Vortices in holographic in university

Vortices in Holographic Superconductors (SCs) & Superfluids (SFs) Boundary UV CFT3 Óscar Dias STAG research centre ( moving from IST, Lisbon ) Horizon IR defect CFT3 Based on: OD, Gary Horowitz, Nabil Iqbal & Jorge Santos,  arXiv:1311.3673 New frontiers in dynamical gravity, DAMTP, Cambridge, March 2014 Fs (r, Fn (r,2 T ) + ↵|4 | +1 | 2µ |is 0an + Abelian |( Higgs ihr2model 2eA) | T ) = 2m of A coupled to M and 2|( ihr2m2eA) ➙ TAccommodating the correspondence (r, ) = Fn (r, T ) + ↵| SCs | + in|Minimize | +gauge/gravity | (3) ⇤ ⇤ ⇤ F , ! + , ! + : potentials for s 2m systems having di↵erent charge q and • Ginzburg–Landau (GL) theory: 〈O 〉correspond to quadratic potential with a mass for the(3) Ein For d having properties: 2 SC wavefunction Φ has 2order parameter in [31] 2eA) We will work probe limit of [31], r,(r,TT) )=+F↵|n (r, | ihr | + 2eA) |( |ihr | (3) +in the B (2)which applie T ) + ↵| | + | + | | + |( n 2m 2µ its equilibrium2value is zero above T and increases gradually below T ⇤ ⇤ ⇤ 2m c c Minimize F , F / = : s s 1/2 large In this limit the gravitational system is approximated by ' Fs|, | ⇠!⇢ ⇤ ,+ =Minimize | |e , C Minimize Fs , ! + , 1/2 = | |e'e, | | ⇠ ⇢C ! ,' is macroscopic ! + : phase SC (1)(4) ⇤ T/T : in a Schwarzschild black hole geometry, (4) A+ with thec ba M and ⇤ (Cooper pair charge 2e, mass m), ' e is macroscopic SC phase (1) We label boundary indices by µ, ⌫, · · · , and bulk indices by M, N, · · · , w Notably 2this gravitational system is dual to a conformal field theory; ⇤ ⇤ ⇤ |( ihr 2eA) | + ↵ + | | = |Ψ|: GL free ! energy+density for! a SC T= Tc for small2 expansion parameter expanded 4around mize F , , + : (4) s Fs (r, T1) = Fn (r, T ) +|(↵|ihr | +2m | | +| + ↵ |(broken ihr (3) so the(5) the chemical conf 2eA) +by both | |2eA) = 0| potential and the temperature, ! 2 2m 2↵ + |2m | |( ihr 2eA) | + =|(0 ihr F (r, T ) = F (r, T ) + ↵| | + | | + s n in what follows 2m 2m ! 2(5) 2eA) | + B 2µ 10 2 (2) Fs (r, T ) = Fn (r, T ) + ↵| |2 + | |4 + |( ihr 2eA) | + B (2) 2m 2µ0 Minimize Fs , Fs / = : (5) r ⇥ B = µ0 J ) µ0 J = r2 A (London gauge: r ⇥ A = 0) & Minimize Fs , A ! A + A : 2 |( ihr 2eA) | + ↵ + | | = Minimize (4) ) µ20 J = 2m r A (London gauge: r ⇥ A = 0) & Fs , A ! A + A Ginzburg–Landau eq I: r A (London ! A 2eA) + A :| (6) F (r, Tgauge: )=F r (r,⇥ T )A+= ↵|0)|2 & + Minimize | |4 + Fs ,|( Aihr |( ihr r⇥ s n 2eA) F| s (r, + ↵T ) + = B = µ0 J ) µ0 J = (6) (3) 2 | =0 (6) 2eA) | F|n (r, | |4 +2m |( ihr (3) T ) + ↵| | + r A (London gauge: r⇥ 2mA = 0) & Minimize Fs , Fs / A = : e ⇤ e ⇤J = [ ( ihr 2eA) + c.c.] J⇥= [ ( ihr Ginzburg–Landau eq II (7) n m 2eA) + c.c.] ⇤ m G e ⇤ )+( +3 = `(µ m⌫) (D + 3✏ + ✏¯ ⇢ + ⇢¯) ( J+=4 +[ 3⌧ ¯ +⇤⌧c.c.] ⇡ ¯ ) (D + 4✏ + 3⇢) ( ihr⇤ 2eA) ⇤ ↵ (7) London gauge: r ⇥ A = 0)Minimize & Minimize ! A, + A!: Fs , F! +(7) : (4) s ,mA + o Note that GL eqs follow from Abelian Higgs model (Klein-Gordon eq for charged scalar): `µ `⌫ ( + + ↵ ¯ ⌧ ) ( + + 3⌧ ) mµ m⌫ (D + 3✏ ✏¯ ⇢) (D + 4✏ + 3⇢) (r i ) • Z  +c.c p 1 ab