Under The Dome: Doped holographic superconductors with broken translational symmetry
PPrepared for submission to JHEP
Under The Dome
Doped holographic superconductors with brokentranslational symmetry
Matteo Baggioli, a,b
Mikhail Goykhman ca Institut de F´ısica d’Altes Energies (IFAE), Universitat Aut`onoma de Barcelona,The Barcelona Institute of Science and Technology, Campus UAB,08193 Bellaterra (Barcelona), Spain b Department of Physics, Institute for Condensed Matter Theory, University of Illinois,1110 W. Green Street, Urbana, IL 61801, USA c Enrico Fermi Institute, University of Chicago,5620 S. Ellis Av., Chicago, IL 60637, USA
E-mail: [email protected],[email protected]
Abstract:
We comment on a simple way to accommodate translational symmetrybreaking into the recently proposed holographic model which features a superconduct-ing dome-shaped region on the temperature-doping phase diagram. a r X i v : . [ h e p - t h ] O c t ontents The original holographic model of [1, 2] successfully describes the physics of super-conducting (SC) phase transitions within a strongly coupled regime. This is achievedby introducing a charged scalar field into the finite-temperature, finite-density AdS-RN black-brane setting. The charged scalar field is stated to be the bulk dual of thecondensate of the boundary charge carriers and it represents indeed the natural or-der parameter for the SC phase transition. When the system is in the normal phase,the bulk scalar is identically trivial; on the contrary in the SC phase, the bulk scalardevelops a non-trivial profile. This model however describes just one face of a vari-ety of phenomena exhibited by real-world materials. For the AdS/CMT field to bea practically-oriented endeavor, one should augment the framework of [1, 2] by cou-pling it to additional sectors, with the intention to account for more of a non-trivialexperimentally observed physics.The present work is motivated by the recent paper [3], which follows in this directionby building a model which exhibits a normal, superconducting, anti-ferromagnetic andstriped/checkerboard phases on the doping-temperature plane. It is interesting thatthe superconducting phase of [3] appears within a dome-shaped region in the middle ofthe phase plane as in actual High-Tc superconductors. However the normal phase of[3] possesses an infinite DC conductivity, a property which it shares with the original– 1 –olographic superconductor [1, 2] and which does not allow it to be labeled as a propermetallic phase. Unlike the infinite DC conductivity of the superconducting phase,which is a result of the condensation of the charge carriers, an infinite DC conductivityin the normal phase is a straightforward consequence of the translational invariance ofthe boundary theory, which needs to be relaxed.Translational symmetry breaking mechanisms in holographic models have recentlyreceived plenty of attention in the literature, in order to mimic more realistic con-densed matter situations. Breaking the bulk diffeomorphism invariance via introducinga graviton mass [4] is an efficient way of achieving it. Massive gravity theories can beformulated covariantly in terms of the Stuckelberg fields and such a construction resultsin a finite DC conductivity [5]. The latter model has been considered in combinationwith the holographic superconductor setting in the recent literature, which includes[6–8].In this note we point out that the recent model [3], featuring a holographic SCdome, can be further improved by coupling it to a neutral scalar sector, governed by ageneral Lagrangian as in [9] and responsible for the breaking of translational symmetry.The resulting holographic superconductor can be studied in spirit of [8]. Its normalphase possesses a finite DC conductivity. Our results prove that the superconductingdome of [3] continues to exist once the translational symmetry has been broken.Using the non-linear model proposed in [9] we are able to describe a normal phasewith conductivity decreasing upon lowering of the temperature. The resulting sys-tem exhibits three phases on the temperature-doping plane: superconducting, normalmetallic and normal pseudo-insulating. The phases are essentially distinguished by theDC conductivity: infinite in the superconducting phase, decreasing with temperaturein the metallic phase, and growing with temperature in the pseudo-insulating phase.The rest of this paper is organized as follows. In the next section we set up themodel which we study in this paper. We describe the normal phase in section 3, wherewe also construct the metal/pseudo-insulator phase diagram on the temperature-dopingplane, for the model governed by a non-linear Lagrangian for the neutral scalars. Insection 4 we determine the critical temperature and doping for which the normal phasebecomes unstable towards the development of a scalar hair. This signals a supercon-ducting phase transition, which we confirm in section 5 by solving numerically for thewhole background and calculating the temperature dependence of the charge conden-sate v.e.v. We discuss our results in 6. In appendix A we collect the equations ofmotion for the whole background that used in the paper.– 2 –
