Striped instability of a holographic Fermi-like liquid
aa r X i v : . [ h e p - t h ] N ov CCTP-2011-18
Striped instability of a holographic Fermi-like liquid
Oren Bergman, ∗ Niko Jokela, , † Gilad Lifschytz, ‡ and Matthew Lippert § Department of PhysicsTechnion, Haifa 32000, Israel Department of Mathematics and PhysicsUniversity of Haifa at Oranim, Tivon 36006, Israel Crete Center for Theoretical PhysicsDepartment of PhysicsUniversity of Crete, 71003 Heraklion, Greece
Abstract
We consider a holographic description of a system of strongly-coupled fermionsin 2 + 1 dimensions based on a D7-brane probe in the background of D3-branes. The black hole embedding represents a Fermi-like liquid. We study theexcitations of the Fermi liquid system. Above a critical density which dependson the temperature, the system becomes unstable towards an inhomogeneousmodulated phase which is similar to a charge density and spin wave state.The essence of this instability can be effectively described by a Maxwell-axiontheory with a background electric field. We also consider the fate of zero soundat non-zero temperature. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Introduction
In the last few years AdS/CFT has been used to analyze systems in which condensed-matter-like phenomena occur. Among them are superconductivity and superfluidity(for a review see [1]), non-Fermi liquids [2,3], metamagnetism [4,5], the quantum Halleffect [6–13], and more. In this note we provide another example, namely that of astriped phase of fermionic matter at non-zero density.Some of the aforementioned phenomena have been investigated in the bottom-upapproach, in which one does not start with a well-defined boundary theory, but ratherwith a toy model in the bulk which seems to capture the essence of the phenomena.Here we take the top-down point of view and study the D3-D7’ model, a particular(2+1)-dimensional system consisting of a D7-brane intersecting a stack of D3-braneswith
N D = 6. We use the probe approximation, in which the D7-brane is treatedas a probe in the near-horizon background of the D3-branes. The low-energy andweak-coupling spectrum has charged states that are purely fermionic. It thereforeseems like a good place to start looking for interesting phenomena.The D3-D7’ system was originally proposed in [14] as a holographic model forstrongly interacting fermions in 2+1 dimensions. In [7, 11] this model was used toexplore the phases of the system at non-zero temperature and charge density in thebackground of a magnetic field. In [11, 12] it was shown that the system exhibits agapped phase similar to a quantum Hall state at fixed ratios of the magnetic field tothe charge density.Here we analyze the system in the ungapped phase and without a magnetic field.This system resembles a Fermi liquid in that the resistivity at low temperature scalesas T . However, the heat capacity at low temperature does not follow the Fermiliquid formula.We analyze the four-dimensional bulk quasi-normal modes, which correspond tothe poles in the bi-fermion two-point functions of the three-dimensional boundarytheory, including the current-current correlators. In particular, we will demonstratethat the system becomes unstable to the formation of a striped phase when theratio of the charge density to the squared temperature is above a critical value. Thisinstability can be effectively described by a U (1) gauge field interacting with an axionfield in a constant background electric field.Striped phases are known to exist in condensed matter systems. Examples includecharge density waves and spin density waves (for reviews see [15, 16]), but thereare more complicated situations as well. What is common to them is that someobservable of the system is modulated in one direction and hence the name stripedphase. Holographic striped phases in 3 + 1 dimensions were analyzed in [17–20].Note added: As this paper was being completed [21] appeared which finds a similarinstability in a different (2+1)-dimensional holographic model.2 An instability in Maxwell-axion theory