a † 2 S = d x g F F 2(D )(D ) 2V (| | )  Z ab 2eA) aA↵= + (London A:= 0)hORG & |(Minimize F , F / : (8) ihr | + | | = s s milar for gauge: spin s r =⇥ +2 ( ) 240 p µ⌫ 2m S= dx g Fab F ab 2(Da )(Da )† 2V (| |2 ) (5)(8) A simple holographic system with this structure begins with a classical field theory living • an asymptotically anti-de Sittercontext spacetime spatial dimensions ) Under the Toindiscuss SCs in the holographic addwith bulk3 gravitational AdS(AdS background standard Z holographic dictionary, the conserved current j µ (x) is mapped to a dynamical U (1) Z p ⇥ ⇤⇤ p 1bulk, while ab the scalar operator gauge field A (x, z) in the gravitational is mapped = dM d xx Fab 2(D (|(| |2|)2 )to SS = g R+ Fab 2(Daa )(D )(Daa ))† † (x)2V 2V abF L a bulk scalar field T (10) (9) (x, z) carrying charge q under the gauge field AM Note z is the radial coordinate of AdS4 Placing the system at nonzero temperature corresponds to adding T > Tc : < Tc : to the bulk spacetime a black hole whose horizon is a two-dimensional plane extended Z in boundary potential corresponds to imposing ⇤a ⇥ p )directions 6Adding a chemical Hairy BH ( spatial HHH SC phase ab S= dx g R+ F F 2(D Planar )(Da RN-AdS )† 2V BH (| |2 ) ab a boundary condition on the bulkLgauge field A = µ at the boundary of AdS4 As found t in [30, 31], if the charge q and scaling dimension sufficiently large drives the bulk scalar field Floating of lie in certain range, taking µ Boundary to condense through the Higgs mechanism, r of outside the horizon so that the black hole developscondensate scalar “hair” of + + + + normal phase (10) scalar field Horizon There are many examples of quantum theories with a low-temperature superfluid phase + + + + +is an+Abelian + Higgs + model of A =andhOicoupled to the Einstein gravity, with di↵erent which admit such a gravitational description [32, 33] A universal bulk description for them M systems having di↵erent charge q and potentials for 〈O 〉correspond to quadratic potential with a mass for For definiteness we will choose a having scaling dimension asunstable RN-AdS=BH to scalar if in [31] We will work in the probe limit of [31], which applies when the charge q ofcondensation is large In this limit the gravitational system is approximated by an Abelian Higgs model of µ > µ4d BF AM and in a Schwarzschild black hole geometry, with the backreaction of AM and on Gubser,! Hartnoll, Herzog, Horowitz ( HHH ),! but We label boundary indices by µ, ⌫, · · · , and bulk indices by M, N, · · · , with AM = (Aµ , Az ) Horowitz, Roberts,! Notably this gravitational system is dual to a conformal field theory; however, conformal symmetry is µ2d NH (µ,q) < µ2d BF T/Tsymmetry conformal c OD, Monteiro, Reall, Santos broken by both the chemical potential and the temperature, so the will plan no role a critical temperature Tc , spontaneously breaking2 the global 2 U (1) sy L ⇥ ⇤ L dt + d⇢ p phase transition 4➙ Normal / SC ab a the † even in absence aF (✓) chemical potential ds +H dx g R+ Fab F 2(Da )(D ) a superfluid 2V (|= cos |2phase ) 2of (9) system into ✓ ⇢ ( At = ) ↵ ZZ L A simple holographic system with this⇤ ⇤structure begins with a cla ⇥ p p | = + + · · · (11) ◆ z=0 ab the functions ab aa in † † the metric 2 {F (✓), G(✓), H(✓)}, + F Both x F 2(D )(D ) 2V (| | (10) r r dd⌘44x g R + F 2(D )(D ) 2V (| |) ) (9) ab aa ab µ in an asymptotically anti-de Sitter spacetime with spatial dimens † , ⌘L= (12) µ V standard holographic dictionary, the conserved current j (x) is mapp ⇥ ⇤ p• Mexican hat potential: ⇥ ⇤ p F1 F abab 2(D )(Daa )†† 2V (| 2|2 ) ΦUV = 0(10) g R+ ! ab a ✓ab F ◆ gauge field (x, z) while the scalar oper dx g R +2 F 2(Da )(D ) AM2V (|in the | ) gravitational bulk, (10) {F) (✓), G(✓), H L ⌘ µ L V(Φ ~ Λ UV UV ! † field (x, z) carrying charge q under a⌘bulk scalar the gauge field A V (⌘) = ⌘ µ , = (12) Z A' (✓), (✓) ! V ⇥ ⇤ p Z temperature ΦIR =1 ! S= dx g 3R +† Fab F abcoordinate 2(Daof AdS )(D4a Placing )† 2Vthe(|system |2 ) at nonzero (10) L d xOfield O (13) hole whose horizon to the bulk spacetime a black is IR a )two-dime decayof scalar bdry ! Sbdry V(Φ ~ ΛIR •SAsymptotic ↵ ! |z=0 = ↵ + +Z+ · · · 1in boundary spatial directions Adding a chemical (11) potential corr † ! r r S ! S  d xO O (13) field A2 the bounda bdry bdry boundary condition ds on 2the bulk gauge µ⇢at | = + + · · · (11) t = z=0 = dt + d⇢ + d' = ⇢ ! @AdS + r r in [30, 31], if the charge q and scaling dimension of lie in ! ✓ = SS2 = ⌘) = ⌘ µ 0.5 L2 V 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 sufficiently large drives the bulk scalar field to condense through ◆ Robin BCs: < µ ~ g Find < T > ~ g(3) Horizon Horizon Previously in the Probe limit: Previously in the Probe limit: • Albash, Johnson; Albash, Johnson; • Montull, Pomorol, Silva; • Maeda, Montull,Natsuume, Pomorol, Silva; Okamura; Maeda, Natsuume, • Kachru, Sachdev; Okamura; • Bao, Kachru, Sachdev; Harrison; • Bao, Harrison; Simp = lim⇡ d✓ H(✓)G(✓) ✓for Maxwell field ➙ Distinguishing & NSF vortices: cos BCs ✓⇤ ! 2SC2G 0 • CFT has a gauge coupling: SF case: gc = ab F Z= gc J S = ⇢s EA⇠ ⇢s =0 dR n ⇠ ⇢s n log φ|z=0 R ⇒ SF BC: • Jφ creates the vortex cos2 ✓ b d3 x (r✓)2 • Global U(1): not gauged (just rotational sym) Z Do not want external applied field • d✓ SC case w/ J=0: gc → ∞ • Gauged U(1) ✓ ◆ field Dynamical RIR•cutof f SC R⇒ core BC: ∇aFab=0 Jφ|z=0 =0 • Aφ creates the vortex A' = A(0) ' + J' z + , R → ∞: 2xTr HHH R → ∞: 2xTr HHH + Aφ (z) B B ➙ T=0, near-horizon & field theory considerations: • Stand ard Higgs Conventional SFs has few low-energy excitations: mech anism at T=0, there is single gapless Goldstone mode associated with spontaneous symmetry breaking Conventional SCs does not even have this mode: it is eaten by the dynamical photon • However, typical holographic SFs or SCs have many gapless dof: their gravity dual has BH horizon at low T • Vortex is localised point in the UV CFT3 directions It becomes a bulk cosmic string, carrying Φmag down to bulk horizon where it interacts with IR dof This interaction is described by an IR CFT3 ( ≠ UV CFT3 ): • We