Published for SISSA by Springer Received: June 17, 2014 Revised: August 19, 2014 Accepted: November 4, 2014 Published: November 14, 2014 J Linear resistivity from non-abelian black holes H E P 1 1 Christopher P. Herzog, Kuo-Wei Huang and Ricardo Vaz ( C.N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy, 2 Stony Brook University, Stony Brook, NY 11794, U.S.A. E-mail: [email protected], 0 [email protected], [email protected] 1 4 Abstract: Starting with the holographic p-wave superconductor, we show how to obtain ) a finite DC conductivity through a non-abelian gauge transformation. The translational symmetry is preserved. We obtain phenomenological similarities with high temperature 0 cuprate superconductors. Our results suggest that a lattice or impurities are not essential 6 to produce a finite DC resistivity with a linear temperature dependence. An analogous 6 field theory calculation for free fermions, presented in the appendix, indicates our results may be a special feature of strong interactions. Keywords: Holography and condensed matter physics (AdS/CMT), AdS-CFT Corre- spondence ArXiv ePrint: 1405.3714 Open Access, (cid:13)c The Authors. doi:10.1007/JHEP11(2014)066 Article funded by SCOAP3. Contents 1 Introduction 1 2 The p-wave holographic superconductor 2 2.1 Action 2 2.2 Superfluid solution and phase diagram 4 3 Conductivity for AdS /CFT 8 4 3 J 3.1 Nonabelian gauge transformation 8 H 3.2 Normal phase conductivity 9 E 3.3 Superfluid phase 13 P 4 Analytic conductivity for AdS /CFT 15 5 4 1 5 Discussion 17 1 ( A SU(2) conductivity for free fermions 18 2 0 1 1 Introduction 4 ) Uncovering the mechanisms of high temperature superconductivity [1] has been one of the 0 great challenges of theoretical and experimental physics. Strong interactions are believed 6 to play an important role, rendering the conventional approach due to Bardeen, Cooper, and Schrieffer (BCS) [2] inadequate. An effective way to capture universal properties of 6 high temperature superconductors is certainly of great interest. The AdS/CFT correspon- dence [3–5], which works by mapping certain strongly interacting field theories to classical theories of gravity, has proved to be a powerful tool for understanding strong interactions more generally. Optimistically, one might hope that AdS/CFT might be able to provide some hints of the mechanisms underlying high temperature superconductivity. The first holographic superconductors were constructed in refs. [6–8] where the black hole develops scalar hair at the phase transition. In refs. [9–11], the scalar was omitted and a non-abelian SU(2) gauge field was introduced, and the order parameter for the phase transition is the set of nonabelian global SU(2) currents. This non-abelian model provided a connection to a p-wave superconductor. While the initial p-wave papers focused on systems with two spatial dimensions, some corresponding analytic results were obtained for three spatial dimensions in ref. [12] based on an analytic solution of the zero mode for the phase transition for an SU(2) gauge field in AdS first observed in ref. [13]. As 5 recognized in ref. [7], these holographic systems are perhaps more accurately described as superfluids because the U(1) that would be associated with the photon is treated as a global symmetry. But for many questions, the distinction may not be that important. – 1 – Oneofthemostdistinctivefeaturesofhigh-temperaturecupratesuperconductorsisthe linear temperature dependence of the DC electrical resistivity at optimal doping. However, a na¨ıve search for this effect in a dual gravitational model meets an immediate