RReplica Instantons from Axion-like Coupling
Kantaro Ohmori ∗ Simons Center for Geometry and Physics, SUNY (Dated: January 21, 2021)We find a phenomenon in a non-gravitational gauge theory analogous to the replica wormhole in a quantumgravity theory. We consider a reservoir of a scalar field coupled with a gauge theory contained in a region withboundary by an axion-like coupling. When the replica trick is used to compute the entanglement entropy fora subregion in the reservoir, a tuple of instantons distributed across the replica sheets gives a non-perturbativecontribution. As an explicit and solvable example, we consider a discrete scalar field coupled to a 2d pure gaugetheory and observe how the replica instantons reproduce the entropy directly calculated from the reduced densitymatrix. In addition, we notice that the entanglement entropy can detect the confinement of a 2d gauge theory.
Introduction—
One of the most significant recent develop-ments in the study of quantum gravity is the island conjecture[1, 2] and the replica wormhole [3, 4] that reproduces theconjecture. These were used to derive the Page curve of theentropy of the radiation from a black hole, and are consideredto be a key milestone to resolve the information paradox.In this letter, we find an analog in a non-gravitational gaugetheory for the replica wormhole. This might indicate that somepart of the black hole evaporation process has an analogy ingauge theory and can be studied in it. In particular, we performthe exact computation for 2d pure gauge theories. There,we observe that the entanglement we consider can detect theconfinement of the theory.The state | ψ (cid:105) considered in this letter is artificial, without aclear experimental meaning. Finding a more physical setup,preferably with some connection or analog to the black holephysics, is remained for future studies.The replica wormhole is a wormhole connecting differentsheets of the replicas in the replica trick computation of entan-glement entropy. Such a configuration should be consideredif the gravitational path-integral involves the summation overthe topology of the geometry. In a non-gravitational theory,instead of the topology of the geometry, the topology of thegauge group bundle can be nontrivial on the replica manifold.The configuration of interest is that an instanton is presenton one of the replica sheets, at the same time an anti-instantonappears on another. Under the presence of an axion-like cou-pling, the total instanton number on the whole replica manifoldis constrained to be zero, so that such a pair, or in general a tu-ple, of instantons causes effects not local to any one of sheets.Therefore, we regard this non-local distribution of instantonsas an analog of the replica wormhole. The analogy between awormhole and a pair of instantons is also noticed in [5]. General setup—
Consider a scalar field φ coupled to a Z -valued topological number density c ( A ) of a gauge field A ona spacetime X : i (cid:90) X φ c ( A ) . (1)For example, c ( A ) can be the instanton density Tr F ∧ F if thespacetime dimension is four, and the coupling is that of theaxion.[6] We assume that the manifold X has a boundary and YX | ψ (cid:105) Σ (a) (cid:101) Y (cid:101) X B (b)
FIG. 1: (a) A spacetime geometry defining the pure state ψ .The shaded region is where the gauge theory lives and thewhite region is the reservoir of scalar field. (b) A single sheetof the replica manifold R n .is embedded into a manifold Y of the same dimensions. Whilewe make φ live over the whole manifold Y , the gauge field A (and matter fields charged under the gauge group) is containedinside the submanifold X . We impose the Dirichlet boundarycondition on A on ∂X , i.e. A | ∂X = 0 . We regard the scalar φ outside of X as a reservoir, and consider the entropy ofa subregion of the reservoir. This configuration mimics thesetup used in the gravity theories.To prepare a state, we let Y has the spatial boundary Σ thatintersects with X , and we consider the state | ψ (cid:105) defined bythe Euclidean path integral over Y (See Fig. 1a).