aa r X i v : . [ h e p - t h ] D ec Prepared for submission to JHEP
Renyi entropy of the critical O ( N ) model Anirbit
Department of Applied Mathematics and Statistics, Johns Hopkins University
E-mail: [email protected]
Abstract:
In this article we explore a certain definition of “alternate quantization” forthe critical O ( N ) model. We elaborate on a prescription to evaluate the Renyi entropyof alternately quantized critical O ( N ) model. We show that there exists new saddles ofthe q -Renyi free energy functional corresponding to putting certain combinations of theKaluza-Klein modes into alternate quantization. This leads us to an analysis of trying todetermine the true state of the theory by trying to ascertain the global minima amongthese saddle points. ontents N (critical) spherical non-linear sigma model 5 F q at different saddles? 125 Conclusion 176 Acknowledgement 18A The branch-cut of the logarithm 18 A.1 log( z + a ) about ia z + a ) about − ia A physical system is imagined to be separated into two parts A and B and let ρ bethe density matrix of the full system. One defines the “reduced density matrix” of thesystem A as ρ A = − T r B [ ρ log ρ ]. Given a physical system X , T r X is supposed to meana trace taken in a basis of quantum states which are localized in system X . Then thevon-Neumann entropy corresponding to ρ A is S A = − T r A [ ρ A logρ A ] and this is defined asthe “entanglement entropy” of the system A . When these quantities are evaluated in anyQFT then these have non-singular (believed to be “universal”) parts which are believed tobe independent of the regularization prescription and are believed to contain data about anearby CFT (“critical point”). – 1 –ut the structure of these universal terms of the entanglement entropy appear to havea strong dependence on the dimensionality of the system, • For 1 + 1 CFTs if the system A is a system of length x and B is the complement ofthat in R then one can show that S = c log xa where a is a short-distance cut-off and c is the central charge (the universal data!) of the CFT. (here the “=” is meant toindicate equality upto terms non-singular in the a → S = A c log ξa and A is the number ofboundary points of A and ξ is the correlation length. (one is imagining the A andthe B to be composed of intervals each of whose lengths is larger than ξ ) • For CFTs in dimensions 1 + ( d >
1) it is conjectured that the leading contributionto the entanglement entropy S takes the form of what is called the “area law”. Thismeans that the most divergent piece of S is f ( a ) A a d − where A is the area of theboundary of the region A and and f is some function of the short-distance cut-off a . This believed “area law” on one hand means that the entanglement entropy islocalized on the boundary of A and on the other hand it also means that there isnothing universal about this leading (most divergent) piece of S . One believes thatat least in the case of 1 + 2 CFTs when the boundary between A and B is a closedsmooth manifold the difference (say ∆ S ) = S − f ( a ) A a d − is something universal (andhopefully geometric). Also the expectation is that this ∆ S remains universal evenwhen the theory is moved away from the critical point. In the case of critical O ( N )model in 2 + 1 dimensions this universality is seen in both N and ǫ = 3 − d expansionsand the universality of this ∆ S is expected to extend into the range 1 < d <
4. [1] • On deforming away from the critical point it is conjectured that the this entangle-ment entropy should change as, lim L →∞ S = f ( a ) (cid:0) La (cid:1) d − + r (cid:16) Lξ (cid:17) d − where ξ is thecorrelation length and L is some length scale of the entangling surface. • These conjectures about the universal parts also extend to the Renyi entropies definedas S q = − q log (cid:2) T r A [ ρ qA ] (cid:3) (and then S = lim q → S q ). Now one expects that ondeformation from a CFT the correlation length is seen by the S q as, lim L →∞ S q = f q ( a ) (cid:0) La (cid:1) d − + r q (cid:16) Lξ (cid:17) d − (the coefficients f and r now have a q dependence) Just QFT considerations go only so far but with the help of AdS/CFT [4], Ryu-Takyanagiproposed a more detailed structure for the universal parts of the entanglement entropy[8–10]. Let A be a d − d + 1 dimensionalspace-time at the boundary of the bulk space-time. The CFT of interest lives on thisboundary and the bulk is asymptotically locally AdS d +2 . Let γ A be the minimal surfacein the bulk which has A as its boundary. Then the Ryu-Takyanagi conjecture states that, S A = Area ( γ A )4 G d +2 N . This conjecture obviously reminds one of the Bekenstein-Hawking entropyformula and the two things indeed seem related in various ways. [11–14]– 2 –t needs to be emphasized that the Ryu-Takyanagi conjecture uses only a tiny amountof the full power of Maldacena’s duality. In particular in scenarios of AdS /CF T whereit has been extensively checked one needs to take the limit of large central charge on theCFT side and by virtue of the Brown-Henneux formula we understand that to be a “semi-classical” limit. It is an important question to understand the possible generalization ofthe Ryu-Takyanagi conjecture at finite central charge.