Euclidean wormhole in the SYK model
EEuclidean wormhole in the SYK model
Antonio M. Garc´ıa-Garc´ıa
1, * and Victor Godet
2, † Shanghai Center for Complex Physics, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, Netherlands
We study a two-site Sachdev-Ye-Kitaev (SYK) model with complex couplings, and identify a lowtemperature transition to a gapped phase characterized by a constant in temperature free energy. Thistransition is observed without introducing a coupling between the two sites, and only appears afterensemble average over the complex couplings. We propose a gravity interpretation of these results byconstructing an explicit solution of Jackiw-Teitelboim (JT) gravity with matter: a two-dimensionalEuclidean wormhole whose geometry is the double trumpet. This solution is sustained by imaginarysources for a marginal operator, without the need of a coupling between the two boundaries. As thetemperature is decreased, there is a transition from a disconnected phase with two black holes tothe connected wormhole phase, in qualitative agreement with the SYK observation. The expectationvalue of the marginal operator is an order parameter for this transition. This illustrates in a concretesetup how a Euclidean wormhole can arise from an average over field theory couplings. * [email protected] † [email protected] a r X i v : . [ h e p - t h ] N ov I. INTRODUCTION
Wormholes are geometric shortcuts that connect distant points in spacetime. Their role in the Euclideanpath integral has been hotly debated in the literature [1–10]. In the context of holography, an importantpuzzle is that geometries connecting two boundaries indicate that the partition function of the dual fieldtheory does not factorize [7]. To address this problem, one could simply decide not to include these config-urations. However, it has been recently realized that it is only after including Euclidean wormholes that onegets results consistent with the interpretation of black holes as ordinary quantum systems. This has beenshown explicitly in Jackiw-Teitelboim gravity [11, 12], a two-dimensional theory of gravity capturing thelow-energy dynamics of near-extreme black holes [13–15], for the spectral form factor [16, 17], for correla-tions functions [18], and for the fine-grained entropy of evaporating black holes [19–23]. The factorizationpuzzle suggests that the gravitational path integral requires some form of ensemble averaging, whose originremains mysterious. We refer to [24–38] for recent discussions on this issue.This problem doesn’t arise in Lorentzian signature because non-traversable wormholes have horizons,which make them consistent with the factorization of the field theory dual. In this case, wormholes areinterpreted as coming from quantum entanglement [39, 40]. It has also been recently shown that thesewormholes can be rendered traversable by introducing a double trace coupling between the boundaries[41, 42]. In particular, Maldacena and Qi [43], see also [44, 45], have described an eternal traversablewormhole solution in JT gravity with a double trace deformation, and argued that a dual picture consists intwo copies of a Sachdev-Ye-Kitaev (SYK) model [13, 46–55] weakly coupled by a one-body operator. Thesystem undergoes a first order transition at finite temperature from the wormhole to a two black holes phasewhich can also be characterized by spectral statistics [56].In this paper, we propose a Euclidean version of this story which doesn’t involve an explicit couplingbetween the boundaries. In section II, we show that the free energy of a two-site SYK model with complexcouplings, obtained by exact diagonalization of the Hamiltonian, undergoes a low temperature transition toa gapped phase similar to that of the wormhole of [43], and which arises only after an ensemble averageover couplings. In section III, we describe a Euclidean wormhole solution of JT gravity plus matter, whichdoes not require a coupling between the two boundaries, but is instead sustained by imaginary sources.We compute the free energy and show that this system undergoes a similar phase transition from a hightemperature phase with two black holes to a low temperature wormhole phase, in qualitative agreementwith the SYK behavior. We end with a discussion and conclusions in section IV.
II. SACHDEV-YE-KITAEV WITH COMPLEX COUPLINGS
We study two uncoupled non-Hermitian SYK models with complex couplings composed by N Majoranafermions in (0 +
1) dimensions with infinite range interactions in Fock space. One of the copies is called Left( L ) with Majoranas denoted ψ L . The other copy is called Right ( R ) with Majoranas denoted ψ R . Majoranasin each copy are governed by an SYK Hamiltonian with complex couplings with the left Hamiltonian beingthe complex conjugate of the right one: H L = N / (cid:88) i , j , k , l = ( J i jkl + i κ M i jkl ) ψ L , i ψ L , j ψ L , k ψ L , l H R = N / (cid:88) i , j , k , l = ( J i jkl − i κ M i jkl ) ψ R , i ψ R , j ψ R , k ψ R , l (1)where κ is a real positive number, { ψ A , i , ψ B , j } = δ AB δ i j ( A , B = L , R ) and J i jkl , M i jkl are Gaussian distributedrandom variables with zero average and standard deviation (cid:113) (cid:104) J i jkl (cid:105) = (cid:113) (cid:104) M i jkl (cid:105) = √ / N / , see [49, 50].We note that both Hamiltonians are non-Hermitian and there is no explicit coupling between them.The combined system Hamiltonian H = H L + H R (2)has a complex spectrum with complex conjugation symmetry: if E n is an eigenvalue, its complex conjugate E ∗ n is also an eigenvalue. We shall see that despite the fact that the two copies are decoupled, the combinedsystem after ensemble average shares many of the properties expected from a Euclidean wormhole.We compute the spectrum by exact diagonalization techniques with N ≤
34. The spectrum for a singledisorder realization is depicted in Fig. 1 for di ff erent values of κ which is the only free parameter of themodel. For κ =
1, the eigenvalues are distributed in the complex plane with an ellipsoid shape with axisof similar sizes. Not surprisingly, for small κ = .