In [17] Nakamura, Ooguri, and Park demonstrated that five-dimensional Maxwell-Chern-Simons theory becomes unstable in the presence of a background electric fielddue to a non-zero momentum mode that becomes tachyonic. As we will now show, asimilar instability can occur in a four-dimensional Maxwell-axion theory, which hasthe Lagrangian density L = − F IJ F IJ + 12 ( ∂ I Φ) − m Φ + α ǫ IJKL Φ F IJ F KL . (1)The equations of motion are given by( (cid:3) + m )Φ − αǫ IJKL F IJ F KL = 0 (2) ∂ I F IJ − αǫ IJKL ∂ I (Φ F KL ) = 0 . (3)Expanding around a background with F = F (0) and Φ = 0, the linearized equationsfor the fluctuations f IJ and ϕ are given by( (cid:3) + m ) ϕ − αǫ IJKL F (0) IJ f KL = 0 (4) ∂ I f IJ − αǫ IJKL ∂ I ( ϕF (0) KL ) = 0 . (5)Consider a constant background electric field in the z -direction F (0)03 = E . The equa-tions for the fluctuations become( ∂ µ ∂ µ + ∂ i ∂ i + m ) ϕ − αEf = 0 (6) ∂ µ f µν + ∂ i f iν = 0 (7) ∂ µ f µj + ∂ i f ij − αEǫ ij ∂ i ϕ = 0 , (8)where µ, ν take values in (0 ,
3) and i, j take values in (1 , ǫ jk ∂ k and using the Bianchi identity gives another equation involvingonly f and ϕ , ( ∂ µ ∂ µ + ∂ i ∂ i ) f − αE∂ i ∂ i ϕ = 0 . (9)For plane wave solutions of the form ϕ ( x, t ) = ¯ ϕe − i ( ωt − kx ) , f ( x, t ) = ¯ f e − i ( ωt − kx ) , (10)we obtain the dispersion relation ω = k + 12 m ± √ m + 64 α E k . (11)Therefore, there is a tachyonic mode in the range 0 < k < √ α E − m . For abackground magnetic field, on the other hand, there is no instability. We will find asimilar instability in the four-dimensional bulk theory of the D3-D7’ system. This willbe interpreted as an instability of the boundary theory at non-zero density towardsthe formation of a striped phase. 3 Review of the D3-D7’ model
A holographic model for fermions in 2+1 dimensions can be constructed using a D7-brane and D3-branes that intersect in 2+1 dimensions. Since
N D = 6 in thiscase, the massless spectrum of the D3-D7 open strings consists only of the desiredfermions. The separation between the D7-brane and D3-branes in the transversedirection corresponds to the fermion mass. We decouple the fermions from the closedstring modes by focusing on the near-horizon limit of a large stack of N D3-branes andtreating the D7-brane as a probe. According to the usual gauge/gravity duality, thedynamics of the probe D7-brane in the near-horizon D3-brane background capturesthe physics of the (2+1)-dimensional fermions interacting with a strongly-coupled(3+1)-dimensional gauge theory.
We begin with the near-horizon background of the non-extremal D3-branes: L − ds = r (cid:0) − h ( r ) dt + dx + dy + dz (cid:1) + r − (cid:18) dr h ( r ) + r d Ω (cid:19) (12) F = 4 L (cid:0) r dt ∧ dx ∧ dy ∧ dz ∧ dr + d Ω (cid:1) , (13)where h ( r ) = 1 − r T /r and L = √ πg s N α ′ . For convenience, we work in dimen-sionless coordinates, e.g. , r = r phys /L . This background is dual to N = 4 SYM theoryat a temperature T = r T / ( πL ). We parameterize the five-sphere as an S × S fiberedover an interval: d Ω = dψ + cos ψ ( d Ω (1)2 ) + sin ψ ( d Ω (2)2 ) ( d Ω ( i )2 ) = dθ i + sin θ i dφ i , (14)where 0 ≤ ψ ≤ π/