[7] We take asubregion B of Σ outside of X , and consider the entanglemententropy of the reduced density matrix ρ B = Tr B c | ψ (cid:105) (cid:104) ψ | ,where B c is the complement of B in Σ .The convenient way of computing the entanglement entropyis the replica trick. We let (cid:101) Y denotes the double Y ∪ Y , where Y is the orientation reversal of Y and the two manifolds areglued along the spatial boundary Σ . A replica sheet is ˜ Y withthe cut along the subregion B of Σ (See Fig. 1b). (cid:101) X in thefigure is the double X ∪ X of X glued along X ∩ Σ , and is acodimension-0 submanifold of Y . The n -th replica manifold R n is constructed by glueing n copies of the replica sheetalong the cut B . Then, the trace of the n -th power of ρ B isexpressed by the partition functions: Tr ρ nB = Z [ R n ] Z [ R ] n , (2)with which the Rényi entropy S ( n ) B is defined by S ( n ) = 11 − n logTr ρ nB . (3) a r X i v : . [ h e p - t h ] J a n FIG. 2: A replica instanton configuration for n = 3 . Thedouble circle in the leftmost sheet denotes an instanton withinstanton number 2, and each of the crosses in the middle andthe rightmost sheets is a unit anti-instanton.The von Neumann entropy S vN B is the limit lim n → S ( n ) B .If the scalar φ has the shift symmetry broken only by theinteraction (1), the topological number (cid:82) X c ( A ) is effectivelyconstrained to be zero, since the integration over the constantmode of φ gives the delta function δ ( (cid:82) X c ( A )) . On a replicamanifold R n , however, the constant come of φ is constantover the whole of R n , and other modes are suppressed by thekinetic term. This constant mode only constrains the total sum of topological numbers: n (cid:88) i =1 k i = 0 , (4)where k i = (cid:82) (cid:101) X i c ( A ) is the topological number on the i -th copy of (cid:101) X . We write the contribution to Z [ R n ] fromthe configurations with topological numbers ( k , · · · , k n ) by Z [ R n , ( k , · · · , k n )] , so that Z [ R n ] = (cid:88) k , ··· ,k n (cid:80) i k i =0 Z [ R n , ( k , · · · , k n )] . (5)The term other than R (0 , ··· , n has a nontrivial topology of thebundles and give non-perturbative corrections. We call such aconfiguration “replica instantons". See Fig. 2) for an example.Note that because of the constraint (4), the non-perturbativegauge theory contribution does not factorize into the productof the contributions from each copy (cid:101) X i , and thus it is notlocalized in any one of the copies. Discrete scalar reservoir—
So far we have been assumingthat the scalar φ takes continuous values. To compute thereplica partition function, we have to do the path-integral of thescalar field on the replica manifold, which is in general not easy.To simplify the computation, in this letter we instead make thescalar field takes the discrete values: φ = 0 , πq , · · · , ( q − πq (mod 2 π ) for some q . In other words, φ is a Z q -valued field.This can be achieved either by imposing a potential cos( qφ ) with a big coefficient to a periodic scalar φ , or by consideringthe lagrangian qφF (cid:48) and integrating the auxiliary gauge field F (cid:48) (not to be confused with the gauge field on X ) out. Such adiscrete scalar field defines a topological field theory, and theHilbert space of the topological theory has dimension q andis spanned by the coherent states | (cid:96) (cid:105) , (cid:96) = 0 , · · · , q − with e i φ | (cid:96) (cid:105) = e π i (cid:96)/q | (cid:96) (cid:105) .With the discrete scalar φ , the path-integral over the replicamanifold is replaced by a finite sum over Z q . We assume that the boundary condition of φ on the boundary of B (and atthe infinity of ˜ M if it is non-compact) is Neumann so that thesummation over the Z q remains.