Criteria For Superfluid Instabilities of Geometries with Hyperscaling Violation
CCriteria For Superfluid Instabilities of Geometrieswith Hyperscaling Violation
Sera Cremonini ♠♠ Department of Physics, Lehigh University,Bethlehem, PA, 18018 USA
Li Li ♣ (cid:7) ♣ Crete Center for Theoretical Physics, (cid:7)
Crete Center for Quantum Complexity and Nanotechnology,Department of Physics, University of Crete, 71003 Heraklion, Greece (Dated: September 26, 2018)
Abstract
We examine the onset of superfluid instabilities for geometries that exhibit hyperscaling violation andLifshitz-like scaling at infrared and intermediate energy scales, and approach AdS in the ultraviolet. Inparticular, we are interested in the role of a non-trivial coupling between the neutral scalar supporting thescaling regime, and the (charged) complex scalar which condenses. The analysis focuses exclusively on un-stable modes arising from the hyperscaling-violating portion of the geometry. Working at zero temperature,we identify simple analytical criteria for the presence of scalar instabilities, and discuss under which condi-tions a minimal charge will be needed to trigger a transition. Finite temperature examples are constructednumerically for a few illustrative cases. a r X i v : . [ h e p - t h ] J un ontents I. Introduction and Summary of Results
II. Setup III. Background geometry
IV. Effective mass and superfluid instability windows τ = 2( m −
1) 17B. Non-scaling case, τ (cid:54) = 2( m −
1) 19
V. Effective Schr¨odinger Potential and Instabilities
VI. Numerics
Acknowledgments References . INTRODUCTION AND SUMMARY OF RESULTS
The recent efforts to use holography to probe strongly coupled quantum systems have led tonew insights into the possible instabilities of a variety of gravitational solutions. One of the primeexamples is that of charged black holes in Anti de Sitter (AdS) space, which have been understoodto be unstable to the formation of scalar hair – thanks to attempts to realize the spontaneousbreaking of an abelian gauge symmetry in gravity [1], and develop a holographic description ofsuperconducting phases [2, 3]. For reviews of holographic superconductors we refer the reader to e.g. [4–7]. Other notable examples include the spontaneous breaking of translational invariance andthe onset of spatially modulated instabilities, which have been identified in a number of geometries(see [8–12] for some of the early papers) and have potential applications to e.g. QCD and condensedmatter systems with striped phases. We have seen growing interest in constructing gravitationalsolutions that exhibit a variety of broken symmetries, with significant attention recently given torealizing holographic lattices through the (explicit) breaking of translational invariance (see e.g. [13–21]).In this paper we revisit the question of scalar field instabilities associated with geometries thatexhibit hyperscaling violation θ and non-relativistic scaling z , with the ultimate goal of reachinga more complete understanding of low temperature superconducting phase transitions in the dualsystems. We will work with gravitational solutions which are hyperscaling violating and Lifshitz-like at infrared (IR) and intermediate energies, and asymptote to AdS in the ultraviolet (UV). Suchgeometries are well known to arise in Einstein-Maxwell-dilaton theories, and are supported by aneutral scalar subject to a rather simple potential. We require AdS asymptotics to ensure that thedual field theory is conformal at the UV fixed point – so that the violation of hyperscaling andrelativistic symmetry is generated at lower energies – and thus can rely on the standard holographicdictionary. We stress that we are only interested in phase transitions that are triggered in thehyperscaling violating regime itself, since in full generality they are much less understood thantheir AdS counterpart.Charged scalar field condensation on non-relativistic backgrounds that don’t respect hyper-scaling has been studied in a number of settings (see e.g. [22–24] but the list is by no meansexhaustive), although typically for specific values of the scaling exponents z and θ or in somewhat Strictly