propose: Use formalism of defect CFT to describe interaction of a heavy, point like object (e.g a vortex) with IR CFT3 Defect breaks translational invariance of the conformal group UV CFT3 • Construct the Near-Horizon solution of the T=0 configuration on the boundary of which the defect IR CFT3 lives IR defect CFT3 ➙ Near-Horizon (T=0) &A IR defect CFT unctions in the metricsolution {F (✓), G(✓), H(✓)}, (✓) are functions of ✓ only ' (✓), 3 ✓ ◆  L 2 2 L dt + d⇢ ds2 =group F (✓) (defect) invariance of the conformal IR 2breaks translational 2 ✓ SO(3,2): cos ds = + d✓ + sin ✓d' 2 cos ✓ ⇢ dt2 + d⇢2 + H(✓)d✓ +G broken SO(2,1) 𝗫 SO(2) ⇢2 (21) • Vortex Both core: the functions the4:metric G(✓), H(✓)}, symmetries: [ Vortex θ=0 ∂inAdS θ=π/2{Fis(✓), conf to AdS S1 (✓), ] (✓), G(✓), H(✓)} 𝗫A • Most general element with these{F  ✓ ' ◆ A ' (✓),2 (✓) dt + d⇢ (23) L 2 {F (✓),(22) G(✓), H(✓)} ds2 = F (✓) + H(✓)d✓ + G(✓) sin ✓d' ! cos2 ✓ ⇢2 A' (✓), (✓) he !functions in the metric F (✓), G(✓), H(✓), A' (✓), (✓) are functions of ✓ only ! ✓ 2 ◆ dtworldline: + d⇢ nontrivial 2is sym group 2of a CFT extending 2 along vortex CFT1 lives on the(24) defect ✓ • SO(2,1) ds@AdS = dt + d⇢ + ⇢ d' = ⇢ + d' dt 2 ⇢ ds2@AdS = 2 2 dt + d⇢ + ⇢ d' =1 ⇢ Defect is mapped to AdS2 boundary and BCs therein Phase of vortex should wind around S • ∃ AdS2 endows bulk solution with Poincaré L2IRa = L2 horizon at ρ → ∞ ! ! Simp = lim⇡ ! ! ✓⇤ ! ⇡L2IR 2GN ✓Z ✓⇤ sin ✓ p d✓ H(✓)G(✓) cos ✓ Z ✓⇤ ? +d ⇢2 (25) There is an entropy associated with this horizon, which extends from θ = to θ = π/2: sin ✓ d✓ cos ✓ ◆ Poincaré H ρ= ∞ Bdry (26) ρ=0 (θ=π/2) • Horizon ⋂ conformal bdry AdS2 𝗫 S1 at θ = π/2: it’s a bulk minimal surface that hangs down from bdry Simp computes entanglement entropy via the Ryu-Takayanagi prescription • Minimal surface wraps the S1 that surrounds the defect on the bdry So Simp is boundary or impurity entropy of the defect with its surrounding ( ➙ Near-Horizon solution (T=0) & IR defect CFT3 • Scalar curvature of the T = horizon is large near the core: Defect is breaking the translational invariance of the 2xTr HHH near-horizon solution Signals a “bubble of RN-AdS horizon” (carrying Simp) sticking out of the usual Poincaré horizon UV CFT RH 30 25 IR defect CFT3 20 15 • SC (Φ ≠ 0) Poincare horiz • SSC sea = when T→0 10 0 Proper distance from vortex core alar curvature of the T = horizon as a function of proper distance from the vortex ge positive peak near the core denotes a “bubble of Reissner-Nordstr¨ om horizon” • mag RN-AdS Bubble • B focused here • SRN ≠ when T→0 (Φ=0) ➙ SC vortex (full geometry @ any T) vs Simp (NH solution T=0 ) • Confirming we have the correct Entropy of the ΔS full NH geometry: vortex solution (that extends to UV) as a function of Temperature as T ↘: vortex causes RN horizon to bubble out as T → approaches Simp as T →0 1.0 ΔS = Svortex - SNO vortex 0.8 0.6 ΔS imp ( NH; T=0 ) S n = (winding #) 0.4 0.2 0.0 0.0 n = (winding #) 0.1 0.2 0.3 T Κ 0.4 0.5 0.6 Temperature FIG 5: Full entropy di↵erence (defined