1, although the spectrum still has an ellipsoid shape, itis mostly concentrated close to the real axis. More information is revealed, see lower plots in Fig. 1, inthe spectral density of the real and imaginary parts of the eigenvalues after 1000 disorder realizations. Thespectral density of the real part seems to be qualitatively similar to that of the SYK model with real cou-plings. The imaginary part shows by contrast a sharp peak at zero and a relatively small depression aroundzero energy. A similar peak is observed for other values of κ , though its strength, as expected, diminishesas κ increases. We do not have a clear understanding of why some eigenvalues have zero imaginary parteven in the bulk of the spectrum while there is some level repulsion around zero. The latter is likely related Antonio M. Garc´ıa Garc´ıa thanks Zhenbin Yang for pointing out section 5.6 of Ref. [43] where the relation between SYK modelswith complex couplings and Euclidean wormholes is mentioned and for suggesting the model (2) -0.3-0.2-0.1 0 0.1 0.2 0.3-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -0.008-0.006-0.004-0.002 0 0.002 0.004 0.006 0.008-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ρ r e a l ( E ) E ρ i m a ( E ) E FIG. 1. Top: Complex spectrum of the combined Hamiltonian (2) for N =
30 and, κ = κ = . ff erent scale of each figure. Bottom: Spectral density of the real (left) and imaginary (right) part of theeigenvalues for κ = N =
34 after average over 45 disorder realizations. The real part looks qualitatively similar tothat of a single SYK model with real couplings. Indeed, it agrees well (solid line) with the analytical prediction, see(24) of [53], valid everywhere except in the tail of the spectrum. The best fitting is close to the analytical prediction for2 N Majoranas. Regarding the imaginary part, it is characterized by a peak at zero energy followed by a suppressionfor small energies whose origin at present we do not understand well. to conjugation symmetry of the spectrum that e ff ectively acts as a chiral symmetry inducing level repulsionaround states with zero (imaginary) energy. Real eigenvalues are not beneficial for the establishment ofthe wormhole phase because the absence of an imaginary part prevents the cancellations necessary for itsexistence. Likely, this is not important as the mentioned cancellations only occur in the infrared limit of thespectrum where the number of real eigenvalues, other than the ground state, is small. An important featureof the spectrum is that the ground state E is always real for any κ or strength of disorder.We can now proceed to the calculation of the thermodynamic properties. Interestingly, because complexconjugation is a symmetry of the spectrum, the partition function of the combined system Z ( β ) = Tr e − β H isreal. In order to reduce statistical fluctuations, we carry out an ensemble average and compute the resultingquenched free energy: (cid:104) F ( T ) (cid:105) = − T (cid:10) log Z ( β ) (cid:11) , (3) -0.8-0.7-0.6-0.5-0.4-0.3-0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 < F ( T ) > T N = 30, κ = 1.00N = 30, κ = 0.75N = 30, κ = 0.50 -0.8-0.7-0.6-0.5-0.4-0.3-0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 < F ( T ) > T N = 34N = 30N = 26
FIG. 2. Left: Free energy after ensemble average of 300 disorder realizations for N =
30 Majoranas and di ff erentstrengths of the imaginary part κ . Right: Free energy for κ = ff erent N ’s. where β = / T . Results depicted in Fig. 2 for di ff erent values of κ show a surprising result. The free energyis constant for su ffi ciently low T and there is a rather abrupt change at a certain critical temperature thatsuggests the existence of a first order transition. We note that a sharp transition only occurs in the N → ∞ limit. Although numerically it is hard to reach beyond a comparatively small value of N ∼
34, it is stillimportant to assess the magnitude of finite N e ff ects in the range of available sizes. Results depicted in theright plot of Fig. 2 show that these e ff ects are relatively small, consistent already with 1 / N corrections. Asis expected, these perturbative finite N e ff ects tend to increase the free energy.Roughly speaking, the location of the kink, which corresponds to the critical temperature, seems toscale approximately as κ . A precise determination is di ffi cult because for small κ (cid:28) N e ff ect or might signal the existence of aminimum value of κ , even in the large N limit, for a gap to be formed. For large κ (cid:29)
1, the free energy isinitially constant but display modulations (not shown) at intermediate temperatures. At present, we do nothave an explanation for this intermediate phase. We can only say that it does not seem to be a statisticalfluctuation that will go away in the large N limit or if more disorder realizations are considered.This free energy is very similar to both the free energy of the eternal traversable wormhole studied in[43] and to that of the Euclidean wormhole described in the next section. A constant free energy signalsthe existence of a gap in the spectrum that separates the ground state from higher excitations of the system.The existence of the gap can be explained by the combined e ff ect of a complex spectrum, the mentionedcomplex conjugation symmetry, and ensemble average as follows.Writing the spectrum of H as, E n = a n + ib n where a , b are real numbers, using the fact that if E n is aneigenvalue then E ∗ n is also an eigenvalue and that the ground state E is real, we write the partition functionfor a given disorder realization as, Z ( β ) = e − β E + (cid:80) n cos( β b n ) e − β a n , assuming no degeneracies. After -0.4-0.38-0.36-0.34-0.32-0.3-0.28-0.26-0.24-0.22 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 < F ( T ) > TN dis = 1N dis = 3N dis = 10N dis = 100N dis = 300 FIG. 3. Dependence of free energy of the Hamiltonian (2) on the number of disorder realizations N dis for N =
30 and κ =
1. A flat free energy in the low temperature limit that signals a gapped spectrum and a wormhole phase is onlyobserved after ensemble average with a large number of disorder realizations. ensemble average, the free energy becomes (cid:104) β F (cid:105) = − (cid:42) log e − β E + (cid:88) n cos( β b n ) e − β a n (cid:43) . (4)In the high temperature limit β →
0, the argument of the cosine function is always small and cos ∼ ff ectively irrelevant and the spectrumcorresponds to that of two identical, uncoupled, systems, in this case two SYK models with real couplings.Assuming that a gravity dual picture still applies, this region corresponds to a phase with two black holes.In the opposite limit β → ∞ , corresponding to low temperatures, the cosine becomes a highly oscillatingfunction whose features depend on the form of the imaginary part of the spectrum. From the results depictedin Fig. 1, the imaginary part of the spectrum seems to vary greatly, especially for intermediate κ , even foreigenvalues close to the ground state. In any case, the variations of the imaginary part are much faster thanthose of the real part.The e ff ect of the ensemble average is to e ff ectively suppress the contribution of eigenvalues in the lowerpart of the spectrum, allowing the opening of a gap above the ground state E . This leads to a free energyvery similar to the one recently reported for the eternal traversable wormhole [43]. The main di ff erences isthat in our setup, there is no explicit coupling between the two copies.We now address in more detail the importance of ensemble average in our results. It has been recentlyargued [21, 22] that Euclidean wormholes play a crucial role to obtain the Page curve [57] expected from theunitarity of the black hole evaporation process. The price to pay is that this requires the existence of somekind of ensemble average of mysterious origin. In Fig. 3, we depict results for the free energy after ensembleaverage with an increasing number of disorder realizations. For a small number of disorder realizations, thefree energy in the low temperature limit is far from being flat. Peaks and oscillations of di ff erent frequenciesare observed. Only after performing the average with a comparatively large number of realizations, the freeenergy becomes completely flat in this limit which is a signature of a gapped spectrum and a wormhole. Itcould be argued that, for a much larger number of Majoranas, which for N ≥
36 is numerically expensive,spectral average on a single realization of disorder is enough to flatten the free energy but at present we donot have evidence of this. Moreover, even if we could reach a much larger value of N , we believe that if noensemble average is carried out, it would be necessary to perform a spectral average in a small window tosmooth out fluctuations.In the next section, we will see that the free energy of this SYK model is strikingly similar to that of asolution of JT gravity plus matter, a Euclidean wormhole with the geometry of the double trumpet. Thisled us to propose that the low-temperature phase that we observe in the SYK model should be interpretedas a Euclidean wormhole. Therefore, with the present evidence, ensemble average in the field theory dualis a crucial ingredient to reproduce the expected features of a Euclidean wormhole, as expected from fac-torization arguments. Finally, let us comment on the advantage of exact diagonalization techniques withrespect to the solution of Schwinger-Dyson equations in the large N limit. There, the ensemble average iscarried out earlier in the derivation of the equations and therefore it is not possible to determine its exactrole in observing the gap at low temperature. In contrast, exact diagonalization displays without ambiguitythe central role played by the ensemble average. III. THE DOUBLE TRUMPET SOLUTION
We have observed that the SYK model with complex couplings has a low temperature phase whichbehaves like a wormhole. In this section, we propose a gravitational interpretation of this phase, as a novelEuclidean wormhole solution of JT gravity plus matter, where the crucial ingredient is the introduction ofimaginary sources for a marginal operator.The theory we consider is JT gravity with a massless scalar field, described by the action S = S JT + S matter , (5)where we have S JT = − S π (cid:34) (cid:90) d x √ g R + (cid:90) d τ √ h K (cid:35) − (cid:90) d x √ g Φ ( R + − (cid:90) d τ √ h Φ ( K − , (6) S matter = (cid:90) d x √ g ( ∂χ ) . (7)JT gravity is a two-dimensional theory that captures the low-energy dynamics of higher-dimensional near-extreme black holes [14, 58–61]. The parameter S is interpreted as the extremal entropy and is taken to belarge. In the Euclidean path integral [17], the first term in (6) gives the topological contribution e − S (2 g + n − to a geometry of genus g with n boundaries, so a large value of S suppresses higher genus contributions.The solution we will describe has the geometry of the double trumpet: a Euclidean geometry with R = − ds = ρ (cid:16) d τ + d ρ (cid:17) , − π ≤ ρ ≤ π , τ ∼ τ + b , (8)which is just global AdS with periodically identified Euclidean time. The proper length of the geodesicat ρ = b and is the only parameter of the geometry. In JT gravity, we also have a left and rightinverse temperatures β L and β R , defined as the periods of the respective boundary times, see Fig. 4. FIG. 4. Left: Double trumpet geometry corresponding to the wormhole phase. Right: Hyperbolic disks correspondingto the disconnected phase with two black holes. We observe a low temperature transition between these two phases.