The system
In this section we set up the holographic system which we will be studying in thispaper. We follow closely the conventions of [3]. We consider the following bulk degreesof freedom: the metric g µν , two U (1) gauge fields A µ , B µ , the complex scalar field ψ ,and two neutral scalars φ I , I = x, y . Here x, y are spatial coordinates on the boundary.We will denote the radial bulk coordinate as u . The boundary is located at u = 0, thehorizon is located at u = u h .We want to describe a system of charge carriers, coexisting with a media of impu-rities. The density of the charge carriers is denoted by ρ A and is dual to the gauge field A µ while the density of impurity ρ B is dual to the gauge field B µ . The quantity x = ρ B /ρ A (2.1)is called the doping parameter and represents the amount of charged impurities presentin the system [3].The boundary system exists in a superconducting phase when the Bose condensateis formed. The vacuum expectation value of the condensate is the order parameterfor the superconducting phase transition and it is described holographically via thecomplex scalar field, ψ . The latter is charged w.r.t. the U (1) A gauge field [1, 2].We introduce explicit translational symmetry breaking into the system by couplingit to a sector of neutral and massless scalars φ I with spatial dependent sources φ I = α x I , I = x, y [5]. In this paper we will be making use of a generalized action for thosescalars introduced in [9].The total action of the model is written as: S = 116 π (cid:90) d x √− g (cid:18) R + 6 L + L c + L s (cid:19) (2.2)where we fixed the cosmological constant Λ = − /L , and denoted the Lagrangiandensities for the charged sector [3], and the neutral scalar sector [9] as: L c = − Z A ( χ )4 A µν A µν − Z B ( χ )4 B µν B µν − Z AB ( χ )2 A µν B µν (2.3) −
12 ( ∂ µ χ ) − H ( χ )( ∂ µ θ − q A A µ − q B B µ ) − V int ( χ ) (2.4) L n = − m V ( X ) . (2.5)Here the A µν and B µν stand for the field strengths of the gauge fields A µ and B µ respectively. Following [3] we decomposed the charge scalar as ψ = χe iθ . We alsodefined: X = 12 g µν ∂ µ φ I ∂ ν φ I . (2.6)– 3 –he most general black-brane ansatz we consider is: ds = L u (cid:18) − f ( u ) e − τ ( u ) dt + dx + dy + du f ( u ) (cid:19) , (2.7) A t = A t ( u ) , B t = B t ( u ) , (2.8) χ = χ ( u ) , θ ≡ , (2.9) φ x = α x , φ y = α y . (2.10)The corresponding equations of motion are provided in appendix A. The temperatureof the black brane (2.7) is given by: T = − e − τ ( uh )2 f (cid:48) ( u h )4 π . (2.11)In the rest of the paper we will be considering: V int ( χ ) = M χ . (2.12)Solving the χ e.o.m. near the boundary u = 0 one obtains χ ( u ) = C − ( u/L ) − ∆ + C + ( u/L ) ∆ , where ( M L ) = ∆(∆ − C − is the source term, which one demandsto vanish, and C + is the v.e.v. of the dual charge condensate operator, C + = (cid:104)O(cid:105) . The∆ is equal to the scaling dimension of the operator O . Following [3] in this paper wefix the scaling dimension to be ∆ = 5 / In the normal phase the charge condensate vanishes, and the charged scalar field istrivial, χ ≡
0. Solving the background equations of motion we obtain τ ≡
0, alongwith: f ( u ) = u (cid:90) uu h dy ρ A (1 + x ) y + 4 ( mL ) V ( α y ) − y , (3.1) A t ( u ) = ρ A ( u h − u ) , B t ( u ) = ρ B ( u h − u ) . (3.2)The temperature in the normal phase is given by: T = 12 − ρ A (1 + x ) u h − mL ) V ( α u h )16 πu h . (3.3)Using the membrane paradigm one can calculate analytically the DC conductivity inthe normal phase [10, 11]. Its value for a general neutral scalars Lagrangian V is givenby [9]: σ DC = 1 + ρ A u h m α ˙ V ( u h α ) . (3.4)– 4 –n particular for the linear Lagrangian: V ( X ) = 12 m X , (3.5)we recover the set-up of [5]. Choosing a non-trivial V ( X ) one can incorporate moreinteresting physics. As it was pointed out in [9] for certain V ( X ) one can realize botha pseudo-insulating and a metallic phases, characterized by the following peculiartemperature dependences of the DC