speaking, the dual theory consists of a condensate breaking a global U (1) symmetry, so the description is ofa superfluid rather than a superconductor. However, considering the limit in which the U (1) symmetry is ”weaklygauged”, we can still view the dual theory as a superconductor. In the present paper we will not distinguishbetween the two terminologies. ∼ B ( φ ) | Ψ | between the neutral scalar φ that determines the background and the chargedscalar Ψ that condenses. We will obtain analytical instability criteria – attempting to be generic,to the extent that it is possible – and highlight the role of B ( φ ) on the onset of the superfluid phasetransition. Since B ( φ ) contributes to the effective mass of the charged scalar, it is intuitively clearthat it will affect the condensation process – enhancing it or impeding it depending on its sign andits radial profile. Throughout the paper we will adopt the choice B ( φ ) ∼ e ˆ τφ in the hyperscalingviolating regime, with ˆ τ an arbitrary constant.To probe the onset of the formation of scalar hair, we are going to focus on the linearizedperturbation of the charged scalar Ψ around the unbroken phase. To obtain the linearized equationof motion for Ψ it suffices to know the structure of the charged scalar couplings up to quadraticorder – such leading terms are enough to compute the temperature at which the unbroken phasebecomes unstable to scalar hair. One should keep in mind, however, that the nonlinear details ofthe couplings could affect the order of the phase transition and the thermodynamics, as has beenstressed in [25].Our instability analysis will be done in two complementary ways. After setting up the modeland the background in Sections II and III, we will inspect the behavior of the effective mass M eff of the charged scalar in Section IV, and in particular, the conditions under which it becomessufficiently negative. In Section V we will then recast the linearized perturbation of the chargedscalar in Schr¨odinger form, and perform a more detailed instability analysis by examining whetherthe effective Schr¨odinger potential V Schr is sufficiently negative to support bound states (for studiesof instabilities in terms of an effective Schr¨odinger potential see e.g. [3, 26, 27]). To complementthe intuition developed from examining M eff and V Schr , one should also analyze the structure of IRperturbations of the charged scalar, to ensure that they can indeed support a scalar condensate.As we will see, this can rule out regions of parameter space for which M eff and V Schr may beambiguous. For simplicity, our analytical arguments are developed working at zero temperature,and are meant to serve as guidance for a more detailed finite temperature analysis. Still, we believethat they capture all the essential physics of their low temperature counterpart, as we confirm inour numerical section VI, in a few illustrative cases. We leave a more thorough finite temperatureanalysis to future work.We will find many similarities with the standard holographic superconductor setup, but alsosome crucial differences. As in [1–3], two distinct mechanisms can lead to the condensation of ascalar in these background geometries. The gauge field contribution to the effective mass M eff
4f Ψ is always negative and can become large enough to make it energetically favorable for thesystem to undergo a superfluid phase transition. Similarly, a negative coupling B ( φ ) can drive M eff to become appreciably negative, thus facilitating the transition. Since the latter process canhappen even at zero charge, it allows neutral scalars to condense – and it is of course the analogof violating BF bounds in AdS.What is novel in the models we consider here is the rich behavior associated with the possibleprofiles of the coupling B ( φ ), and its effect on the interplay between the two instability mechanisms.In particular, the condensation process is highly sensitive to the specific way in which B ( φ ) scalesas compared to the { z, θ } background geometry – qualitatively new behavior will be seen when theeffective mass term B ( φ ) ψ does not respect the scaling of the charged scalar kinetic term (here ψ denotes the modulus of the complex scalar Ψ). We should note that the role of a coupling ∼ B ( φ ) ψ in hyperscaling violating backgrounds was already discussed by [24], although in a slightly differentcontext. Choosing the coupling so that B ( φ ) ψ scales as ∼ ( ∂ψ ) , the authors noted the presenceof a minimal charge needed to form a condensate, and raised the question of whether it could be auniversal feature. Here we will address this point working with general classes of { z, θ } geometriesand couplings B ( φ ) ∼ e ˆ τφ , and show that this is not generally the case – there is a somewhat largeparameter space where neutral scalars can condense. We will also identify the cases in which weexpect to see a minimal charge. As we will see, the existence of the latter will be sensitive to thedetailed behavior of B ( φ ). Again, we find some crucial differences with the standard holographicsuperconductor setup , that can be traced to the non-trivial scaling properties of the coupling B and the background itself. A. Summary of Results
We work with the Lagrangian given in (5), so that the dual field theory has d spatial dimensions.To respect the scaling of the potential V ( φ ) ∝ e − βφ and gauge kinetic function Z ( φ ) ∝ e αφ of thehyperscaling violating background, we have taken the coupling between the two scalars to be ofthe form B ( φ ) ∼ e ˆ τφ or, in terms of the holographic radial coordinate r , B ( r ) = B r τ , (1)with τ an arbitrary constant. However, see [25, 28] for additional ways to modify the effective mass of a charged scalar in the IR
AdS region. M eff ( r ) = ˜ L (cid:104) B r τ − m − − Q r dn (cid:105) + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) , (2)and then in Section V by examining when the effective Schr¨odinger potential V Schr ( r ) = r m − (cid:20) B r τ − m − − Q r dn + 14 ˜ L dn (cid:16) dn + 4 m − (cid:17)(cid:21) (3)develops negative regions which can support the existence of bound states. Here Q is proportionalto the charge of Ψ, ˜ L is a length scale defined in the main text and the parameters { m, n } are m = zd − θzd − θ , n = d − θzd − θ . (4)With our choice of coordinates the IR is located at r = 0, while r = r tr will denote the transitionscale between the non-relativistic, hyperscaling violating solution and the UV AdS region.The possible sources of instability are now apparent. Superfluid phase transitions are gener-ically triggered by a sufficiently large charge term ∝ Q , driving M eff imaginary and V Schr negative. A negative and suitably large contribution from the coupling ∝ B will have the sameeffect, and is responsible for the formation of a condensate even when Q = 0. Moreover, theinterplay between the two terms can lead to interesting behaviors, depending on how τ comparesto the exponents m and n . While these expressions were obtained at zero temperature, they areexpected to capture the key aspects of the finite temperature behavior. This is shown for a fewillustrative cases in Section VI.The main features that have emerged from this analysis are the following: • In these hyperscaling violating backgrounds the gauge field term ∼ Q r dn always decreasestowards to IR (as r → n >
0, as discussed in the main text. • The competition between the contributions coming from the U (1) gauge field and the realneutral scalar is very sensitive to the way in which B ( φ ) scales compared to the background,in particular to whether τ is larger or smaller than 2( m − • Simplifications occur for the scaling choice τ = 2( m − B ( φ ) ψ scaling in the same way as the kinetic term ( ∂ψ ) . In this case the only radialdependence of M eff comes from the charge term, and in the deep IR one obtains generalizedBF bounds analogous to those in AdS. 6 In the scaling case τ = 2( m −
1) neutral scalars will condense when B is sufficiently negative.There will otherwise be a minimal charge Q min needed to trigger the condensation, as inthe standard AdS case. However, here bound states are supported near the transition region r ∼ r tr to AdS, and instabilities are therefore associated with the “effective UV” of the { z, θ } geometry, and not with its IR. • When the scaling τ is arbitrary the behavior is more complex:(i) For B < τ < m −