difficulty: the translation invariance in a gravitational model implies momentum conservation. The charged particles then cannot dissipate their momentum and the conductivity is infinite in the DC limit. Two obvious ways of breaking translation invariance, which real world materials take advantage of, are impurities and a lattice. In an AdS/CFT context, enor- mous effort has recently gone into adding impurities and a lattice to gravity models. (See refs. [14, 15] for early papers on the subject.)1 Inthispaper,wepresentanewmethod,employinganon-abeliangaugetransformation, J that allows us to obtain a finite DC conductivity without breaking translation symmetry. H Moreover, we find the DC resistivity has a linear temperature dependence close to the E superconducting phase transition. Our most interesting result is the DC resistivity plot in P figure 5. Our results suggest that a lattice or impurities are not necessary to produce a finite DC resistivity with a linear temperature dependence. 1 The organization of the paper is as follows: in section 2, we start from an action that 1 is holographically dual to a p-wave superconductor. We next discuss the phase transitions ( for these systems with two and three spatial dimensions. In section 3, we compute the 2 DC conductivity numerically as a function of temperature, using our non-abelian gauge 0 transformationmethod. Motivatedbythefactthatthecupratesuperconductorsarelayered materials, we will focus our numerical results on the AdS model. In Sec 4, we use the 1 4 same idea to obtain an analytic form of a finite DC conductivity in AdS5 using the solution 4 given in [13]. We discuss some generalizations in the last section. An appendix discusses ) the DC conductivity calculation for free fermions transforming under a global SU(2). 0 6 2 The p-wave holographic superconductor 6 2.1 Action Our p-wave holographic superconductor has a dual description via the following gravita- tional action for a non-abelian gauge field Fa with a cosmological constant Λ: µν 1 Z √ 1 Z √ S = dd+1x −g(R−2Λ)− dd+1x −gFa Faµν . (2.1) 2κ2 4e2 µν For the moment, we keep d arbitrary as we will study both the d = 3 and d = 4 cases. Our gauge field is the curvature of the connection Aa: µ Fa = ∂ Aa −∂ Aa +fa AbAc , (2.2) µν µ ν ν µ bc µ ν where fa are the structure constants for our Lie algebra g with generators T such that bc a [T ,T ] = if cT . We will take g = su(2) where T = σ /2, σ are the Pauli spin matrices, a b ab c a a a 1A finite DC limit can also be achieved holographically by decoupling the charge current from the momentum current, for example by setting the total charge to zero [16] or by holding the background metric fixed in a “probe limit” [7, 17]. – 2 – and the structure constants are f = (cid:15) . (The indices a,b,c,... are raised and lowered abc abc with δa.) b The equations of motion for the gauge field that follow from this action (2.1) are D Faµν = 0 which can be expanded as µ ∇ Faµν +fa AbFcµν = 0 . (2.3) µ bc µ Einstein’s equations can be written (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) 1 1 1 1 G ≡ R + Λ− R g − 2Fa Faλ − Fa Faλρg = 0 . (2.4) µν 2κ2 µν 2 µν 4e2 λµ ν 2 λρ µν J One well known solution to these equations in the case of a negative cosmological H constant, Λ = −d(d − 1)/2L2, is a Reissner-Nordstrom black hole with anti-de Sitter E space asymptotics. This solution describes the normal phase of the holographic p-wave P superconductor. The only nonzero component of the vector potential is2 1 A3 ≡ φ(u) = µ+ρud−2 . (2.5) 1 t ( Thus we are using only a U(1) subgroup of the full SU(2) gauge symmetry; this