[8] Another advantage to usethe Z q valued φ is that we can use a topological number definedonly modulo q as c ( A ) in the coupling (1). Accordingly, the“path-sum" of φ imposes the constraint (4) modulo q .As the field φ now does not fluctuate, the evaluation of thepartition function Z [ R n ] reduces to the partition function ofthe gauge theory: Z [ R n , ( k , · · · , k n )] = q (cid:89) i Z gauge [ ˜ X, k i ] (6)where Z gauge [ ˜ X, k ] denotes the partition function of the gaugetheory on (cid:101) X in which only the bundles with (cid:82) (cid:101) X c ( A ) = k mod q are summed over. The factor q comes from the sum-mation over the values of φ . Substituting (6) into (5), weget Z [ R n ] = q − (cid:88) (cid:96) =0 ( Z ( (cid:96) ) gauge [ (cid:101) X ]) n , (7)where Z ( (cid:96) ) gauge [ (cid:101) X ] is the discrete Fourier transform of Z gauge [ (cid:101) X, k ] : Z ( (cid:96) ) gauge [ (cid:101) X ] = (cid:88) k Z gauge [ (cid:101) X, k ] e π i k(cid:96)/q . (8)This is also the partition function of the gauge theory withtopological interaction π(cid:96)q (cid:82) c ( A ) added. Thus the trace of n -th power of the reduced density matrix is Tr ρ nB = (cid:80) (cid:96) (cid:16) Z ( (cid:96) ) gauge [ (cid:101) X ] (cid:17) n (cid:16)(cid:80) (cid:96) Z ( (cid:96) ) gauge [ (cid:101) X ] (cid:17) n . (9) Direct computation of ρ B — The result (9) suggests the sim-ple form of the density matrix: ρ B = 1 (cid:80) (cid:96) Z ( (cid:96) ) gauge [ (cid:101) X ] diag( Z (0) gauge [ (cid:101) X ] , · · · , Z ( q − gauge [ (cid:101) X ]) . (10)We can directly obtain this density matrix. For a fixed value of φ = (cid:96) , the path-integral of the gauge theory over the manifold X in Fig. 1a defines a state | X (cid:105) (cid:96) whose norm is Z ( (cid:96) ) gauge [ (cid:101) X ] .The state | ψ (cid:105) is | ψ (cid:105) = (cid:88) (cid:96) | X (cid:105) (cid:96) ⊗ | (cid:96) (cid:105) . (11)To define the reduced density matrix, we have to define a map[9] f : H Σ → H B c ⊗ H B . (12)In the replica computation we used the Neumann boundarycondition at the boundary of B . With the boundary condition,the Hilbert space H B is spanned by the coherent states | (cid:96) (cid:105) B .Likewise, the Hilbert space H φB c of the region B c prejectedto the φ -sector is also spanned by the coherent states | (cid:96) (cid:105) B c .Then, the map f is f : | (cid:96) (cid:105) (cid:55)→ | (cid:96) (cid:105) B c ⊗ | (cid:96) (cid:105) B . (13)Therefore, the reduced density matrix is ρ B = 1 (cid:104) ψ | f † f | ψ (cid:105) Tr H Bc f | ψ (cid:105) (cid:104) ψ | f † = 1 (cid:80) (cid:96) (cid:48) Z ( (cid:96) (cid:48) ) gauge [ (cid:101) X ] (cid:88) (cid:96) Z ( (cid:96) ) gauge [ (cid:101) X ] | (cid:96) (cid:105) B (cid:104) (cid:96) | B . (14)which is (10).
2d abelian gauge field—
First, we take the gauge group tobe U (1) and set the topological number to be the monopolenumber: c ( A ) = F . We follow the analysis in [10]. For afixed scalar field φ = π(cid:96)q , the theory is the U (1) gauge theorywith the theta angle θ = π(cid:96)q .If the theory is put on an interval I with Dirichlet boundarycondition and the temporal gauge A = 0 is imposed, thetheory reduced to the quantum mechanics of the holonomy G ( t ) = (cid:82) I A with the action πL (cid:90) d t (cid:18) g ˙ G + θ π ˙ G (cid:19) , (15)where L is the length of the interval and g is the couplingconstant. The Hamiltonian and the energy levels are H = 12 g πL (Π G − θL ) , (16) E j = 12 g L π (2 πj − θ ) , (17)where Π G is the canonical momentum for G and j ∈ Z . Wewrite the corresponding normalized eigenstate by | j (cid:105) whichalso diagonalize Π G : Π G | j (cid:105) = g ( j − θ π ) . We take the themanifold X in Fig. 1a to be a rectangular, X = [ − t , × [0 , L ] where the first factor is regarded as the time direction and thestate | ψ (cid:105) is prepared at t = 0 . As we impose the Dirichletboundary condition at {− t } × [0 , L ] , the holonomy G ( − t ) is zero. As the energy eigenstate | j (cid:105) diagonalizes Π G , the stateat t = − t , which has eigenvalue zero of G , is | t = − t (cid:105) = (cid:88) j | j (cid:105) . (18)Thus, the space time X and the theta angle θ = π(cid:96)q preparesthe state | X (cid:105) (cid:96) = (cid:88) j e − a (cid:48) ( j − (cid:96)q ) | j (cid:105) , (19)where a (cid:48) = g Lt is the dimensionless combination of thearea of X and the coupling g . As a 2d pure gauge theory isinvariant under area-preserving diffeomorphisms, this states does not depends on the shape of X other than the area. Thepartition function of the double (cid:101) X is Z ( (cid:96) ) U (1) [ ˜ X ] = (cid:88) j ∈ Z e − a ( j − (cid:96)q ) , (20)where a = 2 a (cid:48) is the dimensionless area of ˜ X .The contribution Z U (1) [ ˜ X, k ] form each topological class ofthe bundles can be obtained by reversing the discrete Fouriertransform (8): Z U (1) [ ˜ X, k ] = 1 q (cid:88) j ∈ q Z e − aj +2 π i jk = 1 q ϑ ( kq , i aπq ) , (21)where ϑ ( z, τ ) = (cid:80) ˜ ∈ Z e π i( τ ˜ +2 z ˜ ) is the elliptic theta func-tion. In particular, Z U (1) [ ˜ X, is the partition function of the U (1) gauge theory with the coupling g/q and θ = 0 . In otherwords, the gauge group of Z U (1) [ ˜ X, is the extension → Z q → (cid:93) U (1) → U (1) → . (22) (cid:93) U (1) is isomorphic to U (1) as a group, but from now on wedistinguish it from the original U (1) gauge group of the theorybefore coupling to φ . The partition function Z U (1) [ ˜ X, k ] isthen interpreted as the partition function of the (cid:93) U (1) gaugetheory with the background B of the Z q subgroup of the U (1) one-form symmetry [11] as B = k PD[ ˜ X ] , where PD[ ˜ X ] isthe Poincaré dual of the fundamental class [ ˜ X ] . Summing over k effectively does the quotient of the gauge group from (cid:93) U (1) to U (1) .Using the modular transformation of the theta function: θ ( τz , − τ ) = √− i τ e πτ i z θ ( z, τ ) , we can rewrite (21) as Z U (1) [ ˜ X, k ] = √ π √ a (cid:88) ˜ ∈ Z e − π a ( q ˜ − k ) , (23)which is entirely non-perturbative in a ∝ g for k (cid:54) = 0 . There-fore, we have explicitly checked that the replica instanton con-tributions are non-perturbative.The entanglement von Neumann entropy S vN for q = 2 , , is plotted in Fig. 3. In all cases the entropy is log q when a = 0 and it becomes zero when a → ∞ . The value log q is merelyfrom the reservoir; the entanglement entropy of an interval (ina circle) in the TQFT of Z q valued scalar is log q as the state f (cid:80) (cid:96) | (cid:96) (cid:105) is the maximally entangled state among the q states.As the Euclidean time t increases, the state | X (cid:105) (cid:96) evolves underthe Hamiltonian (16). When t is large, the lowest energyeigenstate contained in each | X (cid:105) (cid:96) dominates. Such a state for (cid:96) (cid:54) = 0 is considered as the state of confining string betweenthe probe charge (cid:96) particle of the (cid:93) U (1) gauge theory.[10] Asthe pure 2d gauge theory is confined, the confining string hasa nonzero tension, and thus the norm of | X (cid:105) (cid:96) for (cid:96) (cid:54) = 0 ,compared to that of | X (cid:105) , vanishes as a → ∞ . In otherwords, the confinement of the gauge theory purifies the reduced q = = =
420 40 60 80 100a0.20.40.60.81.01.21.4 S vN FIG. 3: The von Neumann entanglement entropy for U (1) gauge theory coupled to Z q scalar reservoir for q = 2 , , and . The value at a = 0 is log q which is the entanglemententropy of the reservoir. zero instantononly with instanton pairfull result20 40 60 80 100 a0.20.40.60.81.01.2 S ( ) FIG. 4: The Rényi entanglement entropy S (3) B for q = 3 . Thedotted line does not include the contribution from a triple ofinstantons, while the dashed line does not include theinstantons at all. The solid line is the full result.density matrix ρ B into the state | (cid:105) B (cid:104) | B in the a → ∞ limitand the entropy gets zero, as shown in the plot.To illustrate the effect of replica instantons, we consider theRényi entropy with n = 3 in the case of q = 3 . The explicitform of (5) in this case is Z [ R ] = 3( Z U (1) [ ˜ X, + 6 (cid:32) (cid:89) k =0 Z U (1) [ ˜ X, k ] (cid:33) + Z U (1) [ ˜ X, + Z U (1) [ ˜ X, ) . (24)The first term is the zero-instanton contribution, while thenext term is from the configuration with a pair of an instantonand an anti-instanton. The last two terms are from a triple ofinstantons or anti-instantons. As q = 3 , the total instantonnumber is constrained only by modulo 3 and thus these config-urations contribute. The Rényi entropy S (3) with and withoutreplica instanton contributions is plotted in Fig. 4. When allthe contributions are included, the entropy approaches zerowhen a → ∞ as ρ B becomes pure, while without some of theinstanton contributions it does not reach zero.
