JHEP08(2017)134 Springer July 13, 2017 May 27, 2017 : : August 14, 2017 August 29, 2017 ment scenario : : e, Spacetime Sin- Revised Received to find a string theory Accepted Published Ontario, ntario, Published for SISSA by https://doi.org/10.1007/JHEP08(2017)134 ] (BBL). We argue that the singularity persists in a super- 1 . 3 1705.08560 The Authors. c Black Holes in String Theory, Gauge-gravity correspondenc

We study the effects of supersymmetry on singularity develop , [email protected] Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada E-mail: Department of Applied Mathematics,London, University Ontario of N6A Western 5B7, O Canada Department of Physics andLondon, Astronomy, Ontario University N6A of 5B7, Western Canada gularities ArXiv ePrint: symmetric extension of theembedding of BBL the model. singularity mechanism. The challenge remains Keywords: Open Access Article funded by SCOAP in holography presented in [ Abstract: Alex Buchel holography Singularity development and supersymmetry in JHEP08(2017)134 4 6 7 ]. 1 3 23 24 10 12 14 17 18 20 3 23 13 22 , 2 . Bulk tenuate ], while their 4 ymmetry to shear and bulk stable horizons 2 uge theory [ they determine the convergence odes ng, non-hydrodynamic modes ns. ctive boundary hydrodynamics g theory correspondence [ e — a slightly perturbed horizon ibe boundary gauge theory states ]. 7 um configuration. If the initial per- , 6 on [ – 1 – — perturbed bulk horizon relaxes via quasinor- 0. 1 < ) ω ]. We call horizons that relax via hydrodynamic modes that at breaking fluctuations ]. Typically, dissipative effects in the hydrodynamics (due 12 5 – 8 3.2.2 Unstable sBBL dynamics 3.2.1 Dynamics of symmetric sBBL sector and its linearized s 2.3.1 QNMs2.3.2 and linearized dynamics of Fully nonlinear DG-B evolution model of DG-B model Even when not excited, non-hydrodynamic modes are important as I.e., the quasinormal modes with Im( A.2 sBBL model A.1 DG-B model 3.1 Phase diagram3.2 and QNMs Dynamics of sBBL horizons 2.1 Grand canonical2.2 ensemble (a review) Microcanonical2.3 ensemble Dynamics of DG-B model 1 2 in space-time domain orholographic via dual positively with (dynamically) gapped stable quasinormal horizonswhich m descr are stable with respect to sufficiently small fluctuatio viscosities) lead to an equilibrationin of a a dual gravitational gauge description theory settlesturbation stat to away an from equilibri thermalparticipate equilibrium in is sufficiently the stro equilibrationmal process modes [ Static horizons are duallong-wavelength near-equilibrium to dynamics thermal encode states theof of effe the theory the [ boundary ga properties of the all-order effective hydrodynamic descripti Horizons are ubiquitous in holographic gauge theory/strin 1 Introduction 4 Conclusions A Numerical setup 3 Supersymmetric extension of BBL model Contents 1 Introduction 2 DG model in JHEP08(2017)134 , ]. ]. h on 17 13 16 , 15 canonical vely gapped non- ] also diverges within us, one might wonder curs in a constrained y breaking a discrete 20 ty is discussed in [ this unstable mode. ] (BBL): both the pres- , inhomogeneous phase of d , we or ) [ zon, associated with the calar potentials do occur 1 ] (DG) in grand canonical ess of the bulk scalar field ]). Additionally, the exotic 19 c framework matches both energy density (or temper- el realizing the dynamics of 21 ase transition towards a new mmetry breaking instability when a hydrodynamic mode 21 t as it precludes evolution of [ Moreover, it was argued that rge curvatures in the vicinity ry [ ly unstable mode. table equilibrium state for the phic horizons was identified: r in a finite time with respect sion that the system evolves to here is more to the story. In [ ture, is robust. us symmetry breaking in a mean- inhomogeneous equilibrium states ] occurs in real string theory holographic 18 , 17 – 2 – , 1 ] was extensively studied in [ 18 , 17 ] that there is no weak cosmic censorship conjecture violati 1 we present a supersymmetric extension of the BBL model, whic ] was identified in supergravity model [ 3 17 the equilibrium properties and dynamics of DG model in micro 2 Alternatively, a horizon can be de-stabilized when a positi ]. In section 3 16 ] for a recent discussion. , 14 15 while there is aature), linearized instability triggered below by somesymmetry, a critical non-hydrodynamic mode spontaneousl there is no candidate equilibrium state with a condensate of Above classification of (un)stable horizons in a holographi Unfortunately, construction of a top-down holographic mod The BBL model is not a top-down holographic construction. Th Translationary invariant horizon can suffer an instability See [ • • 3 ] a new instability of the translationary invariant hologra the gravitational and the field theory intuition. However, t BBL model remains open: whilestudy the in DG section model is “exotic” in gran once the potential isin bounded. supersymmetric On top-down holographic the modelscritical other phenomena (as hand, of e.g., [ unbounded in s ensemble. In this paper we partly address above questions. a singularity, violating the weak cosmic censorship conjec whether the phenomenon discovered inexamples. [ A particular aspectpotential. of It the was argued model in is [ the unboundedn to the boundary theory.non-equilibrium density The of area thea boundary density gauge finite of theo boundary the time. apparentinstabilities towards hori This any latter finite entropy observationas density is spatially well significan — inphase space other (spatial words, homogeneity despite and isotropy), the the fact conclu that BBL dynamics oc ence of the linearizedevolution instability below and the the absence criticalitythe of was gravitational the confirmed system sui dynamically. evolvesof to the a horizon, region asymptotically of turning arbitrary such la region singula field approximation — the system undergoesstable a phase second-order with ph a finite density condensate of the original In this case,the the system. expected end-pointhydrodynamic of mode the becomes evolution unstableThis is when instability a one realizes a lowers new holographic energy dual ( of a spontaneo 18 The dynamics of the model [ ensemble and show that itas in realizes [ a standard spontaneous sy in a system becomes unstable. An example of such an instabili JHEP08(2017)134 i etry (2.2) (2.5) (2.6) (2.1) (2.7) (2.3) µν spect of the  ons of 1 c F µν 2 V,   1 0 ]) is: − F ¯ A 21 µν +  + ]) is: 1 1 21 F µν central charge A  0 1 el are further highlighted ons ∆ of the CFT opera- ) ; (2.4) . ¯ µν , F γ neous and isotropic states σ  A 2 B 1 model in [ / , F µν A .  ions, with the effective actions − sinh 0 model in [ ¯ )) = 8 superconformal gauge theory 4 e of the symmetry breaking oper- 1 1 2 F + cosh DG φ ) = 1 2 L h − σ N − γ ) 2 2 γ O σ  ) 1 2 − = 3 V, A cosh , √ ∂γ d 4 ( − 1 2 cosh( φ i γ ) = ∆( 2 1 κ dx 2 192 4 1 2 4 γ µν − – 3 –  − 2 (2 + cosh(2 = O M 2 0 ]. This sBBL model does not have an equilibrium sinh 2 ) Z c equal charged scalars − ¯  F 1 2 18 0 (DG-A model) are − 1 = ¯ κ , ∂γ 2 A } equal charged scalars ( ) µν V 2 (2 cosh 17 ) = ∆(  = − 1 2 , σ 1 1 − ∂φ 2 γ 1 B ( F / − A O = , γ A 2 − 1 h ) − 2 γ ) V ∆( R γ { σ ∂σ , allowing for different patterns of spontaneous global symm 1 2 DG 2 ( 7 S S 4 1 SO(8) global symmetry within different consistent truncati = sinh sinh( B − ⊂ 1 1 2 2 − at a finite temperature and a finite with re R R 7 − − 1 2 DG S L = × A 4 − . Some technical details are delegated to appendix 4 DG is a four dimensional gravitational constant related to the AdS L dual to bulk scalars κ “DG-A” model Lagrangian (2 + 2 with with “DG-B” model Lagrangian (4 ] the authors studied equilibrium states of O For the maximally supersymmetric quantization the dimensi • • 21 = 11 supergravity on boundary gauge theory at the ultraviolet fixed point as and tors where 2 DG model inIn microcanonical [ ensemble state below the criticality,evolve towards and, asymptotically as divergent expectation in valu ator. BBL The model, similarities and itsin differences homoge section of the BBL and sBBL mod breaking. We consider the following two consistent truncat D exhibits the exotic phenomenon of [ dual to to a diagonal U(1) JHEP08(2017)134 ) } R 0 0 ¯ ¯ q A , (2.9) (2.8) 1 (2.10) (2.14) (2.11) (2.15) (2.12) (2.13) q { = 1 for U(1) 1 µ , ,  ) 2 2 ) below) and 4 a consistent truncation , − dx , ) r the bulk gauge field ) 3 ( 2.14 . 2 + librium states of the CFT − 2 O − 2 1 r γ r ( , , ( + ) . dx ) O 4 ↔ 3 2 O m r − 2 − 1 (1) + ). Imposing the supersymmetric r global symmetry; the bulk scalars r + r σ ( ( + 1 R + symmetry: r (1) r O q O =  σ 2.17 ) 2 i + r + + Z σ 2 (1) ( . 2 = φ . 1 0 g σ  r σ , γ ¯ q 2 µ dr hO 1 2 -symmetry neutral. r 2 (1) ≡ 2 = = R σ + ) = 1 2 2 1 0 1 → − φ + is γ ¯ a – 4 – , σ , 2 O 2 + 1(2) ) σ = γ 3 dt (1) ∆( i 1 − ) ], we set the chemical potential + 2 γ γ 1(2) r r ( γ ( , σ 21 = g 0 O dt , dt , a 2 ) + 1(1) i ¯ r ) ) A i γ ( + γ r r β  ( ( 2 1(1) − 0 1 2 hO (1) γ e r a a i r 1 → − 2  γ 2 r 0 = ¯ = 2 2 ¯ charge 1, while A (DG-B model) is = 0 1 r 1 U(1) charge densities, see ( − ¯ 2 i A A R φ γ × -even sector produces DG-B model with the identification = ). Following [ = = 1 + 2 2 4 R Z is the bulk gauge field dual to U(1) g β ds 1 2.10 ), A 2.7 have the U(1) is a standard asymptotic-AdS radial coordinate (see ( φ r Note that DG-A model is invariant under the = 0, i.e., spatially homogeneous and isotropic thermal equi and 0 i ¯ Its truncation to a 2.1 Grand canonical ensembleWe focus (a on review) DG-A model;as the explained construction for in the ( DG-B model is with the following asymptotic expansions at infinity: where quantization ( determine the U(1) and to the bulk scalar The background metric ansatz takes form we identify the expectation values of the dual operators γ symmetry, and for the other global U(1) (holographic dual to In both models are represented by a bulk geometry with the asymptotics: µ JHEP08(2017)134 , ) s . 2.13 (2.18) (2.19) ( (2.20) (2.17) (2.21) (2.16) (2.22) 4 . is 0 ¯ q B , − − h , the 5485552(5) r = . , . E 1 DG 0 µ ). Assuming that ¯ ρ , ··· = 0 ··· ] in grand canonical = c T 1 ) determine the ther- 2.17 = 0 , 17 ) + ) + = , h 0 h , the entropy density T n [ ¯ r r ρ