conductivity: dσ DC dT > − insulating , dσ DC dT < . (3.6)A transition between these two states happens at critical temperature T which isdetermined by: dσ DC dT = 0 ⇒ ˙ V ( u h α ) − u h α ¨ V ( u h α ) = 0 . (3.7)Solving for the horizon radius u h in terms of the temperature (3.3), and plugging itinto (3.7), we obtain the phase transition line T ( m, α, x ). Solution to (3.7) exists fora non-trivial choice of the Lagrangian V ( X ). In this paper we will consider: V ( X ) = X + X . (3.8)For this V ( X ), fixing x = 0, α = O (1), one obtains a phase diagram on the ( m, T )plane, with the pseudo-insulating phase occupying a compact corner region around theorigin of the phase plane [9]. This property has been embedded in the holographicsuperconductor phase diagram in [8].In this paper, following [3], we are interested in a phase structure on the ( x , T )doping-temperature plane. Therefore we fix m and α and determine the critical temper-ature T ( x ). For the model (3.8) it is possible to achieve a compact pseudo-insulatingregion around the origin of the ( x , T ) plane, see figure 1. To determine whether a boundary system exhibits a superconducting phase one canconsider a normal phase of the bulk system and see whether it becomes unstable towardsdeveloping a non-trivial profile of the scalar χ ( u ). This approach assumes that the A DC conductivity increasing with temperature is reminiscent of an insulating behavior, althoughit is ubiquitous in the considered holographic model that the zero-temperature conductivity is alwaysnon-vanishing, as pointed out recently in [12]. – 5 –
I metal xT Figure 1 . Phase diagram of the model (3.8) with m = 1, α = 1, showing the pseudo-insulating phase, characterized by the DC conductivity behavior σ (cid:48) DC ( T ) >
0, and the metallicphase, characterized by the DC conductivity behavior σ (cid:48) DC ( T ) < corresponding superconducting phase transition is of the second order, which shouldbe checked separately by solving the whole system away from the regime of a small χ ,which we do in section 5. In this section we solve the linearized equation of motion for χ in the normal phase background. Following [3] we define the following expansion ofthe couplings: H ( χ ) = n χ , Z A ( χ ) = 1 + a χ , Z B ( χ ) = 1 + b χ , Z AB ( χ ) = c χ . (4.1)and define the U (1) A,B charges to be q A = 1 , q B = 0 . In this section we use scalingsymmetry and set the charge carriers density ρ A = 1, and express the impurity densityin terms of the doping parameter ρ B = x .A natural place to start searching for superconductor is at zero temperature. Whenthe temperature is zero, the infra-red limit of the bulk geometry is AdS × R , wherethe scale of the AdS is given by: L = 2 L u h f (cid:48)(cid:48) ( u h ) . (4.2)The effective mass of the scalar χ can be read off from its linearized equation of motion– 6 –nd is given by: M eff = 12 f (cid:48)(cid:48) ( u h ) (cid:2) f (cid:48)(cid:48) ( u h ) (cid:0) M L ) − ( a + 2 c x + b x ) u h (cid:1) − n u h ( q A + x q B ) (cid:3) . (4.3)The system becomes unstable towards developing a non-trivial χ ( u ) profile if the BFbound for the scalar χ is violated in the AdS , namely M eff L < − , or more specifi-cally:(2 M − u ( a + 2 c x + b x )) (6 + m (( α u ) ˙ V − V )) − n u ( q A + q B x ) < , (4.4)where dot stands for derivative of V w.r.t. its argument and u for the radial positionof the extremal horizon, T ( u )=0. We also set L = 1.To obtain a superconducting dome on the temperature-doping plane ( T, x ), oneneeds to fix the parameters of the model in such a way that zero-temperature super-conducting instability appears in an interval [ x , x ], between two positive values x , ofthe doping parameter. In the context of instability analyses tuning the model amountsto a choice of the coefficients a , b , c , n , appearing in the expansion (4.1). In [3] thespecific model determined by the parameters: a = − , b = − , c = 143 , n = 1 . (4.5)has been extensively studied, and it was pointed out that in the interval x ∈ [ x , x ], x (cid:39) . x (cid:39) . χ violatesthe AdS BF bound.Now let us consider the model (4.5) but with the translational symmetry brokenby the neutral scalars with the linear Lagrangian (3.5). We observe that for α (cid:54) = 0 theinstability persists, although the ‘depth’ of the AdS BF violation becomes smaller,and therefore we expect the corresponding critical temperature of the superconductingphase transition to be lower. This is as to say that the breaking of translational sym-metry unfavores the SC instability. We plot the α -dependence of the boundary