[16–20]Using the Ryu-Takyanagi conjecture in the case of a spherical entangling surface one be-lieves that the divergent and the universal parts of the entanglement entropy for a CFTin odd D space-time dimensions is of the form, S = s D − (cid:0) Rǫ (cid:1) D − + ... + s Rǫ + s + O ( ǫ )and for D even, S = s D − (cid:0) Rǫ (cid:1) D − + ... + s (cid:0) rǫ (cid:1) + s L ln (cid:0) Rǫ (cid:1) + ˜ s + O ( ǫ ). The s andthe s L are the expected universal terms. s is proportional to the partition function of theEuclidean CFT on S D . s L is proportional to the central charge defined from the coefficientof the Euler density in the trace anomaly. An argument of Casini, Huerta and Myers [21], shows that the reduced density matrix ρ A for the special case of a spherical entangling surface of radius R is the same as thethermal partition function of the CF T d on a hyperbolic cylinder ( H d − × R ) of radius R at atemperature T = πR . Then the entanglement entropy becomes the thermal entropy of theEulcidean CF T d on H d − × S where the circle is of radius πR and the H d − continues to beof radius R . By an extension of the same argument Smolin, Myers, Jung and Yale [24] haveshown that similarly Renyi entropies can also be written as, S q = πqR − q ( F ( πR ) − F ( πqR )),where F ( T ) is the free energy of the CF T d on the same H d − × R of radius R at atemperature T . If one considers a scalar field on
AdS d +1 (we will assume it to be unit radius unless statedotherwise) then one can show that if the field decays towards the boundary as z ∆ ( z asdefined in the usual Poincare patch coordinates) then the only way the field can be nor-malizable is if ∆ > d . But one knows that the unitarity bound of a CF T d is ( d − and by AdS/CF T this ∆ is supposed to be the dimension of the boundary operator dual to thisbulk scalar field. So naively there seems to be a gap between these two facts.So one asks if it is possible to somehow change the bulk action for the scalar field toget the normalizable bulk solutions to asymptote as powers less the d . This is achieved bythe Klebanov-Witten form [5] of the scalar action gotten by dropping the total derivativefrom the bulk scalar action. (modulo the subtleties with the Gibbons-Hawking term if thespacetime has a boundary) Then the new achieved lower bound on the asymptotic powersof the normalizable modes matches the CF T d unitarity bound of d − – 3 –ut a more curious thing happens once this new shifted Klebanov-Witten action isused in the bulk. Initially the two possible boundary asymptotics for the bulk scalar fieldwere ∆ ± = ( d ± √ d + 4 m ) and only one of them was above the normalizablity enforcedlower bound on the bulk asymptotics. But once this lower bound falls to d − one sees thatthere now appears a mass range ( − d , − d + 1) where both the possible solutions for ∆are above the lower bound. ( − d is what is called the Breitenlohner-Freedman bound)So if one is sitting in this sweet spot of mass range where both the solutions are abovethe unitarity (boundary)/normalizability (bulk) bound one is tempted to ask if there is acontinuous deformation on the boundary which RG flows between one fixed point to theother. And that exists and they are given by “double-trace” deformations of the form O where O is an operator of dimension in the range ( d − , d ). This is a “relevant” defor-mation being done in the UV so that the it affects the IR without affecting the UV. Thedeformation strength needs to vanish in the UV and go to infinity at the IR where it willhit the fixed point.Klebanov and Witten go on to show that these two CFTs (when both exist!) have gener-ating functionals related by Legendre transforms. When the bulk scalar goes as z ∆ + it isdetecting the IR fixed point on the boundary and when it goes as z ∆ − it is detecting theUV fixed point on the boundary. So ∆ + is the conformal dimension of the dual operatorin the IR and ∆ − in the UV. (one has ∆ + + ∆ − = d )One can expand this discussion beyond just scalar fields in the bulk and consider spin-s (Bosonic) fields in the bulk. In the following we briefly sketch the general ideas followingthe beautiful paper, [6] For s ≥ AdS d +1 will be ∆ = s + d − z for the (free) spin-s fields are z − s +(∆ − =2 − s ) and z − s +(∆ + = s + d − (these are quite distinct from the s = 0 case described earlier!). So whenthe bulk spin-s field is asymptoting as ( − s +∆ − ), clearly the dual operator in the boundaryCFT (sitting at its UV fixed point) has to have conformal dimension of ∆ − and the mostnatural such choices are the spin-s gauge-fields. And similarly at the IR fixed point thedual operator will have a conformal dimension of ∆ + , which is the unitarity bound andthe dual operator will be a spin-s conserved current.One wants to understand if the two scenarios of having boundary conserved currents inthe IR and boundary spin-s gauge fields in the UV can be connected by a double-tracedeformation induced RG flow. One thinks of deforming the boundary theory in the IRby a double-trace term of the form, R ( J ( s ) ) , where J ( s ) is a spin-s operator of dimension∆ > d . So its an irrelevant deformation being done in the boundary in the IR. This willinduce a RG flow which will gives rise to an UV fixed point at which the dimension of J ( s ) will be ∆ − = d − ∆ + O (1 /N ). (thinking of a boundary theory made up of N com-plex scalar fields on S d ) So now specializing to the case d = 3 and recalling the unitaritybound for s ≥ − ∆ ≥ s + 1. Combining this with our condition for the deformation to be irrelevant we– 4 –ave, < ∆ ≤ − s . Its clear that this condition can’t be satisfied for s ≥ d = 3 and ∆ = s + 1 and then J ( s ) inthe UV becomes a