2, 0 ≤ θ i ≤ π , and 0 ≤ φ i < π . As ψ varies, the sizes of the two S ’s change. At ψ = 0 one of the S ’s shrinks to zero size, and at ψ = π/ S shrinks. The S × S at ψ = π/ The D7-brane extends in t, x, y, r and wraps the two two-spheres. The D7 embeddingis usually taken to be characterized by ψ ( r ) and z ( r ). However, excitations aroundthis embedding are tachyonic. To cure this we turn on fluxes through the two two-spheres labeled by the parameters f and f . With the correct choice of f and f , onegets a stable embedding. We also consider a non-zero charge density, by including thetime component of the worldvolume gauge field a ( r ), and a background magnetic4eld b . The D7-brane action has a DBI term given by S DBI = − T Z d x e − Φ q − det( g µν + 2 πα ′ F µν )= −N Z dr r q (4 cos ψ + f ) (cid:0) ψ + f (cid:1) ×× s(cid:0) r hz ′ + r hψ ′ − a ′ (cid:1) (cid:18) b r (cid:19) , (15)and a CS term given by S CS = − (2 πα ′ ) T Z P [ C ] ∧ F ∧ F = −N f f Z dr r z ′ ( r ) + 2 N Z dr c ( r ) ba ′ ( r ) , (16)where N ≡ π L T V , and c ( r ) = ψ ( r ) − sin (4 ψ ( r )) − ψ ∞ + sin(4 ψ ∞ ). Notethat c ( r ), and therefore ψ ( r ), plays the role of an axion in this model. Indeed, wewill find a modulated instability of the type described in the previous section.The asymptotic behavior of the fields is given by ψ ( r ) ∼ ψ ∞ + mr ∆ + − c ψ r ∆ − (17) z ( r ) ∼ z + f f r (18) a ( r ) ∼ µ − dr , (19)where the boundary value ψ ∞ and the exponents ∆ ± are fixed by the fluxes f , f :( f + 4 cos ψ ∞ ) sin ψ ∞ = ( f + 4 sin ψ ∞ ) cos ψ ∞ (20)∆ ± = − ± s f + 16 cos ψ ∞ −
12 cos ψ ∞ f + 4 cos ψ ∞ . (21)The parameters m and c ψ correspond to the “mass” and “condensate” of the fun-damental fermions, respectively, and µ and d to the chemical potential and chargedensity, respectively. Throughout the paper we will choose the fluxes f , f such that∆ + = − − = − f and fix f to satisfy this condition. For the MN embedding that describes the gapped phase, one must also add a boundary term S boundary = 2 N c ( r min ) ba ( r min ). However, we will only discuss BH embeddings, for which theboundary term vanishes. The physical charge density is given by d physical = 8 π L α ′ T d . .3 Embeddings There are two kinds of embeddings in general: “black hole” (BH) embeddings, inwhich the D7-brane crosses the horizon, and “Minkowski” (MN) embeddings, in whichthe D7-brane ends smoothly outside the horizon. The latter exist only for f = 0 or f = 0 and for a fixed charge density to magnetic field ratio (filling fraction). Herewe will only consider BH embeddings with a vanishing magnetic field b = 0. For the fluctuation analysis, it is convenient to use the coordinate u ≡ r T /r . Theboundary is now at u = 0, and the horizon is at u = 1. The embedding equations ofmotion now read − u r T z ′ ( u ) ≡ ¯ z ′ ( u ) = − f f h ( u )ˆ g ( u ) − u r T a ′ ( u ) ≡ ¯ a ′ ( u ) = ˆ d h ( u )ˆ g ( u ) u ∂ u (ˆ g ( u ) ψ ′ ( u )) = h ( u )2ˆ g ( u ) ∂ ψ G ( u ) , (22)where for convenience we have defined ¯ a ′ and ¯ z ′ and the prime now denotes differen-tiation with respect to u . We have also defined ˆ d = dr T . The functions G ( u ) and ˆ g ( u )are defined slightly differently than in [12]: G ( u ) = ( f + 4 cos ψ )( f + 4 sin ψ )ˆ g ( u ) = hu s ˆ d u + G − f f h hu ψ ′ . (23)The presence of the horizon imposes the boundary condition ψ ′ (1) = − ∂ ψ G (1)ˆ d + G (1) . (24)The mass associated with a given solution is given byˆ m ≡ mr ∆ + T ≡ u ∆ + ( ψ ( u ) − ψ ∞ ) (cid:12)(cid:12)(cid:12) u → . (25)The BH embedding corresponds to the metallic phase of the model. In particular,it has a longitudinal conductivity at zero magnetic field given by [11] σ xx = N π r − T q d + r T G (1) . (26)At low temperature and non-zero charge density σ xx ∼ dT , (27) The