1) the coupling makes V Schr and M eff more and morenegative as the IR is approached. Thus, neutral scalars will condense generically,without having to tune the size of B , unlike in the standard AdS case. The instabilityis now associated with the IR of the geometry, and there is no minimal charge.(ii) In all other cases a minimal charge seems to be needed to trigger the phase transition.A particularly interesting case corresponds to B < τ > m − Q min exists independently of how large | B | is tuned to be, unlike in the standard AdS story. • The choice τ − m −
1) = 2 dn is also special, since the coupling and charge contributionsto M eff and V Schr scale in the same way, ∝ r dn (cid:2) B − Q (cid:3) :(i) For B > Q there will never be a phase transition triggered in the IR hyperscalingviolating region, no matter how large the charge is.(ii) For B < Q we expect to have a condensate, as long as the effective mass can becomenegative enough near r tr , where r dn attains its largest value. Thus, one can trigger atransition by varying B across the critical value Q . However, there will always be aminimal charge, no matter how negative B is. • The transition scale r tr between the hyperscaling violating geometry and the AdS regionplays a crucial role in controlling the onset of the instability and the value of the minimalcharge. II. SETUP
We want to examine D = d + 2 dimensional Einstein-Maxwell-dilaton theories coupled to acomplex scalar field Ψ, L d +2 = R −
12 ( ∂φ ) − Z ( φ ) F µν F µν − V ( φ ) − C ( φ ) (cid:0) | D Ψ | + B ( φ ) | Ψ | (cid:1) , (5)7hich is charged under the U (1) field A µ , so that D µ Ψ = ( ∂ µ + iqA µ )Ψ. For now we allow fortwo arbitrary couplings C ( φ ) and B ( φ ) between the neutral and the charged scalars. The formerresults in a non-canonical kinetic term for Ψ and will be set to one shortly, the latter acts as aneffective mass for Ψ and will be the focus of our discussion.Einstein’s equations for (5) are given by R µν + 12 Z ( φ ) F µρ F ρν − ∂ µ φ ∂ ν φ − C ( φ ) [ D µ Ψ( D ν Ψ) ∗ + D ν Ψ( D µ Ψ) ∗ ]+ 12 g µν (cid:20)
12 ( ∂φ ) + V ( φ ) − R + Z F + C ( φ ) (cid:0) | D Ψ | + B ( φ ) | Ψ | (cid:1)(cid:21) = 0 , (6)where, writing the charged scalar as Ψ = ψe iθ , we have | D Ψ | = (cid:2) ( ∂ψ ) + ψ ( ∂θ + qA ) (cid:3) . (7)The gauge field equation of motion is1 √− g ∂ µ (cid:0) √− gZF µν (cid:1) = 2 C q A ν | Ψ | + i q C [Ψ ∂ ν Ψ ∗ − Ψ ∗ ∂ ν Ψ]= 2
C q A ν ψ + 2 C q ψ ∂ ν θ , (8)while the neutral scalar obeys (cid:3) φ = ∂V∂φ + 14 ∂Z∂φ F + ∂C∂φ | D Ψ | + ∂ ( C B ) ∂φ | Ψ | . (9)Finally, the real ψ and imaginary θ parts of the charged scalar satisfy1 √− g C ∂ µ (cid:0) √− g C∂ µ ψ (cid:1) = (cid:2) ( ∂θ + qA ) + B (cid:3) ψ , (10) ∂ µ (cid:2) √− g C ψ ( ∂ µ θ + qA µ ) (cid:3) = 0 . (11)We take the phase of the charged scalar to vanish, θ = 0. This solves the equation of motion (11)when the gauge field is purely electric, A = A t ( r ) dt and no fields depend explicitly on time. Thecharged scalar equation of motion then becomes1 √− g C ( φ ) ∂ µ (cid:0) √− g C ( φ ) ∂ µ ψ (cid:1) = (cid:2) q A µ A µ + B ( φ ) (cid:3) ψ . (12)While the non-canonicality function C ( φ ) could contribute to the instabilities in an interestingway – it clearly affects the scaling behavior of the charged scalar, and hence the scaling dimension Superconducting/Superfluid instabilities in a theory with non-canonical couplings has been considered recently ontop of soft wall backgrounds in [29].
8f the dual operator – here for simplicity we will neglect it and set C = 1, focusing instead on therole of the B ( φ ) coupling. We then see that (12) becomes (cid:3) ψ = m eff ψ , (13)with the effective mass given by m eff = q A µ A µ + B ( φ ) = − q A t | g tt | + B ( φ ) . (14)As in the case of the standard holographic superconductor [1–3], the condensation of the chargedscalar field will depend on the interplay between the two contributions to its effective mass, onecoming from the coupling B ( φ ) and the other from the charge. However, we will see that theadditional dependence on the neutral scalar – and in particular, the fact that the profile of B ( φ )will depend on the holographic radial coordinate and can be chosen to scale in different ways – willlead to some interesting differences. III. BACKGROUND GEOMETRY