The double trumpet geometry is not an ordinary solution of JT gravity plus matter. The crucial additionalingredient that we use here is to turn on imaginary sources for a massless scalar field. The motivation forthis comes from the SYK story described in the previous section. There, a wormhole-like phase appearsafter adding an imaginary part to the couplings. We can interpret that as a deformation of the left and rightHamiltonian by an operator δ H L = − i κ M L , δ H R = i κ M R , M ≡ N / (cid:88) i , j , k , l = M i jkl ψ i ψ j ψ k ψ l . (9)We aim to investigate the e ff ect of a qualitatively similar deformation on the gravity side. For that purpose,we deform the JT gravity action by a scalar operator O on each side δ S = − ik (cid:90) S L d τ O R ( τ ) + ik (cid:90) S R d τ O L ( τ ) , (10)where the source is ik on the left boundary and − ik on the right boundary. Using the standard AdS / CFTdictionary, this should correspond to fixing the asymptotic value of some bulk scalar field χ .To obtain a wormhole solution in JT with matter, it turns out that we should take O to have conformaldimension ∆ = χ is a massless scalar field in the bulk. This is explained in Appendix B. Thecondition we impose is then lim ρ → π/ χ = ik , lim ρ →− π/ χ = − ik , (11)and we will see that this makes the wormhole solution possible. To be clear, we are not claiming that thisfollows from a precise holographic duality. It must be understood as a setup in Euclidean JT gravity plusmatter, which is inspired by the previous SYK results, and turns out to provide a qualitatively similar picturein gravity. The parameter k in gravity is analogous to κ in SYK but we do not aim to establish a precisequantitative correspondence between the two. A. Equations of motion
Let us now solve the equations of motion. The scalar field χ satisfies the massless wave equation (cid:3) χ = . (12)Together with the condition (11) coming from the imaginary sources, this fixes the classical solution to be χ = ik π ρ . (13)We are finding here an imaginary solution for a real scalar field χ because of the complex sources. InLorentzian signature, imaginary sources are unphysical as they lead to violations of the averaged null energy0condition. Indeed, they can be used to construct traversable wormholes in AdS without a non-local couplingbetween the boundaries, in contradiction with the “no-transmission principle” [62] (see [63] for a relateddiscussion). In Euclidean signature, these imaginary sources do make sense. They can be defined by analyticcontinuation from real sources in the thermal partition function and can be used to study the statisticalmechanics of a physical system. For example, imaginary chemical potentials have been useful in studyingthe phase diagram of gauge theories [64, 65].We now impose the equation of motion for the JT dilaton which is ∇ µ ∇ ν Φ − g µν (cid:3) Φ + g µν Φ + (cid:104) T χµν (cid:105) = . (14)The stress tensor of χ decomposes into a classical and a quantum piece (cid:104) T χµν (cid:105) = T class µν + T quantum µν . (15)The classical piece is obtained by evaluating T µν on the solution (13), which leads to T class µν = ∂ µ χ∂ ν χ − g µν ( ∂χ ) = k π − . (16)This generates negative energy in the bulk because the solution for χ is pure imaginary. What wouldnormally be positive energy for real sources becomes negative energy for imaginary sources.In addition to this classical piece, there is also a quantum stress tensor due to the Casimir energy of χ .Although this piece is negligible in the regime where we will compare with the previous field theory results,it becomes important at higher temperature, where it destabilizes the wormhole. The quantum stress tensorcan be computed explicitly, using the Green function method and point splitting, as detailed in AppendixA. The result is T quantum µν = − π g µν − E ( b ) π − , (17)where we have introduced the quantity E ( b ) ≡ (cid:88) n ≥
14 sinh ( nb ) > , (18)whose properties are given in the Appendix A.We can now solve for the JT dilaton using (14). The general solution is given by Φ = (cid:32) k π − E ( b ) (cid:33) (1 + ρ tan ρ ) + η ρ + π , (19)1where η is an arbitrary constant. The existence of the wormhole requires that Φ → + ∞ at both boundaries.Expanding this solution at the two boundaries gives Φ ∼ ρ → + π (cid:32) k π − E ( b )2 + η (cid:33) π − ρ + . . . , (20) Φ ∼ ρ →− π (cid:32) k π − E ( b )2 − η (cid:33) π + ρ + . . . . (21)The position of the left and right boundaries are determined by imposing the boundary condition Φ = ¯ φ r /(cid:15) .We should also impose the boundary conditions for the metric ds ∼ ρ → π du R (cid:15) , ds ∼ ρ →− π du L (cid:15) , (22)which tell us how u L and u R are related to τ . The left and right temperatures are defined as their periods u L ∼ u L + ¯ φ r β L , u R ∼ u R + ¯ φ r β R rescaled by ¯ φ r for convenience. This gives β L = b k π − E ( b ) − η , β R = b k π − E ( b ) + η . (23)We see that b and η are determined by the boundary conditions: they are fixed by the choice of temperatureson each side. The condition for the existence of the wormhole is that β L , β R ≥
0. In the following, we willset the asymmetry parameter η to zero so that we consider a situation where β L = β R . The temperature T = T L = T R is then T = k π b − E ( b )2 b . (24)At large b , this is positive since we have E ( b ) ∼ e − b which becomes negligible. Hence, this gives a consistentwormhole solution. From the expression of the temperature, we see that the existence of the solutiondepends crucially on the pure imaginary sources. Real sources would change the sign of k and preventthe solution to exist. Moreover, we see that the Casimir energy contributes negatively (since E ( b ) > ff ect which is described in more detailed below.It’s interesting to note that the operator O acquires an imaginary expectation value (cid:104)O L (cid:105) = ik π , (cid:104)O R (cid:105) = − ik π , (25)which can be read o ff from the solution for χ using the AdS / CFT dictionary. This is a consequence of theimaginary gradient in the solution for χ . We will see below that this expectation value is an order parameterfor the phase transition.It’s interesting to note that the marginality of the operator O is important for the wormhole solution tobe possible. Indeed, an operator O with ∆ (cid:44) The formulas for independent T L and T R can be obtained by replacing T → ( T L + T R ) / B. Thermodynamics
We now compute the free energy of the wormhole solution. The partition function islog Z = − b T + bk π + log Tr e − bL . (26)The first term comes from the classical contribution of the Schwarzian action that describes the two trumpetgeometries [17] in conventions where 8 π G N =