fluctuations of the MN embeddings (at zero temperature) were studied in [12]. C v ∼ T d , which is different from a regular Fermi liquid. In this section we will study the fluctuations of all the fields around the BH embeddingdescribed in the previous section. Due to rotational invariance, we can restrict tofluctuations propagating only in the x -direction; we thus consider fluctuations of theform f ( u ) e − iωt + ikx . We rescale the fluctuations δ ˆ z ≡ r T δz, δ ˆ a t,x,y ≡ δa t,x,y r T , δ ˆ e x ≡ δe x r T ≡ kδa t + ωδa x r T , (28)and define ˆ ω ≡ ω/r T and ˆ k ≡ k/r T . This will remove r T from all the fluctuationequations. It is worth recalling here the relationship to the physical momenta ω phys = πT ˆ ω and k phys = πT ˆ k . We also define the functionˆ A ( u ) ≡ h ( u ) u ψ ′ ( u ) + h ( u ) u − ¯ z ′ ( u ) − ¯ a ′ ( u ) . (29) Expanding the action to second order in fluctuations, we obtain a set of coupled,linearized equations for the fluctuations. The equation of motion for δψ is given by (cid:18) − h gu (cid:18) ∂ ψ G − G ( ∂ ψ G ) (cid:19) + u ∂ u [ˆ gψ ′ ∂ ψ log G ] (cid:19) δψ = − u ∂ u (cid:20) ˆ g ˆ A (cid:0) hu − ¯ z ′ − ¯ a ′ (cid:1) δψ ′ (cid:21) + ˆ gu h h − (cid:0) hu − ¯ z ′ (cid:1) ˆ ω + (cid:0) hu − ¯ z ′ − ¯ a ′ (cid:1) h ˆ k i δψ − ˆ g u ∂ ψ log G ¯ z ′ δ ˆ z ′ + ˆ g ¯ z ′ ψ ′ h h − ˆ ω + h ˆ k i δ ˆ z − u ∂ u (cid:20) ˆ gh ˆ Au ψ ′ ¯ z ′ δ ˆ z ′ (cid:21) + ˆ du ∂ ψ log Gδ ˆ a ′ t + u ∂ u " ˆ dhu ˆ A ψ ′ δ ˆ a ′ t − ˆ k ˆ du ψ ′ δ ˆ e x − i ˆ k a ′ sin (2 ψ ) δ ˆ a y . (30)7he equation of motion for δ ˆ z is given by − ˆ gh ¯ z ′ ψ ′ (cid:16) − ˆ ω + ˆ h ˆ k (cid:17) δψ + u ∂ u (cid:20) ˆ gh ˆ Au ¯ z ′ ψ ′ δψ ′ − ˆ g u ∂ ψ log G ¯ z ′ δψ (cid:21) =+ ˆ gh (cid:16) − (cid:0) hu ψ ′ (cid:1) ˆ ω + (cid:0) hu ψ ′ − ¯ a ′ (cid:1) h ˆ k (cid:17) δ ˆ z − u ∂ u (cid:20) hu ψ ′ − ¯ a ′ ˆ A ˆ gu − δ ˆ z ′ (cid:21) + ˆ k ˆ d ¯ z ′ δ ˆ e x − u ∂ u " ˆ dh ˆ Au ¯ z ′ δ ˆ a ′ t . (31)The equation of motion for δ ˆ a t is given by u ∂ u ˆ H + ˆ d ˆ k (¯ z ′ δ ˆ z − u ψ ′ δψ ) − ˆ k ˆ gu h (cid:0) hu − ¯ z ′ + hu ψ ′ (cid:1) δ ˆ e x − i ˆ ku δ ˆ a y ∂ u [2 c ( u )] = 0 . (32)whereˆ H ( u ) = ˆ gu ˆ Ah (cid:0) hu − ¯ z ′ + hu ψ ′ (cid:1) δ ˆ a ′ t − ˆ d ∂ ψ log Gδψ + ˆ dhu ˆ A ψ ′ δψ ′ − ˆ dh ˆ Au ¯ z ′ δ ˆ z ′ . (33)The equation of motion for δ ˆ a x is given byˆ d ˆ k ˆ ω (¯ z ′ δ ˆ z − u ψ ′ δψ ) − ˆ ω ˆ gu h (cid:0) hu − ¯ z ′ + hu ψ ′ (cid:1) δ ˆ e x + u ˆ ω ∂ u h ˆ gu ( − δ ˆ e ′ x + ˆ kδ ˆ a ′ t ) i − i ˆ ωu δ ˆ a y ∂ u [2 c ( u )] = 0 . (34)Finally, the equation of motion for δ ˆ a y is given by i ˆ k (2 ψ )¯ a ′ δψ + iδ ˆ e x u ∂ u [2 c ( u )] − ˆ gu h (cid:0) hu − ¯ z ′ + hu ψ ′ (cid:1) ˆ ω δ ˆ a y + ˆ gu h ˆ A ˆ k δ ˆ a y − u ∂ u (cid:2) ˆ gu δ ˆ a ′ y (cid:3) = 0 . (35)There is also a constraint coming from the gauge choice a u = 0: − ˆ ω ˆ H + ˆ k ˆ ω ˆ gu ( − δ ˆ e ′ x + kδ ˆ a ′ t ) = 0 . (36) In this subsection we will briefly describe how the above equations of motion are solvedto find quasi-normal modes. Although it is true that in some cases the equations ofmotion decouple from one another, we do not separately discuss those special cases.The method that we will describe here works for any number of coupled or decoupleddifferential equations. 