black hole 2 solution requires only an abelian gauge symmetry. The line element for this black hole solution has the form 0 ds2 −f(u)dt2+d~x2 du2 = + (2.6) 1 L2 u2 u2f(u) 4 where the warp factor is ) f(u) = 1+Q2(cid:18) u (cid:19)2d−2−(cid:0)1+Q2(cid:1)(cid:18) u (cid:19)d (2.7) 0 u u h h 6 and the charge Q has been defined as 6 r d−2 Q ≡ λρud−1 , (2.8) h d−1 where we have defined the dimensionless parameter κ λ ≡ , (2.9) eL controlling the back reaction on the metric. The horizon is located at u = u , and the h Hawking temperature is d−(d−2)Q2 T = . (2.10) H 4πu h Our gauge potential (2.5) is well defined globally, at both the horizon and the boundary, provided µ ρ = − . (2.11) ud−2 h 2Thenotationρismeantoevokeachargedensity. TheactualchargedensityaccordingtotheAdS/CFT dictionary would be ρ˜=−(d−2)ρ/e2. – 3 – 2.2 Superfluid solution and phase diagram Increasing the chemical potential µ or equivalently decreasing the Hawking temperature, this black hole is well known [10, 11] to undergo a phase transition to a state (dual to the superconducting state) with a nontrivial profile for A1 ≡ w(u) . (2.12) x We need to reconstruct these results here as we will be exploring the conductivity close to the phase transition line. Depending on the value of λ, this phase transition can be either J first or second order [18, 19]. In the case of a second order phase transition, the location H is given by the existence of a nontrivial zero mode solution for w with regular boundary E conditions at the horizon, w(u ) < ∞, and Dirichlet boundary conditions at the conformal h P boundary, w(0) = 0. The differential equation that must be solved is 1 ud−3f(fu3−dw0)0 = −φ2w, 1 ( where φ is given by (2.5), f by (2.7), and f0 ≡ ∂ f. 2 u Tofindthelocationofthefirstorderphasetransition, weneedtoworkharderandfind 0 a numerical solution for the condensed phase. Following [18], we choose a metric ansatz 1 4 ds2 1 (cid:18) dx2 (cid:19) du2 = −f(u)s(u)2dt2+ +g(u)2d~y2 + . (2.13) ) L2 u2 g(u)2(d−2) f(u) 0 We will be interested in d = 3 or d = 4 dimensions, and the relevant differential equations 6 can be expressed by 6 (cid:18) φ0 (cid:19)0 g2(d−2)w2 ud−3s = φ, ud−3s f !0 ud−3 sg2(d−2)fw0 φ2 = − w, g2(d−2) ud−3 fs ud (cid:18)sf(cid:19)0 λ2 u3(φ0)2 d = − , (2.14) s ud d−1 s2 u (g0)2us λ2u3g2(d−2) (cid:18) w2φ2(cid:19) s0 = −(d−2) − (w0)2s+ , g2 d−1 f2s (cid:18) (g0)2(cid:19) (cid:18)w2φ2 (cid:19) (d−1) g00− = −λ2u2g2d−3 −(w0)2 g f2s2 g0 (cid:18) λ2u4(φ0)2 d(cid:19) −(d−1) 1+ − . u (d−1)fs2 f To find the superfluid phase, we require these differential equations to have the follow- – 4 – ing u = 0 expansions in d = 3: w2µ φ = µ+ρu+ 1 u4+O(u5), (2.15) 12 w µ2 w = w u− 1 u3+O(u4), (2.16) 1 6 w2λ2 s = 1− 1 u4+O(u6), (2.17) 8 w2λ2 g = 1+g u3+ 1 u4+O(u6), (2.18) 3 8 1 f = 1+f u3+ λ2(w2+ρ2)u4+O(u6) ; (2.19) J 3 2 1 H and d = 4: w2µ E φ = µ+ρu2+ 2 u6+O(u8), (2.20) 24 P w µ2 w = w u2− 2 u4+O(u6), (2.21) 1 2 8 1 2w2λ2 s = 1− 2 u6+O(u8), (2.22) ( 9 w2λ2 2 g = 1+g u4+ 2 u6+O(u8), (2.23) 4 9 0 2λ2 f = 1+f4u4+ 3 (w22+ρ2)u6+O(u8) . (2.24) 1 4 At the horizon, we demand f(u ) = 0 = φ(u ) while the remaining functions w, s, and h h g should all be finite. We proceed to solve the differential equations (2.14) by means of a ) shooting method. If we expand the functions near the horizon we find that there are only 0 four independent coefficients, {φh,wh,sh,gh}, where φh ≡ φ0(u = u ), wh ≡ w(u = u ), 1 0 0 0 1 h 0 h 6 and similarly for the other two. The method then consists in choosing boundary data 6 {φh,wh,sh,gh} and (numerically) integrating the differential equations. Once done, we 1 0 0 0 