2d pure
P SU ( N ) gauge theory— The above analysis iseasily generalized to a gauge group G with a non-trivial fun-damental group. Here we in particular set G = P SU ( N ) .In this case, the topological quantity c ( A ) is taken to be the(generalized) 2nd Stiefel-Whitney class w ( A ) ∈ H ( ˜ X, Z N ) of the bundle, which is the obstruction for a P SU ( N ) bundleto be lifted to a SU ( N ) bundle. As w is Z N valued, q has to PSU ( )
10 20 30 40 50a0.10.20.30.40.50.60.7 S vN FIG. 5: The von Neumann entanglement entropy for
P SU (2) = SO (3) gauge theory coupled to Z scalar.divide N for the coupling (1) to be consistent. Here we take q = N . The partition function Z ( (cid:96) ) P SU ( N ) [ ˜ X ] is [12] Z ( (cid:96) ) P SU ( N ) [ (cid:101) X ] = (cid:88) R,N ( R )= (cid:96) (dim R ) e − aC ( R ) , (25)where the sum is over the SU ( N ) irreducible representations R with N -ality N ( R ) = (cid:96) , and C ( R ) is the quadratic Casimir.For the exact results on 2d pure gauge theory, see [13, 14]. Thecontribution from a fixed topological bundle is Z P SU ( N ) [ (cid:101) X, k ] = 1 N (cid:88) R (dim R ) e − aC ( R )+2 π i kN ( R ) N . (26)This partition function should be understood as the partitionfunction of SU ( N ) (which is the Z N -extension of P SU ( N ) )gauge theory with k units of one-form symmetry background.The entropy S vN B is plotted in Fig. 5. As in the U (1) case,the entropy approaches to zero as a → ∞ , indicating theconfinement of the theory. Discussion—
In this letter we have discussed the entangle-ment entropy in a gauge theory coupled with the reservoir ofa scalar field by the axion-like coupling (1). We pointed outthat instantons distributed across the replica sheets give non-perturbative corrections to the entanglement entropy, which isanalogous to the replica wormholes in gravitational theories.The explicit computation is done for 2d pure gauge theoriescoupled with discrete scalars. It is noted that the entropy canbe used to detect the confinement in 2d gauge theories. Therelationship between the entanglement and confinement is alsopointed out in [15] in the holographic context.It is desirable to do a computation with continuous scalars,where we expect more dynamical phenomena. Also havinga more physically meaningful set up, hopefully with directanalogy to the black hole physics would be interesting. Itwould also be interesting to consider the large- N limit of the2d P SU ( N ) model and its interpretation as a string theory.The entanglement entropy in a large- N
2d Yang-Mills theoryis considered in [16].Finally, we would like to point out that the "replica instan-ton" has been considered from 90’s, e.g. [17], in the spin-glassliterature. The replica method is used to compute the disor-dered free energy, and the "instanton" correction representsthe "island" of ordered phase appears due to the fluctuation ofthe coupling. It would be significant to find the connectionbetween this phenomenon and the replica wormholes.
Acknowledgements—
KO thanks Zohar Komargodski andDouglas Stanford for useful discussions and comments. ∗ [email protected][1] G. Penington, JHEP , 002 (2020), arXiv:1905.08255 [hep-th].[2] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, JHEP , 063 (2019), arXiv:1905.08762 [hep-th].[3] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, (2019),arXiv:1911.11977 [hep-th].[4] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, andA. Tajdini, JHEP , 013 (2020), arXiv:1911.12333 [hep-th].[5] K. Yonekura, (2020), arXiv:2011.11868 [hep-th].[6] We can consider a sigma model that admits a theta term insteadof a gauge theory. The general argument of this letter applies tothat case.[7] The geometry might admit a quench picture; the gauge field A is frozen due an additional Higgs field with a large potential, butin the region X a huge positive mass-squared is added to theHiggs field. [8] The boundary condition at the edge of the subregion in thereplica manifold is regarded as a choice of UV regularization ofthe entanglement entropy [9]. In the context of topological fieldtheory, the same boundary condition was used in [16, 18].[9] K. Ohmori and Y. Tachikawa, J. Stat. Mech. , P04010(2015), arXiv:1406.4167 [hep-th].[10] Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seif-nashri, (2020), arXiv:2008.07567 [hep-th].[11] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, JHEP ,172 (2015), arXiv:1412.5148 [hep-th].[12] The Dirichlet boundary condition forces the holonomy alongthe boundary to be zero. The same effect can be caused byidentifying the boundary to a point. Thus the partition functionon ˜ X is the same as that on a sphere with the same area.[13] S. Cordes, G. W. Moore, and S. Ramgoolam, Nucl. Phys. BProc. Suppl. , 184 (1995), arXiv:hep-th/9411210.[14] G. Aminov, Phys. Rev. D , 105017 (2020), arXiv:1911.03494[hep-th].[15] I. R. Klebanov, D. Kutasov, and A. Murugan, Nucl. Phys. B , 274 (2008), arXiv:0709.2140 [hep-th].[16] W. Donnelly, S. Timmerman, and N. Valdés-Meller, JHEP ,182 (2020), arXiv:1911.09302 [hep-th].[17] V. S. Dotsenko, Journal of Physics A: Mathematical and General , 2949 (1999).[18] W. Donnelly and G. Wong, JHEP10