1. The corresponding one-loop term proportional to log T is always negligible in our discussion and will be ignored. The second term is the on-shell action of theclassical matter solution (13). The third term is the one-loop contribution from the matter, which can bederived as follows. The path integral of the scalar field χ on a cylinder of width π and circumference b givesthe thermal partition function Z χ cylinder = Tr e − b ( L − ) = η ( e − b ) , (27)where η ( q ) is the Dedekind eta function. We can perform a Weyl transformation to the double trumpet. Itfollows from the discussion in [43] that the e ff ect of the Weyl anomaly is to shift the ground state energy byremoving the − , so we end up with Z χ = Tr e − bL = e − b / η ( e − b ) . (28)This formula is also a consequence of the computation of the quantum stress tensor in Appendix A.The saddle point of (26) with respect to b leads to the relation T = k π b − E ( b )2 b , (29)where we used that E ( b ) = + ∂ b log η ( e − b ) from the identities given in Appendix A. Note that E ( b ) = (cid:104) L (cid:105) b is the thermal expectation value of L at the inverse temperature b . It is a nice consistency check that thisexpression of T matches with (24) obtained from the explicit solution. a. Large wormhole regime. Let’s first consider a regime where b is large. In this regime, we have E ( b ) ∼ e − b so the second term of (29) becomes negligible as soon as b becomes relatively large. We thenhave the saddle point b ∗ = k π T which leads to the free energy F WH = − T log Z = − k π , when E ( b ∗ ) (cid:28) k (30) Without imaginary sources and for a general matter CFT, the expression (29) becomes T = −(cid:104) L (cid:105) b / (2 b ) which is negative because L is a positive operator. This shows that for any matter CFT, Casimir energy alone can never support the double trumpet geometry. - - - - FIG. 5. Plot of the free energy of the wormhole and the black holes at low temperature. The solid black line representsthe free energy of the system. It corresponds with that of the wormhole, characterized by a constant negative freeenergy, at su ffi ciently low (high) temperatures. As temperature increases, we observe a first order transition separatingthe wormhole from the two black hole phase. This qualitatively matches the behavior seen in the SYK with complexcouplings. Notice that there is also another solution at smaller b whose free energy is always larger. We use k = S = . This formula is valid when E (cid:16) k π T (cid:17) (cid:28) k which holds at su ffi ciently low temperature since we have E (cid:16) k π T (cid:17) ∼ T → e − k / ( π T ) . We see that the wormhole has constant negative free energy at low tempera-ture, as is expected from a gapped system. This is the regime where we expect that JT gravity and the SYKmodel share similar features.Our gravitational boundary conditions consist of two circles of length β = / T and Dirichlet boundaryconditions for the scalar field χ given in (11). Besides the double trumpet solution, we can also have twodisconnected black holes ( i.e. two hyperbolic disks). In a black hole, the source for the scalar field doesn’tcontribute to the on-shell action, because the classical solution is just χ = const. The one-loop contributiongives a contribution that depends on (cid:15) and gets renormalized away. As a result, the free energy of the twoblack holes is the same as in pure JT gravity: F BH = − T log Z = − S T − π T . (31)The physical free energy is then the minimum F = min ( F WH , F BH ) , (32)and there is a phase transition at the critical temperature T c ∼ k (cid:28) S k π S . (33)4 - - - - - - - - - - - - FIG. 6. Log-log plot of the free energy of the di ff erent solutions. We plot the disconnected contribution of the twoblack holes (in blue). It dominates at high temperature until the phase transition at T = T c (dashed vertical blue line)after which the large wormhole solution (in orange) becomes favorable. There is also a small wormhole solution (ingreen) which has higher free energy. Both wormhole solutions only exist below a maximal temperature T max (dashedvertical red line) and we have T max > T c . The dashed orange line corresponds to the classical approximation wherewe don’t include the Casimir energy of the matter, thus obtaining a wormhole at all temperatures. This shows that theCasimir energy has a destabilizing e ff ect which is responsible for the existence of both the maximal temperature andthe small wormhole solution. The free energy is plotted in Fig. 5, and matches the qualitative behavior of the SYK model displayedin Fig. 2. The classical solution χ = const in the black hole phase implies that the expectation value of themarginal operator vanishes: (cid:104)O L (cid:105) = , (cid:104)O R (cid:105) = , (34)This shows that the expectation value of O is an order parameter for the transition from the two black holesto the wormhole. It is zero in the black hole phase, and becomes non-zero in the wormhole phase, wherewe have (25). The non-zero expectation value at one boundary is a direct consequence of the source at theother boundary so this can be seen as a field theory incarnation of the geometric connection in the bulk.It was recently argued [66] that we should include replica wormholes in the computation of the freeenergy as we are interested in the quenched free energy − T (cid:104) log Z (cid:105) rather than the annealed one − T log (cid:104) Z (cid:105) .We show in Appendix C that in our setup, the additional wormholes give a negligible correction.5 - - FIG. 7. Log-log plot of the size b of the wormhole as a function of the temperature. We use k = S = . b. General analysis. Let’s now analyze the solution more generally, beyond the regimes where JTgravity is well approximated by SYK. Firstly, the wormhole solution only exists if T ≥
0. Since E ( b ) is adecreasing function of b , we see that this implies that there is a minimal value of b defined implicitly by E ( b min ) = k π . (35)The formula (29) also implies that there is a maximal value T max of the temperature beyond which thewormhole solution disappears. This value is attained when ∂ b T = ≤ T ≤ T max . In this range, there is actually two values of b that give the same T : the large wormholethat was discussed previously but also a small wormhole at a much smaller value of b . This is the result ofa competition between the classical negative energy coming from the imaginary sources, and the positivequantum Casimir energy of the matter. It is here a classical e ff ect that sustains the wormhole while aquantum e ff ect destabilizes it. It can be checked that the smaller wormhole has always higher free energythan the large one, so that it never dominates in the canonical ensemble. The main thermodynamic featuresof the system are summarized in Fig. 6 which describes the free energy of the di ff erent branches as a functionof temperature and Fig. 7 that depicts the dependence of the wormhole size b with temperature.6 IV. DISCUSSION