1 ··· T 2.16 q 3 1 y density − − E µ − he dual gravitational back- , ρ r r ) + ( , ( = h , r RN 1 (1) (1) h h ) and ( β σ , − ] with a radial coordinate const 2Ω = 0 , r = + + i − ( 22 } , we have asymptotic expansions in 1] 2.15 σ 1 , ,h h = (0) (0) h h , ρ (1) 0 r (0 hO β σ a ), ( T,µ 2 h  { ]: =

∈ 2 h = = = ¯ 2 1 πr r 21 r µ 0 2.12 4 , the temperature RN 5234403(5) ¯ : a . , P Ω = 2 } i 0 ), ( φ − – 5 – in the system separating phases with nonzero ¯ ρ 1 + ≥ = 0 , , x : c  . As its grand potential density Ω c 1 B c h 2.11 T T 3 ρ h r − RN black brane. r T , σ RN r { = , s 4 T i ≡ hO DG 2 ≥

≡ 2 / ··· 2 1 1 = 2 Ω γ x T AdS ρ µ (0) h E β , , β hO 0 there is a phase of the CFT with zero condensates ) + − h e = r ··· ··· i , (1) h , − 1 T > g γ 2 1 r ) + ) + ( µ ) 2 h h π h 12. hO h r r r 2 (1) i,h − √ γ is related to the energy density of the state, see ( − / πr − 2 h = 2.13 1 8 r r r + µ ( m ( U(1) charge density 12 ,h 1739106(2) ≥ . (0) (1) 1 i,h (1) × h g a γ = h r = 0 (1) = = = T 0 as i R 1 c 1 g m , T γ a > U T µ 2 , the grand potential density Ω = ) E − h where ensemble. and a grand potential density Ω There is a critical temperature At any temperature condensates ( A phase with i.e., model DG-B, exists at DG-B model realizes the exotic discovered i 1 2 r We use numerical shooting method developed in [ = This is just an electrically charged • • • 4 = − E r and the modynamic properties of an equilibrium CFT state, the energ a regular Schwarzschild horizon is located at P ( where parameter Parameters in asymptotic expansions ( to relate the boundaryground. and We the reproduce near-horizon the results asymptotics reported of in t [ JHEP08(2017)134 1 = T/µ del } RN (2.23) (2.24) (2.25) charge 0.20 Ω , R is B − A 0.19 − DG Ω DG , A g grand potential 3 1 − 0.18 , modification needed /µ DG , eous symmetry break- Ω Ω ) 6= 0 { 0.22 0.24 0.26 0.28 0.30 0.32 2 ------0.17 0 ); now we require . − ¯ ρ . 1 r in model DG-A (green curve) ( 1 1 ] in grand canonical ensem- , 2.11 of the spontaneous symmetry O ] to confirm that model DG-B T/µ 0.16 17 nism in both DG-A and DG-B ical evolution with fixed energy + , ssignment of dimensions of bulk oper- T/µ 21 sities Ω = 0 µ =const = 0 } 0.15 1 = ¯ i 6 σ 0 are adjusted to keep fixed U(1) T,µ ¯ a } { ] hinges upon the persistency of exotic