The instability we are interested in is associated with the formation of charged scalar hairaround the normal unbroken black brane background in which ψ is zero. In the vicinity of thetransition point at which scalar hair begins to develop, the value of ψ should be very small, andbackreaction negligible. As a result we can treat the charged scalar as a perturbation on top of thebackground solution which interpolates between asymptotic AdS and a Lifshitz-like, hyperscalingviolating region that extends into the IR. More precisely, the latter geometry extends over therange r IR < r < r tr , with AdS describing the remaining r tr < r < r UV portion of the spacetime.Thus, r tr denotes the transition scale between the two regimes, while r IR and r UV correspond tothe IR and UV endpoints of RG flow. In what follows we set the charged scalar field to zero, andfocus entirely on the background geometry. A. The hyperscaling violating background solution
We begin by discussing the non-relativistic { z, θ } scaling solutions that range over the infraredand intermediate part of the geometry. It is well known that such solutions can be generatedin the class of models (5) by taking the scalar potential and gauge kinetic function to be simpleexponentials, Z ( φ ) = Z e αφ , V ( φ ) = − V e − βφ , (15)9here V and Z are arbitrary positive constants. Black brane solutions are then given by [30–33] ds = ρ θd (cid:18) − f ( ρ ) dt ρ z + L ρ dρ f ( ρ ) + d(cid:126)x ρ (cid:19) ,f ( ρ ) = 1 − (cid:18) ρρ h (cid:19) d + z − θ , L = ( d − z − θ )( d + z − θ ) V ,α = 2 dκ − d − θdκ , β = 2 θdκ , κ = 2( θ − d )( θ − dz + d ) d , (16)and are supported by the following scalar and gauge field profiles φ = κ ln( ρ ) , A = a ρ θ − z − d f ( ρ ) dt , a = (cid:115) z − Z ( d + z − θ ) . (17)In the extreme limit the metric reduces to ds = ρ θd (cid:18) − dt ρ z + L dρ ρ + d(cid:126)x ρ (cid:19) , (18)and is known to suffer generically from curvature and null singularities. Moreover, the logarith-mically running scalar φ ∼ ln ρ tends to drive the bulk gravitational theory to strong or weakcoupling, depending on whether the gauge field is chosen to describe a magnetic or an electric field.Although a possible resolution comes from turning on a temperature, the presence of instabilitiesin these systems generically indicates that there may be additional ground states. Possible IRcompletions of these scaling geometries have been discussed e.g. in [34–43].There are some disadvantages to using the ρ radial coordinate adopted above. For example ,whether the IR is located at ρ = 0 or ρ → ∞ depends on the values of { z, θ } . Also, in thesecoordinates in order to recover the standard AdS × R d extremal solution to Einstein-Maxwell theory(with constant φ ) one must take the limit z → + ∞ with θ finite, which makes a direct comparisonto the standard holographic superconductor (in which AdS plays a crucial role) cumbersome. Toavoid some of these difficulties we will choose to work with a new radial coordinate r , in termsof which the IR in the zero temperature solution is always located at r = 0. By performing thefollowing transformation, ρ = r d θ − dz , ρ h = r d θ − dz h , a = ˜ a , κ = 2 θ − dzd ˜ κ,z = 2 m − m + n − , θ = d ( m − m + n − , ˜ L = L ( m + n − , (19) In order to have an unambiguous IR one must require (cid:0) θd − z (cid:1) (cid:0) θd − (cid:1) >
0, which simply ensures that the ( t, (cid:126)x )components of the metric scale in the same way with ρ . ds = − r m ˜ f ( r ) dt + ˜ L dr r m ˜ f ( r ) + r n d(cid:126)x , ˜ f ( r ) = 1 − (cid:16) r h r (cid:17) m + dn − , A = ˜ a r m + dn − ˜ f ( r ) dt, φ = ˜ κ ln( r ) , ˜ L = ( m + ( d − n )(2 m + dn − V , ˜ a = (cid:115) m − n ) Z (2 m + dn − , ˜ κ = 2 dn (1 − n ) ,α = 2(1 − m − dn )˜ κ , β = 2(1 − m )˜ κ , (20)where r h denotes the location of the horizon. We have traded the scaling exponents { z, θ } forthe two parameters { m, n } . In terms of these, the AdS × R d geometry is obtained by choosing m = 1 , n = α = β = 0, while geometries that are conformal to AdS × R d correspond to m + n = 1with m (cid:54) = 1 , m (cid:54) = 1 /