One of the main motivation of this paper is to shed light on the holographic interpretation of Euclideanwormholes. We have shown how thermodynamic properties consistent with a wormhole can arise fromaveraging over complex couplings in the SYK model, and we have proposed a gravity interpretation as aEuclidean wormhole solution of JT gravity plus matter sharing similar properties.We have studied the free energy of two copies of a system with complex conjugated Hamiltonians. Thisis also equal to the real part of the free energy of a single copy:2 Re (cid:104) F (cid:105) = − T (cid:104) log Z ¯ Z (cid:105) . (36)From the point of view of a single system, our wormhole can be seen as “real part wormhole” whichconnects the system to its complex conjugate in the gravitational computation of Re F . The gravitationalcomputation of the free energy also involves replica wormholes [66] but they are negligible in our setup asexplained in Appendix C.The mechanism that makes the Euclidean wormhole possible is the introduction of imaginary sources fora marginal operator, which provides the negative energy necessary to sustain the wormhole. We emphasizethat adding imaginary sources is perfectly consistent in Euclidean signature; it is akin to probing a systemat imaginary chemical potential which, for example, has been useful to study QCD [64, 65]. We haveidentified a low temperature phase, where the wormhole dominates, and in which the marginal operatorcondenses. The non-zero expectation value of this operator is linked to the existence of the wormhole, sinceit vanishes in the black hole phase and becomes non-zero in the wormhole because of the source on theother boundary. This can be seen as a field theory diagnosis of the geometric connection in the bulk.Our wormhole is di ff erent from the eternal traversable wormhole of [43], which makes sense as aLorentzian geometry, and requires an explicit coupling between the boundaries. It’s also possible to intro-duce such an explicit coupling in our setup, in addition to the imaginary sources, and it would be interestingto study the resulting solutions.In this paper, we have seen that imaginary sources can be used to support Euclidean wormhole solutions.As the comparison with SYK suggests, these sources might be related to some averaging procedure. Itwould be interesting to see whether other Euclidean wormholes can be constructed using this idea. Oursolution extends to JT gravity with an additional gauge symmetry [67, 68] or including the gravitationalU(1) symmetry described in [69]. It should be possible to study multi-boundary solutions for which a dualSYK setup might exist. Euclidean wormholes have also found recent applications in AdS holography[70–74] and it would be interesting to explore solutions in higher dimensions [38].7Euclidean wormholes have played an important role in the study of eigenvalue statistics in pure JTgravity and in its field theory realization as a random matrix model [16–18, 75]. The conclusion of theseworks is that JT gravity is quantum chaotic. Level statistics of quantum chaotic systems are described byrandom matrix theory. For the spectral form factor, a useful observable in spectral analysis, a signature ofquantum chaos is the presence of a ramp for su ffi ciently long times that saturates at the Heisenberg time. Inpure JT gravity, the spectral form factor is related to a double trumpet geometry which is not a solution ofthe equations of motion. An explicit evaluation of the path integral gives the contribution (cid:104) Z ( β L ) Z ( β R ) (cid:105) JTconn = π √ β L β R β L + β R + higher genus , (37)which is actually the universal answer for a double-scaled random matrix integral. Replacing β L = β + it and β R = β − it leads to a contribution that grows linearly with t for t (cid:29) β and accounts for the ramp in thespectral form factor. The same computation does not work for JT gravity plus matter as the integral over b has a divergence at small b , seen here from the fact that E ( b ) ∼ b → π b . If we add imaginary sources,we can get around this issue by using our wormhole solution as a saddle point in this path integral. Thissuggests that Euclidean wormhole solutions, of the type constructed here, might be useful in studying theeigenvalue statistics of gravitational theories for which we cannot perform the full path integral. We leavethis interesting question and the comparison with SYK level statistics to a future work.Euclidean wormholes have also been important in understanding better the fine-grained entropy of evap-orating black holes, where replica wormholes were crucial in obtaining an answer consistent with unitarity[21, 22]. It has also been argued that a gravitational replica trick is already needed for the computation ofthe free energy [66]. We have implemented this replica trick in Appendix C to compute the free energy ofour gravity setup and shown that the additional wormholes only give a negligible correction. It would beinteresting to see whether wormhole solutions similar to the one described in this paper could be used assaddle points in the replica computation of the free energy in situations where they give large corrections,for example at very low temperatures.On the field theory side, a non-Hermitian Hamiltonian is superficially related to a loss of probability con-servation and a non-unitary evolution as most eigenvalues are complex and therefore have a finite lifetime.From the point of view of one of the two systems, this could be interpreted as the observation of particlescoming in and coming out which is typical of an open quantum system. Although this is an appealing pic-ture, it cannot really explain why a gap is observed. Ensemble average is a key ingredient for the formationof the gap at low temperatures so just a spectrum with imaginary eigenvalues is not enough to reproduce theobserved phenomenology. As we mentioned earlier, another outstanding feature is that for su ffi ciently hightemperature the non-Hermitian e ff ects related to a complex spectrum becomes irrelevant. This suggests that8the addition of an imaginary part in the SYK couplings, and the subsequent ensemble average, is just an ef-fective way to describe quantum tunneling between two Hermitian SYK’s, and produces a gap between theground state and the first excited state as in a double well potential. Typically, this is qualitatively modeledby an explicit coupling between the two SYK’s, as in the case the eternal traversable wormhole. However,we are interested in understanding better the issue of factorization, so we want to study configurations withan explicit coupling between the two systems.Assuming that complex couplings and ensemble average are necessary ingredients to model tunnelingwithout an explicit coupling, it would be interesting to determine the conditions on the field theory Hamil-tonian that lead to this type of wormhole behavior. It seems that some form of randomness is required aswormhole-like features are only seen after an ensemble average. However, it is unclear whether infinite-range and strong interactions, as in the SYK model studied here, are also a necessary requirement. ACKNOWLEDGMENTS
AMG was partially supported by the National Natural Science Foundation of China (NSFC) (Grantnumber 11874259) and also acknowledges financial support from a Shanghai talent program. AMG ac-knowledges illuminating correspondence with Zhenbin Yang, Jac Verbaarschot, Dario Rosa and Yiyang Jia.VG acknowledges useful conversations with Charles Marteau.