hO 0 1 ¯ 0.14 µ 1 , RN 1 ]. as a function of , Ω µ ) as a function of , } 16 { – 6 – T/µ 2 ≤ i , − 1 . As its grand potential density Ω γ A r c = 0 15 ( − T i 6 0.20 hO ), are presented in figure O 2 γ DG ≤ + Ω T hO 1 2.20 r q 0.19 = + i 6 1 1 γ i µ 1 0.18 γ (dashed blue curve) hO , = hO 1 ), as well as the vanishing charge for the remaining U(1) (in mo a 1 0.20 0.15 0.10 0.05 0.35 0.30 0.25 q 0.17 . ]. } 16 , σ (red curve) 0.16 i , , γ { 15 condensate in various phases of the CFT and the correspondin [ i . Left panel: expectation value 5 1 0.15 γ A phase with i.e., model DG-A, exists at DG-A model realized a standarding holographic in dual grand of a canonical spontan ensemble [ To study DG-A/B models in microcanonical ensemble, the only hO We verified that this remains true for supersymmetry breaking a • 5 0.14 (green curve) density (parameter { and model DG-B (red curve). Right panel: grand potential den where the “chemical potential” parameters models in microcanonical ensemble leadbreaking to standard picture is the change of the boundary conditions on gauge fields in ( In previous section we reproducedindeed some exhibits of the exotic resultsble. thermodynamics of [ in The the singularityphase spirit structure mechanism in of identified microcanonical [ ensemble, indensity i.e., and [ in charges. dynam Unfortunately, the instability mecha 2.2 Microcanonical ensemble densities close to criticality, see ( Figure 1 ators dual to JHEP08(2017)134 (2.26) (2.27) = 1. 1 µ 2 / 3 1 /ρ E ) with . The dashed blue 2 2.20 . Interestingly, it is he dominant phase of ure crit e. The latter curve ends ) that is realized in the . E 1.4 ) nd model DG-B (red curve) in model DG-A (green curve), < ensemble. Here we describe 4 / π E 1 ergy density. 1 nsemble ( E 2 he model. We confirm that the 3 , ion of the spontaneous symmetry s/ρ . ce is irrelevant in so far as we represent 2 4 / / — such a choice will put the phase 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 3 3 1.2 2 3 1 2 ( = = 1 q 448503(6) . − 1.0 = – 7 – = 1 1 extremal , ρ crit 6 RN

2 / E ) 3 1 0.8 1 ρ s ρ , 2 / E as a function of the energy density 1 3 ρ s 0.6 ( ), the symmetry unbroken phase becomes unstable at 2.20 0.4 . Entropy density Phase diagram in microcanonical ensemble is presented in fig Because we are discussing states in the CFT, the precise choi 6 at the extremal RN solution: curve represents symmetry unbroken phase — the RN black bran the data in dimensionless quantities. In agreement with ( transition numerically close to the one in grand canonical e dynamics of spatially homogeneous andcriticality isotropic in states the of model t isbreaking. a standard holographic realizat microcanonical description of the CFT below the2.3 critical en Dynamics ofIn DG-B model the previousthe section CFT we below argued the critical that energy DG-B density in model microcanonical describes t the exotic phase of the (model DG-B Figure 2 DG-A). For numerical analysis we set DG-B (red curve) and the RN black brane (dashed blue curve). Two symmetry broken phases, i.e., modeldominate DG-A (green entropically curve) a the symmetry unbroken phase for JHEP08(2017)134  2 ) 1 (2.30) (2.29) (2.28) (2.31) f + d 2 f − ) 2 2 f V. , f 1 ) + ′   − 2 f d ) ′ 2 f , 1 )) 2 1 f f in the effective action µν 2 f 1  , f  1 2 2 f − 1  f − + ( ′ 2 ) 2   F dx ) 2 2 ′ ′ 2 f 2 2 ′ 2 f f f + (1 + f ′ 1 − ′ : 2 + µν 2 1 f ) } f  d ( 2 2 + ( 1 1 , dx (ln Σ) (ln Σ) (1 + f − a + 2  2 1 ′ F 1 tions, we take φ, θ ) 2 2 f f 2  1 ′ f 1 { 2 f 2 ( 2 f f )  − 1 − + 2 f 4 1 2 ) + φ . , f (1 + ) ′ 2 2 2 2 f ′ ′ 1 d 1 1 )) ) 2 t, r + ( f f − f 2 f f 1 f 1 ) 1 − ( ( + f 2 1 f 2 2 2 , 2 f 1 2 1 ) t, r sinh 2 f 2  1 ′ + 1 ′ ′ 2 f ( f f f 2 1 ′ 2 1 f i f d f f f f , , 2 f (1 + iθ 2 in lieu of 1 2 A + + − ′ 2 ) + Σ( ) ) e (1 + − f f ′ ′ i 2 − f 2 2 ′ 1 − Σ 2 2 = 2 − ) f 1 + (1 + dt f ) f f f ≡ i 2 − – 8 – 2 2 2 − 2 ) ) + 2 ′ f 1 2 − + 1 ′ ∂θ f + + − 2 f 2 ′ 1 a 2 f if 1 + 2 2 2 (ln Σ) (ln Σ) 1 f 1 1 t, r ( ) f 1 ′ f φ f 1 ( f f ′ 1 1 f 1 4 ( 1 + f + 2 ′ 2 2 f (1 + a 1 2 2 A f (1 + ′ 2 ′ 2 ( 1 2  f ′ ′ f f 1 1  f 2 ( f f 2 − f f dt , f 2 − − 2 1 f (1 + (1 + ( ( + + sinh ′ ′ f f 1 2 ) 1 1 Σ 2 2 2 2 2 2 2 2 dr 1 1 f f + a a + + Σ f f f − ( A f f 2 1 d d 2 t, r 1 2 Σ 2 f f ( + + + f ) Σ f 3Σ + Σ 2 + 2 + 2 1 2 1 dt (ln Σ) 2 2 2 + + 1 1 1 Σ + Σ + f f Σ 1 + 1 + Σ a 1 ′ 1 ′ 2 ′ 1 − ∂φ f f f d d + + a a + + + a ′ ( Σ 2 1 + 2 2 1 d d = = 2 d d + + d f f ′ ′ 2 1 ′ − 2 1 1 2 4 − Σ Σ + 1 + 2 1 + 2 1 + f 1 1 f + 2 − f f A ): ′′ ′′ R ds ′′ 1 + ′ + Σ − − + + + − + 2 1 d A a d ′ + 2.5 Σ (ln Σ) = + Ad 0 = Σ 0 = 0 = 0 = 0 = B d 2 − − DG Σ + Σ L ′ + 2 + d d We begin with undoing the implicit bulk gauge symmetry fixing 0 = 0 = leading for the following equations of motion Assuming translational invariance along the spatial direc of DG-B model ( We further introduce a pair of scalar fields together with the constraint equations: JHEP08(2017)134 (2.34) (2.38) (2.37) (2.33) (2.35) (2.36) (2.32) , Σ A + 3) , 1 2 . Following the 2 , ) 1 f 2 ¨ f − 2.2 1 , r 1 + 2 ( , correctly reproduce f 2 1 O f , − . + ) 1 , ection 2 2 → ∞ ) 2 f ¨ t f . t ( 1 r + 2(2 → ∞ 1 , + } . 2 2 1 1 r . A face — which in our case is the , 2 1 1 f a , 1 . with the energy density and the ) f 2 rameters , as t ( 1 − f , ( } ) )Σ 1 , a 1 r 2 1 1 ) + 1 ) 2 2 , 2 , a t 1 a i, 1 1 , 1 − , f ( 2 ∼ f f 2 r ˙ − ˙ ) λ 1 ( f , a + t − ,A a 1 1 ( , = ) solution of the equations of motion, − = 0 2 O 2 ′ 1 1 + 4 2 , λ A f a 2 1 2 1 f 2 ′ { + , ˙ ) ρ − t 2 A ) at late times, + , f 2 ( , , f ) ) 1 t 1 1 ) ) 1 , t , → ∞ , − , ( ) become equivalent to energy and charge con- 3 3 2 ( 1 2 1 1 ˙ – 9 – 1 f )Σ + ( 1 − − f 2.38 f , i, , ρ r , f 1 ˙ r r 4 1 2 , f f , 1 f ( ( − + f 1 2.35 2 1 2 , + − d A 2 f r O O 1 ) 2 = 0 d ( + 2 2 1 t , , f 2 λ ˙ 2 + + ( 2 − E f ˙ , f ) ) ) = 1 , f , ) ) 2 t t 1 1 1 2 , t t ( ( = , − + 2 , − ) imply that only five of them are independent: 1 ( ( . The constraint equations are preserved by the evolution f 1 1 2 2 2 1 , ˙ r 1 1 r 2 2 ) + 2 i, E f , , r r f 1 ( f f 1 2 1 − 1 f , f , 2 2 2 f f O 2 1 2 2.32 f f λ , f A ∂ f + 2 f − { + + )) + as follows: + + 1 t ) , + d d + ( t ) ) ) 2 ( 1 2 t t ( + 2 t t 1 1 λ f i, , ( ∂ ( ( 2 λ 1 1 2 a f 1 r − , r r 1 1 , , , , f + 1 2 1 ˙ ≡ ˙ 2 1 ( + f f Σ r a ˙ f f λ + 2 1 2 r + , + 2 2 d d = = = ) − f = ( ′ 1 + 2 ′ 1 1 2 1 1 a , ] we used bulk gauge symmetry to require Σ a 1 A f a Σ = f 1 2 ˙ relate to + 2 and 23 a A Σ( 2 r + i, A ∂ 1 f 1 2 a 0 = ˙ 0 = 7 ≡ ′ + ′ − d The general asymptotic boundary ( Following [ 7 0 = The last two constraints in ( the equilibrium thermodynamics of DG-B model discussed in s As we will see, the boundary conditions ( provided given by equations provided they are satisfiedAdS at boundary. a given timelike sur servation, i.e., is characterized by seven (generically time-dependent) pa charge density as Restricting to static configurations, we identify Dynamically, the constraint equations ( where JHEP08(2017)134 ), 1 − 2.13 (2.42) (2.41) (2.40) (2.39) (2.44) (2.46) (2.43) (2.45) . uations , + 4) ) in time. symmetry 3 A.1 )) x R ( mx 2.43 g dg + + 4 t energy and charge 4 R er + 1) x 3 . ω 2 1 , q λx . ) RN black brane, which , ( + 4 + cos( x ) x 4 CFT d in appendix taneous U(1) 2 1 ) 1 λ x t q m tion, which in our case will q ( = 0 I . 4 ω 2 AdS e 2.42 ldf + 4  =1 + 4) − r ) ) ) 2 1 .

