move along circle the gets bulk squashed radial along direction. the Radius of the cylinderblue is curve is for The sub-leading piece ofanisotropic. the Near solution the is boundary a the consequence extremal of surface the can fact be that represented the as bulk (see figure metric is Figure 12 JHEP01(2014)137 = 2 3 (6.1) (6.2) (5.9) (5.13) (5.10) (5.11) (5.12) dx + 2 2 dx : via , , ) } u 2 3 ( , r 1 ) will be more involved since h . This constraint sets a lower , 4 2 ≤ B div } | u . 2 2 ). We can consider the case of ) 2 1 ∪ πr c r x } 3 b + B r . 2 A u ( ( 2 r ( + c ≤ ) L S r . Note that this case falls under the O π S . 2 1 2 2 2 u 2 ≤ ( x x + N 0 = 2 3 ρ c x ≤ + l div → ∞ | 2 1 s 2 2 2 π + ) r x by ). We are taking this surface to estimate a possible L | 2 2 | du A

= ) ( x ) ) A 5.4 ) = finite | 2 b u 2 S ( – 21 – u ) ρ 2 , x 3 , x = 3 1 1 a with “area” A, B u x , x x 1 π 2] with ( ( ( 2 div I | { { 14 x Z ) , & ( b , L/ { = = B L r u 2 3 2 s ∪ = B A = g L/ πR A A u ( − 2 [ S to draw a sharp curve in the otherwise fuzzy plane to investigate = ∈ Region Region 1 A x is not an exact solution of ( . In this case, the computation of pretend 3 b u . -plane and define the region < r = r } 2 u 3 . The bulk interpolating minimal area surface can now be parametrized by 2 , x < r 2 dθ x 2 1 = 4 SYM case: { r ρ To proceed we define the corresponding polar coordinate in the plane + N Note that ). With this, the area functional is given by 2 u 14 ( lower bound on category where we what the classical bulk RT-surface yields. dρ ρ Now let us discuss potentiallyin a the more intriguing case. Let us consider constructing a circle We also imagine that Thus it is again possible toentropy construct as a evaluated well-defined using quantity derived the from RT-formula. the entanglement 6 More general shapes: noncommutative cylinder there are many candidate minimalthe area surfaces. However, it is easy to check that just like and hence where 5.2 Mutual information Here we will merely argueindependent that quantity mutual information above is athree still minimal concentric a circles well-defined, length-scale finite of and various radii cut-off and consider the two sub-regions as: bound for the value for the radius of the cylinder This is reminiscent ofstrip. the similar constraint we encountered earlier for the rectangular than a degenerate surface at JHEP01(2014)137 ∼ 2 (6.3) (6.4) /a 0 ρ = 10. The a b u for r ) and all the other U- solution to the equation 6.6 0 (6.5) → ) ) is taken to infinity. The vertical , a . 0 → ∞ b L u ) 3 best possible ). This means that we allow u ua u ( ( 1 g h 6.5 4 = 1 at the boundary which is the hallmark u   ) + u 2 2

( ) 0 /a ρ u 0 ) ) ( – 22 – 0 u L u ρ ( ( 0 ρ ρ s 3 ) with u u = (  0 g L d du + r With this condition, it can be shown that near the boundary ) = u ( 15 ρ is a dimensionless quantity. 2 ) that does not satisfy the condition ( /a 0 ρ 6.3 . Extremal surfaces for the noncommutative cylinder for different To investigate this case further we will find out the Note that (1) at the boundary. 15 of a well-behaved solution. of motion ( O is not a solution, indicatingsuch there a solution is existed, no it well-behaved would solution mean for this case. Note that, had where Now it can be checked that red line denotes the location of the boundary. which yields the following equation of motion Figure 13 black line corresponds toshaped the solutions leading asymptotically approach behavior this of solution the as critical ( solution ( JHEP01(2014)137 . 13 (6.9) (6.6) (6.7) (6.8) (6.10) , which is α that do not = b , u a c b . One can check u r = . , u ,  ) with boundary con- b 3 u  6.3 12 + ) represents the profile of → ∞ , r > r u u 1 3 b )  ( c 12 u u 2 5 c a X ( r 0  2 ).  2 O a L ( + 2 ) behaves in the following way: ) is an attractor to all these solutions , ρ 8 N , where . u , it is clear that there is a transition at b 6.3 ∼ O 8 6.6 1 = u b , for this solution we obtain a ) r u 14  8 u ) = 0 3 b ( 2 → c 0 3