2. Finally, in the extreme limit the metric and gauge field reduce to ds = − r m dt + ˜ L dr r m + r n d(cid:126)x , A = ˜ a r m + dn − dt , (22)with the scalar field maintaining its log form. The temperature and entropy density associatedwith these black brane solutions are given by T = | m + dn − | π ˜ L r m − h , S = 14 G N r dnh , (23)so that the thermal entropy can be seen to scale like S ∼ T dn m − ∼ T d − θz , (24)which can be interpreted as describing a system in which the degrees of freedom occupy an effectivenumber of dimensions ∼ d eff = d − θ .In addition to the background geometry (20), in which the gauge field flux is non-trivial, thetheory we are considering admits another type of hyperscaling violating solution for which A t For completeness we include the expression for { m, n } in terms of the original exponents, m = zd − θzd − θ , n = d − θzd − θ . (21) However, note that when m = 1 and n = 0 (i.e., the AdS case) the above transformation fails, because ( z, θ, L )are not well defined. In the special case m = 1 /
2, or equivalently z = 0, the temperature is independent of r h . ds = − r n ˜ f ( r ) dt + ˜ L dr r n ˜ f ( r ) + r n d(cid:126)x , ˜ f ( r ) = 1 − (cid:16) r h r (cid:17) (2+ d ) n − , A t = 0 , φ = ˜ κ ln( r ) , ˜ L = dn ((2 + d ) n − V , ˜ κ = 2 dn (1 − n ) , β = 2(1 − n )˜ κ , (25)which can be considered as the special case of (20) with m = n , or equivalently z = 1. Thesesolutions are characterized entirely by the hyperscaling violating exponent θ . We will refer to themas IR neutral throughout the text. B. The asymptotic AdS solution
To adopt the standard holographic dictionary and ensure a UV CFT, we would like to embedthese solutions in AdS space. This can be easily done by modifying the scalar potential V ( φ )appropriately, so that the neutral field φ can settle to a constant value φ UV at the boundary. Morespecifically, the effective scalar potential V eff = V ( φ ) + Z ( φ ) F will have to be chosen so thatit admits an extremum at the UV fixed point, V (cid:48) eff ( φ UV ) = 0. It will suffice to add a secondexponential to (15), so that V ( φ ) → − V e − βφ + V e γφ , as done e.g. in [31, 43]. The transition scale r tr to AdS is then determined by the location at whichthe new term in the potential begins to dominate over the original V term. The new exponentialwill then determine the properties of the AdS UV background solution. In the numerical studiesof Section VI we will work for convenience with a scalar potential V ∼ cosh φ . However, more general choices can easily be implemented. Furthemore, since we are interested inidentifying the instabilities that arise solely from the hyperscaling violating region of the geometry,any term in the potential which dominates only in the UV will not affect the main discussion ofthis paper.
C. Constraints on the parameter space of the scaling exponents
The allowed parameter space of the scaling exponents { m, n } (or equivalently { z, θ } ), can berestricted by imposing a number of physical constraints, which will ensure that the background12an be taken to describe a well-defined ground state. Here we focus on the IR charged solutionand exclude the AdS geometry for the sake of greater clarity.(a) By inspecting the form of the metric note that in order for the solution (20) to be real, weshould demand ˜ L > , ˜ κ > , ˜ a > , (26)from which we obtain n (1 − n ) > , ( m + ( d − n )(2 m + dn − > , m − n m + dn − > , (27)or alternatively in terms of z and θ ,( θ − d )( θ − dz + d ) > , ( d − z − θ )( d + z − θ ) > , ( z − d + z − θ ) > . (28)For the IR neutral case (25), the last relation must be set to zero, i.e. z = 1 or m = n .(b) To have an unambiguous IR we should require the ( t, (cid:126)x ) components of the metric scale in thesame way with r in (22), which means m n > . (29)The location of the IR depends on where the ( t, (cid:126)x ) metric elements vanish. Inspecting (27)one finds that m > n >