Appendix A: Quantum stress tensor in the double trumpet
In this section, we compute the quantum stress tensor of a massless scalar in the double trumpet geom-etry. We consider a free boson described by the action S = (cid:90) d x √ g ( ∂χ ) , (A1)and whose stress tensor is given by T µν = ∂ µ χ∂ ν χ − g µν ( ∂χ ) . (A2)To compute its expectation value, we use the point-splitting method so that (cid:104) T µν ( x ) (cid:105) = lim x (cid:48) → x (cid:32) ∂ µ ∂ (cid:48) ν G ( x , x (cid:48) ) − g µν ∂ µ ∂ (cid:48) ν G ( x , x (cid:48) ) (cid:33) , (A3)where G ( x , x (cid:48) ) is the Green function. See [63, 76] for examples of application of this method in relatedcontexts. We will first perform the computation on a cylinder described by the coordinates ds = d ρ + d θ , − π ≤ ρ ≤ π , θ ∼ θ + b . (A4)9We will then use a Weyl transformation to the double trumpet. The Green function can be defined as thesolution of the equation (cid:3) x G ( x , x (cid:48) ) = − δ ( x − x (cid:48) ) . (A5)Using the following mode decomposition G ( ρ, θ ; ρ (cid:48) , θ (cid:48) ) = (cid:88) m , m (cid:48) ∈ Z G ( ρ, m ; ρ (cid:48) , m (cid:48) ) e i π ( m θ + m (cid:48) θ (cid:48) ) / b , (A6)the equation (A5) becomes (cid:32) ∂ ρ − π m b (cid:33) G ( ρ, m ; ρ (cid:48) , m (cid:48) ) = − b δ ( ρ − ρ (cid:48) ) δ m + m (cid:48) , (A7)with the boundary conditions G ( ρ, m ; ρ (cid:48) , m (cid:48) ) | ρ = ± π = G ( ρ, m ; ρ (cid:48) , m (cid:48) ) | ρ (cid:48) = ± π = . (A8)This can be solved using the discontinuity method, as explained in [76] for the standard Casimir e ff ectbetween two plates. At the end, we find G ( ρ, θ ; ρ (cid:48) , θ (cid:48) ) = − (cid:88) m ∈ Z sinh( π mb ( ρ + − π )) sinh( π mb ( ρ − + π ))2 π m sinh( π mb ) e i π m ( θ − θ (cid:48) ) / b , (A9)which allows us to obtain (cid:104) T ρρ (cid:105) = − (cid:88) m ∈ Z π mb tanh (cid:16) π mb (cid:17) , (cid:104) T θθ (cid:105) = (cid:88) m ∈ Z π mb tanh (cid:16) π mb (cid:17) . (A10)The sum over m can be regularized using the Poisson resummation formula, which allows us to write S ( x ) ≡ (cid:88) m ∈ Z m tanh( mx ) = − π x (cid:88) n ∈ Z ( π nx ) . (A11)We regulate the divergence at n = b → + ∞ , we recover the known result forthe strip. The final formula is (cid:104) T θθ (cid:105) = −(cid:104) T ρρ (cid:105) = c π − c π E ( b ) , (A12)where have multiplied by an overall c and defined E ( b ) ≡ (cid:88) n ≥
14 sinh ( nb ) . (A13)We can now compute the stress tensor in the double trumpet using the formula (cid:104) T µν (cid:105) = (cid:104) T µν (cid:105) cyl + (cid:104) T µν (cid:105) anomaly , (A14)where the formula for the anomalous stress tensor is given in [77]. This finally gives in the ( τ, ρ ) coordinates (cid:104) T µν (cid:105) = − π g µν − E ( b ) π − . (A15)0 a. Some properties of the function E ( b ) . We give some properties of the function E ( b ) which appearedin the above computation. Firstly, we have E ( b ) > E ( b ) decreases with the following asymptotics: E ( b ) ∼ b → π b , E ( b ) ∼ b → + ∞ e − b + O ( e − b ) . (A16)After defining q = e − b , we can write E ( b ) as E ( b ) = (cid:88) n ≥ q n (1 − q n ) = (cid:88) n ≥ (cid:88) m ≥ mq nm = (cid:88) m ≥ mq m − q m =
124 (1 − P ( q )) , (A17)in terms of the function P studied by Ramanujan [78], which is related to the Dedekind eta function P ( q ) = q ddq log η ( q ) , η ( q ) ≡ q / (cid:89) n ≥ (1 − q n ) . (A18)There is no simple analytic expression for E ( b ), but it can be expressed in a complicated way in terms ofelliptic integrals [79]. Appendix B: Massive bulk fields
We have constructed a wormhole solution using a massless scalar field in the bulk, corresponding to anoperator O with conformal dimension ∆ =
1. In this Appendix, we consider massive fields and explain why ∆ =
Φ = ¯ φ r (cid:15) , ds = du (cid:15) . (B1)Satisfying both in the double trumpet (8) requires that at leading order Φ ∼ ρ → π/ C π − ρ , (B2)where C is some constant. Now, take χ to be a general massive scalar field satisfying the equation( (cid:3) − m ) χ = , m = ∆ ( ∆ − . (B3)The pure imaginary sources give the condition χ ∼ ρ → π ik (cid:18) π − ρ (cid:19) − ∆ , χ ∼ ρ →− π − ik (cid:18) π + ρ (cid:19) − ∆ . (B4)This fixes the classical solution for χ which can be written explicitly in terms of hypergeometric functions.We will focus on the low-temperature regime where the Casimir energy of χ can be ignored, which isthe relevant regime for comparison with SYK. There, the equation of motion for the JT dilaton is ∇ µ ∇ ν Φ − g µν (cid:3) Φ + g µν Φ + T χµν = , T χµν = ∂ µ χ∂ ν χ − g µν (cid:16) ( ∂χ ) + m χ (cid:17) . (B5)1Taking the trace of this equation gives ( (cid:3) − Φ = − ∆ ( ∆ − χ . (B6)From (B2), we can verify lim ρ → π ( (cid:3) − Φ = finite constant , (B7)where the constant depends on subleading terms in Φ . This is only compatible with ∆ = χ ∼ (cid:16) π − ρ (cid:17) − ∆ − diverges for ρ → π . This shows that a wormhole solution can be obtained in this wayonly for ∆ = Appendix C: Quenched versus annealed in gravity
As pointed out in [66], the gravitational free energy must be computed with a replica trick because weare interested in the quenched, rather than annealed, free energy. This means that the naive answer forthe free energy ( i.e. the annealed one) might be incorrect in situations where quantum gravity e ff ects areimportant. For a system with boundary B , we should compute the gravitational path integral P ( B m ) whoseboundary is m copies of B , analytically continue in m and use the formula (cid:104) log Z ( B ) (cid:105) = lim m → m ( P ( B m ) − . (C1)We take B to be the system with two circle boundaries: one is labeled + and has a source + ik while the otheris labeled − and has the source − ik . The wormhole solution can only connect a + circle to a − circle. Wetake the temperature to be not too small so that taking S to be large allows us to ignore the higher genustopologies. A typical contribution to P ( B m ) is drawn in Fig. 8.Denoting by Z the black hole contribution and by Z the wormhole contribution, we find P ( B m ) = m (cid:88) k = k ! (cid:32) mk (cid:33) Z k ( | Z | ) m − k = Z m exp (cid:32) | Z | Z (cid:33) (cid:90) + ∞| Z | / Z dt t m e − t . (C2)The e ff ect of the replica trick is to introduce a permutation factor k !, which takes into account wormholesconnecting boundaries belonging to di ff erent copies of B . Indeed, removing this factor leads to P ( B m ) = ( | Z | + Z ) m and gives the annealed result. The expression1 m ( P ( B m ) − = exp (cid:32) | Z | Z (cid:33) (cid:90) + ∞| Z | / Z dt (cid:32) m ( Z m t m − (cid:33) e − t (C3)makes it possible to take the m → (cid:104) log Z ( B ) (cid:105) = log Z + e | Z | / Z (cid:90) + ∞| Z | / Z dt log t e − t . (C4)2 FIG. 8. Computation of P ( B m ) in the replica trick for the free energy. At large temperature T > T c , there are only disconnected contributions and we obtain the free energy oftwo black holes. At small temperature T < T c , we have Z (cid:29) | Z | and (C4) gives (cid:104) log Z (cid:105) = log Z − γ , ( T < T c ) . (C5)where we used (cid:82) ∞ dt log t e − t = − γ with γ ≈ .