8irst of all, we are interested in normalizable solutions with infalling boundaryconditions for the different fields. By inspection, one finds that all the fields, exceptfor δ ˆ a t , will have the same infalling behavior as u → δψ, δ ˆ z, δ ˆ e x , δ ˆ a y ∼ (1 − u ) − i ˆ ω (37) δ ˆ a t ∼ (1 − u ) − i ˆ ω . (38)We separate out this leading singular behavior. For example, δψ = (1 − u ) − i ˆ ω δψ reg ,where δψ reg is regular at the horizon.We are interested in finding those solutions for which all the fluctuations have anormalizable behavior near the AdS boundary. Generically choosing boundary condi-tions at the horizon and shooting towards u → δψ reg , δ ˆ z reg , δ ˆ e x,reg , δ ˆ a y,reg , δ ˆ a t,reg )to solve, but they are subject to a constraint. Imposing the constraint for this systemis equivalent to imposing it on the boundary conditions at the horizon and satisfyingall the other equations of motion. Thus, there are two alternative routes one canchoose for taking the constraint into account: 1) make use of the constraint to solvefor δ ˆ a ′ t in terms of all the other fields or 2) only impose the constraint on the horizonboundary conditions for the temporal gauge field. Both of the routes are equivalent,but we found it numerically faster to follow the second path. Therefore, we will notlose any equations but make sure to take the constraint into account. According tothe determinant method, we choose a set of linearly independent boundary conditionsat the horizon, say, { δψ reg , δ ˆ z reg , δ ˆ e x,reg , δ ˆ a y,reg } = { (1 , , , , (1 , , , − , (1 , , − , , (1 , − , , } . (39)As mentioned above, the boundary condition for δ ˆ a t,reg at the horizon is set by theconstraint: By expanding the constraint as a power series around the horizon, atzeroth order, one finds a condition for δ ˆ a t,reg (1) in terms of others (this condition isequivalent to the relationship one would find from the δ ˆ a t equation of motion (32)).The derivatives at the horizon are set by the equations of motion. For example, δψ ′ reg (1) is found by expanding the δψ equation of motion (30) around the horizon,and δ ˆ a ′ t,reg (1) is set by the boundary conditions and the derivatives of the other fields.Finally, for any given momentum we solve the set of differential equations fourtimes, corresponding to the four different boundary conditions in (39). The interesting Notice that the different fall-off guarantees that the temporal component of the gauge fieldvanishes at the horizon, as should be the case when the thermal circle pinches off. u ∆ + δψ Ireg u ∆ + δψ IIreg u ∆ + δψ IIIreg u ∆ + δψ IVreg δ ˆ z Ireg δ ˆ z IIreg δ ˆ z IIIreg δ ˆ z IVreg δ ˆ e Ix,reg δ ˆ e IIx,reg δ ˆ e IIIx,reg δ ˆ e IVx,reg δ ˆ a Iy,reg δ ˆ a IIy,reg δ ˆ a IIIy,reg δ ˆ a IVy,reg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u → (40)at the AdS boundary. For a given ˆ k , one then begins to scan over the complexvalued energy ˆ ω until a zero of the determinant is found. Once this is the case, oneconcludes that a normalizable solution has been found. There is a linear combinationof the boundary conditions giving the desired normalizable solution, for which allfluctuations vanish at the AdS boundary.In practice, we start with a limit of the parameters such that the equations decou-ple and consider separately the different fluctuations. The quasi-normal modes areidentified as the values of (ˆ ω, ˆ k ) where the contribution to the determinant changessign. The accuracy of these positions is therefore determined by the resolution of thescan, which in our case is at least 10 − . Away from this limit the determinant iscomplex in general, and we look for positions at which its absolute value is smallerthan some “numerical zero”. The “numerical zero” is chosen so as to maintain thesame accuracy as in the decoupled limit. Generically, the equations for the fluctuations are all coupled. However, there arecertain limits of the parameters ( ˆ d, ˆ m, ˆ k, f ) where some of the equations decouplefrom the rest. This enables us to associate, even in the range of parameters wherethe equations are coupled, certain modes to particular fields at least by name. Butthe reader should not get confused; when the system is in the range of parameterswhere the equations are coupled, this naming may be meaningless.Let us start with the case where ˆ m = 0 and ˆ d = 0. For this particular case ψ ′ = 0 and ¯ a ′ = 0, so the equation for δ ˆ e x decouples from the rest. For δ ˆ e x we finda hydrodynamical mode ( i.e. , ˆ ω (ˆ k → →