3 t t + x R ) q + 1) ( ( 2 ( + 1) + ω ) 1 1 1 x , , , f ) to find 2 t + mx 2 1 t, r ( λx ( f f 1 λ λx  ≡ , x ( 4( 2 Σ( ′ r 1 ) = 2.44 , + 1) f + m ))  ) d t x + t, x (4 ( ( + 16) λx O ( 2 x ( λ arctan 2 f ) ≡ t m  dg ( + 6) + 4 + ) , specifying the dual ) = 1 4 (ln , ) = r } d 1 – 10 – dt 1 , r λx − , f t, r , f 1 ) are employed to evolve such data ( → ) t, x = ( q )) λ Σ( r , a = 0 ˙ + 4 θ  x 1 t 2 r = ( + 6) + (4 + 2.31 ( A ∂ x g i i λ { 2 f φ →∞ ) + λ ldf r ( t = 0, we provide the bulk scalar profiles, ) takes form: hO + 4 2 t, r + = lim λ , a R ( t x 2 1 2 ω λ A A.2 ). Eqs. ( µ λ ( ) = 2 + sin( λ ) is identified with t ), ( ldfλx 2.36 ∂ t, x t + (4 ( ) and relation to the expectation value of the dual operator ( I 2 ω A.1 σ is the symmetry breaking QNM frequency, we linearized the eq 2.25 e 0 = 4 + 1)( dg I ) 2.29 , ), ( on the gravitational background ( x in ( − ) i ω ( λx i 2 x ( 1 f f ( + I µ 2.30 ω R − ldf dg 4 ) is the residual radial coordinate diffeomorphisms paramet ω t + − ( ≡ ′ ( is determined from the stationarity of the horizon ( = λ ) = ) x λ × ω ldf ( ′ = 1: t, x To initialize evolution at A symmetry unbroken phase of the boundary CFT is a Further details of the numerical implementation can be foun g ( r 1 0 = f in notations ( be which can adjusted to keep the apparent horizon at a fixed loca breaking and the linearized dynamics in DG-B model. Finally, Introducing for the scalars Additionally, 2.3.1 QNMs andWe linearized discuss dynamics here the of spectrum DG-B of model QNMs associated with the spon we identify where where densities according to ( field redefinition ( along with the constant values of JHEP08(2017)134 2 / 2 3 1 / 1 1 5 /ρ tρ – E 4 2 ) (2.47) (2.48) 25 1 + 1) − + 3) at critical 2.38 alar fields . 1 1.48 λx λx µ ( ). Left panel . Note that + 4) critical energy 20 2 . Figures 3 3 3 λ + 3 + 1)) 2.33 k brane horizon re 2 mx x + 8 2 λx 15 1.46 ) on RN black brane λ + 4 ( (2 . ldf 1 x 4 ) ), see ( t, x q )) 2 t ( x x ( i λ 2 ( 1 m 1 f , q + 2 1 10 O . f 0) following from ( + + 3 1.44 + must reproduce at late times crit ldf λ + (16 x E → R 2 1 2 (symmetric phase is stable); right alue ω q ). x 5 2 ( 1 2 + 1) + 4) / 2 1 1 crit x − 3 + 1) E 2.27 976328 1 λx /ρ . 1.42 , = ( I 1 mx + 4 λx x f ω ( ( 1 = 0 λ x q 01538 0.10 0.05 0.05 0.02 0.01 0.01 0.02 0.03 dg . 2 - - - - E , see ( / I 2 3 1 / ω + 4 1 1 crit 4 = 1 + 16) /ρ E , dg . The red dot (left panel) represents tρ dg – 11 – 3 E ) E − E < x + 6) + 4 ) )) ( λ E mx m 25 O symmetry breaking fluctuations at vanishing spatial mo- λx + + R boundary conditions ( + 3 + 4 λ 1.48 4 ldf x ) with QNM spectrum presented in figure , 2 2 1 20 − x + q )) 2 ( I 2 AdS λ 2.46 ω + 3) + (16 m ( x 2 1 2 x λx ldfλx + 2 15 1.46 2 = + 4 λ x ); the vertical red dashed line (right panel) represents the λ 2 1 + 3 q ldf 2 x 2.20 + (4 + 1))( + 2 10 + 16) λ ( 1.44 λx ). 3 ( x 2 λmx R ldfdg ], we verify our dynamical code for DG-B by “turning off” the sc λ mx 2.27 ω 5 2 1 (4 / ) are solved subject to regularity at the location of the blac x 1 1 + 2 . Linearized dynamics of the symmetry breaking scalars . The spectrum of the U(1) ((4 ′ 1 /ρ 1.42 , + 2 + (16 + (16 × 2.47 dg 1 R 0, signalling the instability, once f ω As in [ The spectrum of the symmetry breaking QNM is presented in figu > = 1), and the asymptotic 0.10 0.06 0.08 0.02 0.02 0.04 0.04 - - 0 = 1.386 1.380 1.382 1.384 I x Eqs. ( panel describes spontaneous symmetry breaking at ω backreaction on the geometry.the Scalar quasinormal evolution behaviour in ( this case ( corresponds to dynamics at energy density density, see ( energy density, see ( Figure 4 Figure 3 mentum as a function of the energy density background is captured by the evolution of the expectation v JHEP08(2017)134 2 2 / / 1 1 1 1 6 tρ tρ crit E nding (2.50) (2.49) < 25 E 80 ). 20 eous symmetry (left panels — . The late time determined from 2.46 1 , φ 2 2 crit 60 f O E ) we expect + 15 > 1 , 2 1 E f 2.46 40 q ponds to dynamics at energy for 10 gy densities we find 2 = and the linearized dynamics roach to equilibrium is rather ) ] this is expected in the sector t i ( lose to transition. φ 12 1 , 2.2 20 2 5 hO . f 7 i y breaking operator i + − φ φ 2 . ) 10 t t hO I hO ( ω . 1 , 0.10 0.06 0.08 0.02 0.04 0.20 0.15 0.10 0.05 e