0. Therefore the IR is located at r = 0.(c) To resolve the deep IR singularity of the geometry (22), we require the temperature deformationto be relevant, following the discussion of [30, 44]. This corresponds to the following constraint,2 m + dn − > , or equivalently d ( z + d − θ ) zd − θ > , (30)which however is already imposed by (27) and (29).(d) We would like the geometry to have positive specific heat. From the scaling of the entropywith temperature, we should demand dn m − > , or equivalently d − θz > , (31)which implies that m > since n is positive.13he allowed parameter range once we combine all the conditions above is given by (cid:104) < m (cid:54) , < n < m (cid:105) , [ m > , < n < . (33)For completeness we include the final { z, θ } parameter space in terms of the original ρ coordinateused in (16), IR located at ρ → z < , θ > d ] , IR located at ρ → ∞ : [1 < z (cid:54) , d + θ < dz ] , [ z > , θ < d ] . (34)One can easily check that Null Energy Condition is automatically satisfied. IV. EFFECTIVE MASS AND SUPERFLUID INSTABILITY WINDOWS
Having introduced the properties of the background geometry we will be working with, we arenow ready to examine under what conditions the charged scalar field can condense. For simplicitywe will treat ψ as a perturbation on top of the hyperscaling violating solutions we have justdiscussed, and neglect the effects of backreaction. Since we are zooming in on the transition pointat which scalar hair begins to form – the onset of the instability – the ψ scalar is going to be verysmall, and ignoring backreaction should be a good approximation.In this section we are going to approach the question of instabilities by asking what we canlearn from the structure of the effective mass (14) of the charged scalar, m eff = − q A t | g tt | + B ( φ ) , (35)and focus entirely on unstable modes which arise from the hyperscaling violating region of thegeometry, r IR ≤ r ≤ r tr . We will obtain simple analytical instability conditions which include, inthe most tractable cases, generalizations of the well-known BF bound for AdS space. Althoughwe work for simplicity at zero temperature, we expect these conditions to capture all the essentialfeatures of the finite temperature phase transition (as long as the temperature is not too large).Indeed, this will be confirmed by the analysis of section VI, where we will revisit the intuitiondeveloped here by performing numerical studies in the background of finite temperature solutions. For the IR neutral background (25), the parameter range reads12 < ( m = n ) < . (32) B ( φ ) and A µ A µ ,with the two contributions to the effective mass competing against each other when B is positive,and otherwise enhancing each other. It will be the structure of the coupling B ( φ ) between the twoscalars – and in particular, how it scales compared to the { z, θ } background – that will be at theroot of the key differences with the standard AdS story.Indeed, if we want to ensure that the mass term B ( φ ) ψ scales in the same way as the kineticterm ( ∂ψ ) , the coupling B must be chosen appropriately , as discussed e.g. in [45]. More precisely,in the hyperscaling violating portion of the geometry the kinetic term scales as( ∂ψ ) ∼ ˜ f ( r ) r m r ψ ∼ r m − ψ , (36)where in the last expression we have switched off the temperature by taking ˜ f ( r ) = 1. Thus, inorder for the B ψ mass term to respect this scaling one needs B ( φ ) ∼ r m − ⇒ B ( φ ) ∼ e m − κ φ . (37)The gauge field contribution to the effective mass of the scalar, i.e. q A ψ , will generically scaledifferently, in particular q A µ A µ ∼ − q r m + dn − , (38)and will agree with (37) only when n = 0, or equivalently θ = d , which is outside the allowedparameter space (33) and (34) of interest here . In this paper we will take the coupling to be ageneric power law (an exponential function of φ ), B ( r ) = B r τ , (39)with the case preserving the scaling of the kinetic term corresponding to τ = 2( m − ⇒ B = B r m − = B r θdz − θ . (40) The additional coupling C ( φ ), which we set to unity, would not affect the relative scaling between the kinetic andmass terms but it would change the overall scaling of the term C ( | D Ψ | + B | Ψ | ) in the action. It may be worth examining this case separately, as it could lead to a qualitatively different behavior.