0) associated with charge conservation,which is purely imaginary and behaves asˆ ω = − iD ˆ k + . . . , ˆ k ∼ , (41)where D is the diffusion constant. For low ˆ k , this is the important mode since ithas the lowest imaginary value, corresponding to the collective mode with the longestlifetime. However, as ˆ k is increased this mode meets another mode which initiallystarted at ˆ ω (ˆ k = 0) ∼ − i . After the modes meet they become a pair of complexmodes with both real and imaginary parts; the real parts have the same magnitudebut opposite signs. We depict this in Fig. 1 (left panel).10 k `- - Ω` d ` Figure 1: Left: Quasi-normal modes for δ ˆ e x at ˆ d = 0, ˆ m = 0 and ψ ∞ = π/ p ˆ d . Notice the linear behavior for smalltemperatures, ˆ d ≫ d = 0, but still ˆ m = 0, the longitudinal vector and the δ ˆ z scalar arecoupled but the quasi-normal mode structure (if ˆ ω is not too large) is quite similar tothe zero-density case. There is a purely imaginary hydrodynamical mode that meetsat some finite ˆ k another purely imaginary, but non-hydrodynamical, mode, and forlarger ˆ k there are two complex modes. This transition from purely imaginary modesto complex modes can also be regarded as a transition from a hydrodynamical regimeto a collisionless regime (see [25] for a similar phenomenon in a (3+1)-dimensionalboundary theory). The values of both | ˆ ω | and ˆ k at the meeting point decrease withincreasing ˆ d ; see the left panel of Fig. 2. k `- - - Ω` k `- - - - - Ω` Figure 2: Left: Quasi-normal mode for the coupled ( δ ˆ e x , δ ˆ z ) system at ˆ m = 0, ˆ d = 10 and ψ ∞ = π/
4. Right: Imaginary part of the quasi-normal mode of the δ ˆ e x , δ ˆ z system,with ˆ m = 0, ˆ d = 5 and ψ ∞ = π/ d = 0 is shown on the right ofFig. 2. The imaginary part decreases as ˆ k increases and then increases for very largeˆ k towards zero. As ˆ d is increased, the minimum of Im ˆ ω shifts to larger ˆ k .Indeed, this mode has the properties of the holographic zero sound mode [23, 26].As T → d (and therefore ˆ d → ∞ ), the purely diffusive mode disap-pears completely and the complex pair of modes start at ˆ k →
0, with an approximate11ispersion relation ˆ ω = v ˆ k − ia ˆ k + . . . . (42)A calculation of v , for all f , gives v ∼ √ in the large ˆ d limit, in agreement with [23].In this model, in addition to the quantum attenuation at zero temperature, thereis very large damping for low momentum at non-zero temperature, turning the zerosound mode to a purely dissipative mode.However, as we will see in the next subsection, these results can only be trustedup to some critical value of ˆ d , beyond which there is an instability towards a stripedphase. In this section we show that at large enough ˆ d , an instability occurs. This instabilityoccurs only at some non-zero k phys , similar to what was found in [17] but now in a(2+1)-dimensional theory. This indicates that the Fermi liquid is unstable and thatthe true ground state of the (2+1)-dimensional theory will be a striped phase, similarto a charge density and spin wave. k `- - - - Ω` k `- - Ω` Figure 3: Lowest quasi-normal mode for δ ˆ a y (left) and δψ (right), with ˆ d = 0 , ˆ m = 0and ψ ∞ = π/ d = 0 and ˆ m = 0. In this case the transverse gauge