1 . 2 2 1 f / / symmetry is spontaneously broken at 1 1 1 1 2.2 − p R tρ tρ – 12 – i ∝ = φ i fit φ 25 hO I, ω I,QNM hO to compare with the QNM behaviour, see ( ω 80 the symmetric phase is the dominant one. As figure

fit I, 20 ω crit ) and at late times and comparing with the expected decay/growth , except for the evolution of E t 4 ≡ ( . The red dashed line represents the energy density correspo 5 (right panels — unstable case). From ( 1 60 I , > 1 ω crit indicate that U(1) 15 f E E crit (symmetric phase is stable); right panel describes spontan E 20 in this case ) — as emphasized in [ < 2.3.1 crit & 40 E 976328 E 10 . T = 0 01538 . E 20 5 = 1 . Same as in figure . Fully nonlinear evolution of DG-B model. Left panel corres i i E φ φ equilibration t hO hO 0.10 0.06 0.08 0.02 0.04 0.02 0.14 0.12 0.10 0.08 0.06 0.04 microcanonical analysis of the model in section asymptotic (equilibrium) expectation value of the symmetr breaking at dynamics allows to extract in the model, while at Figure 6 2.3.2 Fully nonlinearMicrocanonical evolution of analysis DG-B of model the DG-B model in section stable case ) and density discussed in section of the gauge theory responsible for the critical behaviour c from the earlier QNM computations at the corresponding ener present the evolution of presents, we find that thisslow is ( indeed the case. Notice that app Fitting the results in figure Figure 5 JHEP08(2017)134 ) ] in 2.7 (3.6) (3.4) (3.7) (3.5) (3.1) (3.3) (3.2) 16 , 15 . 2 χ 2 g φ 1 2 rsymmetry imposes d order EOMs derived + 2 symmetry variation of a χ . l operators (e.g., see ( , dual to a supersymmetric W  +  2 2 2 ∓W ial symmetry breaking [ φ . , dx 1 2 2 j = bosonic + Φ W − L ] to a “supersymmetric” model. As 2 1 µ 3 , dr 3 dA : even though DG-B model exhibits ∂ + 18 i dx − − , Φ R 2.2 j + − P µ = 1 2 17 W Φ ∂ 2 , , ∂  ∂ kin } 1 dt j i k BBL L γ ] in grand canonical ensemble, it represents − W Φ Φ W Φ P ∂ ∂ − ∂ ∂ = { 17 ) – 13 – r √ ij ij ( ij 4  A  K 1 2 1 , dx − e − of the scalar fields coupled to gravity is restricted to − 4 2 bosonic K ) + relates to the central charge of the boundary theory as = K L  M  2 κ Z ∂χ = kin ( ± dr 2 1 bosonic L κ 4 1 P = = L i 2 4 − = Φ 2 dr S ds ) d ] the bosonic part of the effective action takes form ∂φ ( 25 4 1 , − 24 = kin L ), and the Lagrangian from the Lagrangian; specific quantization for the scaling dimensions for the dua because flow equations for scalars are of the first order, supe for DG-A model). the first order RG flow equations are consistent with the secon In four-dimensional gauged supergravity holographically We confirm the conclusion reached in section Recall, for the BBL model we have: 2.2 • • boundary QFT [ Two comments are in order: Assuming the metric ansatz, in ( We would like to generalize BBLwe construction show, [ this is rather easy to achieve. an exotic thermodynamicsthe discovered well-known in holographic [ realizationmicrocanonical of ensemble. the spontaneous 3 Supersymmetric extension of BBL model supersymmetric RG flow equationsthe are fermions: obtained from the super where the gravitational constant and the potential is determined from the (real) superpotent where the kinetic term is JHEP08(2017)134 ]. ), 18 2.2 (3.9) (3.8) , (3.12) (3.13) (3.11) (3.10) parity ). The 0. The 17 φ 2 , 3.6 1 Z g < ow [ ) is a massive 2  , 4 3.1 2 φ χ 2 ]; as a result, numer- φ 1 )) , w equations ( + 531)  + 351 g 2 g , 36 ion directly obtained from 3.12 near coupling φ 0 2 ) with a central charge ( ies in bulk scalar potentials, 372 + articular, we can borrow the 4 χ ≥ , del, there is a single nonlinear − earity of the BBL model, but 1 r ]. For a supersymmetric gener- 24 ) are trivially unbounded from (see ( , 2 . , dual to the bulk scalar O g AdS  r 0 18 φ W 2 + 780 ) is closer to that of DG-B model , BBL O + 4 160 . 6= 0 explicitly breaks the P , n . 4 17 χ ], appropriate for the supersymmetric ) A.2 − φ + Λ χ ( n . 4 17 − 3 φ 6 104 < g < 2 CFT χ ↔ − + 540 χ n + 1071) 2 + φ ) = 1 φ φ r 4 g ( 77 2 ): 1 φ O – 14 – O √ 84 −→ H gχ 12960 (compare to 3 3.1 + − ∆( 160 104 2 − 520 CFT g + 6 BBL χ 2 H sBBL 21 + 27 ) is different from the one used in [ P χ 208 P 1 2 − , dual to the bulk scalar − = 45 r ( 3.12 1352 1 4 + O χ 2 − 2 φ sBBL 6 φ 4 1 116812800 P φ − 1 9 1600 16848 . Explicitly, ], the holographic dual to gravitational sBBL model ( = 1 + g BBL ). We present the results only — for detailed discussion foll 1 − − P W = 3.12 sBBL P are consistent with the corresponding supersymmetric RG flo symmetry of the effective action ( by a relevant operator QFT obtained as a deformation of the CFT (dual to with Λ being the deformation mass scale. Λ Similar to [ We explicitly verified that the second order equations of mot • sBBL latter is necessary for the exotic phase structure of [ Notice that the quantization ( below when holographic renormalization for theexpressions models for is the the thermodynamic same.quantization quantities ( In in p [ ical implementation of sBBLrather model than (see the original appendix BBL model. 3.1 Phase diagramSince and BBL QNMs and sBBL models differ by higher-order nonlinearit supersymmetry imposes the following quantization on alization of the model, we introduce the superpotential leading to Note that the potential is unbounded from below for the nonli L Notice that additional terms in i.e., the sBBL scalardiffers for potential higher captures order nonlinear thecoupling interactions. leading constant As nonlin in BBL mo JHEP08(2017)134 E 3 crit 2 3 E κ Λ Λ 8 and 2 2 2 (red g κ − 2 − 4 3 2 1 itical one = = 2.5 -symmetry 2 g g Z E 5 3 2 - κ Λ ve vanishing real 2 2.0 10 10 - = 0, of sBBL model as 1.5 e blue curve represents i i 15 hO - . 1.0 8 20 respect to linearized → ∞ - e the exotic phases at 0.5 3 ) f QNM in sBBL model that connect Λ ω / Λ nset of this instability as a function of E 25 - Re ( 1 7 6 5 4 3 2 6 . 30 - crit E E E 3 3 2 2 κ κ Λ Λ > 2 2 – 15 – E 10 4 -symmetric phase, i.e., with 2 2.5 symmetry breaking QNM in sBBL model at Z 2 Z 8 2.0 ) )) (right panel). black brane in the limit of the 2 4 Λ 2 π 3.10 (left panel). As the energy density is decreased below the cr 1.5 sym 6 s (2 AdS E / (see ( 1.0 sym g ) 4 s 2 2 Λ κ π 0.5 8 (green curve) at vanishing spatial momentum. These QNMs ha 0.5 3.0 2.5 2.0 1.5 1.0 ) (2 ω − / 2 Λ = Im ( sym g 0.5 0.5 1.0 1.5 2.0 2.5 s - - - - - . Phase diagram of sBBL model in microcanonical ensemble. Th . Left panel: spectrum of . Entropy density 2 κ 2.5 2.0 1.5 1.0 0.5 3.0 0.5 2 correspondingly, which exist only for - − , the symmetric phase becomes perturbatively unstable with = crit part. The blue curvesto (both standard panels) ∆ represent = 1 the QNM spectrum of o breaking fluctuations. The criticalthe nonlinear energy coupling density for the o curve) and E a function of energy density Figure 7 Figure 8 g Figure 9 the symmetric phase of the system; the green and red curves ar JHEP08(2017)134 . 2 7 8 φ } → 8 − − 3 − E Λ = = , (3.14) (3.16) (3.15) 2 g g {− is trivially = . g sBBL The blue curves 8 P d symmetry break- 10 ith results in figure s is the case for the l as the critical energy ished by the symmetry 44726(8) . = 0, and the symmetry densities in the symmet- holographic renormaliza- 2 (red curve) and tive imaginary part, when , a function of the coupling i i − 3 ], is the fact that they have = 0 d imaginary parts, typically with 8 = hO 18 hφ − g . − = g ) in sBBL model the (vacuum) energy 0. Additionally, the value √