pointsof the IR instability region, x , ( α ), in figure 2.The critical temperature T c of a second-order phase transition can be determined bystudying the dynamics of the scalar χ ( u ), considered as a probe in a finite-temperaturenormal phase background. We are looking for a maximal value of the temperature forwhich the source coefficient C − of the near-boundary expansion of the field χ vanishes.Consider the model with the linear Lagrangian (3.5). It is interesting to observe howthe critical temperature depends on the magnitude α of the translational symmetry– 7 – .0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.2 α x α x Figure 2 . The boundaries of the zero-temperature IR instability region on the doping linefor the model (4.5), with the translational symmetry broken by the neutral scalars with theLagrangian (3.5). x = α T c Figure 3 . Critical temperature T c ( α ) for the model (4.5), with the translational symmetrybroken by the neutral scalars with the Lagrangian (3.5). Here the doping is fixed to be x = 2 . breaking. In accordance with our expectations from the zero-temperature instabilityanalyses we observe a decrease of the critical temperature with α , as shown in figure 3.Now let us fix the value of α and plot the critical temperature as a function of thedoping parameter x , see figure 4. The breaking of translation symmetry preserves thesuperconducting dome structure exhibited by the model (4.5), and merely diminishesa little the critical temperature.Now let us consider the model with translational symmetry broken by neutral– 8 – = α = SC metal x T c Figure 4 . Phase diagram in the ( T, x ) plane for the model (4.5) coupled to the neutralscalars with the Lagrangian (3.5). We compare the case α = 0 of [3], and the system withbroken translational symmetry, at α = 1. SCPI metal xT Figure 5 . Phase diagram in the ( T, x ) plane for the model (4.5) coupled to the neutralscalars with the Lagrangian (3.8). We fixed α = 0 . m = 1. scalars governed by the non-linear Lagrangian (3.8). We fix α = 0 . m = 1 and deter-mine the critical temperature T c ( x ). In figure 5 we combine this with the temperature T ( x ) of the metal/pseudo-insulator phase transition (MIT), described in section 3, andobtain the full phase diagram of the system with the superconducting phase enclosedinside a dome.This means that even if momentum dissipation unfavores the SC phase it is still– 9 –ossible to achieve a SC dome-shaped region as in actual High-Tc superconductors andhaving a normal phase with a finite DC conductivity. This is the main result of ourpaper. In the previous section we studied the instability of the normal phase (3.1)-(3.2) towardsdevelopment of a non-trivial profile of the scalar χ ( u ). Observing an instability attemperature T = T c on its own is not sufficient for the conclusion that the systemexhibits a phase transition at this point. Indeed, the instability analyses relies on theassumption that the phase transition is a continuous second-order phase transition. Todetermine whether this is actually the case, one should calculate behavior of the orderparameter as a function of temperature, and make sure the continuous critical point isnot shielded by a first-order phase transition.In our case we need to solve numerically five background equations of motion forthe model (2.2). These equations are provided in appendix A. The order parameter (cid:104)O(cid:105) ( T ) is read off as the coefficient of the sub-leading term in the near-boundaryexpansion of the χ ( u ). The methodology of a numerical solution for the background ispractically identical to the one we performed recently in [8], which the interested readeris encourage to consult for the details. The only new subtlety now is that we need tokeep the doping parameter x fixed. With care this can be achieved, for example, usingthe FindRoot function in
Mathematica , now applied to solve for the A (cid:48) t ( u h ), B (cid:48) t ( u h ) foreach fixed χ ( u h ), such that both the source C − of the field χ at the boundary vanishes,and the x is fixed.We use scaling symmetry of the background equations of motion to fix u h = 1. Weplot the condensate as a function of temperature in the linear model (3.5), for α = 0(the case of [3]), and α = 1; and in the non-linear model (3.8), for α = 0 . m = 1.We checked numerically the free energy of the system for the cases analyzed andwe found that the SC phase whenever present is favoured. The interested reader canfind details in [3] or [8]. In this paper we described a straightforward generalization of the holographic super-conductor model proposed in [3]. An important feature pointed out in [3] revealsthat introducing a non-trivial coupling between the order parameter and the gaugefields one can achieve an enclosure of the superconducting phase inside a dome-shapedregion on the doping/temperature plane as in actual High-Tc superconductors. We– 10 – = α = T < O > T < O > Figure 6 . Condensate for the model (4.5), with the doping fixed to x = 2, with brokentranslational symmetry. Left:
The linear model (3.5) with α = 1, plotted next to thetranslationally-symmetric system α = 0. Right:
The non-linear model (3.8) with α = 0 . m = 1. The condensate is measured in units ρ / A , the temperature is measured in units ρ / A . have expanded the model of [3] introducing a simple momentum dissipation mecha-nism through a sector of neutral and massless scalars, breaking translational symmetry[5]. The conclusion is that in a generic situation the superconducting dome of [3] sur-vives the translational symmetry breaking and can be equipped with a normal phasefeaturing a finite DC conductivity.In the case of a non-linear Lagrangian for the neutral scalars [9] the normal phasecan be further split into two phases, distinguished by the sign of the first temperaturederivative of the DC conductivity. When this sign is negative, the system behaves likea metal, when it is positive, it resembles an insulator. We pointed out that for a genericchoice of translational symmetry breaking parameters an insulator occupies a compactregion in the corner of the ( x , T ) plane. The total resulting phase diagram exhibitsthree phases: metal, superconductor, and pseudo-insulator.The main result of this paper is to show that the SC dome-shaped region built in [3]can be completed with a simple momentum dissipation mechanism and embedded in anormal phase region featuring a finite DC conductivity. This represents a further steptowards reproducing holographically the phase diagram for High-Tc superconductors.It would be interesting to incorporate the translational symmetry breaking frame-work into the whole phase diagram, constructed in [3], which also includes normalferromagnetic and stripe/checkerboard phases. It would also be interesting to calcu-late the AC conductivity and study the collective excitation in the non-linear V model,pointed out in [9], and further investigated in the holographic superconductor of [8].Because of the non trivial couplings between the various sectors and the two gauge– 11 –elds we expect interesting features to appear.Finally it would be great to generalize the model to account for a real insulating normalstate with σ DC ( T = 0) = 0. In order to do so one has to break the assumptions of [12];results in this direction are coming soon [13]. Acknowledgments
M.B. would like to thank the University of Illinois and P.Phillips for the warm hospi-tality during the completion of this work. M.B. acknowledges support from MINECOunder grant FPA2011-25948, DURSI under grant 2014SGR1450 and Centro de Excelen-cia Severo Ochoa program, grant SEV-2012-0234 and is supported by a PIF grant fromUniversitat Autonoma de Barcelona UAB. M.G. would like to acknowledge supportfrom the Oehme Fellowship.
A Background equations of motion
The equations of motion following from the action (2.2) for the ansatz defined in (2.7)are: (cid:0) u χ (cid:48) − τ (cid:48) (cid:1) f + 2 u e τ H ( q A A t + q B B t ) = 0 , u f (cid:48) − (12 + u χ (cid:48) ) f − u e τ H ( q A A t + q B B t ) f ( u ) + 12 − L (2 m V + V int ) − e τ u (cid:0) Z AB A (cid:48) t + B (cid:48) t (2 Z AB A (cid:48) t + Z B B (cid:48) t ) (cid:1) = 0 ,Z A (2 A (cid:48)(cid:48) t + τ (cid:48) A (cid:48) t ) + Z AB (2 B (cid:48)(cid:48) t + τ (cid:48) B (cid:48) t ) + 2 χ (cid:48) ( ˙ Z A A (cid:48) t + ˙ Z AB B (cid:48) t ) − q A H q A A t + q B B t u f = 0 ,Z B (2 B (cid:48)(cid:48) t + τ (cid:48) B (cid:48) t ) + Z AB (2 A (cid:48)(cid:48) t + τ (cid:48) A (cid:48) t ) + 2 χ (cid:48) ( ˙ Z B B (cid:48) t + ˙ Z AB A (cid:48) t ) − q B H q A A t + q B B t u f = 0 ,χ (cid:48)(cid:48) + (cid:18) f (cid:48) f − u − τ (cid:48) (cid:19) χ (cid:48) − L u f ˙ V int + e τ u f (cid:16) ˙ Z A A (cid:48) t + ˙ Z B B (cid:48) t +2 ˙ Z AB A (cid:48) t B (cid:48) t (cid:17) + e τ ˙ Hf ( q A A t + q B B t ) =0 . Here dot stands for a derivative w.r.t. the scalar χ , and prime stands for a derivativew.r.t. the radial coordinate u . References [1] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a HolographicSuperconductor,” Phys. Rev. Lett. , 031601 (2008) [arXiv:0803.3295[hep-th]]. – 12 –
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