later in (5.7)) as a function of T /( ) for q L = Squares LG = ➙ (❨ FULL geometry )❩ SC results: type I vs type II • Ginzburg–Landau parameter: LG = • ⇠ (17) LG = ⇠ London penetration depth: how quickly B falls off (17) ⇠ !theType Coherence length: how quickly  disturbances of ⟨O⟩ II fallSC off LG > p 1 p LG < ! Type I SC LG > p ! Type II SC 1 ? ? p Landau-Ginzburg theory, thethe threshold between thethe twotwo is at precisely = = 2p Thus wewe LG 2theory, Landau-Ginzburg threshold between is at precisely LG Thus conclude that thisthis ratio of correlation lengths should bebe correlated with vortex stability conclude that ratio of correlation lengths should correlated with vortex stability Gravitational results: LG < p ! Type I SC Type Type I SC: q L = II SC: q L = 0.8 0.8 0.4 0.4 SO(3, 2) ! SO(2, 1) ⇥ SO(2) B RB R 0.6 0.6 SO(3, 0.3 0.3 2) ! SO(2, 1) ⇥ SO(2) (18) R R R: distance (@ bdry) to vortex core ⇥ L IRR 0.4 2 2 0.4 B B R ds = dt + dR + R d' + dz 0.2 0.2 z2 R R ⇥ ⇤ LIR 2 2 2 ds = dt + dR + R d' + dz 0.2 0.2 (19) 0.1 0.1 z 0.0 0.0 0 Ξ 1 Ξ Λ Λ 2 3 4 5 0.0 0.0 0 Λ Λ Ξ 1 Ξ R = ⇢ sin ✓ 3 4 zR=R ⇢ cos ✓ 5 R =R⇢ Rsin ✓ = ⇢II, cosfield ✓B(R) (20) 14 14 Profile magnetic field andand order hO(R)i forfor a single with qLqL == 1 FIG Profile ofztype magnetic B(R) order parameter hO(R)i a single vortex with Holographic SCs can beFIG either type Iofor depending onparameter the scalar charge vortex (left(left panel) andand qL qL = 3=(right panel) ⇠ are found from exponential fitsfits and measure thethe rate panel) (right panel) andand ⇠ are found from exponential and measure rate Probe limit is q → ∞ so misses it 2field of fall-o↵ of magnetic field andand order parameter, respectively of fall-o↵ of magnetic order parameter, respectively Conventional wisdom assumed all holographic SCs are type II [ Umeh 2009 for earlier suggestion I / II ] ➙ SC results: vortex thermodynamics & stability • Applied magnetic field penetrates the condensate sample and creates region of normal phase with flux ! Bdry ! • Domain wall z (DW) separating normal / SC phase costs energy H ! ! • For type I SCs, DW costs positive E: to minimize cost system creates a single large lump of Normal phase n=2 vortex should be energetically favoured over two n=1 vortices ! • For type II SCs, DW costs negative E: to maximize DW length, system tries to creates as many vortices with Normal phase as possible (eventually a Abrikosov lattice of vortices is favoured) n=2 vortex should be unstable to fragmentation into two n=1 vortices • For ! agreesthat with our from studying LG Finally, for qL =This 3, we again have verified both theexpectations entropy and free energy di↵erences type I SCsthan( qL qL= 2, and 1.9continue ), n=2to vortex energetically over two n=1 are larger favor the is n= vortex over thefavoured two n = vortices This again agrees with our expectations from studying LG p1 ≳ Microcanonical (fix E) 0.45 ! 0.40 ! ! ! ! ! 