Models of Phase Stability in Jackiw-Teitelboim Gravity
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PHYSICAL REVIEW D 100, 124061 (2019) Models of phase stability in Jackiw-Teitelboim gravity 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 dilaton and the Abelian 1-form where we analyze the asymptotic structure as well as the phase stability corresponding to charged black hole 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 xÆ ¼ t Æ 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 Æ Æ Æ Æ 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- Æ 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 124061-2 MODELS OF PHASE STABILITY IN JACKIW-TEITELBOIM … PHYS. REV. D 100, 124061 (2019) ∂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).