scanthespaceofsolutionsinsearchoftheoneswiththerightboundaryvalues,asin(2.15)– (2.19) and (2.20)–(2.24), in particular obeying w(u = 0) = 0. Note that we are picking the boundary metric to be Minkowski such that g(u = 0) = s(u = 0) = 1. We have used this shooting method to find solutions in d = 3 and d = 4 and the goal of this section is to plot the corresponding phase diagrams.3 Once we have the solutions with the appropriate asymptotic behaviour, the first thing we can plot is the order parameter hJ1i as a function of the temperature. From the x AdS/CFT dictionary, we know that hJ1i ∼ w and hJ1i ∼ w in d = 3 and 4, respectively. x 1 x 2 The temperature can be read from the periodicity of the time-like direction, and for this phase it reads sh (cid:18) λ2 (φh)2(cid:19) T = 0 d− 1 . (2.25) 4π d−1 (sh)2 0 3Ref.[18]wasthefirsttostudythep-wavesuperconductorwithback-reactioninAdS . Ref.[19]provides 5 acorrespondingdiscussioninAdS . In[20],theAdS caseisalsostudied. However,therangeofparameters 4 4 where the first order phase transition occurs is not fully explored. Our phase diagrams agree with figure 2 of [19] and figure 8 of [21]. – 5 – 0.30 0.25 0.20 w1(cid:144) Μ2 0.15 0.10 0.05 0.00 0.6 0.7 0.8 0.9 1.0 1.1 1.2 J T(cid:144)Tc H Figure 1. The order parameter hJ1i∼w for different values of λ, in d=3. The curve λ=0.4< E x 1 λ3d (dot-dashed, green) corresponds to a second order phase transition. The curve λ = 0.8 > λ3d c c P (dashed, purple) corresponds to a first order phase transition. The curve λ ∼ λ3d ∼ 0.62 (solid, c orange) passes through the critical point separating the first and second order transitions. 1 1 The form of the curve hJ1i as a function of the temperature immediately tells us whether ( x we are looking at a first or second order phase transition. 2 Below we plot w1/µ2 as a function of T/Tc for λ = 0.4, λ = 0.8, and for the crit- 0 ical λ where the transition goes from second to first order, which we estimate to be at 1 λ3d = 0.62±0.01. In the dot-dashed green plot we see a solution with a non-vanishing c 4 condensate emerges below a certain critical temperature. In the dashed purple one we see ) that below a certain temperature there are two different solutions with non-vanishing w , 1 and the transition is first order because the superfluid solution becomes thermodynami- 0 cally preferred starting at a non-zero value of the order parameter. The results in d = 4 6 share the same qualitative profile, as shown in [18], and the critical point can be estimated, 6 λ4d = 0.365±0.01. c In order to identify the critical temperatures (which we have already used in figure 1) and draw the phase diagram, we need to look at the free energy. The field theory stress tensor is [22]: 1 √ (cid:20) (cid:18) d−1(cid:19) (cid:21) Tν = lim −γ −Kν + K + δν , (2.26) µ u→0 κ2 µ L µ where K = 1(n + n ) is the extrinsic trace of a constant u surface, γ is the µν 2 µ;ν ν;µ µν corresponding induced metric, nµ is an inward pointing unit vector normal to the surface, and K = Kµ. We obtain µ Ld−1 T = (d−1)f , (2.27) tt 2κ2 d Ld−1 T = (f +2d(d−2)g ), (2.28) xx 2κ2 d d Ld−1 T = (f −2dg ) i = 1,...d−2. (2.29) yiyi 2κ2 d d – 6 – 0.002 0.002 Ts 0.000 -0.002 T2 0.000 T2 DW -0.004 DW T1 Μ3 Μ3 -0.006 -0.002 -0.008 -0.010 -0.004 -0.012 0.01 0.02 0.03 0.04 0.05 0.06 0.006 0.008 0.010 0.012 0.014 T(cid:144)Μ T(cid:144)Μ Figure 2. The difference between the free energies of the superfluid and normal phases in d = 3, for λ=0.4 (left) and λ=0.8 (right). The phase transitions occur at T (left), and T (right), and J 2 1 are of second and first order, respectively. On the right, T and T mark spinodal points in the 2 s H phase