as a function of the energy density ry. 3 3.6 = 0 > We discuss dynamics of sBBL model crit ambiguous. However, for the super- dx 3 E 9 sym 4 8) 2 Λ E in the holographic renormalization. s κ − M green 4 2 ∂ & = Z M ∂ g ∝ ( red

, ) – 16 – crit ), such term is not allowed, as it will violate the ω E 3.12 Re( ) = 2: a finite boundary counterterm r 0 and 42618(8) O . counter,finite 0 < (left panel). While this phase is thermodynamically stable L : the symmetric phase with ), where the gravitational scalar potential − = 0. 7 χ 2) i 6 = i − symmetry breaking quasinormal modes in sBBL model is pre- 3.11 2 2 hO − = ) is shown in the right panel. ↔ − as a function of energy density for Z = g g ), these modes must disappear from the spectrum in the limit χ ( 8

3.10 for two representative values of the coupling constant crit crit 3.16 3 E ]. E in ( 2 Λ 3.2 26 κ g with 0, it is perturbatively unstable with respect to a linearize 2 )[ ω > crit E Im( E is the induced metric on the cut-off surface , i.e., when sBBL approaches its UV (conformal) fixed point. 2 is presented in figure 2 sym < ∂ h ∂s ∼ − Because of ( (green curve) at vanishing spatial momentum. In agreement w scalar quantization with ∆( ∞ symmetric quantization, i.e., ( The unusual feature of these modes, first observed in [ Notice that Notice that the energy densitydensity) of can the be symmetric negative. phasetion (as Typically, make finite wel the counterterms definition in of the energy density ambiguous. Thi ing fluctuations. The criticalconstant energy for the instability as renders the definition of the energy density unbounded from below. vanishing real part, i.e., (right panel), these modes becomeE unstable i.e., have a posi energy conservation Ward identity. So,ric the phase negative of energy the sBBLin model are section unambiguous. property under There are two equilibrium phases of the sBBL model, distingu is within the range ( sented in figure The spectrum of broken phase with The entropy density of the symmetric phase E − ) Recall that QNMs of a holographic dual to a CFT have both real an √ It is easy to verify that for the supersymmetric RG flows ( • • • ω 8 9 10 Re( density vanishes, as required by the boundary CFT supersymmet JHEP08(2017)134 > en E (3.17) tion of ly large odels in ain results and , symmetry break-  2 . Once again, these se (blue curve) and 2 reaking fluctuations Z 1 9 r y than the symmetry ; (3.18) Λ and the expectation  i ) , t O 11 ), the general asymptotic ( r  + , figure 3 increases sufficiently far over 1 symmetric equilibrium phase r r µ hO 3.12 of the instability towards crit 2 E ; there is no end-point for the 3  ale in the boundary QFT. Of course, E / Z , 1 ) + O 3 crit t > − ( E + ˙ E λ ows we measure all dimensionful quantities 2 , , and three dynamical variables 1 r − QF T } ) = 2 2  t )) ) ( , µ 2 1, and adjusted its precise value for convenience. t t 1 1 2 r ( ( p 1 ∼ λ  { p 2 1 ) t O is modified, see ( . Notice that as ( + – 17 – 3 λ , p φ + 2 2 of the dual ) r 1 − t Λ r ( ) 2 1 CFT . t 3 ) O ( p / t 1 2 ( 1 8  p 1 − 5 1 p r − + ˙ 8 1 r  2 = 2 p ) − , there is a ”level crossing” (left panel), and the red and gre 2 O t ) p ( t crit + ( λ + ( E λ ) ) + t t 4 ( ] for further implementation details. Because the quantiza ( r + 2 r 1 4 1 2 r q r p ) solution of the equations of motion is now given by . These parameters have the following interpretation: = = = dual to the bulk scalar ) are identified with the deformation mass scale represent the sBBL QNM mode, which connects at asymptotical } t ) Σ = A φ χ ( r 8 we established that the equilibrium physics of BBL and sBBL m t 1 ( O p → ∞ , λ r 3.1 ) t ( and 4 2 , q p value of the relevant operator new phases have lowerthus entropy never density dominate than the the dynamics symmetric of pha the system. energies to ∆ = 1 QNM of the in figure the corresponding curve QNMs cease to dominatein the the relaxation system of — the symmetry it b is governed by the blue curve QNM. As in BBL model, in sBBL model new phases with spontaneous ing appear in the microcanonical ensemble once ) We now study the dynamics of the sBBL model. We highlight the m t . These new phases are exotic — they have lower entropy densit The precise value of Λ is not important — it sets the overall sc • • ( 1 11 crit p { boundary ( the operator microcanonical ensemble is identical: in both models the care should be takenwith to respect keep to Λ time-independent. Λ.We verified In In that what numerical the foll evolution results we are used independent Λ of our Λ-choice. It is characterized by two constants instability and the ‘hairy’ phases bifurcate from the onset In section 3.2 Dynamics of sBBL horizons E becomes unstable below some critical energy density preserving phase and thus never dominate dynamically. refer the reader to [ JHEP08(2017)134 , 3 Λ t (3.22) (3.21) (3.23) QF T 30 ) and the try break- fact that the 3.18 as follows 25 3 tion, which in our . 20 , QF T of the dual er ] are collected in ap- denoted by a dashed red i 1  e = 0 r O 4 1 | 15 O r Λ =1  r /

