PHYSICAL REVIEW D 100, 124061 (2019)

Models of phase stability in Jackiw-Teitelboim

Arindam Lala* Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla, 4059 Valparaiso, Chile † Dibakar Roychowdhury Department of Physics, Indian Institute of Technology Roorkee, Roorkee, 247667 Uttarakhand, India

(Received 4 October 2019; published 31 December 2019)

We construct solutions within Jackiw-Teitelboim (JT) gravity in the presence of nontrivial couplings between the and the Abelian 1-form where we analyze the asymptotic structure as well as the phase stability corresponding to charged solutions in 1 þ 1 D. We consider the Almheiri-Polchinski model as a specific example within 1 þ 1 D JT gravity which plays a pivotal role in the study of Sachdev-Ye- Kitaev (SYK)/anti–de Sitter (AdS) duality. The corresponding vacuum solutions exhibit a rather different asymptotic structure than their uncharged counterpart. We find interpolating vacuum solutions with AdS2 in the IR and Lifshitz2 in the UV with dynamical exponent zdyn ¼ 3=2. Interestingly, the presence of charge also modifies the black hole geometry from asymptotically AdS to asymptotically Lifshitz with the same value of the dynamical exponent. We consider specific examples, where we compute the corresponding free energy and explore the thermodynamic phase stability associated with charged black hole solutions in 1 þ 1 D. Our analysis reveals the existence of a universal thermodynamic feature that is expected to reveal its immediate consequences on the dual SYK physics at finite density and strong coupling.

DOI: 10.1103/PhysRevD.100.124061

I. INTRODUCTION AND MOTIVATIONS For the last couple of years, there has been a systematic For the last couple of decades, there had been consid- effort toward unveiling the dual gravitational counterpart erable efforts toward a profound understanding of the of the SYK model. A hint came from the JT dilaton gravity 1 1 – underlying nonperturbative dynamics in large N gauge in þ D [32 36] based on which disparate dual gravity – theories using the celebrated AdS=CFT framework [1–3]. models have been proposed [37 44] along with several – Nonetheless, to date there exist only a few examples where interesting extensions [45 49]. this duality can actually be tested with precise accuracy. The original SYK/AdS duality deals with Majorana In other words, one can exactly solve the spectrum on both fermions for which neutral dual gravity models are enough sides of the duality albeit they are strongly interacting. In to consider. This has been the line of analyses for most of the the recent years, an example of this kind has emerged models so far. However, very recently charged SYK models where the spectrum of the 0 þ 1 dimensional strongly have been constructed in [50,51] whose dual gravitational counterpart has been proposed to be given by the 2D interacting Sachdev-Ye-Kitaev (SYK) model can be solved 1 exactly using large N techniques whose dual counterpart effective gravity action of the following form [50]: has been conjectured to be the Jackiw-Teitelboim (JT) Z   Φ2 model in 1 þ 1 D [4–31]. Apart from being exactly 2 pffiffiffiffiffiffi 2 2 Zð Þ μν S2 ∼ d x −g Φ RþVðΦ Þ− F Fμν : ð1Þ solvable, the SYK model exhibits maximal chaos together D 4 with an emergent conformal symmetry at low energies which therefore provides a reliable platform to test the The last term in the above action (1) represents the holographic correspondence. nontrivial coupling between the dilaton and the Abelian 1-form. This interaction term can be viewed as an effective coupling which can be obtained as a result of dimensional *[email protected]; [email protected] † reduction from the 3 1 D version of the theory [50]. [email protected]; [email protected] þ Equation (1) without the gauge field is precisely the form Published by the American Physical Society under the terms of of the action considered in [37] with the potential VðΦ2Þ the Creative Commons Attribution 4.0 International license. linear in dilaton. In the present analysis, we choose to work Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1See Appendix A for details.

2470-0010=2019=100(12)=124061(16) 124061-1 Published by the American Physical Society ARINDAM LALA and DIBAKAR ROYCHOWDHURY PHYS. REV. D 100, 124061 (2019) with two specific forms of the dilaton coupling, namely, 2 2 2 F ∂UðΦ Þ Φ2 ∼ Φ2 2 Φ2 ∼ −Φ 0 R − Φ2 − ; Zð Þ ð Þ and Zð Þ e together with the choice ¼ 2 ∂Φ2 ð3bÞ of the dilaton potential VðΦ2Þ as given in [37]. ffiffiffiffiffiffi Based on the classical gravity computations, we con- 0 ∂ p− Φ4 μν struct charged 2D black hole solutions in the two afore- ¼ μð g F Þ: ð3cÞ mentioned models. Our analysis reveals that the presence of charge substantially modifies the asymptotic symmetries of Let us now consider the conformal gauge the space-time, namely, converting it to a two-dimensional 2 2ω þ − − asymptotic Lifshitz geometry which otherwise would ds ¼ −e ðx ;x Þdxþdx ; ð4Þ have been an AdS2 geometry. We further compute the free energy and explore the thermodynamic phase stability of where x ¼ t z. In this light-cone gauge the equations of the obtained solutions. For both the models, we observe a motion can be written as universal thermodynamic feature of phase stability at sufficiently low temperatures and finite density. 4∂ ∂ Φ2 − 2ω Φ2 − 2Φ4 −2ω 2 0 þ − e Uð Þ e Fþ− ¼ ; ð5aÞ The organization of the paper is as follows: In Sec. II,we 2 2 2 propose our first model with ZðΦ Þ ∼ ðΦ Þ . Considering 2 2 ∂ ∂ Φ − 2ð∂ ωÞð∂ Φ Þ¼0; ð5bÞ linear potential for the dilaton potential we explore the bounds on the potential which makes the space-time 2ω ∂ Φ2 asymptotically AdS. In Secs. III and IV we comment on e Uð Þ −2ω 2 2 4∂ ∂−ω − þ 2e Φ F − ¼ 0; ð5cÞ the vacuum structures as well as the black hole solutions þ 2 ∂Φ2 þ both for an asymptotically flat and asymptotically AdS ∂ χ 0 space-time, respectively. In Sec. V we construct perturba- ¼ ; ð5dÞ tive solutions (in charge, Q) to our model and analyze the pffiffiffiffiffiffi underlying geometry associated to both the vacuum and as where we have defined χ ¼ −g Φ4Fþ−. Notice that our well as the charged black hole solutions. This is supple- analysis differs from that of [37] in the sense that there is no mented with the study of the phase stability of derived fully decoupled equation of motion for ωðzÞ. As a result we solutions using the standard background subtraction 2ω obtain a different solution for the conformal factor e ðzÞ. method [52]. In Sec. VI we repeat our analysis for the In the next step we would like to consider the static exponential dilaton coupling. Finally, we conclude in solutions. In order to do so, we revert back to the (t; z) Sec. VII where we mention the possible implications of coordinates and use the following ansatz for the gauge our findings on the corresponding SYK counterpart. field:

II. EXAMPLE I: QUADRATIC COUPLING 1 1 − 0 We start with the Einstein-Maxwell-dilaton action of the Aþ ¼ 2ðAt þAzÞ;A− ¼ 2ðAt AzÞ;Aμ ¼ðAt; Þ: ð6Þ following form: Z   In these coordinates (5a)–(5d) can be expressed as ffiffiffiffiffiffi 1 − 2 p− RΦ2 − Φ2 − Φ2 2 2 S ¼ d x g Uð Þ ð Þ F 1 4 2 00 2ω 2 4 −2ω 02 Z ðΦ Þ þ e UðΦ Þþ Φ e At ¼ 0; ð7aÞ pffiffiffiffiffiffi 2 − dt −γΦ2K; ð2Þ ðΦ2Þ00 − 2ω0ðΦ2Þ0 ¼ 0; ð7bÞ where F ¼ dA is the Maxwell 2-form field, Φ is the dilaton 2 and UðΦ Þ is the dilaton potential. Notice that we have ∂UðΦ2Þ 2ω00 e2ω − e−2ωΦ2A02 0; added the Gibbons-Hawking-York boundary term [53,54] þ ∂Φ2 t ¼ ð7cÞ in the above action, where γ is the determinant of the K induced metric on the boundary and is the trace of the Φ2A00 − 2A0ðΦ2ω0 − ðΦ2Þ0Þ¼0: ð7dÞ extrinsic curvature [55]. In the subsequent analysis we set t t the AdS length scale L ¼ 1 and 16πG ¼ 1. We can rewrite (7a) in the following form: The equations of motion can be written as   1 1 1 Φ2 − −2ω Φ2 00 − Φ4 −4ω 02 0 ∇ ∇ − □ Φ2 Φ4 ρ − 2 Uð Þ¼ e ð Þ 2 e At : ð8Þ ¼ð μ ν gμν Þ þ 2 FμρFν 4 F gμν

1 2 Let us now consider a constant dilaton profile: − gμνUðΦ Þ; ð3aÞ 2 2 2 Φ ðzÞ¼Φ0. In this case substituting (8) in (7c) we obtain

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∂U Φ2 B. Charged black hole solutions 2ω00 −2 2ω ð Þ ¼ e 2 : ð9Þ ∂Φ 2 2 Φ ¼Φ0 We choose to work with metric ansatz of the following form: The Ricci scalar (R), on the other hand, can be written as 2 − 2 −1 2 ds ¼ fðzÞdt þ f ðzÞdz : ð18Þ ∂ Φ2 R 2 Uð Þ ¼ 2 ; ð10Þ ∂Φ 2 2 The only component of the Maxwell field strength tensor Φ ¼Φ 0 can thus be written as where we have used (9). Q Now, in order for the space-time to have AdS2 asymp- F ; tz ¼ Φ4 ð19Þ totics we must have the following criteria: ðzÞ

∂ Φ2 where Q is an integration constant which we can identify as Uð Þ 0 2 < : ð11Þ the charge of the black hole, and we have used (3c) in order ∂Φ Φ2 Φ2 ¼ 0 to derive (19). With the metric (18) the remaining equations of motion (3a) and (3b) can be written as III. GRAVITY IN ASYMPTOTICALLY FLAT 1 + 1 D 2 2 Q 2 A. Minkowski vacuum f0ðzÞðΦ Þ0 þ þ UðΦ Þ¼0; ð20Þ 2ðΦ2Þ2 In this section, we would like to consider the vacuum 1 1 solution corresponding to þ D dilaton gravity. In order Q2 to do so we consider the following metric: 2fðzÞðΦ2Þ00 þ f0ðzÞðΦ2Þ0 þ þ UðΦ2Þ¼0; ð21Þ 2ðΦ2Þ2 ds2 ¼ −dt2 þ dz2: ð12Þ Q2 ∂UðΦ2Þ f00 z − 0: ð Þ Φ6 þ ∂Φ2 ¼ ð22Þ From the gauge equation of motion (3c) we can write ðzÞ

Q From (20) and (21) it is easy to check that F : tz ¼ Φ4 ð13Þ ðΦ2Þ00 ¼ 0; ð23Þ It is easy to check that the scalar equation of motion (3b) and one of the trace equations of motion corresponding to and as a result the dilaton becomes constant at the the metric, (3a), lead to the following form of the dilaton boundary z ¼ 0. potential: As a next step, we determine the metric coefficient fðzÞ with particular choices of the dilaton potential UðΦ2Þ 1 2 Φ2 φ Φ2 − Q where we finally set ¼ 0 ¼ const. Uð Þ¼ 2 Φ4 : ð14Þ 1. Case I: The Almheiri-Polchinski model [37] On the other hand, the remaining trace equation of [UðΦ2Þ =C− AΦ2] and the Callan-Giddings-Harvey- motion can be written expressed as Strominger (CGHS) model [56] [UðΦ2Þ = − AΦ2] 1 Q2 1 From (22) we obtain Φ2 00 U Φ2 0: ð Þ þ 4 Φ4 þ 2 ð Þ¼ ð15Þ 2 00 Q f ðzÞ¼A þ 3 ; ð24Þ Substituting (14) into (15) we obtain φ0

ðΦ2Þ00 ¼ 0; ð16Þ whose solution may be written as      1 2 whose solution may be written as 1 − z 1 − Q fðzÞ¼ A þ 3 zzH : ð25Þ zH 2 φ0 Φ2 ¼ bz þ a; ð17Þ Using (25), the Hawking temperature of the black hole where a and b are arbitrary integration constants. can be obtained as

124061-3 ARINDAM LALA and DIBAKAR ROYCHOWDHURY PHYS. REV. D 100, 124061 (2019) rffiffiffiffiffiffiffiffiffiffi 1 ∂ − gtt (i) the Hawking temperature TH ¼ 4π z gzz z z       ¼ H   1 1 B Q2 1 1 2 T ¼ A þ þ z2 − 1 ; ð33Þ Q 2 − 1 H 4π 2 φ2 φ3 H ¼ A þ 3 zH : ð26Þ zH 0 0 4πzH 2 φ0 (ii) extremal value of charge The extremal limit, in which the Hawking temperature    vanishes, is characterized by the extremal value of the 2 B Q2 − A φ3; charge given by e ¼ 2 þ φ2 0 ð34Þ zH 0   2 2 − φ3 (iii) the specific heat Qe ¼ 2 A 0: ð27Þ zH ∂S C ¼ T W H ∂T On the other hand, by using the Wald formalism, the H     4πφ3 1 B Q2 3Q2 −1 entropy of the black hole is given by [57] 0 1 − A z2 B : ¼ 2 2 þ φ2 þ φ3 H þ 2φ zH 0 0 0 4πΦ2 SW ¼ : ð28Þ ð35Þ

The thermodynamic stability of the black hole is Like in the previous example, it is easy to check that determined by computing the corresponding heat the condition for extremality implies the thermody- capacity (C): namic instability in black holes.     ∂S 8πφ4 1 Q2 IV. VACUUM SOLUTIONS WITH AdS C T W 0 1 − A z2 ; 2 ¼ H ∂ ¼ 3 2 2 2 þ φ3 H ð29Þ ASYMPTOTICS TH Q zH 0 Unlike the previous example, here we discuss the where we have used (26) and (28). possibilities on vacuum solutions with AdS2 asymptotics. 0 For nonextremal black holes one must have TH > We show that solutions with AdS2 asymptotics are indeed which leads to the following condition: possible both for the constant as well as the running dilaton   profiles. 1 2 Q 2 1 A þ 3 zH > : ð30Þ 2 φ0 A. Solution with constant dilaton We first construct solutions with constant dilaton Substituting (30) in (29) we note that the specific heat is 2 2 Φ ¼ Φ0. always negative. This suggests that the black holes (25) in an asymptotically flat space-time are indeed unstable and 1. Case I: e2ω = 1 decay through . z2 In this case from (7a) we may write