577 is the Euler gamma constant. This shows that the e ff ectof the replica wormholes is to lower the entropy of the wormhole by γ . It gives the low temperature freeenergy (cid:104) F (cid:105) = F WH + γ T , ( T < T c ) , (C6)where F WH = − k /π is the free energy (30) of the wormhole solution. The second term represents thecorrection from the replica wormholes and can be neglected because T < T c (cid:28) k . Hence, in our gravitysetup, the quenched and annealed free energy are equal up to negligible corrections.3 [1] George V. Lavrelashvili, V.A. Rubakov, and P.G. Tinyakov, “Disruption of Quantum Coherence upon a Changein Spatial Topology in Quantum Gravity,” JETP Lett. , 167–169 (1987).[2] S.W. Hawking, “Quantum Coherence Down the Wormhole,” Phys. Lett. B , 337 (1987).[3] Steven B. Giddings and Andrew Strominger, “Axion Induced Topology Change in Quantum Gravity and StringTheory,” Nucl. Phys. B , 890–907 (1988).[4] Steven B. Giddings and Andrew Strominger, “Loss of incoherence and determination of coupling constants inquantum gravity,” Nuclear Physics B , 854 – 866 (1988).[5] Steven B Giddings and Andrew Strominger, “Baby universe, third quantization and the cosmological constant,”Nuclear Physics B , 481 – 508 (1989).[6] Sidney R. Coleman, “Why There Is Nothing Rather Than Something: A Theory of the Cosmological Constant,”Nucl. Phys. B , 643–668 (1988).[7] Juan Maldacena and Liat Maoz, “Wormholes in AdS,” Journal of High Energy Physics , 053–053 (2004).[8] Nima Arkani-Hamed, Luboˇs Motl, Alberto Nicolis, and Cumrun Vafa, “The string landscape, black holes andgravity as the weakest force,” Journal of High Energy Physics , 060–060 (2007).[9] Thomas Hertog, Brecht Truijen, and Thomas Van Riet, “Euclidean axion wormholes have multiple negativemodes,” Phys. Rev. Lett. , 081302 (2019), arXiv:1811.12690 [hep-th].[10] Arthur Hebecker, Thomas Mikhail, and Pablo Soler, “Euclidean wormholes, baby universes, and their impacton particle physics and cosmology,” Front. Astron. Space Sci. , 35 (2018), arXiv:1807.00824 [hep-th].[11] Roman Jackiw, “Lower dimensional gravity,” Nuclear Physics B , 343 – 356 (1985).[12] Claudio Teitelboim, “Gravitation and hamiltonian structure in two spacetime dimensions,” Physics Letters B , 41 – 45 (1983).[13] Kristan Jensen, “Chaos in ads holography,” Phys. Rev. Lett. , 111601 (2016).[14] Juan Maldacena, Douglas Stanford, and Zhenbin Yang, “Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space,” Progress of Theoretical and Experimental Physics , 12C104 (2016).[15] Julius Engels¨oy, Thomas G. Mertens, and Herman Verlinde, “An investigation of ads2 backreaction and holog-raphy,” Journal of High Energy Physics , 1–30 (2016).[16] Phil Saad, Stephen H. Shenker, and Douglas Stanford, “A semiclassical ramp in SYK and in gravity,” (2018),arXiv:1806.06840 [hep-th].[17] Phil Saad, Stephen H. Shenker, and Douglas Stanford, “JT gravity as a matrix integral,” (2019),arXiv:1903.11115 [hep-th].[18] Phil Saad, “Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity,” (2019),arXiv:1910.10311 [hep-th].[19] Geo ff rey Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” JHEP , 002 (2020),arXiv:1905.08255 [hep-th].[20] Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP , 063 (2019), arXiv:1905.08762 [hep-th].[21] Geo ff Penington, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang, “Replica wormholes and the blackhole interior,” (2020), arXiv:1911.11977 [hep-th].[22] Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini,“Replica wormholes and the entropy of hawking radiation,” Journal of High Energy Physics (2020),10.1007 / jhep05(2020)013.[23] Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “The en-tropy of hawking radiation,” (2020), arXiv:2006.06872 [hep-th].[24] Andreas Blommaert, Thomas G. Mertens, and Henri Verschelde, “Eigenbranes in Jackiw-Teitelboim gravity,”(2019), arXiv:1911.11603 [hep-th].[25] P. Betzios, E. Kiritsis, and O. Papadoulaki, “Euclidean Wormholes and Holography,” JHEP , 042 (2019),arXiv:1903.05658 [hep-th].[26] Donald Marolf and Henry Maxfield, “Transcending the ensemble: baby universes, spacetime wormholes,and the order and disorder of black hole information,” Journal of High Energy Physics (2020),10.1007 / jhep08(2020)044.[27] Jason Pollack, Moshe Rozali, James Sully, and David Wakeham, “Eigenstate Thermalization and DisorderAveraging in Gravity,” Phys. Rev. Lett. , 021601 (2020), arXiv:2002.02971 [hep-th].[28] Mark Van Raamsdonk, “Comments on wormholes, ensembles, and cosmology,” (2020), arXiv:2008.02259[hep-th].[29] Panagiotis Betzios and Olga Papadoulaki, “Liouville theory and Matrix models: A Wheeler DeWitt perspective,”JHEP , 125 (2020), arXiv:2004.00002 [hep-th].[30] Steven B. Giddings and Gustavo J. Turiaci, “Wormhole calculus, replicas, and entropies,” (2020),arXiv:2004.02900 [hep-th].[31] Andreas Blommaert, “Dissecting the ensemble in JT gravity,” (2020), arXiv:2006.13971 [hep-th].[32] Tarek Anous, Jorrit Krutho ff , and Raghu Mahajan, “Density matrices in quantum gravity,” SciPost Phys. , 045(2020), arXiv:2006.17000 [hep-th].[33] Yiming Chen, Victor Gorbenko, and Juan Maldacena, “Bra-ket wormholes in gravitationally prepared states,”(2020), arXiv:2007.16091 [hep-th].[34] Lorenz Eberhardt, “Partition functions of the tensionless string,” (2020), arXiv:2008.07533 [hep-th].[35] Douglas Stanford, “More quantum noise from wormholes,” (2020), arXiv:2008.08570 [hep-th].[36] Jacob McNamara and Cumrun Vafa, “Baby Universes, Holography, and the Swampland,” (2020),arXiv:2004.06738 [hep-th].[37] Mohsen Alishahiha, Amin Faraji Astaneh, Ghadir Jafari, Ali Naseh, and Behrad Taghavi, “On free energy fordeformed jt gravity,” (2020), arXiv:2010.02016 [hep-th].