mass scale Λ ( i , identified with the expecta- identically vanishes) is shown ) e r O χ 3 10 χ t, r ) = − O Σ( ; (3.19) r ; (3.20) ng-down of the expectation value to its QF T , r 5 , i + ) ) d of the boundary µ t t 3 |hO ( 4 ( = 0 i E Λ λ ≡ − t ln ( 5 hO 15 10 20 - + - - - ) = r E t, r ) = 3 Λ 2 t → t ( κ Λ – 18 – Σ( , χ 4 r 2 35  q r  ∂ r 1 ) 30  symmetric sector of sBBL model. The expectation value of ) settles to its equilibrium value ). φ t, r 2 O ( , specifying the dual Z } = 0, we provide the bulk scalar profiles, A 25 3.20 t ( ) = , µ + 2 E t , r p ∂ 20 { -symmetry breaking irrelevant operator  2 = 0 Z t ( 15 φ = 1: r 10 Λ . An important check on the consistency of the evolution is the 5 / . i . Typical relaxation in ) is the normalizable coefficient of the bulk scalar r 10 ) is the residual radial coordinate diffeomorphisms paramet t ing fluctuations t is related to the conserved energy density ( A.2 ( 4 hO which can adjusted tocase keep will the be apparent horizon at a fixed loca tion value of the q λ µ 1.5 1.0 0.5 To initialize evolution at Some details for the modification of the numerical code of [ • • • operator (dual to the bulk scalar r Figure 10 along with the values of equilibrium value. line (left panel). Right panel shows a characteristic QNM ri 3.2.1 Dynamics of symmetric sBBL sector and its linearized symme initial state energy density pendix O Typical evolution in the symmetric sector (the bulk scalar in figure JHEP08(2017)134 10 crit E Λ Λ t t (3.25) (3.24) < . From value of E ue at the 30 0. In fact, 3.1 for ), computed ≥ i i 25 crit 30 E hO sym s 20 ynamics of strongly value ) ]. Because a derivation 15 20 42978(1) . 27 , ectly used in the numerical sym s =1 -equilibrium entropy density 10 t of parameters as in figure mputed in section = 0 r ng [ (Λ egative, i.e., ˙

| / 2 E 4 ) ) 10 Λ 5 χ . / he entropy production rate is always i + sym i ˙ d s =1 r δ 8 and

( sBBL |hO 2 + ( − symmetric sector of sBBL model. The red P ) 2 2 ln ) 2 0.00002 0.00002 0.00004 5 0.00004 5 = - - - − t, r φ Z g + Σ( d Λ Λ ( t t 2 – 19 – π ′ κ 2 35  2 35 Σ ). ) = 30 ). The right panel presents numerical error in computation t ( 2 s π 3.26 30 κ 2 3.25 25 25 ) = . t ( 20 ˙ s 3.1 20 evolves at asymptotically late times to its equilibrium val (right panel). r 15 ) . The red dashed line (left panel) indicates the equilibrium O crit 2 15 E 11 Λ | π 4 > 10 10 Λ (2 E / / i i 5 5 sym . Evolution of the entropy density in . Linearized dynamics of the symmetry breaking expectation s |hO 2 ) involves gravitational bulk constraint equations not dir κ ln 5 15 10 20 - - - 0.35 0.30 0.25 0.45 - 0.40 One of the advantages of the holographic formulation of the d 3.25 ), associated with the area density of the apparent horizon, t ( which can be analytically proven to be non-negative followi corresponding energy density (here we take of ( the entropy density atthe the plot corresponding it is energy clear density, that co the entropy production rate is non-n of the entropy production rate, see ( The evolution of the entropy density with time for the same se expectation value of is shown in figure using the gravitational bulk equations of motion we find independently in section dashed line indicatespositive the during equilibrium the value evolution, (left see ( panel). T (left panel) and Figure 12 Figure 11 interactive gauge theories iss a natural definition of the non JHEP08(2017)134 8 as − and Λ = t s (3.27) (3.28) (3.26) crit g E Once the 20 and presents the collect results 12 crit 14 E – 15 13 ) exponentially with . . crit =1 E r with the QNM spectrum 42978(1)

= const = 24). Different 10 etry breaking dynamics of . , the entropy density  same initial conditions, but 4 r 2 ) crit crit discussed above. O = 0 E E χ AdS 0. Figures , k on the code is the vanishing of , after a brief non-linear regime, i + breaking fluctuations decay (left K E > < 42978(1) , d > . ). O 5 2 crit E E operators in sBBL model with sBBL E Z Λ + ( crit =1) r P / = 0 3.26 2 E > 2 i O t,r ) r ( 8 and E (recall − E φ

, , − + 5 6 4 hO 5 4 3 2 1 and d − − abcd ( i = AdS 8 in the bulk scalar potential, we study next 10 10 R ′ O Λ  g − K t ( 2 – 20 – . Different color coding represents different spatial abcd Σ = . R crit

20 g E 1 − 10, the linearize = 2 − < h & symmetry breaking fluctuations. Figure Σ K 2 E d Λ dt fit t Z ) 15  QNM ) ω evaluated at the apparent horizon, monitors the quantity ( , we now present fully nonlinear dynamical results. We focus 2 ω π h κ 2 ) or grow (right panel: 11 Im( K Im( 3.2.1 : ≡ crit