2 B2 2 2. Case II: Magnetic branes [58],UðΦ Þ = 2 − AΦ 2 Φ 02 − Φ2 At ¼ 4 4 Uð 0Þ; ð36Þ The equation of motion corresponding to the metric fðzÞ z Φ0 can be expressed as whereas (7b) is satisfied trivially. Notice that in the above 2 2 equation UðΦ0Þ must be negative. Using the relation 00 B Q f ðzÞ¼A þ 2 þ 3 ; ð31Þ ω0 − 1 we can write (7c) as φ0 φ0 ¼ z

2 1 ∂U Φ2 where we have used (22). The general solution to the above ð Þ − 2Φ2 02 0 2 þ 2 2 z 0At ¼ : ð37Þ equation (31) is given by z z ∂Φ 2 Φ0      z 1 B Q2 Finally, using (36) we obtain the following relation: fðzÞ¼ 1 − 1 − A þ þ zz : ð32Þ z 2 φ2 φ3 H   H 0 0 ∂ Φ2 Φ2 Uð Þ −2 1 − jUð 0Þj 2 ¼ 2 ; ð38Þ ∂Φ 2 Φ Proceeding in the same line of analysis as in the previous Φ0 0 case I, the thermodynamic quantities may be found as follows: which leads to the following bound to the potential:

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Φ2 whose solution may be formally expressed as jUð 0Þj 1 2 < : ð39Þ Φ0 2 b1 Φ ¼ − þ a1: ð47Þ z 2. Case II: e2ω = 1 sinh2 z It is interesting to note that the dilaton diverges near the In this case the equation for the gauge field can be boundary, z ≈ 0. expressed as In a similar way, considering the conformal factor as e2ω ¼ sinh−2z, sin−2z, the solutions to the dilaton can be 2 02 − Φ2 found as At ¼ 4 4 Uð 0Þ; ð40Þ Φ0 sinh z Φ2 − 2 ¼ b2 coth z þ a2; ð48Þ from which we again conclude that UðΦ0Þ < 0. Substi- 2ω tuting e in (7c) it is trivial to check that one arrives at the 2 Φ ¼ −b3 cot z þ a3; ð49Þ same constraint conditions (38) and (39). respectively. Interestingly, both these solutions diverge near 2ω 1 ≈ 0 3. Case III: e = sin2 z the boundary, z . In this case the equation for the gauge field can be Notice that the divergence of the dilaton profile (47)–(49) written as near the boundary (z ≈ 0) is a generic feature of the AdS space-time; see for example [59] and references therein. 2 However, as far as the present analysis is concerned, one of 02 − Φ2 At ¼ 4 4 Uð 0Þ; ð41Þ Φ0 sin z the solutions (47) has an interesting consequence from the perspective of the SYK/AdS duality. One of the interesting 2 2ω from which it is evident that UðΦ0Þ < 0. Substituting e facets of this duality is that it allows us to relate the SYK in (7c) we once again arrive at the same constraint degrees of freedom to the underlying dynamics of the AdS2 conditions (38) and (39). counterpart, [11,42–44]. We can consistently set a1 ¼ 0 in 2 Thus we observe that the above constraint conditions (47) by demanding that the dilaton Φ vanishes as we probe (38) and (39) are universal for all three cases considered; deep IR (z → ∞). This observation is in fact consistent with 2 i.e., they are the same irrespective of the choice of the thestronglyinteracting(J ∼ 1=Φ ≫ 1)natureofthedual coordinate systems. SYK model in which we are mostly interested [42–44].

B. Solutions with running dilaton V. GENERAL SOLUTIONS: A PERTURBATIVE Let us consider the metric of the following form: APPROACH In this section, we adopt perturbation techniques in order 1 2 − 2 2 to find the general solutions to the equations of motion ds ¼ 2 ð dt þ dz Þ: ð42Þ z (3a)–(3c) and determine the metric of the space-time. In order to perform our analysis, we consider a running With this choice of metric the gauge equation of 2 dilaton where the dilaton is a function of z only: Φ ¼ motion (3c) leads to Φ2ðzÞ. We also consider the dilaton potential as2 [37] Q F ¼ : ð43Þ Φ2 − Φ2 0 tz Φ4z2 Uð Þ¼C A ;C;A>: ð50Þ

The metric equation of motion can be expressed as In order to obtain solutions to the metric as well as the dilaton equations of motion we expand the above entities as 1 Q2 1 U 1 Q Φ2 0 − − Φ2 0 a perturbation in the ð Þ charge , namely, zð Þ 4 Φ4 2 Uð Þ¼ ; ð44Þ 2 2 2 2 Φ ðzÞ¼Φ 0 ðzÞþQ Φ 1 ðzÞþ; ð51aÞ 1 Q2 1 ð Þ ð Þ z2 Φ2 00 z Φ2 0 U Φ2 0: ð Þ þ ð Þ þ 4 Φ4 þ 2 ð Þ¼ ð45Þ ω ω 2ω ðzÞ¼ ð0ÞðzÞþQ ð1ÞðzÞþ: ð51bÞ Substituting (44) into (45) we obtain 2 2 Φ2 00 2 Φ2 0 0 See Appendix B regarding the black hole solution with z ð Þ þ zð Þ ¼ ; ð46Þ A<0.

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2 The Maxwell field strength tensor is given by the C4 1 − 2z − 4z þ 2ð1 þ 2zÞ · logð1 þ 2zÞ ωvacðzÞ¼ þ : solution of (7d) which may be written as ð1Þ z 8zð1 þ 2zÞ Qe2ω ð61Þ F ≡ A0 ¼ − : ð52Þ zt t Φ4ðzÞ Thus the metric for the vacuum can be expressed as On the other hand, the equation of motion for the dilaton (7b) can be recast in the following form: 2 2ω 2 2 2 ð0Þ 1 2 ωvac − ds ¼ e ð þ Q ð1Þ Þð dt þ dz Þ: ð62Þ ðΦ2Þ0 ¼ Ce˜ 2ω; ð53Þ Let us now analyse the IR and UV behaviors of the whose solution may be written as Z solution (62). (i) In the IR region z → ∞ the behavior of the metric is Φ2 ˜ 2ω ¯ ðzÞ¼C e dz þ C; ð54Þ given by ˜ ¯   where C and C are arbitrary integration constants. 1 Q2 In the following we note down the equations of motion e2ω ≃ 1 − þ Oðz−3Þ; ð63Þ z2 2 up to leading order in the perturbative expansion. Substituting (51b) into (7c) we find which thereby leads to an emerging AdS2 geometry. 2ω 0 00 ð0Þ (ii) The UV (z ¼ 0) behavior of the metric is found to be OðQ Þ∶ 0 ¼ ω 0 − e ; ð55Þ ð Þ of the following form:

2 00 2ω 0 2ω 0 2 −3 OðQ Þ∶ 0 ¼ 2ω − 4e ð Þ ω 1 − e ð Þ ðΦ Þ : ð56Þ ð1Þ ð Þ ð0Þ 2 1 2 2 2ω ≃ Q 1 8C − Q O e 3 ð þ 4Þþ 2 þ ðzÞ; ð64Þ On the other hand, substituting (51a) into (7a) we find 4z z 3