spin-s gauge field of dimension 2 − s and then its not a gauge singlet andhence not an observable and hence doesn’t raise any obvious reason to comply with theunitarity bounds. Then in the UV we have a conformal spin-s gauge field theory. Anotherspecial case is when s = 0 and then the unitarity bound is d − and hence for the UV tobe unitary one needs 3 − ∆ > − and hence ∆ > .One wants to understand the change in the free-energy between the UV and the IR fixedpoints under such a double-trace transformation and the argument clearly needs to bedifferent depending on whether ∆ = s + 1 or ∆ = s + 1 (restricting to d = 3). Thenreproduced from both the bulk as well as from the boundary in [6] it has been argued thatthe free-energy change is given by, • When ∆ = s + 1, F ( s ) UV − F ( s ) IR = − (2 s + 1) π Z ∆ dx ( x −
32 )( x + s − x − s − Cot ( πx ) • When ∆ = s + 1, F ( s ) gauged UV − F ( s ) free IR = (4 s − s logN + O ( N ) N (critical) spherical non-linear sigma model Let us focus on the ( φ ) field theory. We are thinking of this as the free scalar field theorybeing deformed in the UV by a double-trace term of the form δS UV ∼ R d xJ (0) J (0) where J (0) = P Ni =1 φ i φ i is a scalar singlet operator of scaling dimension ∆ = 1 in the UV. It isknown that for a double-trace deformation like this in the UV to flow to to a IR fixed pointone needs the operator to have UV dimensions in the range ( d − , d ) and here one seesthat this will satisfy the condition for d = 3. For such a deformation it has been shown forthe free energy ( F = − log Z ) both from the bulk (a Vasiliev theory on AdS ) as well asfrom the boundary that δF = F UV − F IR = − ξ (3)8 π + O ( N )We now try to go further than this and try to understand the Renyi entropies at thelarge- N critical point of this theory. This system is potentially interesting because here atthe large- N critical point one can mainatin a dual gravity description and also get quan-tum corrections in the boundary. This can be believed because the large- N limit of this φ theory is exactly equal to the large- N spherical non-linear sigma model to all orders in N and hence if the former is critical so is the later. And we have that for critical non-linearsigma model the central charge is proportional to the dimension of the target manifoldwhich in this case is S N − . – 5 –ence the large- N critical theory has its central charge increasing (linearly with N )and hence one can believe that a consistent gravity truncation should holographically existin the bulk.The recent papers by Maldacena, Aitor, Faulkner [3] and Hartman [17] have made thispoint amply clear that Ryu-Takyanagi conjecture is a (semi)classical phenomenon in the(boundary) bulk. Ryu-Takyanagi is a shadow evidence that there might be a holographicpicture out there. That the conjecture is classical in the bulk is obvious right away sincenothing more than classical gravity is used there. On the boundary side the thing is semi-classical in a way that has been made clear in Hartman’s recent paper - that one needsto take a double limit whereby the primary operator dimension and the central charge areboth very large but at a constant ratio. And large central charge is basically suppressingthe quantum effects - as can be evidenced from say Brown-Henneux like arguments.We write the partition function of the non-linear sigma model as, Z q = Z [ Dφ ( x )][ Dλ ( x )] exp[ − S ( φ, λ )] (3.1)where S ( φ, λ ) = 12 g Z H × S q d d x √ g [( ∂ µ φ ) + λ ( x )( φ − N )] (3.2)and φ is a N component vector field and λ is a Lagrange multiplier which is con-straining the field to be on a S N − defined as φ = N . We are using the imagery of a φ representing a classical spin of size √ N and the coupling g is the loop expansion parameter.As necessary for the calculation of the q − Renyi entropy the action integral is being doneon H × S q (where H is the 2 − dimensional hyperbolic plane or Euclidean AdS and S q isa circle with the standard round metric on it but where the angular parameter goes from0 to 2 πq )One might want to make it explicit that the Lagrange multiplier field of λ ( x ) is implement-ing a point-wise constraint on the base manifold as in, R Dλ exp[ − g ] R d d x √ gλ ( x )( φ ( ~x ) − N ) ∼ Q x δ ( φ ( ~x ) − N )Hence doing the Gaussian integral over φ we have, Z q = Z Dλ exp[ − N T r log[ − ∂ + λg ]] exp[ N g Z d d x √ gλ ] (3.3)Now we look for uniform saddles of the kind, λ ( x ) = m − . Here we think of m asthe conformal mass on the branched manifold and we are measuring it in terms of its gapfrom the BF bound on H of unit radius i.e − .– 6 –o at the large-N saddle we have, Z q = exp[ − N T r log[ − ∂ + m − g c ] + N g c V ol ( H × S q )( m −
14 )] (3.4)We note that from here on we make it explicit that the coupling constant g is tunedto its flat-space critical value of g c and that its only the large-N theory that is critical andits the critical theory which has been put on the branched manifold. The main goal ofthe following analysis will be to see if and there can exist such a conformal mass for thisflat-space critical theory lifted to the branched manifold.Now we have for the free-energy F q as, F q = N T r H × S q log[ − ∂ + m −
14 ] − N g c V ol ( H × S q )( m −
14 ) (3.5)For a circle factor the eigenfunctions are e inzq (thinking of z as the coordinate aroundthe circle). The volume of this circle is 2 πq . We continue with the conventions from [6]about the measure and eigenvalues of the scalar harmonics on H and hence we have,2 F q N V ol ( H ) = X n ∈ Z Z µ> dµ tanh( π √ µ )4 π log( µ + m + n q ) − πqg c ( m −