[38] Jordan Cotler and Kristan Jensen, “Gravitational Constrained Instantons,” (2020), arXiv:2010.02241 [hep-th].[39] Juan Maldacena, “Eternal black holes in anti-de sitter,” Journal of High Energy Physics , 021–021 (2003).[40] Juan Maldacena and Leonard Susskind, “Cool horizons for entangled black holes,” Fortsch. Phys. , 781–811 (2013), arXiv:1306.0533 [hep-th].[41] Ping Gao, Daniel Louis Ja ff eris, and Aron Wall, “Traversable Wormholes via a Double Trace Deformation,”JHEP , 151 (2017), arXiv:1608.05687 [hep-th].[42] Juan Maldacena, Douglas Stanford, and Zhenbin Yang, “Diving into traversable wormholes,” Fortsch. Phys. ,1700034 (2017), arXiv:1704.05333 [hep-th].[43] Juan Maldacena and Xiao-Liang Qi, “Eternal traversable wormhole,” (2018), arXiv:1804.00491 [hep-th].[44] Dongsu Bak, Chanju Kim, and Sang-Heon Yi, “Bulk view of teleportation and traversable wormholes,” JHEP , 140 (2018), arXiv:1805.12349 [hep-th].[45] Jaewon Kim, Igor R. Klebanov, Grigory Tarnopolsky, and Wenli Zhao, “Symmetry Breaking in Coupled SYKor Tensor Models,” (2019), arXiv:1902.02287 [hep-th].[46] J.B. French and S.S.M. Wong, “Validity of random matrix theories for many-particle systems,” Physics LettersB , 449 – 452 (1970).[47] O. Bohigas and J. Flores, “Two-body random hamiltonian and level density,” Physics Letters B , 261 – 263(1971).[48] Subir Sachdev and Jinwu Ye, “Gapless spin-fluid ground state in a random quantum heisenberg magnet,” Phys.Rev. Lett. , 3339–3342 (1993).[49] Alexander Kitaev, “A simple model of quantum holography,” KITP strings seminar and Entanglement 2015program, 12 February, 7 April and 27 May 2015, http: // online.kitp.ucsb.edu / online / entangled15 / .[50] Juan Maldacena and Douglas Stanford, “Remarks on the sachdev-ye-kitaev model,” Phys. Rev. D , 106002(2016).[51] Antal Jevicki, Kenta Suzuki, and Junggi Yoon, “Bi-local holography in the syk model,” Journal of High EnergyPhysics , 1–25 (2016).[52] Antonio M. Garc´ıa-Garc´ıa and Jacobus J. M. Verbaarschot, “Spectral and thermodynamic properties of thesachdev-ye-kitaev model,” Phys. Rev. D , 126010 (2016).[53] Antonio M. Garc´ıa-Garc´ıa and Jacobus J. M. Verbaarschot, “Analytical spectral density of the sachdev-ye-kitaevmodel at finite n ,” Phys. Rev. D , 066012 (2017).[54] Dmitry Bagrets, Alexander Altland, and Alex Kamenev, “Sachdev–ye–kitaev model as liouville quantum me-chanics,” Nuclear Physics B , 191–205 (2016).[55] Dmitry Bagrets, Alexander Altland, and Alex Kamenev, “Power-law out of time order correlation functions inthe syk model,” Nuclear Physics B , 727 – 752 (2017).[56] Antonio M. Garc´ıa-Garc´ıa, Tomoki Nosaka, Dario Rosa, and Jacobus J. M. Verbaarschot, “Quantum chaostransition in a two-site sachdev-ye-kitaev model dual to an eternal traversable wormhole,” Phys. Rev. D ,026002 (2019).[57] Don N. Page, “Information in black hole radiation,” Phys. Rev. Lett. , 3743–3746 (1993).[58] Ahmed Almheiri and Joseph Polchinski, “Models of ads2 backreaction and holography,” Journal of High EnergyPhysics , 1–19 (2015).[59] Ahmed Almheiri and Byungwoo Kang, “Conformal Symmetry Breaking and Thermodynamics of Near- Extremal Black Holes,” JHEP , 052 (2016), arXiv:1606.04108 [hep-th].[60] Pranjal Nayak, Ashish Shukla, Ronak M. Soni, Sandip P. Trivedi, and V. Vishal, “On the Dynamics of Near-Extremal Black Holes,” JHEP , 048 (2018), arXiv:1802.09547 [hep-th].[61] Alejandra Castro and Victor Godet, “Breaking away from the near horizon of extreme Kerr,” SciPost Phys. ,089 (2020), arXiv:1906.09083 [hep-th].[62] Netta Engelhardt and Gary T. Horowitz, “Holographic Consequences of a No Transmission Principle,” Phys.Rev. D , 026005 (2016), arXiv:1509.07509 [hep-th].[63] Ben Freivogel, Victor Godet, Edward Morvan, Juan F. Pedraza, and Antonio Rotundo, “Lessons on EternalTraversable Wormholes in AdS,” JHEP , 122 (2019), arXiv:1903.05732 [hep-th].[64] Andre Roberge and Nathan Weiss, “Gauge Theories With Imaginary Chemical Potential and the Phases of { QCD } ,” Nucl. Phys. B , 734–745 (1986).[65] Philippe de Forcrand and Owe Philipsen, “The QCD phase diagram for small densities from imaginary chemicalpotential,” Nucl. Phys. B , 290–306 (2002), arXiv:hep-lat / , 186 (2020), arXiv:1912.12285 [hep-th].[69] Victor Godet and Charles Marteau, “New boundary conditions for AdS ,” (2020), arXiv:2005.08999 [hep-th].[70] Henry Maxfield and Gustavo J. Turiaci, “The path integral of 3D gravity near extremality; or, JT gravity withdefects as a matrix integral,” (2020), arXiv:2006.11317 [hep-th].[71] Alexander Maloney and Edward Witten, “Averaging Over Narain Moduli Space,” (2020), arXiv:2006.04855[hep-th].[72] Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, and Amirhossein Tajdini, “Free partition functions andan averaged holographic duality,” (2020), arXiv:2006.04839 [hep-th].[73] Jordan Cotler and Kristan Jensen, “AdS gravity and random CFT,” (2020), arXiv:2006.08648 [hep-th].[74] Alexandre Belin and Jan de Boer, “Random Statistics of OPE Coe ffi cients and Euclidean Wormholes,” (2020),arXiv:2006.05499 [hep-th].[75] Antonio M. Garc´ıa-Garc´ıa and Salom´on Zacar´ıas, “Quantum jackiw-teitelboim gravity, selberg trace formula,and random matrix theory,” (2019), arXiv:1911.10493 [hep-th].[76] K.A. Milton, The Casimir e ff ect: Physical manifestations of zero-point energy (2001).[77] Lowell S. Brown and James P. Cassidy, “Stress Tensors and their Trace Anomalies in Conformally Flat Space-Times,” Phys. Rev. D , 1712 (1977).[78] B.C. Berndt, Ramanujan’s Notebooks , dl. 1 (Springer New York, 1985).[79] Bruce C. Berndt, “Ramanujan’s theories of elliptic functions to alternative bases,” in