diagram. E P The indices of T are raised and lowered with the Minkowski metric tensor η = µν µν (− + ···+), and with some care the coefficients f and g can be extracted from the 1 d d numerical solution near the boundary. To compute the on-shell action S = R Ldd+1x, 1 bulk we note that ( √ Ld−1 (cid:18) fs (cid:16)g(cid:17)0(cid:19)0 L = −2 −gGy − . (2.30) y κ2 ud−2g u 2 0 We need to add counter-terms to regulate the divergences at u = 0, 1 1 Z (cid:18) (d−1)(cid:19)√ S = Sbulk+ κ2 K + L −γddx, (2.31) 4 ) where the second term is the Gibbons-Hawking term. We find that in general, S = os 0 T Vol/T where Vol is the spatial volume and T is the temperature. The free energy is yiyi 6 then defined as Ω = −T S /Vol. An interesting feature, as discussed in [23] and confirmed os by our numerics, is that g = 0. As a consequence, the stress tensor is spatially isotropic, 6 d and we can study the object Ωe = fd to determine the nature and location of the phase transition. Infigure2, weplotthedifferencebetweenthefreeenergiesofthesuperfluidand normal phases, ∆Ωe/µ3, as a function of T/µ, for the same values we chose above, λ = 0.4 and λ = 0.8, in d = 3. Once again the behaviour in d = 4 is entirely analogous. We can see a clear difference between the first and second order transitions. In figure 2 (left), wecanidentifythesecondorderphasetransitiontemperatureT . Infigure2(right), 2 in contrast, we can identify instead a first order phase transition temperature T , as well 1 as two spinodal temperatures T and T . s 2 By repeating this calculation for different values of λ, we determined numerically how these special temperatures depend on λ and constructed a phase diagram for the holo- graphic p-wave superconductor in the T/µ-λ plane. See figure 3 for d = 3 (left) and d = 4 (right). Before the critical point λ , the blue line in figure 3 signals a second order phase c transition from the normal to the superfluid phase, with a non-vanishing expectation value for the condensate emerging below the critical temperature, as we see in figure 1 (dot- dashed green). To the right of the critical point, there is a first order transition signalled by the red line. It is of first order because, as we can see in figure 1 (dashed purple) and 2 – 7 – 0.07 0.08 00..0065 TTT12s 0.06 TTT12s 0.04 T(cid:144)Μ T(cid:144)Μ 0.04 0.03 0.02 0.02 0.01 0.00 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Λ Λ Figure 3. Phase diagrams of the p-wave superconductor in (2 + 1) (left) and (3 + 1) (right) J dimensions. For λ<λ3d ∼0.62 (resp. λ<λ4d ∼0.365) , there is a second order transition at the c c H solid blue line. For λ > λ3d,4d the phase transition is first order and occurs at the solid red line. c There are two additional spinodal lines (dashed, blue and black), corresponding to temperatures E identified in figure 2. The phase diagram on the right coincides with figure 8 in [21]. P 1 (right), one of the solutions with non-vanishing condensate becomes thermodynamically 1 favoured for a value hJ1i =6 0. Beyond the critical point, the blue and black dashed lines x ( represent spinodal lines, with the temperatures clearly identifiable in figure 2 (right). It bears mentioning that our numerical procedure in the superfluid phase gets harder, and 2 less reliable, once the critical temperature gets too close to zero. 0 We note that the blue lines in the phase diagram, both before and after the critical 1 point, can be found in the normal phase through studying the DC conductivity, which we 4 discuss in the next section. ) 3 Conductivity for AdS /CFT 0 4 3 6 3.1 Nonabelian gauge transformation 6 As mentioned