14 ) (3.6)The conformal mass ( m ) of this theory is that value of m for which F q extremizesi.e the value of m for which the following derivative vanishes,2 N V ol ( H ) ∂F q ∂m = X n ∈ Z π Z µ> dµ tanh( π √ µ ) µ + m + n q − πqg c (3.7)But this infinite sum is term by term divergent! So we need to come up with a regu-larization scheme for this and that is to (1) first sum over the infinite KK modes and then(2) to take the saddle-point of the difference free-energy, F q ( m ) − qF (0). The intuitionbeing that at q = 1 the theory is on flat-space where the conformal mass vanishes and thephysically relevant part of the free-energy of the critical theory on the q − branched man-ifold is its difference from q copies of the flat-space free-energy of the “same” critical theory.For some a, q ∈ R we have the zeta-function regularized identity, X n ∈ Z aa + n q = 2 πq sgn ( a ) coth( πq | a | )Now we use this in the above saddle equation to get,2 N V ol ( H ) ∂F q ∂m = q Z µ> dµ tanh( π √ µ ) sgn ( p µ + m ) coth( πq | p µ + m | ) p µ + m − πqg c (3.8)– 7 –hese integrals are still divergent and we now focus on the physically motivated dif-ference. (and choosing the positive square-root) we have,2 N V ol ( H ) (cid:20) ∂F q ∂m − q ( ∂F q ∂m | q =1 ,m =0 ) (cid:21) (3.9)= q Z µ> dµ tanh( π √ µ ) " coth( πq p µ + m ) p µ + m − coth( π √ µ ) √ µ The above quantity is finite! One can flow the R.H.S above as a function of m and seethat (1) it monotonically decreases with increasing m > m < m = 0) and (2) that it has a root in m only for q ≤ q = 1 its theunique root m = 0 whereas for q < x − axis). For all q > O ( N ) model when put on the branchedmanifold doesn’t have a critical point and hence its not a CFT there. So we need to dosome modification to this theory to get critical theories on the branched manifold. Let us take a closer look at the contribution of each KK mode to the above analysis.We remember that to the quantity, NV ol ( H ) ∂F q ( m ) ∂m the n th KK mode contributed the(divergent) quantity,14 π Z µ> dµ tanh( π √ µ ) µ + m + n q = 12 π Z λ> dλ λ tanh( πλ ) λ + m + n q (3.10)Firstly we imagine this to be an integral analytically continued to the complex λ plane.Now we regulate the integral by putting an UV cut-off Λ on the eigen-spectrum. Then inthis complex picture using the parity symmetry in λ we see that the value of this integralis (2 πi times) half the sum of the residues of the integrand in a semicircle of radius Λ aboutthe origin and the semi-circle being closed either in the upper or the lower half plane. Onenotes that on that plane the function λ tanh( πλ ) λ + m + n q has poles at i ( n + ) ( n ∈ Z ) and at ±| q m + n q | and the residues of it at these poles are respectively, − i (2 n +1) π (1+4( n + n − ( m + n q ))) and ± i tan h | q m + n q | π i .In the AdS d +1 what constituted as the choice of standard vs alternate boundary con-ditions was the choice between the boundary scaling of the fields to be either z ∆ ± . Inthe context of this integral, we say that the difference between the two quantization is tobe thought of as the choice between which of the “mass” poles ( ±| q m + n q | ) is to beaccounted for while keeping the semi-circular contour in the upper half-plane.– 8 –o if we are are going to put the n th KK mode in alternate quantization then from thequantity previously calculated, NV ol ( H ) h ∂F q ∂m − q ( ∂F q ∂m | q =1 ,m =0 ) i we first subtract the con-tribution of the n th KK mode in the standard quantization and then add in the contributionin the alternate quantization. This expression is given as, q R µ> dµ tanh( π √ µ ) (cid:20) coth( πq √ µ + m ) √ µ + m − coth( π √ µ ) √ µ (cid:21) + π " − R Λ →∞ λ> ,in dλ λ tanh( πλ ) λ + m + n q + R Λ →∞ λ> ,out dλ λ tanh( πλ ) λ + m + n q (3.11)In the above expression the subscript in/out means whether when interpreted as acomplex integral (along a semi-circular contour in the upper-half plane) which of the poles ±| q m + n q | is taken to contribute respectively. One notes that each of the integralswithout the UV cut-off of Λ are divergent but we realize that a net finite contribution isobtained by first taking at the difference between the integrals with a cut-off and themlater letting the cut-off go to infinity.So we have, R Λ λ> ,in/out dλ λ tanh( πλ ) λ + m + n q = πi "P Λ n =0 − i (2 n +1) π (1+4( n + n − ( m + n q ))) ± i tan ( | q m + n q | π ) = πi i π H − −| r m + n q | + H − + | r m + n q | − H +Λ −| r m + n q | − H +Λ+ | r m + n q | + πi h ± i tan ( | q m + n q | π ) i (3.12)In the above the + / − on the RHS corresponds to the in/out on the LHS respectively.And the H stand for the harmonic number function. Now doing a large Λ asymptoticexpansion of R.H.S about Λ = ∞ we have, Z Λ →∞ µ> ,in/out dλ λ tanh( πλ ) λ + m + n q = πi i π ( − γ + H − −| r m + n q | + H − + | r m + n q | ) − iπ log Λ + O ( ) + πi h ± i tan( | q m + n q | π ) i (3.13)So we see that when the “out” integral is subtracted from the “in” integral the UVdivergence parameterized by Λ will cancel out and we will be left with a regular contribu-tion. Hence we can write the regularized saddle equation for the n th KK mode being inalternate quantization as, – 9 –