8 10 ˙ s Kretschmann scalar E δ 4 without the backreaction on the symmetric sector dynamics. | 4 i i AdS Λ 5 7191(2) / . i hO i . Evolution of expectation values of = 1 |hO ). Recall that for this value of the coupling E ln 5 5 - Taking the value of the coupling We consider first the evolution with 3.15 evolution code, an important (dynamical) consistency chec resolutions of the numerical runs. time. Using thepredictions, linear see fit, figure we compare the decay/growth rates color coding on the plots represents numerical runs from the the Kretschmann scalar 3.2.2 Unstable sBBLHaving dynamics reproduced the phasesBBL diagram model and the in linearized section symm relative to the in the unstable regime, i.e., with panel: The right panel in figure Figure 13 symmetric sector equilibrates, for the evolution of the expectation values of operators on the unstable case only, as for simulations with evolution of the system evolves to symmetric equilibrium configurations in ( the linearized dynamics of the JHEP08(2017)134 i 2 O − ter Λ Λ = t t lution. 2 and (3.29) g as well. − 50 h 20 = K evaluated at g h K 40 0) we do not see 15 > 30 crit E ]. 29 . 10 8 ( n expectation values of ). Here we have a stronger emain finite. − 20 el, instability in sBBL model etschmann scalar ocation points: along with the more dramatic apparent horizon (the entropy tions of the numerical runs. = 3.15 h 4 g 8 in the unstable regime, i.e., with 5 K − 10 AdS operators in sBBL model with Λ ement as in [ = / r i g /K r O h hO blue dashed red green dotted K 5 25 20 15 10 1.8 1.6 1.4 1.2 and and the Kretschmann scalar of the code is needed to answer conclusively i s O Λ Λ 12 , , , t t – 21 – . Different color coding represents different spatial 20 40 60 50 crit        20 E = < 40 E ] might be more suitable to capture high-gradients in the evo 15 28 ], the numerical code always crashes, albeit at a slightly la 1 collocation 30 N 10 20 collect the same data, but for the sBBL coupling constant with the negative critical energy as in ( | ) 4 2 16 Λ crit – 5 Λ / E i π 10 i 15 (2 . Evolution of the entropy density . Evolution of expectation values of |hO s/ 2 . Different color coding represents different spatial resolu . However, contrary to BBL, in sBBL model with ln κ 0454(8) r 8 2 4 6 . 10 - - - - crit - 0.6 0.4 0.2 0.8 O Figures E A finite-difference code as in [ = 1 < 12 E a clear signature of thedensity); divergence moreover, of there the area is density no of obvious the divergence in the Kr resolutions of the numerical runs. with different spatial relation — the different number of coll As for the BBL model in [ and E indication for the divergence of the Kretschmann scalar Even more leverage could be achieved with adaptive mesh refin Clearly, a better (different) implementation whether the entropy density and/or the Kretschmann scalar r time with increasing spatialpersists resolution. in Similar a to nonlinear BBL regime mod — we do not see the saturation i the apparent horizon in dynamics of sBBL model with in the unstable regime, i.e., with Figure 14 Figure 15 JHEP08(2017)134 ] 8, 21 − Λ ] vio- t = 17 g 50 tial trans- evaluated at rium ground h 40 K ]. 30 introduced in [ 16 t another realization of , ction? at the singularity mecha- lopment? fy BBL model to mimic the 15 o a configuration with both etschmann scalar evaluated 20 ere is no run-away instability lly homogeneous and isotropic tions of the numerical runs. time dynamics. Following the . 4 onal dual describes dynamics of 2 in the unstable regime, i.e., with in gauged supergravity consistent crit − AdS 10 E nding relativistic fluid mechanics. = /K g h K 500 ]. 3500 3000 2500 2000 1500 1000 and the Kretschmann scalar ] that consistent truncations studied in [ 30 s 1 2 is physically different from that at Λ t − – 22 – ] in grand canonical ensemble. We studied here An arbitrary weakly curved initial gravitational = 17 50 g 13 . Better implementation of the numerics could answer 40 s ], the BBL holographic model points to a development of the 5 30 20 , any finite entropy density state with broken (boundary) spa 3 ) 2 with spontaneous symmetry breaking, but without an equilib Λ π QF T 10 3 (2 . Evolution of the entropy density s/ 2 . Different color coding represents different spatial resolu QF T κ crit 0.8 0.6 0.4 0.2 ] it was argued that the phenomenological holographic model In this paper we attempted to address two questions: Concerning (1), it was pointed out in [ Concerning (2), we showed that it is straightforward to modi E 1 Potentially related phenomenon was reported in [ < 13 (2) what is the role of supersymmetry on the singularity deve (1) can BBL scenario be realized in a top-down string constru whether dynamics in sBBL model at growth of the entropy density E state below the instability threshold. lational invariance and/or rotationsgravity-fluid can correspondence not [ dominate late and whether this difference is attributed to the sign of in DG models inthe microcanonical mean-field ensemble spontaneous — symmetry breaking rather, mechanism we of found [ ye structure of the bulk scalar coupling to gravity ubiquitous exhibit the exotic thermodynamicsthe of corresponding [ models in details and established that th singularities from regular initial conditions in correspo configuration was shown to evolve,the in divergent a area finite density boundary ofat time, the t the apparent horizon. horizon andnism is the The robust Kr divergence — while ofstates the of the evolution was the area restricted to density spatia signals th lates the weak cosmic censorshipcertain conjecture. The gravitati the apparent horizon in dynamics of sBBL model with 4 Conclusions In [ Figure 16 JHEP08(2017)134 (A.2) (A.4) (A.5) (A.1) (A.3) ) terms in for all the tion of the x ( + 0 O , x . → ∂ , ) t ) , x ) ( ) he equilibrium phase 2 2 nism in string theory. t, x t, x x ( x λ x , ( L and sBBL models are ) A divergent area density of . O e latter might depend on 2 2 g ) Σ( , 1 + x x t ) ( ( 2 ) + , − λ , O t x ) t ( ) + ( } rical code) are needed to firmly x ∂ 1 + . ( O i, f } t, x ) + 2 O oupling 1 + → − Σ( m x + x t, x r =  ]. 1 x + ∂ m, q  2 i 1 Σ( 2 σ , a , dσ ) { 2 f ) t ) ) − t ( + t 2 x = ( 1 2 ( , t, r 1 ) 1 , 2 } ( , + 2 ) f 2 t A f 2 f t, x ( ) + } → { 1 – 23 – + , + − , d ) are enforced with the absence of 2 ] to study dynamics of spatially homogeneous and 1 t Σ 2 ) ) 2 , t, x 1 t ) ∂ 2 ) + , a t ( ) t x ) ). ( , 1 ( Σ( 2.38 ) + Σ( , t = ( 1 1 ) , 1 ( , 2 1 t, x 1 O − f 1 + ( A.4 f x t, x f ) A ( ( σ , A , d + ) { t − 1 2 O x 2 ( Σ + in ( t, x ) , and redefine the fields ( λ t ) { t i + − 1 x ( t x dσ a f , d ∂ ( ˙ ) λ x + t 1 + = 1] ( ≡ i, 1 d − , ) = ) = q λ f m A ), we find the asymptotic boundary expansion [0 = = = = = t, x t, x ∈ i 1 a σ and ˙ f ]. The scalar potential of the “supersymmetric” generaliza 2.38 a r 1 Σ( Σ( dσ x + 25 ∂ ≡ d , x ≡ 24 ′ ) and ( 2.33 Finally, the challenge remains to find embedding of BBL mecha The rest of the code implementation is as in [ where we set Note that the boundary conditions ( asymptotic expansion of A Numerical setup We adapt numerical code developed in [ isotropic states in DG and sBBL models. A.1 DG-B model We introduce a new radial coordinate details of the scalar superpotential, i.e., the nonlinear c BBL model (called sBBLdiagram here) and remains the unbounded linearizedconceptually from identical. symmetry below. Further breaking studies dynamics (and T establish in a whether better BB sBBL nume modelthe also apparent evolves horizon to and a the geometry curvature. with As we pointed out, th fields: maintaining truncations [ Using ( as follows JHEP08(2017)134 (A.6) (A.7) (A.8) ommons , ) x , ( . } O ) t + ( , 2 µ x xλ ) t 2 + ( for the new fields: 2 ry Grants program. 2 1 + p xλ + x ) , 0 t ]. 4 , y of Research & Innovation. ) ( redited. ) ) + 1 x ent of Canada through Indus- 1 + 2 → ( p x x ( O t, x . + SPIRE ) + ) O 2 Σ( 2 IN ) ) + x t + t ][ t, x ( x ( ( 1 x 1 2 p,q,σ,a,dp,dq,dσ 4 O ]. p Σ( 16 q ) ) 1 t 2 − t + ( 1 x ( 2 4 − x ) λ q − 2  = = = } → { t, x ) 2 ) Σ – 24 – t Σ( ( + t, x 1 ) and employ the following field redefinitions: 1 2 p , Σ( Unstable horizons and singularity development in ) , t ) , χ , d ( ) A.1 arXiv:1704.05454 2 1 [ ) + ) λ + , , dq , dσ t, x , 1 8 ) t, x ( ) ) t, x ), which permits any use, distribution and reproduction in ( ) + ) t, x 2 2 ( σ ( + φ , d x x t, x dq q as in ( ( ( ( + µ + t, x t, x 3 3 ( ( x O O  1 x dp a x x dσ p x , q + + (2017) 135 ) = ) + x x 2 , A , d t ) = ) = ) = ) = ) = ) = x 07 A ( 2 2 ( CC-BY 4.0 ˙ Σ ) p λ t t, x t, x t, x t, x t, x t, x O 8 ( ( ( ( ( − − This article is distributed under the terms of the Creative C 1 φ φ χ χ Σ( Σ( + p ) 2 + t + JHEP ) + x ( t d d 2 φ , χ , d , − 8 1 ( ) 1 { ) p t t ( p ), we find the asymptotic boundary expansion ( 1 − λ p − 3.17 = = = = p a σ holography The rest of the code implementation is as in [ dp [1] P. 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