0 2 2ω 2 O ∶ 0 Φ 00 2 ð0Þ 1 − Φ whose leading contribution comes from the first term ðQ Þ ¼ð ð0ÞÞ þ e ð ð0ÞÞ; ð57Þ on the rhs of (64).ThisturnsouttobeaLifshitz2 3 2 2 2ω 2 00 ð0Þ geometry with dynamical exponent z ¼ . OðQ Þ∶ 0 ¼ðΦ 1 Þ − 2e Φ 1 dyn 2 ð Þ  ð Þ  Thus we observe that the vacuum solution interpolates 2ω 2 1 2 −2 4 ð0Þ ω 1 − Φ Φ between Lifshitz2 in the UV and AdS2 in the deep IR. It is þ e ð1Þð 0 Þþ ð 0 Þ : ð Þ 8 ð Þ interesting to note that in the absence of the charge, Q ¼ 0, ð58Þ the geometry is AdS2 for both in the IR as well as the UV. Thus we conclude that the presence of the gauge field modifies the UV asymptotics from AdS2 to Lifshitz2 [60]. A. Interpolating vacuum solutions B. Charged 2D black holes The vacuum solution corresponding to Q ¼ 0 is char- Φ2 ω acterized by [37] The zeroth-order solutions ð0ÞðzÞ and ð0ÞðzÞ are, respectively, given by [37] 2ωvac z 1 ð0Þ ð Þ e ¼ 2 ; ð59Þ z 2 pffiffiffi pffiffiffi   Φ 0 ¼ 1 þ μ cothð2 μzÞ; ð65aÞ 1 ð Þ Φ2vac ðzÞ¼ 1 þ ; ð60Þ ð0Þ 2z 2ω 4μ e ð0Þ ¼ pffiffiffi : ð65bÞ which satisfy the zeroth-order equations (55) and (57), sinh2ð2 μzÞ respectively, for the following values of the constants in ˜ ¯ (54): C ¼ −1=2 and C ¼ 1. Thus for the purpose of our Notice that, in order for the solution (65a) to be 2 present analysis it is sufficient to solve the OðQ Þ equations consistent with the equations of motion (7a) and (7b), of motion, namely, (56) and (58). we must choose the constants appearing in (54) as ωvac ˜ ¯ Using (56) the solution corresponding to ð1Þ can be C ¼ −1=2 and C ¼ 1. expressed as3 It is trivial to check that, with the choice of the constants C˜ and C¯ , (65a) and (65b) are indeed the solutions to the 3 We set the second integration constant C3 to zero. This is to equations of motion (55) and (57). Using (65a) and (65b) ωBH render the on-shell action in Sec. VC finite at the horizon. in (56) the solution for ð1Þ may be written as

124061-6 MODELS OF PHASE STABILITY IN JACKIW-TEITELBOIM … PHYS. REV. D 100, 124061 (2019)   ffiffiffi   ρ ρ pμ ρ BH þ ω ρ C1 ffiffiffi C2 ffiffiffi · log ffiffiffi − 1 ð1Þ ð Þ¼ pμ þ 2pμ pμ − ρ    1 pffiffiffi 2 þ ρ μ2 þ × −2 μ 1 − μ þ þ 4μ3=2ρ · logð1 þ ρÞ 8μ3=2ðμ − 1Þ2 1 þ ρ 1 þ ρ  pffiffiffi pffiffiffi − ρ · logðρ − μÞð1 − 3μ þ 2μ3=2Þþρ · logðρ þ μÞð1 − 3μ − 2μ3=2Þ ; ð66Þ

where C1 and C2 are the integration constants and we have Notice that in writing the second line we have used (67). used the following change in the spatial coordinate [37,45]: Thus the complete solution up to OðQ2Þ can be   expressed as 1 −1 ρ z → pffiffiffi coth pffiffiffi : ð67Þ 2 μ μ 2 2 2 2 Φ ≃ Φ 0 þ Q Φ 1 ð Þ ð Þ Z In the subsequent calculations we set C2 ¼ 0 in order to 1 ρ 2 2 ω ρ ρ obtain a physically meaningful asymptotic structure of the ¼ð þ Þþ Q ð1Þð Þd : ð71Þ space-time. Finally, using (51b), (65b) and (66) the metric (4) We now compute the thermodynamic quantities corre- corresponding to the black hole geometry can be sponding to the above black hole geometry (68). The expressed as corresponding Hawking temperature is given by ds2 ¼ e2ωð−dt2 þ dz2Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi 1 1 pμ 4μ − tt ρρ ∂ 2 2 BH 2 2 TH ¼ g g ð ρgttÞ ffiffi ¼ : ð72Þ ¼ pffiffiffi ð1 þ 2Q ω 1 ðzÞÞð−dt þ dz Þ 2π 4 ρ→pμ π sinh22 μz ð Þ   dρ2 4 ρ2 − μ 1 2 2ωBH ρ − 2 On the other hand, the Wald entropy associated with the ¼ ð Þð þ Q ð1Þ ð ÞÞ dt þ 2 2 : 4ðρ − μÞ black hole can be found as ð68Þ h Z i pffiffiffi 2 S ¼ 4π ð1 þ μÞþ2Q ω 1 ðρÞdρ ffiffi Notice that the above black hole solution (67) has a W ð Þ ρ pμ pffiffiffi ¼ horizon at ρ ¼ μ. On the other hand, the boundary is   pffiffiffi 2 1 þ 4C1μjμ − 1j located at ρ ¼ ∞. Expanding the metric near the boundary ¼ 4π 1 þ μ þ Q pffiffiffi : ð73Þ 4 μjμ − 1j we find ffiffiffi 2 2 p μ 1 8Q C1 4 8Q C1 μ Notice that (73) is ill defined at ¼ . Therefore one can 2ωðδÞ ≃ ffiffiffi − − 4μ O δ e pμδ3 þ δ2 δ þ ð Þ; ð69Þ define black hole solutions for either of the two branches, namely, μ < 1 or μ > 1, which is thereby consistent with where we have changed the variable ρ → 1=δ and taken the the earlier observations [37,45]. limit δ → 0 subsequently. The leading term in the expan- ∼ 1 sion (69) behaves as δ3, which is a signature of an C. Phase stability asymptotically Lifshitz geometry with dynamical critical In order to check whether there is any phase transition 3 exponent zdyn ¼ 2. It is trivial to check that in the limit or crossover between the empty AdS2 and the AdS2 black Q2 → 0 the resulting metric is that of an asymptotically hole one needs to compare free energies between different AdS2 space-time. Our result thus indicates that in the configurations. We substitute (3b) into the action (2) which presence of the gauge field the asymptotic behavior of the finally yields space-time indeed changes from AdS2 to Lifshitz2 [60].   Z 2 Z Next, we turn our attention toward computing the dilaton 2 pffiffiffiffiffiffi F 2 2 pffiffiffiffiffiffi 2 profile for our model. Using (54) we observe that S ¼ − d x −g ðΦ Þ − C − dt −γKΦ : Z 4 Φ2 − 2ω0 ω ð74Þ ð1Þ ¼ e ðzÞ ð1ÞðzÞdz Z 2 ω ρ ρ ¼ ð1Þð Þd : ð70Þ Next we use the equation of motion for the gauge field, (3c), to find the on-shell action as