NV ol ( H ) h ∂F q ∂m − q ( ∂F q ∂m | q =1 ,m =0 ) i ( n,alternate ) = q R µ> dµ tanh( π √ µ ) (cid:20) coth( πq √ µ + m ) √ µ + m − coth( πq √ µ ) √ µ (cid:21) + i − i π ( − γ + H − −| r m + n q | + H − + | r m + n q | ) − iπ log Λ + O ( ) + i tan ( | q m + n q | π ) + i i π ( − γ + H − −| r m + n q | + H − + | r m + n q | ) − iπ log Λ + O ( ) − i tan ( | q m + n q | π ) and this simplifes to, NV ol ( H ) h ∂F q ∂m − q ( ∂F q ∂m | q =1 ,m =0 ) i ( n,alternate ) = q R µ> dµ tanh( π √ µ ) (cid:20) coth( πq √ µ + m ) √ µ + m − coth( π √ µ ) √ µ (cid:21) + tan( | q m + n q | π ) We note that given the effective notion of mass for the n th KK mode its BF bound is − n q (so the standard quantization for it is valid in the range − n q ≤ m ). Further it can be putin alternate quantization for − n q ≤ m ≤ − n q . So the n = 0 mode has the highest BFbound of 0 and hence the n > q modes can never be put in the alternate quantization sincethat will have no common mass range to share with the other modes. Further if the n = q mode is put in alternate then its upper bound will coincide with the lower bound of the n = 0 mode and hence irrespective of the quantization of the other modes only the m = 0situation will be allowed. Hence modulo these considerations, the modes n = 0 , ..., q − q possible scenarios to explore at every q .Hence we write the expression for the saddle point as, NV ol ( H ) h ∂F q ∂m − q ( ∂F q ∂m | q =1 ,m =0 ) i (3.14)= q R µ> dµ tanh( π √ µ ) (cid:20) coth( πq √ µ + m ) √ µ + m − coth( π √ µ ) √ µ (cid:21) + P n = qn =0 s n tan( | q m + n q | π )The parameters s n are put in such that if s n = 1 then it would put the n th KK modein the alternate quantization and if s n = 0 then it would have that mode in the standardquantization. In the above we have let the sum on the RHS to go till the q th mode withthe implicit understanding that is s q = 1 then the only value of m that can be tested asa candidate saddle is m = 0. – 10 –e remember that if s j = 0 then its allowed range is [ − j q , ∞ ) and if s j = 1 then itsallowed range is [ − j q , − j q ]. Thus the valid range of m is the intersection of all theseconstraints (one constraint each for each of s n =0 ,..,q ).It turns out that this integral in the above expression is well-defined and can be numericallyevaluated for specific values of the parameters. Care needs to be taken that the integrandbe supplied to the computer as q R µ> dµ tanh( π √ µ ) (cid:20) √ µ coth( πq √ µ + m ) − √ µ + m coth( π √ µ ) √ µ √ µ + m (cid:21) .Hence one can numerically find the roots of 3.14.For example at q = 2 the allowed ranges and saddles are,n = 0 n= 1 n = 2 allowed m range saddle value of m S S S [0 , ∞ ) no solutionsA S S [0 , ∼ . , ] no solutionsA A S [0 , ] ∼ . q = 3 the allowed ranges and saddles are,n = 0 n= 1 n = 2 n=3 allowed m range saddle value of m S S S S [0 , ∞ ) no solutionsA S S S [0 , ∼ . , ] ∼ . , ] no solutionsA S A S [0 , ] ∼ . , ] no solutionsA A A S [0 , ] ∼ . , ∼ . , ] ∼ . F q so that it can be evaluatedon these saddles for any value of q and thus determine which is the global minima of F q and thus be able to determine the true state of the theory.– 11 – Can we compare the free-energy F q at different saddles? Earlier we had the following expression,2 F q N V ol ( H ) = X n ∈ Z Z µ> dµ tanh( π √ µ )4 π log( µ + m + n q ) − πqg c ( m −
14 ) (4.1)On this we can use the zeta-function regularized identity, X n ∈ Z log( a + n q ) = 2 log(2 sinh( πq | a | ))to write it as,2 F q N V ol ( H ) = 12 π Z µ> dµ tanh( π √ µ )[ πq p µ + m + log(1 − e − πq √ µ + m )] − πqg c ( m −
14 )(4.2)The integral in the above expression is divergent and the divergence is not goingto disappear by an analogous subtraction that was done previously. So for the mo-ment we don’t spend efforts into trying to regularize it but try to understand betterthe contribution of any specific ( n th ) KK mode to F q NV ol ( H ) and that is the integral, R λ> dλ tanh ( πλ )2 π log( λ + m + n q ). (where we have substitute µ = λ ). Let ~n denotethe tuple of integers corresponding to the KK modes being put into alternate quantization.For each n i ∈ ~n we symbolically subtract the contribution of that mode in the standardquantization and add in its contribution in the alternate quantization. So we have,2 F ( ~n,alt ) q N V ol ( H ) = π R µ> dµ tanh( π √ µ )[ πq p µ + m + ln(1 − e − πq √ µ + m )] − πqg c ( m − )+ P n i ∈ ~n lim Λ →∞ h [ − R Λ λ =0 ,standard + R Λ λ =0 ,alternate ] dλ λ tanh( πλ )2 π log( λ + m + n i q ) i In the above expression it is understood that the m featuring on the right is the value ofthe saddle point mass (as calculated for example in the last section for q = 2 ,
3) if it existsfor this combination of ~n being put in alternate quantization. In the following we shallshow that it is possible to give a finite meaning to the P n i ∈ ~n term – 12 –ow we define the two integrals R Λ λ =0 ,standard and R Λ λ =0 ,alternate as follows, • We notice that the integrand is symmetric in changing λ to − λ and hence hence wecan extend the integral to the line segment [ − Λ , Λ] and divide the final answer by .Now we further imagine the integral to be in a complex λ plane so that we are aftergoing from − Λ to Λ we move up on a semicircle of radius Λ centered at the origintill we get close to the y − axis.Now we note that the function log( λ + m + n i q ) needs two branch-cuts with one endat ± i q m + n i q . ( this branch-cut has been explained in details in the Appendix ) Welet the two branch-cuts start at ± i q m + n i q and move up/down the y − axis. Wealso note that the function tanh ( πλ ) has poles all along the y − axis at points i ( n + )for n ∈ Z .Hence once the contour reaches the y − axis we drop down along its right side keepinga distance δ > y − axis and making ǫ > πz ) pole encountered till we are within ǫ of the branch-point i q m + n i q . Then we make a clockwise circular tour around the point i q m + n i q inan ǫ radius circle and then move up the left side of the y − axis keeping a distance δ from it and as before making semi-circular humps around the tanh( πz ) poles. Oncewe are at − δ + i Λ we go on to complete the semicircle of radius Λ around the originthat we started out making. It is understood that ǫ, δ → R Λ λ =0 ,standard = [ − R Λ ,semicircle − R standard key − hole ] with themeanings of the notation being obvious. • Now when we try to define the quantity R Λ λ =0 ,alternate , we do exactly as above ex-cept that instead of turning around and up around the branch-point i q m + n i q we keep going down into the lower half plane till we are ǫ away from the otherbranch-point − i q m + n i q . When we are at z = δ − i ( q m + n i q + ǫ ), we starta counter-clockwise tour around the negative imaginary branch-point and reach thepoint − δ − i ( q m + n i q + ǫ ) along this partial circle. It is clear that we take this circleso as to avoid intersecting the branch-cut which moves down the y − axis starting atthe point − i q m + n i q .Then we continue moving up the y − axis keeping a distance δ from it and mak-ing semi-circular ǫ radius humps around the tanh( πz ) poles (as we also made on theright side of the y − axis while coming down). We move up till we are at − δ + i Λand then we go on to complete the semicircle of radius Λ around the origin that westarted out making. – 13 –ence we have defined R Λ λ =0 ,alternate = [ − R Λ ,semicircle − R alternate key − hole ] with themeanings of the notation being obvious. (we note that the part R Λ ,semicircle is exactlythe same in both the quantization prescriptions)So the expression currently looks like,2 F ( ~n,alt ) q N V ol ( H ) = π R µ> dµ tanh( π √ µ )[ πq p µ + m + log (1 − e − πq √ µ + m )] − πqg c ( m − )+ P n i ∈ ~n lim Λ →∞ h [ [ R Λ ,semicircle + R standard key − hole ] dλ λ tanh( πλ )2 π log( λ + m + n i q ) i + P n i ∈ ~n lim Λ →∞ h [ − R Λ ,semicircle − R alternate key − hole ]] dλ λ tanh( πλ )2 π log( λ + m + n i q ) i = π R µ> dµ tanh( π √ µ )[ πq p µ + m + log (1 − e − πq √ µ + m )] − πqg c ( m − )+ P n i ∈ ~n h π [ R standard key − hole − R alternate key − hole ] dz z tanh( πz ) log( z + m + n i q ) i Hence effectively the remnant quantity is R standard key − hole − R alternate key − hole and init the contributions from the contours above the point i ( ǫ + q m + n i q ) completely cancelout. What remains are these two parts, • There is a contour integral of the function λ tanh( πλ ) log( λ + m + n i q ) along a cir-cular contour along an ǫ circle around the branch-points ± i q m + n i q but avoidingintersecting the branch-cut at both of these.One can imagine both these integrals to be of a holomorphic function on a Riemannsurface such that the integration path doesn’t close to a loop on the branch-cut. Inthat imagination the integration measure “ dz ” scales with ǫ and hence in the limitthese two integrations do not contribute. • (From here on for the sake of ease of notation we denote a i = q m + n i q )The integration path that does contribute is the journey from − i ( a i + ǫ ) to i ( a i + ǫ )upwards and downwards along the right and the left of the y − axis keeping a distanceof δ from it and making semi-circular humps of radius ǫ around the points i ( n + )( n ∈ Z ) that one encounters along the way. Let i ( n iu + ) be such a pole just below ia i and let i ( n id + ) be such a pole just above − ia i . One notes that if { a i } > then n id = − (1 + ⌊ a i ⌋ ) , n iu = ⌊ a i ⌋ . If { a i } ≤ then n id = −⌊ a i ⌋ , n iu = ⌊ a i ⌋ − i ( a i + ǫ ) to iǫ + i ( n iu + ). Weparameterize the upward journey on the right of the y − axis as z R = i ( a i + ǫ ) + δ − it from t = a i − ( n iu + ) to t = 0 and the downward journey on the left as z L = i ( a i + ǫ ) − δ − it for the same range of t in the reverse order. (we rememberthat the direction of travel is upward on the right and downward on the leftbecause of the “-” sign infront of the R alternate key − hole integral) We note thatan extra 2 iπ needs to be added to the log function when it is on the left of the y − axis and above the branch-point i.e on the z L path from t = 0 to t = ǫ . ( thisfactor of iπ has been derived in details in the Appendix ) So the correspondingintegral is, R t = a i − ( n iu + ) z R tanh( πz R ) log( z R + a i ) dz R + R a i − ( n iu + ) t =0 z L tanh( πz L ) log( z L + a i ) dz L + R ǫ z L tanh( πz L )[ i π ] dz L (= 0)If we let ǫ, δ → i ( n id + ) inanti-clockwise direction in a circle of radius ǫ and moves up/down the y − axistill − i ( a i + ǫ ). Let the trip around the n id pole be parameterized as z c = i ( n id + ) + ǫe iφ and on the left below it as z L = i ( n id + ) − iǫ − δ − it and onthe right as z R = i ( n id + ) − iǫ + δ − it . For z L its parameter t goes from 0 to a i + ( n id + ) and for z R its the same range of t but in reverse order. Now weneed to add an extra 2 iπ to log function in the z L integral when it is below thebranch-point i.e for t = a i + ( n id + ) − ǫ to t = a i + ( n id + ). ( this factor of iπ has been derived in details in the Appendix ) So the corresponding integralis, R