u a − u √ ρ ) = u ( 4 b ρ + – 23 – + sub-leading terms = u u ( 4 5 b 1 2 b as c ρ a u r 2 4 ru ), probe the bulk geometry. 15 2 a 2 2 a α ( )  1 + and → ∞ ). From figure  L ra 0 all solutions correspond to the same radius in the boundary. ( 2 u ∼ O q X 14 3 2  N ) r a √ u L = ( 2 0 → ∞ ρ N ) = a div ) = b b | 3 u ) u u 5 ( ( √ 0 or (ii) c A ρ 2 ( ρ S , we recover the familiar area law → = c 0 is the closest approach point that depends on the radius div X | c ) u r > r 1) but finite. A ( =  S u 1) one solution of the equation of motion ( α . For c It is interesting to check the behavior of leading divergence for the U-shaped solutions. It is important to note that the solution in ( Note that we have previously encountered solutions with the following boundary be- There are other U-shaped solutions for this case; we have showed them in figure r  = ua An analytic answer isinvestigate the no issue longer (see available;r plot hence we have used numerical techniques to and hence in the limit Therefore, to make sense ofto the have calculation various of values entanglement of entropylarge and the ( to radius, allow it ourselves is essential to introduce a cut-off the degenerate minimal areaprobe surfaces the in bulk (ii), geometryqualitatively e.g. different at for the the all, noncommutative ones result cylinderthe given and boundary in the by behaviour divergent minimal area mutual surfaces information. with The physics is that for all these solutions at the boundary haviours: (i) the minimal area surface.behaviour for For entanglement the entropy class and of a solutions finite in mutual (i), information. we On obtain the a other familiar hand, area-law These numerical solutions areditions: obtained by solving the equation ( where, It looks like theiniscent entanglement of entropy the in volumerectangular this divergence strip case that or has we the a have commutative encountered cylinder volume geometries before divergence earlier. while and considering is the rem- This solution yields the following divergence Now imposing the boundary condition ( JHEP01(2014)137 ∼ 3; ∼ for ) √ / when δr ra 2 (6.11) (6.12) (6.13) a c ) which → ∞ r ( a r S ra ) = ( u r > r q ( , ) obtained by 0 r c ρ ( S 30 with step size r ) = 1 and a c , r < r r  ( , 25 q 1 > 2 r ) − ra ( 2 3 20 q x 5 3 + ˆ √ 2 2 π ) for the noncommutative cylinder with x 4 ) is a critical solution with A (

S = 6.6 Thus, entanglement entropy 15 ) = ˆ + sub-leading terms . The divergence structure heavily depends on – 24 – 3 r ) and we have a volume divergence. This is also 3 b ˆ x 16 u , . 2 5 2 com 13 2 plays the role of renormalized noncommutivity parameter. ) in ( r ˆ x − noncom r u 2 2 and it can be shown that , ( − a ) 1 c cyl r − 1, indicating an area law behavior. | 0 should be a step function of x ρ 10 cyl ) where ( ra 2 | ∼ ( ) ˆ a A F > p ) ( r r A 2  1 2 2.3 ( S a n 4 b L S ) F + 2 A indicates a volume divergence, Lu (1) function of 3 (see figure ( 2 n N S 5 4 O √ 5 N b 14 / 3 ∼ 2 with √ r u = ( 2 2 (1) function of i L a n div , | n c A > a O F L ) H F ) 2 | , figure S A b c ( N 5 0.7 1.0 0.9 0.8 u 3 and we recover an area law for entanglement entropy. Interestingly, it can r > r S ( 3 0 √ ρ ) is another / ) is an . Variation of entanglement entropy 2 , r < r 2 c ra ra ( ( p q < a . However, RT-prescription provides us with an entanglement entropy ) r Before we conclude, a few comments are in order: let us try to connect these results b ) at the boundary. Solution = 10. For r < r Note that in our case, the parameter 2 u u / a 16 08. For ( ( 2 b 0 0 . with eigenvalues tracing out all a which implies that there is more entanglement for thewith noncommutative the cylinder. discussion of section reminiscent of the volumeρ behavior that wealso have be encountered checked before. that for large For where ρ for u where 1 Figure 14 JHEP01(2014)137 N ) indi- u ( 0 ρ noncommuta- ] holds. In the N 18