124061-7 ARINDAM LALA and DIBAKAR ROYCHOWDHURY PHYS. REV. D 100, 124061 (2019) Z Z ffiffiffiffiffiffi ffiffiffiffiffiffi β ðosÞ 2 p 2 p μν 2 2 black hole, respectively. Moreover, while 1 is fixed by −S ¼ − d x −gC þ d x∂μð −gF ðΦ Þ AνÞ (72), β0 is arbitrary. Thus we can fix β0 by demanding Z ffiffiffiffiffiffi that, at some arbitrary ρ ¼ Λ, the temperature of both p zt 2 2 4 ¼ −C · ðVolÞþ dtð −gF ðΦ Þ AtÞ the configurations should be the same, namely, Z ffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2ωBH þ dt −γKΦ ; ð75Þ β0 e ρ ð Þ ¼ 2ωvac : ð81Þ β1 Int ρ e ð Þ ρ→Λ where (Vol) is the volume term associated with the following two geometries: In the next step, we would like to compute the difference (i) For empty interpolating vacuum we may write down between the volume terms (78) and (76). This may be the volume term as written in the following form: Z Z   β0 Λ 1 vac 2ωvac Δ − BH − vac − dτ dρ e Int ρ ðVolÞ¼ CððVolÞ ðVolÞ Þ ðVolÞInt ¼ ρ2 ð Þ Int 0 0 2 ¼ Cβ1½ðA1 − B1ÞΛ þðA2 − B2ÞΛ −β A ρ Λ ¼ 0 vacð Þj0 ; ð76Þ ffiffiffi ðμÞ p þðA3 − B3Þ − A ðρ → μÞ; ð82Þ where we have used the change of coordinate z → 1 ρ where the individual coefficients are given by and    2 2C 2 2 Q 1 Q ρ ρ ρ 2 ρ A1 ¼ pffiffiffi ; A2 ¼ 2; A3 ¼ ; ð83Þ A ρ − Q 1 − 1 8C − þ μ 2 μ − 1 vac ¼ ð þ 4Þ · log : ð Þ 2 4 2 ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð77Þ Q C1ð1 þ 8C4Þ B1 ¼ pffiffiffi ffiffiffi ; 2 pμ (ii) For the AdS2 black hole the volume term can be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expressed as 1 C1 pffiffiffi B2 ¼ pffiffiffi pffiffiffi ð μð1 þ 8C4Þþ24C1Þ; Z Z 4 2C μð1 þ 8C4Þ β1 ρ∼Λ 1 1 BH − τ ρ 2ωBH ρ ffiffiffi ðVolÞ ¼ d ffiffi d 2 e ð Þ 2 p 0 ρ∼pμ 2ðρ − μÞ B3 ¼ −Q ð1 þ C1 μÞ: ð84Þ ffiffiffi −β AðΛÞ ρ → Λ − AðμÞ ρ → pμ ¼ 1ð ð Þ ð ÞÞ; It is to be noted that in (84) we have simplified the ð78Þ cumbersome expression for B3 by using the following relation: ffiffiffi where pμ C1 ¼ ð1 þ 8C4Þ; ð85Þ 1 ffiffiffi 8 AðΛÞ ρ → Λ 2Λpμ μ − 1 2μ μ − 1 ð Þ¼ 3 2 2 f ð Þ½ ð Þ 2μ = ðμ − 1Þ which is easily derived by setting the coefficient of Λ2 in 2 pffiffiffi þ Q ð1 þ ΛC1 μðμ − 1ÞÞ (82) equal to zero. In addition, using (85), it is easy to check that the coefficient of the Λ term in (82) vanishes. 2 2 3=2 þ Q ðΛ − μÞ½4μ · logð1 þ ΛÞ Let us now consider the second term Π in (75). Using pffiffiffi þð−1 þ 3μ − 2μ3=2Þ · logðΛ − μÞ (52), (68) and (62) this may be written as pffiffiffi Z Z þð1 − 3μ − 2μ3=2Þ · logðΛ þ μÞg Λ 2 2ω 0 ðzÞ 2 −2 Π ¼ Q dτ e ð Þ ðΦ 0 ðzÞÞ dz ð79Þ 0 ð Þ 8   2 1 Λ < −4β0Q ðinterpolating vacuumÞ; and 2þρ 0 ¼   ð86Þ : 2 1 Λ −2β1Q ffiffi ðblack holeÞ; 2 1þρ pμ μ pffiffiffi 4μðμ − 1ÞþQ ½1 þ 4μC1ðμ − 1Þ Að Þðρ → μÞ¼ pffiffiffi : 2 μðμ − 1Þ 4 ð80Þ The choice of normalization (81) for β0 is unique in the sense that the geometry of the hypersurface placed at a radial cutoff ρ Λ → ∞ Λ ¼ ð Þ should be the same both for the vacuum as well as Here, we have introduced a UV cutoff in order to the black hole space-time [61]. This also ensures the consistency make the integrals finite, and β0 and β1 are the periods in the thermodynamic description in the sense of holography as associated with the interpolating vacuum and the AdS2 the limit Λ → ∞ is approached.

124061-8 MODELS OF PHASE STABILITY IN JACKIW-TEITELBOIM … PHYS. REV. D 100, 124061 (2019) where we have substituted (65a) and (65b). Finally, in the We now analyze the behavior of the free energy of limit Λ → ∞ the difference between the second terms can the configuration. In the path integral formulation, the be written as free energy is given by ΔF ¼ −β−1 log Z, where Z is  ffiffiffi 6 Z ≔ −ΔSðosÞ 1 2pμ the partition function and is defined as e . Thus BH vac 2 þ ΔðΠÞ¼Π − Π ¼ −2Q β1 pffiffiffi : ð87Þ the free energy for the present configuration may be 1 þ μ expressed as