t = a i +( n id + ) z R tanh( πz R ) log( z R + a i ) dz R + R a i +( n id + ) t =0 z L tanh( πz L ) log( z L + a i ) dz L + R a i +( n id + ) a i +( n id + ) − ǫ z L tanh( πz L )[ i π ] dz L + R πφ = − π z c tanh( πz c ) log( z c + a i ) dz c (= R πφ = − π z c tanh( πz c ) log( z c + a i ) dz c )gfIf we let ǫ, δ → i ( n iu + ) to i ( ǫ + n id + ). This journey has a repeating structure as in foreach i ( k + ) pole encountered we are going around it anti-clockwise in a circleof radius ǫ and then moving down/up on the left/right of the y − axis (keeping δ away from it) below that circle till we hit the top of the next ǫ circle below iti.e the ǫ circle around i ( k − ). This goes on from k = n iu down to k = n id +1.We parameterize the trip around the pole i ( k + ) as z c = i ( k + ) + ǫe iφ . Thetrip on the right and below it can be given as z R = i ( k + ) − iǫ + δ − it . The tripon the left and below the z c circle is parameterized as z L = i ( k + ) − iǫ − δ − it .For z R , t goes from 1 − ǫ to 0 and for z L it goes through the same values butin reverse order. So the net contribution of this part of the contour is, P n iu k = n id +1 hR − ǫt =0 z L tanh( πz L ) log( z L + a i ) dz L + R t =1 − ǫ z R tanh( πz R ) log( z R + a i ) dz R i + P n iu k = n id +1 hR πφ = − π z c tanh( πz c ) log( z c + a i ) dz c i (= P n iu k = n id +1 hR πφ = − π z c tanh( πz c ) log( z c + a i ) dz c i )Here too in the limit ǫ, δ → F ( ~n,alt ) q N V ol ( H ) = π R µ> dµ tanh( π √ µ )[ πq p µ + m + log(1 − e − πq √ µ + m )] − πqg c ( m − )+ P n i ∈ ~n h π P n iu k = n id [ R πφ = − π dz c z c tanh( πz c ) log( z c + m + n i q )] i (4.3)These φ integrals can be given by the residue theorem as, R πφ = − π z c tanh( πz c ) log( z c + a i ) = 2 πiRes i ( k + ) [ z c tanh( πz c ) log( z c + a i )]= 2 πi ( i ( k + )) π log[ a i − ( k + ) ]= − (2 k + 1) log[ a i − ( k + ) ] (4.4)– 16 –ence the simplified final expression is,2 F ( ~n,alt ) q N V ol ( H ) = π R µ> dµ tanh( π √ µ )[ πq p µ + m + log(1 − e − πq √ µ + m )] − πqg c ( m − ) − P n i ∈ ~n h π P n iu k = n id (1 + 2 k ) log[( m + n i q ) − ( k + ) ] i (4.5)We remind ourselves that for each i index we had defined a i = q m + n i q . If { a i } > then n id = − (1 + ⌊ a i ⌋ ) , n iu = ⌊ a i ⌋ . If { a i } ≤ then n id = −⌊ a i ⌋ , n iu = ⌊ a i ⌋ − P n i ∈ ~n exists only if for the given q and ~n (and hence the saddlepoint m ) is such that the value of q m + n i q is large enough such that there exists anon-zero number of i ( k + ) poles (of tanh( πz )) on the y − axis between ± i q m + n i q . Itturns out that for the q = 2 and q = 3 cases explicitly computed in the previous sectionthis is not the case and hence there is no P n i ∈ ~n term for them!Hence for all the saddles detected at q = 2 and q = 3 the corresponding 4.5 equationhas only the first two terms, the divergent µ -integral and the g c dependent term. Thedivergent µ -integral has its integrand dependent on q and ~n and hence it is tempting toredefine L.H.S of 4.5 to absorb the divergent integral and compare the remaining finiteterms to determine the physical minima. Doing this in the particular case of q = 2 and q = 3 leads to the conclusion that the q-Free-energy density is minimum for whoever hasthe minimum g c term and that is true for whichever saddle has the heaviest mass and hencethat is the true state of the theory. A particularly novel interpretation of the meaning of “alternate quantization” has beenexplored in this writing in the specific context of Renyi entropy of a critical O ( N ). Itseems exciting to further explore the ramifications of this interpretation in other scenariosand to hopefully uncover a possibly more general definition of “alternate quantization”.Immediately it seems interesting to try to generalize this calculation to other more generalsigma models. Even in the context of the specific question explored in this paper, we arefaced with a whole range of unanswered questions which we shall briefly enunciate here.Firstly it should be interesting to understand better as to how the formalism of this articleis related to the analysis in section 6 of the pioneering work [1] by Max A. Metlitski, CarlosA. Fuertes and Subir Sachdev. Secondly, as of now no structure is visible in the table ofsaddle points, 3.2 and 3.2. One would ideally want to have an analytic method of knowingas to which combinations of KK modes being put into alternate quantization producessaddle points and where. This seems like a major question that is currently open here.Thirdly, the interpretation of 4.5 is currently not on a very firm ground. One would hopeto get a cleaner interpretation about how this equation helps us determine the true state of– 17 –he theory. The theory of Renyi entropy has clearly opened up a new and exciting avenue ofresearch in holography and this article hopefully shows how this can provide us a context toexplore this otherwise rare question of understanding ground states of non-supersymmetricQFTs. The author is extremely grateful to Thomas Faulkner for the extensive discussions whichhelped formulate all the crucial techniques in this work. Without the enormous help ofThomas Faulkner this work would have never happened. The author was supported by theUniversity of Illinois at Urbana-Champaign (UIUC) during the writing of the first half ofthis work. A lot of gratitude is due to Max Metlitski for providing a lot of insights andinspiration during the later stages of this work.