0.0001 0.40 > 𝗫 ΔSn=1 ΔSn=2 0.0001 0.35 S n n ∆ F n Κ 0.0003 ∆ F n Κ 0.0003 0.0004 0.0004 0.25 < 𝗫 ΔFn=1 0.0002 0.30 0.30 ΔFn=2 0.0002 S 0.35 0.25 0.0005 0.20 0.20 0.11 0.12 ! • Canonical (fix T) 0.45 ! vortices 0.13 E n 0.14 Κ 0.12 0.13 E n ≲ 0.0005 δF=Fn=2-2Fn=1 0.1 0.15 0.11 F=E-TS 0.2 0.14 0.3 T Κ 0.4 0.5 0.6 0.15 0.1 0.2 0.3 T Κ 0.4 0.5 0.6 Κ For type qL2, the 1.9 n=2isvortex unstable tonfragmentation into: two n=1 vortices FIG.II 16:SCs For qL( = n = 2),vortex (slightly)is favored over two = vortices Left Panel 0.7 entropy0.7di↵erence (5.7) as a function of E/( ) for q L = Disks correspond to n = and FIG 16: Fordi↵erence qL = 2, n = vortex squares to n = Right Panel : the in the free energies, F = is F(slightly) Ffavored as aover n=2 /2 n=1 , 0.015 > Δ Sn=2 0.6 entropy di↵erence function of2T𝗫 /(ΔS ).n=1 0.6 0.015 (5.7) as a function of two n = vortices Left Panel : < 𝗫 ΔFn=1 to ΔF E/( ) for q L = Disks2correspond n =n=2 and squares to n = Right Panel : the di↵erence in free energies, F = Fn=2 /2 Fn=1 , as a We end our discussion of vortex stability with a final comment: we have seen that vortex function of T /( ) 0.010 S 0.5 stability Type II superconductors ∆From our S 0.5 is precisely the distinction between Type I and 0.010 F ∆ F n n analysis it is clear that whether or not a particular superconductor nis Type I Κ n holographic Κ Wedetailed end discussion of vortex stability final comment: we have seen that vortex or Type dynamics, i.e the non-universal ratio with of twoadi↵erent 0.4 our 0.4 II depends on the correlation lengths, which appears to be sensitive (for example) the precise value of0.005 theType II superconductors From our 0.005 stability is precisely the to distinction between Type I and scalar charge While most of the literature on holographic superconductors states that they analysis 0.3 it is clear that whether or not a particular holographic superconductor is Type I 0.3 are Type II [3, 10], this was originally based on the fact that the scalar condensate starts or Type on the i.e.B the non-universal ratio of two di↵erent to condense at a nonzero valueIIof depends the magnetic field.detailed This wasdynamics, interpreted as c2 , the 0.16 0.1 0.2 0.3 0.4 0.5 0.6 0.18 0.20 0.22 correlation lengths, which appears to be sensitive to (for example) precise the 0.1 0.2the 0.3 0.4 value 0.5 of 0.6 0.16 0.18 0.20 0.22 T Κ E T Κ n Κcharge While scalar 35 most ofE the literature on holographic superconductors states that they ➙ SC results: holographic SCs ≠ • Boundary magnetic field Bbdry as a function of Bbdry = Fxy|bdry conventional SCs distance R to SC Vortex core for several Temperatures 0.6 n = (winding #) 0.5 0.5 n=1 0.4 B B 0.4 0.3 0.3 0.2 0.2 0.1 0.0 0.0 T 0.5 0.1 1.0 1.5 R 2.0 2.5 0.0 0.0 0.5 1.0 1.5 R 2.0 2.5 R: distance (@ bdry) to vortex core 11: Boundary profileremains as a function of R,even plotted outside a Rcoremagnetic (~ κ ).field Rcore finite asforT several →0 values of T /( ) • B falls-off expFIG The left panel has n = 1, and the right panel n = Here, disks, squares, diamonds, triangles and with the fall-off density: triangles have of T /(energy ) = 0.029, 