We now calculate the contribution from the third term Σ −1 ðosÞ ΔF ≔−β1 ð−ΔS Þ: ð92Þ in (75). In order to do so, we choose a ρ ¼ const hyper- ρ surface [55]. Let us now define an outward normal n , In Fig. 1 we present the behavior of the free energy ΔF ρ pointing along the increasing ,as between the black hole and the interpolating vacuum as a ΔF 0 1 function of the temperature (72). We observeffiffiffi that < ρ ρ ρ ∼ pμ gρρn n ¼ 1 ⇒ n ¼ pffiffiffiffiffiffi ; ð88Þ for all values of the temperature TH . The free energy gρρ increases asymptotically for sufficiently small values of the temperature and changes its slope at some particular where gρρ is the metric coefficient in (68) corresponding to temperature. Afterwards it continues to decrease. ρ the spatial coordinate . The trace of the extrinsic curvature, In Fig. 2, we plot the black hole entropy S ¼ S ¼ K, can then be written as ffiffiffi W − ΔFffiffi ∼ pμ ffiffiffiffiffiffi Δpμ against temperature TH . In these plots the ∂ p−γ ρ ρ entropy is continuous and increases smoothly as we lower K ¼ n ffiffiffiffiffiffi : ð89Þ p−γ the temperature which rules out the possibility of a first- order phase transition. It is to be noted that this behavior Next, we use (54), (88) and (89) to compute the differ- of entropy is consistent with the Wald entropy of the black ence between the third terms in the Λ → ∞ limit. After a hole in (73). This allows us to conclude that the phase few easy steps, we note down this expression as5 diagram corresponding to the parameter space with μ < 1 is thermodynamically more preferred than that of the μ > 1 ΔΣ ¼ ΣBH − Σvac branch. We comment on the plausible implications of  pffiffiffi β μ μ −4 4μ Q2 such phase stabilities on the dual SYK physics in the 1 −14 2 μ − ð þ þ Þ ¼ Q þ þ 4 2 2 concluding remarks. 8 Q C1 Q C1ðμ − 1Þ pffiffiffi  μ μ Q2 − 4 − − ð Þ 2 4 2 : ð90Þ VI. EXAMPLE II: EXPONENTIAL COUPLING 2C1Q Q C1 In this section we consider the effective 2D gravity After performing a long but simple calculation, the action (1) of the following form: difference in the on-shell action can be expressed as Z   2 pffiffiffiffiffiffi 2 2 λ −Φ2 μν    S ¼ d x −g RΦ − VðΦ Þ − e FμνF pffiffiffi 1 pffiffiffi ed 4 −Δ ðosÞ β −2 μ 2 1 C μ Z S ¼ 1 C þ Q 2 μ − 1 þð þ 1 Þ ð Þ pffiffiffiffiffiffi 2   pffiffiffi þ dt −γ Φ K; ð93Þ 1 4μC1 μ − 1 1 þ 2 μ − ð þ ffiffiffi ð ÞÞ − 2 2 ffiffiffi 2pμ μ − 1 Q 1 pμ ð Þ þ λ  pffiffiffi where is the coupling constant. Also, in our subsequent 1 μ μ −4 4μ Q2 −14 2 μ − ð þ þ Þ analysis we shall choose the potential as [37] þ Q þ þ 4 2 2 8 Q C1 Q C1ðμ − 1Þ pffiffiffi  6 μ μ Q2 − 4 This definition of the partition function arises from evaluating − − ð Þ 2 4 2 ; ð91Þ the path integral for field configurations close to the classical field 2C1Q Q C1 ϕcl which satisfies the classical equations of motion. If S½ϕ is the corresponding action, then the path integral is dominated by where we have used the relation (85). fields ϕ ≈ ϕcl þ δϕ. Expanding the action for such fields we obtain 5 Notice that (90) is cutoff independent. Therefore this adds a δ ϕ 1 δ2 ϕ S½ S½ 2 finite contribution to the free energy. However, this boundary S½ϕ þ δϕ ≃ S½ϕ þ δϕ þ δϕ þ: → 0 cl cl δϕ 2 δϕ2 term (90) does not have a smooth Q limit and is therefore ϕcl ϕcl valid only at finite and nonzero Q. Also note that in the limit → 0 Q one can still havepffiffiffi a finite contribution providedpffiffiffi the Considering the variation of the fields at the boundary vanishes temperature is also low ( μ ≪ 1) such that the ratio μ=Q2 is and neglecting the higher-order terms, the on-shell action is small but finite. approximately given by S½ϕcl.

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0.0002 0.0004 0.0006 0.0008 0.0010 0.0001 0.0002 0.0003 0.0004 0.0005

–2 –2

–4 –4 –6 Q = 0.01 –6 Q = 0.007 –8 –8 –10

–12 –10

FIG. 1. Free energy of the black hole as a function of the temperature for different values of charge Q. We have taken C1 ¼ 1, C2 ¼ 0 and C ¼ 2.

20000 50000

Q = 0.01 40000 Q = 0.007 15000

30000 10000 20000

5000 10000

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 ffiffiffi ΔFffiffi p FIG. 2. Thermodynamic entropy S ¼ − Δpμ of the black hole as a function of temperature μ for different values of charge and C1 ¼ 1, C2 ¼ 0 and C ¼ 2.

Φ2 − Φ2 2 2 00 0 0 2 0 Vð Þ¼C A ;C;A¼ : ð94Þ Φ At − Atð2ω þðΦ Þ Þ¼0: ð96dÞ

The corresponding equations of motion are given by The Maxwell field tensor is the solution to (96d) and can   be expressed as λ 1 0 ∇ ∇ − □ Φ2 −Φ2 ρ − 2 ≡ 0 − 2ω Φ2 ¼ð μ ν gμν Þ þ 2 e FμρFν 4 F gμν Fzt AtðzÞ¼ Qe e : ð97Þ 1 Notice that in the absence of charge (Q) the equations − Φ2 2 gμνVð Þ; ð95aÞ of motion correspond to those of the Almheiri-Polchinski model [37]. This allows us to perform a perturbative λ 2 ∂VðΦÞ expansion of the dilaton as well as the metric as done 0 R e−Φ F2 − ; ¼ þ 4 ∂Φ2 ð95bÞ before and find the corresponding vacuum as well as the (charged) black hole solutions. pffiffiffiffiffiffi −Φ2 μν 0 ¼ ∂μð −ge F Þ: ð95cÞ A. Interpolating vacuum solution In the next step, using the definition of the metric (4) and In the following, we note down the OðQ2Þ equations of ≡ the light-cone coordinates x ðt zÞ along with the motion corresponding to the metric as well as the dilaton: ansatz (6) we rewrite the above equations of motion as λ Φ2 00 2ω 0 2ω 0 0 0 ¼ 2ω − 4e ð Þ ω 1 þ e ð Þ e ð Þ ; ð98Þ λ 2 ð1Þ ð Þ Φ2 00 2ω Φ2 −Φ −2ω 02 0 2 ð Þ þ e Vð Þþ2 e e At ¼ ; ð96aÞ 2 2ω 2 0 Φ 00 − 2 ð0Þ Φ ¼ð ð1ÞÞ e ð1Þ 2 00 0 2 0   ðΦ Þ − 2ω ðΦ Þ ¼ 0; ð96bÞ 2 2ω 2 λ Φ 4e ð0Þ ω 1 1 − Φ e ð0Þ ; 99 þ ð Þð ð0ÞÞþ8 ð Þ 2 ∂VðΦ Þ λ 2 2ω00 þ e2ω þ e−Φ e−2ωA02 ¼ 0; ð96cÞ ∂Φ2 2 t while Oð1Þ solutions are given by (59) and (60).

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The solution to (98) can be found as Finally, using (51b) we note down the metric (4):    C λ 1 5 1þ1=2z 2 2 2ω 2 2 ω 1 ðzÞ¼ − −2ze ð1þ2z−4z ÞþeEi ; ds ¼ e ð−dt þ dz Þ ð Þ z 24z 2z   dρ2 ð100Þ ¼ 4ðρ2 − μÞð1 þ 2Q2ωBHðρÞÞ −dt2 þ : ð1Þ 4ðρ2 − μÞ2 where the exponential integral function is given by ð106Þ Z ∞ −t e Clearly, in these coordinates the horizon of the EiðzÞ¼−P dt : ð101Þ pffiffiffi −z t black hole is located at ρ ¼ μ. On the other hand, the behavior of the metric (106) near the boundary ρ → ∞ may Finally, we express the metric (4) as be found as