A The branch-cut of the logarithm
The logarithm function can be thought to have a branch-cut along the negative x − axiswhich mathematically can be stated as, lim θ →± π Im [ log ( re iθ )] = ± π (though θ = ± π isgeometrically the same location). Now we want to do this same analysis for the function log ( z + a ). A.1 log( z + a ) about ia Consider the function log( z + a ) while taking z around in a small circle of radius r < a about ia for a >
0. We parameterize this roundtrip as z = ia + re iφ . Hence we arelooking at the function f ( φ ) = log (( ia + re iφ ) + a ). This when expanded out becomes, f ( φ ) = log (( r cos 2 φ − ar sin φ ) + i ( r sin 2 φ + 2 ar cos φ ))Let φ = π + ǫ , i.e when this trip is just crossing the positive y − axis. Then the realpart of the argument of the logarithm evaluates to − r cos 2 ǫ − ar cos ǫ . This for ǫ → ± is a negative number. Hence the logarithm is being evaluated around its branch-cut whichis the negative x − axis.At the same location the imaginary part of the argument of the logarithm function evalu-ates to − r sin 2 ǫ + 2 ar sin ǫ and this for ǫ → ± is 2 rǫ ( a − r ). Since r < a it follows thatfor ǫ > x − axis branch-cut andfor ǫ < f ( π ) = 2 iπ + f ( π − ). This gives meaning to the statement that one of the branch-cutsof the function f ( z ) = log ( z + a ) starts at ia and goes up the positive y − axis. (so a 2 iπ has to be added to the log function on the left of this branch-cut)– 18 – .2 log( z + a ) about − ia Consider the function log ( z + a ) while taking z around in a small circle of radius r < a about − ia for a >
0. We parametrize this roundtrip as z = − ia + re iφ . Hence we arelooking at the function f ( φ ) = log (( − ia + re iφ ) + a ). This when expanded out becomes, f ( φ ) = log (( r cos 2 φ + 2 ar sin φ ) + i ( r sin 2 φ − ar cos φ ))Let φ = − π + ǫ , i.e when this trip is just crossing the negative y − axis. Then the realpart of the argument of the logarithm evaluates to − r cos 2 ǫ − ar cos ǫ . This for ǫ → ± is a negative number. Hence the logarithm is being evaluated around its branch-cut whichis the negative x − axis.At the same location the imaginary part of the argument of the logarithm function evalu-ates to − r sin 2 ǫ − ar sin ǫ and this for ǫ → ± is − rǫ ( a + r ). Hence it follows that for ǫ > x − axis branch-cut and for ǫ < f ( − π − ) = 2 iπ + f ( − π ). This gives meaning to the statement that one of the branch-cutsof the function f ( z ) = log ( z + a ) starts at − ia and goes down the negative y − axis. (soa 2 iπ has to be added to the log function on the left of this branch-cut) References [1] Max A. Metlitski, Carlos A. Fuertes, Subir Sachdev, “Entanglement entropy in the O ( N )model”, http://arxiv.org/abs/0904.4477 [2] Aitor Lewkowycz, Juan Maldacena, “Generalized gravitational entropy”, http://arxiv.org/abs/1304.4926 [3] Thomas Faulkner, Aitor Lewkowycz, Juan Maldacena, “Quantum corrections to holographicentanglement entropy”, http://arxiv.org/abs/1307.2892 [4] Juan M. Maldacena, “The Large N Limit of Superconformal Field Theories andSupergravity” http://arxiv.org/abs/hep-th/9711200 [5] Igor R. Klebanov, Edward Witten, “AdS/CFT Correspondence and Symmetry Breaking”, http://arxiv.org/abs/hep-th/9905104 [6] Simone Giombi, Igor R. Klebanov, Silviu S. Pufu, Benjamin R. Safdi, Grigory Tarnopolsky,“AdS Description of Induced Higher-Spin Gauge Theory”, http://arxiv.org/abs/1306.5242 [7] Igor R. Klebanov, Silviu S. Pufu, Benjamin R. Safdi, “F-Theorem without Supersymmetry”, http://arxiv.org/abs/1105.4598 [8] Shinsei Ryu, Tadashi Takayanagi, “Holographic Derivation of Entanglement Entropy fromAdS/CFT”, http://arxiv.org/abs/hep-th/0603001 [9] Shinsei Ryu, Tadashi Takayanagi, “Aspects of Holographic Entanglement Entropy”, http://arxiv.org/abs/hep-th/0605073 – 19 –
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