earlier, in the normal phase the only nonzero component of the vector po- tential is A3 ≡ φ(u) = µ+ρud−2. We now consider a nonabelian gauge transformation on t the background: A → Ae= U−1AU +U−1dU , (3.1) where A = −iT Aadxµ and U = exp(iT λa) . (3.2) a µ a We here take a particularly simple isospin rotation λ = (0,θ,0) . (3.3) For this gauge transformation, ! cos(θ/2) sin(θ/2) U = (3.4) −sin(θ/2) cos(θ/2) and a short calculation reveals (cid:16) (cid:17) Ae1,Ae2,Ae3 = (cid:0)A3sinθ,0,A3cosθ(cid:1) . (3.5) t t t t t – 8 – This gauge transformation affects the boundary behavior of A , changing the direction of t the chemical potential and charge density vectors in isospin space. In the context of our holographic application, we assume that the electric field is applied in the 3 direction in the tilde’d coordinate system. More importantly, there will be an angle θ in isospin space between the charge density and the electric field. By tuning θ to a special value θ∗ we will show it renders a DC conductivity finite, σ < ∞. DC One might na¨ıvely guess this goal is achieved when θ∗ = π/2, when the electric field and charge density are orthogonal in isospin space. However, because of the non-abelian terms in the action, θ∗ will depend on temperature, ranging from θ∗ = 0 at the second order phase transition (blue lines in the phase diagrams in figure 3) all the way to a finite J value that interpolates between 0 (at λ = 0) and π/2 (as λ → ∞) in the high temperature H limit. We will explore the special θ∗ in more detail in the following section for d = 3. E 3.2 Normal phase conductivity P We consider fluctuations around the normal phase background, aa and g , and assume 1 x tx they are small. We work in the untilde’d frame in order to keep the background solution 1 as simple as possible, and transform to the tilde’d frame at the end. While we let the ( fluctuations have arbitrary radial dependence, we restrict the time dependence to have the 2 form e−iωt. At linear order, the equation of motion for the metric fluctuation g is tx 0 1 u2∂u(u2gtx) = −2λ2φ0L2a3x . (3.6) 1 4 The equations of motion for the gauge fields are then ) (cid:16) (cid:17) ud−3f∂ fu3−d∂ a3 = (cid:0)−ω2+2λ2(φ0)2u2f(cid:1)a3, (3.7) u u x x 0 (cid:16) (cid:17) ud−3f∂ fu3−d∂ a = −(±ω−φ)2a . (3.8) 6 u u ± ± 6 where we have defined a ≡ a1 ±ia2. ± x x The way in which a finite DC conductivity can be extracted from equations (3.7) and (3.8) can be understood at a schematic level. Given an electric field in the 3 isospin direction, the pole in the DC conductivity at ω = 0 comes from the (φ0)2a3 term in (3.7). x Similarly, for an electric field in the 1 or 2 isospin directions, the pole (this time with the opposite sign) would come from the −φ2a term. By carefully selecting the angle θ, we ± can cancel out one pole with the other. It might be somewhat counter-intuitive that the DC conductivity remains finite in the absence of momentum dissipation, but we should note that while the cancellation of poles leads to a finite DC conductivity in the isospin direction parallel to the electric field, E~ will act to accelerate a current in an orthogonal direction in isospin space4 and also in energy density. At a physical level, the (φ0)2a3 term in (3.7) comes from the mixing of the momentum x and charge currents. Indeed, using eq. (3.6), φ0 can be replaced with g . The −φ2a term tx ± in (3.8) appears because a acts like a charged particle under F3 . The time derivative is ± µν 4Morespecifically,thechemicalpotentialandE~ pickoutaplaneinisospinspace. E~ willacttoaccelerate a current in this plane but orthogonal to E~. – 9 –