2 2ω 2 2 2 ð0ÞðzÞ 1 2 ω − ds ¼ e ð þ Q ð1ÞðzÞÞð dt þ dz Þ: ð102Þ 1=Δ 8 2C 4 1 2ω e Q 6 e ðρÞ ≃ þ pffiffiffi þ þ ; ð107Þ OðΔÞ μΔ3 Δ2 OðΔÞ The behavior of the metric (4) in the IR (z → ∞)is ρ→1=Δ obtained as where we have set C7 ¼ 0 in order to obtain physical     2 1 1 1 boundary conditions in the asymptotic limit of the metric. 2ωðzÞ ≃− λ 2 1 λ 2 O e e Q þ 2 þ e Q þ 3 ; Referring to (107), we may conclude that the geometry is 3 z 4 z z→∞ asymptotically Lifshitz near the boundary of the space-time 3 2 ð103Þ with dynamical exponent zdyn ¼ = . This observation is similar to that observed in Sec. VB. which clearly indicates that the IR behavior of the geometry is AdS2. On the other hand, near the boundary (z ≈ 0) the metric (4) behaves as C. Free energy and phase stability In order to understand the underlying phase structure 2 2Q C5 1 1 associated with the black hole solution (106) we follow the 2ωðzÞ ≃ 1=2z O 1 e 3 þ 2 þ e O þ ð Þ; ð104Þ same line of analysis as in Sec. VC. The on-shell action for z≈0 z z ðzÞ the present model may be calculated as 3 Z whose leading-order term is ∼1=z , which is a signature of ffiffiffiffiffiffi − ðosÞ 2 p− an asymptotically Lifshitz space-time with dynamical Sed ¼ C d x g 3 2 scaling exponent zdyn ¼ = . Thus the geometry interpo- Z λ 2 pffiffiffiffiffiffi −Φ2 2 μν lates between AdS2 in the IR and Lifshitz2 in the UV. This ∂ − 1 Φ þ 2 d x μð ge ð þ ÞF AνÞ observation is similar in spirit to what we have found in the Z λ ffiffiffiffiffiffi earlier example. − 2 ∂ p− −Φ2 Φ2 μν 2 d x μð ge F ÞAν Z B. Black hole solution pffiffiffiffiffiffi − dt −γKΦ2: ð108Þ In order to obtain the charged black hole solution we recall the corresponding zeroth-order solutions (65a) and (65b) and substitute them into (99), which finally yields the first-order correction to the metric: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 5 10 15  pffiffiffi  ρ ρ μ þ ρ ω 1 ðρÞ¼C6 ffiffiffi þ C7 −1 þ ffiffiffi · log ffiffiffi ð Þ pμ pμ pμ − ρ –50 ffiffi 1−pμ λ ffiffiffi pffiffi e p ρþ μ − ffiffiffi ½−2 μe –100 8μpμ Q = 0.007 pffiffi pffiffiffi pffiffiffi þ ρe2 μð μ − 1ÞEiðρ − μÞ pffiffiffi pffiffiffi –150 þ ρð μ þ 1ÞEiðρ þ μÞ; ð105Þ

–200 where C6 and C7 are arbitrary integration constants and EiðzÞ is given in (101). Notice that in finding the above FIG. 3. Free energy of the black hole as a function of the solution we have used the change in coordinate (67). temperature for Q ¼ 0.007. We have taken C6 ¼ 1 and C7 ¼ 0.

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350

300 10 Q = 0.007 Q = 0.007 250 8 200

150 6 100

50 4

0.01 0.02 0.03 0.04 5 10 15

ΔF pffiffiffi FIG. 4. Thermodynamic entropy S ¼ − pffiffi of the black hole as a function of temperature μ. The left panel corresponds to the left pffiffiffi Δ μ half [ μ ∈ ð0; 0.04Þ] of the corresponding curve in Fig. 3 while the right panel shows the remaining right half of the same curve. Here Q ¼ 0.007 and C6 ¼ 1.

The Abelian 1-form in (108) may be replaced as this plot it is evident that there is no first-order phase pffiffiffi Z transition as the S − μ plot is continuous. We further 2ω Φ2 2 A ¼ −Q dz e ð0Þ e ð0Þ ð1 þ OðQ ÞÞ; ð109Þ observe that this behavior of entropy is consistent with that t of the Wald entropy of the corresponding black hole which can be easily computed using (51a) and (54) and is given by where the exponentials in the integrand stand for the zeroth-  pffiffi  1þ μ order solutions to the metric and the dilaton. ffiffiffi λe þ 4C6μ 4π 1 pμ 2 ffiffiffi If we now substitute (97) and (109) into (108), it is easy SW ¼ þ þ Q 4pμ : ð113Þ to check that the second and the third terms in the on-shell action (108) exactly cancel each other. On the other hand, pffiffiffi From Fig. 3 we notice two turning points, μ 1 and the difference in the volume terms may be schematically pffiffiffi pffiffiffi pffiffiffi ð Þ μ ( μ < μ ), at which the sign of the slope expressed as ð2Þ ð1Þ ð2Þ changes. This behavior is reflected in the corresponding ΔðVolÞ¼CððVolÞBH − ðVolÞvacÞ entropy plots in Fig. 4 inffiffiffi whichffiffiffi the entropyffiffiffi increasesffiffiffi  Int  pμ pμ pμ pμ pffiffi β asymptotically both for < ð1Þ and > ð2Þ.In − β D − Dð μÞ − D 0 pffiffiffi pffiffiffi pffiffiffi ¼ C 1 1 2 3 ; ð110Þ the window Δ μ ∼ μ 2 − μ 1 the entropy does not β1 ð Þ ð Þ change considerably. In this window the system falls in whose finite contribution may be expressed as the minimum entropy state whose plausible interpretation in terms of SYK degrees of freedom are discussed in the ffiffi pμ 1 pffiffi Dð Þ ffiffiffi 1þ μ 2λ 4μ 1 C 2 concluding remarks. 2 ¼ 2pμ ½e Q þ ð þ 6Q Þ: ð111Þ VII. CONCLUDING REMARKS Finally, using the definition (92) the corresponding free energy can be expressed as7 In the present paper, we have proposed models of charged solutions within the framework of 1 þ 1 DJT 1 pffiffi gravity. Based on our model computations, we have ΔF − 1þ μ 2λ 4μ 1 C 2 ¼ π ½e Q þ ð þ 6Q Þ; ð112Þ shown that in the presence of nontrivial couplings between the Uð1Þ gauge field and the dilaton the asymp- pffiffiffi where β1 ≡ 1=TH ¼ π= μ is the inverse Hawking temper- totic geometries get substantially modified both for the ature of the black hole, which is easily calculated by vacuum as well as the black hole solutions. In both the substituting the metric components of (106) in (72). examples, the vacuum solutions interpolate between We now plot the free energy (ΔF) of the black hole Lifshitz2 in the UV to AdS2 in the IR. On the other hand, pffiffiffi against the temperature TH ∼ μ (see Fig. 3). Furthermore the black hole solutions turn out to be asymptotically 3 2 in Fig. 4, we show the changes in entropy S ¼ S Lifshitz2 with zdyn ¼ = . pffiffiffi pffiffiffi W (¼ −ΔF=Δ μ) against the temperature TH ∼ μ. From We have further analyzed the stability of black holes in both the models and observed a universal feature in the free 7For the present model, the Gibbons-Hawking-York boundary energy and therefore the entropy of the system. In the first term in (108) does not provide any finite contribution to the free model we have considered the quadratic coupling and in the energy. second model the exponential coupling of the dilaton to the