0.370, 0.495, 0.546, 0.571, respectively • This contrastsinverted ε(R) ∼ e−α(T) R, inverse “energy screening length” 0.7 • Thus @ α(T) → as T → T=0 the vortex sources a 0.6 long-range disturbance in the stress tensor (requires back-reaction), due to its interaction with the IR CFT 0.5 Bmax • Long-range tail demonstrates a difference between Holographic & Conventional SC vortices 0.4 (latter source no long range fields) 0.3 ➙ SUPERFLUID results: • Recall: SC vortex sourced by a boundary magnetic field B ~ ∂RAφ SF vortex sourced by boundary current Jφ (no applied B field) • Boundary Current Jφ|z=0 as a function of distance R to the SC Vortex core for several Temperatures: n=1 Ê 0.8 Jφ|z=0 0.8 Ê Ê 0.6 Jj Ê Jj 0.4 0.6 Ê Ê 0.4 Ê ‡Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê 1.5 Ê Ê Ê Ê ‡ n = (winding #) ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ Ï Ï Ú Ú Ù Ù Jj R Ï Ú Ú Ù Ù R Ê 1.0 Ï Ï Ú Ù Ï Ï Ï Ú Ú Ú Ù Ù Ù Ê ‡ T ‡ Ê ‡ Ê ‡ Ï Ê‡ 0.2 Ï 0.2 ‡ Ê ‡Ï Ï Ê Ï ‡ Ï Ê‡ Ê Ï Ú Ï‡ Ê ‡ ʇ χ ÚÏ Ú Ú Ê‡ ÏÏÊ‡Ú ÚÏ Ú Ê Ï ‡ Ú Ú Ê Ï Ê Ú Ï ‡ ‡Ï Ê ‡ Ê Ê ‡ Ï ÙÚÙÚ Ù ÙÙ ÚÚ ‡Ï ‡ ÙÏ Ï ÚÏ Ù Ê Ï ÚÏ Ù Ú Ù ‡ ÙÚ Ú ÙÚ Ú Ê Ù Ï Ù ‡ Ù ÚÏ Ù Ù Ï Ù Ú Ù Ù 0.0 Ï Ê ‡ Ú Ù Ê ‡ Ú Ê ‡ Ú Ê ‡ Ú Ï ÚÏ ‡ Ï ÚÙ Ê Ï ÚÏ ÙÙ Ù Ù Ù Ú Ï ÙÚ ‡ Ù Ú Ù Ú Ê Ï Ù ‡ Ù Ú Ù Ù Ï Ù Ú Ù Ù 0.0 Ï Ê ‡ Ú Ù ‡ ‡ 1.5 Ï Ú Ù 1.0 Ê Jj Ê 0.5 0.5 Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ï Ê ‡ Ê ‡ Ê ‡ Ï Ê‡‡ Ï Ê Ï‡ Ú ÚÏ Ê Ê ÊÙ ‡ Ê Ï ‡Ú ‡ Ê Ï Ú ‡ Ê ‡ Ê Ú ‡ Ê Ú ‡ Ê ÚÊ ‡ Ù ÏÙÚ ÚÏ ‡ Ê Ù Ï ‡ ÚÏ Ï ÚÏ Ï ÚÏ ‡ Ù Ï Ê ÚÏ Ù ÚÏ Ú Ù ‡ Ï Ú Ù Ú Ù Ù Ï Ú ÚÏ Ù Ù Ù Ù Ù Ù 0.0 Ï Ê ‡ Ú Ù Ù Ù Ê ‡ Ê ‡ Ê ‡ Ï Ú Ú ‡ Ê Ï ‡ Ê ÚÙ Ï ‡ Ê ÚÏ ‡ Ï Ê Ú ‡ Ï Ú ‡ Ï Ê Ï ‡ Ú Ï Ú Ï Ú ‡ Ù Ï Ê Ú Ù Ï Ú Ú Ù ‡ Ï Ú Ù Ú Ù Ù Ï Ú Ú Ù Ù Ù Ù Ù Ù 0.0 Ï Ê ‡ Ú Ù Ù Ù Ù Ù ‡ ‡ ‡ Ï Ï Ï ‡ ‡ ‡ ‡ Ê Ê ‡ ʇ Ê Ê Ê Ê ‡ Ê Ï Ï Ï Ú Ï Ú Ú Ï Ú Ú Ú Ù Ù ÙÚ Ù Ù Ù Ù R R Ú Ù Ï Ú Ù R: distance (@ bdry) to vortex core FIG 17: 17: Boundary current profile of aofsuperfluid vortex as aasfunction of R, for for sev-sevFIG Boundary current profile a superfluid vortex a function of plotted R, plotted eral values of T /( ) The left panel has n = 1, and the right panel n = (both are for ➙ SUPERFLUID results: • • A vortex Z SF low-energy dynamics is S given the action for a Goldstone mode θ: S = ⇢s = ⇢by d x (r✓) s with winding charge n has θ (R → ∞) ∼ n φ ! E ⇠ ⇢s ! • dE = T dS: Z dR n ⇠ ⇢s n2 log R ✓ It has energy: ◆ RIR cutof f Rcore E ⇠ ⇢s IR divergent E ⇒ IR divergent S Z Z (27)2 d3 x (r✓) 2(28) T dR n ⇠ ⇢F n log s R ry Bd ✓ Rcuto Rcor ! Gravity: • Entropy & energy densities decay polynomially as 1/R2 when R → ∞ (in SC case they decay exponentially) ⇒ S & E diverge (∫vol ~ log R) ! ! ! Ê Ê ÊÊ ÊÊ Ê Ê Ê Ê ÊÊ Ê ÊÊ Ê Ê ‡Ê ‡Ê ‡Ê ‡‡‡ ‡ ‡‡Ê ‡‡ ‡Ê ‡Ê ‡Ê ‡Ê Ê ‡‡ ‡Ê ‡ ‡Ê ‡ ‡‡ ‡ ‡Ê ‡‡ ‡‡Ê ‡Ê ‡ ‡Ê ‡ ‡‡Ê ‡ ‡ Ê ‡‡ ÊÊ ‡ Êʇ‡ Êʇ‡ Êʇ‡ 0.01 0.01 ʇ ʇ ʇ ʇ Ê Ê ‡ ‡ Ê Ê ‡ ‡ Ê Ê ‡ ‡ Ê ‡ Ê ‡ 10-4 10-4 Ê ‡ Ê ‡ Ds Ds Ê ‡ Ê ‡ Ê H-kL2H-kL2 Ê Ê ‡ Ê ‡ Ê ‡ Ê ‡ 10-6 10-6 ! ÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊ 0.1 Ê ÊÊ Ê ÊÊÊÊÊ ÊÊÊ ÊÊÊ ‡‡‡‡ ‡Ê ‡‡‡‡ ‡‡ ‡‡‡‡‡‡‡‡‡Ê ÊÊ ‡‡ ‡‡‡Ê ‡‡‡ ‡ ‡‡ ‡ ‡‡‡‡‡ Êʇ‡ Êʇ‡‡ Ê ‡ Ê ‡ Ê ‡ Ê ‡ Ê Ê Ê‡ ʇ ʇ ʇ ʇ ʇ ‡ Ê Ê ‡ 0.001 0.001 Ê ‡ Ê ‡ Ê Ê DE ‡ DE ‡ Ê Ê 3 H-kL H-kL Ê ‡ Ê ‡ 0.1 Ê ‡ Ê Ê 0.01 0.01 0.1 0.1 R R 10 10 100 100 Ê ‡ Ê ‡ Ê Ê -5 10 Ê ‡ ! ! 