]. This stems from the general result obtained limit the noncommutative Yang-Mills theory 17 – 25 – N the RT-prescription, for a region residing entirely on via because the relevant length scale for these calculations is 1. r . c  r ) = r δr/r and hence ( 2 a ], which states that all planar Feynman diagrams in a noncommutative Yang-Mills b Let us conclude by saying that our analysis here does not involve any perturbative There are various directions for future explorations. In this article we have only fo- Let us also note that in the large We have also observed that the role of this additional cut-off is more crucial if we want It is extremely difficult to compute mutual information on this noncommutative plane; u 18 ∼ we alluded to in section 4. field theoretic computation. Ittropy will in a be weakly interesting coupled tomechanical noncommutative consider set-up. field analyzing theory, It or entanglement in is en- a well-known more that elementary noncommutative quantum theories do play an important large temperature limit, however, we do recover the thermalcussed entropy on as the expected. divergent part ofIt entanglement will entropy be for interesting general to shapesfor analyze of and such the understand regions. sub-regions. the For finite thesible part “rectangular of second strip”, the order it entanglement entropy phase will transition also be of interesting entanglement to entropy explore at the finite pos- temperature, which in [ and an ordinary Yang-MillsIn theory this are article the we same, haveactually observed unless receives that there entanglement a is entropy, non-trivial as anand obtained contribution external thus using from momentum. falls RT-formula the outside noncommutativity the even class at of large observables for which the result in [ clear why this violationthan is the necessarily area. of volume-worth, rather than anythingdoes else not bigger differthermodynamics from is the identical in ordinary both cases Yang-Mills [ one for a number of observables, e.g. the to define entanglement entropy, the noncommutative plane.distinctly In peculiar properties, this which case, mayentropy lead the and to also corresponding a result volume-law minimal behaviour inthe area for a violation entanglement surfaces divergent of mutual have an information. area-law It stems is interesting from to the note inherent that non-locality if of the theory, it is not not completely lie onan the additional noncommutative cut-off, plane. whichnoncommutative This torus. in comes the The at bulkentropy corresponding the geometry is leading cost not is divergence of altered realized structure by introducing as noncommutativity. in the entanglement degeneration of the 7 Conclusions In this article we investigated aspectstive of gauge quantum theory entanglement using in a theallows large AdS/CFT us correspondence. to obtain We observed well-defined that entanglement observables the for RT-formula a class of regions, which do r however, strong dependence of the divergent partcates of that the mutual entanglement information entropy defined on inbehaviour the even usual above way, may not yield a cut-off independent is a continuous function of JHEP01(2014)137 ]. , 09 ] , , SPIRE Adv. , IN JHEP ]. Thus it (1999) 032 , ][ 19 09 ]. JHEP hep-th/9711200 ]. , ][ ]. ]. SPIRE ]. IN reduced model as hep-th/9303048 SPIRE ][ [ N SPIRE IN SPIRE ]. IN IN ][ SPIRE [ ][ (1999) 1113 IN ][ SPIRE A large- 38 IN (1993) 666 ][ M theory as a matrix model: a ]. (1947) 38 71 hep-th/9907166 71 [ SPIRE

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