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˜ gauge field. In both the cases, at sufficiently low temper- field Ψ. The gauge fields Aa in (A2) are known as the atures, we observed a turning point after which the Kaluza-Klein vectors. In the subsequent analysis, we free energy falls off sharply (Figs. 1 and 3) which choose to work with the ansatz (6). corresponds to an increase in the entropy below this In the next step, we wish to calculate the 2 þ 1 D Ricci temperature (Figs. 2 and 4). scalar Rð3Þ which is related to the 1 þ 1 D Ricci scalar The existence of minimal entropy at low temperatures Rð2Þ as could be interpreted as the formation of the Bose-Einstein– like condensate8 (BEC) in the dual SYK model which 1 Rð3Þ Rð2Þ − −2βΨ ˜ 2 2β ∂ ∂zΨ − 2β2 ∂ Ψ ∂zΨ possibly leads toward superfluid instabilities at low temper- ¼ 4 e Fμν þ ð z Þ ð z Þð Þ: atures and finite density [67]. We hope to clarify some of these issues from the perspective of the dual SYK physics ðA3Þ in the near future. We now find a relation between the determinants of the two metrics as ACKNOWLEDGMENTS qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi The work of A. L. is supported by the Chilean 3 2 −βΨ −gð Þ ¼ −gð Þe : ðA4Þ FONDECYT Project No. 3190021. D. R. is indebted to the authorities of IIT Roorkee for their unconditional support If we now substitute (A3) and (A4) in (A1), the 2 1 D toward researches in basic sciences. Both the authors would þ action reduces to the following 1 1 D form: like to thank Hemant Rathi, Jitendra Pal and Dr. Arup þ Z qffiffiffiffiffiffiffiffiffiffi   Samanta for useful discussions. Special thanks to Jakob 1 Salzer for his valuable comments on the manuscript. 2 − ð2Þ −βΨ Rð2Þ χ Ψ − −2βΨ ˜ 2 S2D ¼ d x g e þ ð Þ 4 e Fμν ;

APPENDIX A: A NOTE ON DIMENSIONAL ðA5Þ REDUCTION pffiffiffiffiffiffiffi where we set β ¼ γ=2 and γ ¼ 2. In this section, we propose a dimensional reduction Ψ → Ψ β −1 1 1 If we redefine log and use ¼ in (A5) we procedure in order to show that our þ D dilaton gravity recover an action which is similar in spirit to the action (2) models (2) and (93) are indeed effective models of a higher- corresponding to model I. Notice that, in order to obtain the dimensional gravity theory. desired form of the potential, one must set χ ¼ðA − C=ΨÞ, Let us consider the following 2 1 D Einstein-dilaton þ where A plays the role of in the gravity: original 2 þ 1 D gravity model (A1). On the other hand, in Z qffiffiffiffiffiffiffiffiffiffi the limit Ψ ≪ 1 we obtain an action similar to (93) which 3 − ð3Þ Rð3Þ χ Ψ γ ∂ Ψ ∂zΨ S3D ¼ d x g ð þ ð Þþ ð z Þð ÞÞ; ðA1Þ corresponds to that of model II. In this case we set β ¼ −1 and χ ≈−C þðC þ AÞΨ. where Rð3Þ is the three-dimensional Ricci scalar and χðΨÞ is the dilaton potential which includes the cosmological APPENDIX B: BLACK HOLE SOLUTION constant as we see below. Here Ψ ≡ Φ2 of the original WITH A < 0 analysis. In order to obtain a 1 þ 1 D effective action, we dimen- In this Appendix, we discuss the metric solution corre- sionally reduce (A1) along the compact direction θ: sponding to our first model (2) while considering the linear dilaton potential as UðΦ2Þ¼C þ AΦ2. In order to simplify 2 2 −2βΨ θ ˜ a 2 2 dsð3Þ ¼ dsð2Þ þ e ðd þ Aadx Þ ; ðA2Þ the calculations we choose A; C ¼ . Using the perturbation expansion (51a) and (51b),we 2 2ω 1 2 write down the equation of motion (7c) up to leading order where dsð2Þ is the usual AdS2 metric (4) with e ¼ =z . Notice that in (A2) the indices a, b run over the in the expansion as θ ∼ uncompactified directions. Also, we have identified 0 2ω O ∶ 0 2ω00 2 ð0ÞðzÞ θ þ 2π and assumed an Uð1Þ symmetry for the dilaton ðQ Þ ¼ ð0ÞðzÞþ e ; ðB1Þ

2ω 0 ðzÞ 8 2 2ω e ð Þ Notice that the classical solutions in the JT gravity correspond O ∶0 2ω00 4 ð0ÞðzÞω − ðQ Þ ¼ ð1ÞðzÞþ e ð1ÞðzÞ 2 3 : ðB2Þ to large N dynamics in the dual SYK model. Therefore, one ðΦ 0 Þ should interpret the charged condensate as a classical (large N) ð Þ analog of the BEC-like phenomena [62–66] in the dual SYK picture where quantum fluctuations are suppressed because of Similarly, substituting (51a) and (51b) in the dilaton 1=N corrections. equation of motion (7a) we obtain

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0 2 2ω 0 2 4μ OðQ Þ∶ 0 ¼ðΦ Þ00 þ 2e ð Þ ð1 þ Φ Þ; ðB3Þ 2ω ð0Þ ð0Þ e ð0Þ ¼ pffiffiffi ; ðB5Þ cosh2 2 μz 2 2 2ω 2 O Q ∶ 0 Φ 00 2e ð0Þ Φ ð Þ ¼ð ð1ÞÞ þ ð1Þ ffiffiffi ffiffiffi   2 p p ðΦ 0 Þ¼−ð1 þ μ tanh 2 μzÞ: ðB6Þ 2ω 2 1 2 −2 ð Þ 4e ð0Þ ω 1 1 Φ Φ : þ ð Þð þ ð0ÞÞþ8 ð ð0ÞÞ Notice that, in obtaining the solution (B6), we have set ðB4Þ ¯ ˜ 1 C ¼ −1 and C ¼ − 2 in (54). The zeroth-order equations (B1) and (B3) have the Now, using (51a) and (51b) in (B2), the leading-order ωBH following solutions: solution ð1Þ can be expressed as

   ρC ρ ρ 1 BH 8 −1 ω ðρÞ¼ pffiffiffi þ C9 −1 þ pffiffiffi tanh pffiffiffi þ ð1Þ μ μ μ 8μ − 12μ3=2ð1 þ ρÞ pffiffiffi × f−2 μðμ − 1Þð−1 þ μ − ρÞþρð1 þ ρÞ½4μ3=2 · logð1 þ ρÞ pffiffiffi pffiffiffi þð−1 þ 3μ − 2μ3=2Þ logðρ − μÞþð1 − 3μ − 2μ3=2Þ logðρ þ μÞg; ðB7Þ

where C8 and C9 are constants of integration. In writing the position of the boundary of the space-time (B9) is not (B7) we have made the following change in the spatial quite apparent. coordinate: The Hawking temperature corresponding to the above   black hole (B9) can be written down using the formula (72) 1 ρ → ffiffiffi −1 ffiffiffi and is given by z 2pμ tanh pμ : ðB8Þ ffiffiffi pμ Finally, using (51b) and (B5), the metric (4) correspond- TH ¼ 2π : ðB10Þ ing to the black hole can be written as   2 The corresponding Wald entropy can be expressed as 2 2 2 2 dρ ds ¼ 4ðμ − ρ Þð1 þ 2Q ω 1 ðρÞÞ −dt þ : ð Þ 4ðμ − ρ2Þ2   pffiffiffi 2 1 þ 4C8μjμ − 1j ðB9Þ S ¼ 4π 1 þ μ þ Q pffiffiffi : ðB11Þ W 4 μjμ − 1j Clearly, the horizon of the black hole (B9) is located at pffiffiffi ρ ¼ μ. However, from the structure of the solution (B5) Notice that, in writing (B11), we have used (51a) and (54).

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