10-5 ‡ ‡ ‡ Ê Ê 0.01 0.1 0.01 0.1 R 10 100 10 Ê 100 R • But Δs ~ f(T) n2 / R2 with f(T) → as T → ⇒ ΔS SF → ΔSimp as T → (like SC; same NH) • S ~ n2 ⇒ any high winding charge SF vortex is unstable to fragmentation into n = vortices [ agrees with time FIG 19 Entropy density (left panel) and energy density (right panel) as a function of R for the FIG 19 Entropy density (left panel) and energy density (right panel) as a function of R for th vortex superfluid phase Here, disks and squares describe, respectively, isolated vortices with n = vortex superfluid phase Here, disks and squares describe, respectively, isolated vortices with n = and n = (both for qL = 2) At large R, both densities decay polynomially as 1/R2 as described by and n = (both for qL = 2) At large R, both densities decay polynomially as 1/R2 as described by the dashed curves that give the best fit of the asymptotic tails For example, for n = one finds the evolution Adams, Chesler, 1212.0281 Lehner’s ] th theof dashed give the best fitLiu, of the asymptotic tails For& example, for n3= talk one finds curves that ↵ ➙ Take-home messages: • Constructed nonlinear (backreaction) holographic vortices at any temperature T and condensate charge q • Superfluid [ Global U(1); Aφ|z=0 =0 ] & Superconductor [ Gauged U(1); Jφ|z=0 =0 ] vortices • SC vortices can be type I or type II depending on the scalar charge [so far it was thought they were type II] • Type I / II SC classification is correlated with thermodynamic stability • Vortex carries magnetic flux down to horizon where it interacts with IR dof: IR CFT3 ( ≠ UV CFT3 ) • Use formalism of defect CFT to describe interaction of vortex (breaks translational invariance) with • Constructed the associated near-horizon solution @ T=0: • “bubble of RN-AdS horizon” sticking out of the usual Poincaré horizon • There is an entropy (Simp) associated with this bubble horizon • Simp computes entanglement entropy via the Ryu-Takayanagi prescription • Simp is boundary or impurity entropy of the defect with its surrounding • @ T=0 the vortex sources a long-range disturbance in the stress tensor (requires back-reaction), IR CFT3 due to its interaction with the many gapless dof of the IR CFT • Long-range tail demonstrates a difference between Holographic & Conventional SC vortices (latter: no tail)  ... boundary of AdS4 As found t in [30, 31], if the charge q and scaling dimension sufficiently large drives the bulk scalar field Floating of lie in certain range, taking µ Boundary to condense through... Shomer] bulk indices by M, N, · ·T/T · , with c A [Ishisbashi,Wald], [ Faulkner, Horowitz, Roberts ( 2xTr HHH ) ] , indices [Witten], ➙ Adding SC & SF vortices: how to ? ➙ Adding SC & SF vortices: ... SF or SC vortex means that Aφ ≠ • Solve Einstein-deTurck PDEs for { |ψ|, Αφ , gµν } using • Solve Einstein-deTurck PDEs for { |ψ|, Αφ , gµν } [Wiseman] using H Horizon R R Boundary Boundary z z