field fluctuation δ ˆ a y and the embedding scalar fluctuation δψ decouple. The lowest quasi-normalmodes are portrayed in Fig. 3. At ˆ k = 0 the lowest quasi-normal mode has the samevalue of ˆ ω as the second quasi-normal mode of the longitudinal vector (equations (34)and (35) are the same in this limit).For ˆ d > m = 0 the modes δ ˆ a y and δ ˆ ψ couple to one another. Above a criticalvalue of ˆ d , the lowest negative imaginary quasi-normal mode crosses onto the upperhalf-plane, at some value of ˆ k , and becomes a positive imaginary frequency mode, i.e. , an instability (Fig. 4, left panel). The critical value is ˆ d c ≃ .
5. There is a rangein ˆ k for which the frequency is positive imaginary, and at some larger ˆ k it crossesagain back onto the lower part of the complex frequency plane. The dependence ofthis range on ˆ d is shown in Fig. 4 (right panel). In this range the system is driven12owards a striped phase, where both the transverse current and the fermion bi-linearare spatially modulated. The latter, in particular, describes a spin density wave. Thiscan be seen, for example, from the fact that at finite density a spatially modulatedfield ψ couples linearly to the magnetic field through the bulk CS term in (16). k `- - - - - - Ω` d ` k ` d ` Figure 4: Properties of the unstable mode with ˆ m = 0 and ψ ∞ = π/
4. Left: Lowestpurely imaginary mode for the coupled ( δ ˆ a y , δ ˆ ψ ) system. The lowest curve is forˆ d = 5, and the upper curve is for ˆ d = 6. Right: Range of instability; ˆ k min (blue) andˆ k max (red) are plotted against ˆ d . - - - m ` k ` Figure 5: Domains of instability for non-zero mass ˆ m and ψ ∞ = π/ d = 6 (smallerdomain) and ˆ d = 10 (larger domain).The situation is similar when ˆ m = 0. In this case all the fluctuations are coupled,and we still find an instability. As | ˆ m | increases, the critical value of ˆ d grows; seeFig. 5. Since now δ ˆ a t is also involved, one also obtains a charge density wave. As T → k phys ≃ . √ dL . Things are qualitativelysimilar when ψ ∞ = π/
4. For instance when one decreases ψ ∞ from the value π/ d c decreases slightly. 13 Conclusion
The ungapped (BH embedding) phase of the D3-D7’ system has an instability to-wards a striped phase above a critical value of the ratio of the charge density to thetemperature squared. For a non-zero mass the instability involves all the fields, andtherefore one expects that the new phase will be such that all the physical quantities(normalizable modes) associated with these fields are spatially modulated. These in-clude charge, spin, and the transverse currents. In the dual bulk theory the instabilityis the result of an axion-like term in the action. On the other hand, the existence ofthis term is intimately related to the fermionic nature of the boundary theory. Webelieve that striped phases are a general property of such theories. Since the modelshould have a striped phase, one should be able to construct it as a solution of theequation of motion and study its properties.
Acknowledgments
We thank Daniel Podolsky for useful comments and discussions. O.B. is supportedin part by the Israel Science Foundation under grant no. 392/09, and the US-IsraelBinational Science Foundation under grant no. 2008-072. N.J. has been supportedin part by the Israel Science Foundation under grant no. 392/09 and in part at theTechnion by a fellowship from the Lady Davis Foundation. The work of G.L. issupported in part by the Israel Science Foundation under grant no. 392/09. Theresearch of M.L. is supported by the European Union grant FP7-REGPOT-2008-1-CreteHEPCosmo-228644.
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