15t turns out to be convenient to let ψ ( r ) = r − m − dn g ( r ), so that the equation of motion forthe charged scalar (12) can be written in the suggestive form ∂ r (cid:0) r ∂ r g (cid:1) = (cid:20) ˜ L (cid:16) B r τ − m − − Q r dn (cid:17) + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17)(cid:21) g, (41)where we have used (39) and defined Q ≡ ˜ a q . (42)Since the term on the left-hand side of (41) is essentially the AdS d’Alembertian (with the radius L AdS = 1), we can interpret the right-hand side of the equation as defining the analog of aneffective mass in AdS , i.e. M eff ( r ) ≡ ˜ L [ B r τ − m − − Q r dn ] + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) , (43)where the last constant term in terms of the original scaling exponents reads (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) = ( d − dθ + 2 θ )( d − dθ + 2 zd − d )4( zd − θ ) . (44)Indeed, if we set Q = 0, m = 1 and n = τ = 0, we recover the pure AdS × R d case, for which M eff = ˜ L B and the solution to (41) is g ( r ) ∝ r − ± ν , ν = (cid:113) L B , (45)with the AdS BF bound coming from requiring the index ν not to become imaginary, B ˜ L ≥ − . (46)On the other hand, when z (cid:54) = 1, θ (cid:54) = 0 and Q (cid:54) = 0 the effective mass (43) depends genericallyon the radial coordinate, and we lack a sharp local instability criterion, unlike in the simple AdS case. Still, instabilities can be expected to appear if M eff becomes negative enough. Interestingly,even for generic values of the scaling exponents – as long as they fall within the range (33) – theconstant term satisfies (cid:0) m + dn (cid:1) (cid:0) m + dn − (cid:1) > − and remains above the AdS BF bound.Thus, it will be the combination of the charge term ∝ Q and the coupling B which will typicallygenerate a sizable negative contribution to M eff . Indeed, it is apparent from (43) that a scalarfield condensate can form via two distinct mechanisms, as is well known. First, a sufficiently largeand negative B ( φ ) can trigger the transition, allowing even neutral scalars to condense (as alreadyknown from AdS). The second mechanism is the usual negative contribution to M eff coming fromthe gauge field term, which can make it energetically favorable for the charged scalar to condense.16owever, there are some key differences with the usual holographic superconductor setup. First,depending on the scaling behavior of B ( φ ) interesting competitions between the two mechanismscan be generated. More importantly, note that the contribution to (43) from the U (1) gauge fieldbecomes less and less important as the IR is approached , and vanishes at r = 0. Thus, herewe expect instabilities associated with the charge term to be generically localized close to r tr (orpossibly at some intermediate radial distance r at which the effective mass M eff ( r ) has a deepnegative minimum) and not in the IR. As we will see shortly, this behavior can be modified forcertain choices of B , but it is otherwise robust. Below we are going to make the discussion morequantitative by highlighting a few cases, and leave a more detailed analysis to Section V. A. Scaling case τ = 2( m − (i) Neutral scalar:
We consider first the case of a neutral scalar. When Q = 0 the effective mass is just a constant, M eff = ˜ L B + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) , (47)and the ψ perturbation has the power law form ψ ( r ) = r − m − dn g ( r ) = r − m − d n ± ν = r − d ( z + d − θ ) zd − θ ± ν , (48)with the exponent ν given by ν = (cid:113) M eff = (cid:115) L B + d ( z + d − θ ) ( dz − θ ) = ˜ LL (cid:112) B L + ( z + d − θ ) . (49)Requiring the index ν not to become imaginary immediately leads to the non-relativistic, hyper-scaling violating analog of the standard AdS BF bound,4 B L ≥ − ( z + d − θ ) . (50)It can be equivalently expressed in terms of the effective mass, M eff ≥ − , (51) In the
AdS case n = 0, therefore the U (1) gauge field has a finite contribution in the IR. M eff dippingbelow the critical mass saturating the AdS BF bound, M AdS = − /
4, even with genericscaling exponents z (cid:54) = 1, θ (cid:54) = 0. Thus, in these scaling backgrounds we expect a neutral scalarto be able to condense provided the value of B is negative enough to violate (51), as in thesimpler AdS case. Finally, we note that the generalized BF bound (50) was already obtained in [46].(ii) Charged scalar:
When we restore the charge, the effective mass becomes radially dependent, M eff = ˜ L B − ˜ L Q r dn + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) , (52)and the perturbation g ( r ) is a combination of Bessel functions, g ( r ) = c J ˜ ν (cid:32) Q ˜ Ldn r dn (cid:33) + c Y ˜ ν (cid:32) Q ˜ Ldn r dn (cid:33) , (53)with the index ˜ ν of the Bessel functions related to that appearing in (49) through˜ ν = 12 dn ν . (54)Thus, as in the case of vanishing charge, there will be an instability when the mass term ∼ B is sonegative that it violates the generalized BF bound (51), corresponding to the index of the Besselfunctions becoming imaginary.Of course, there is an additional source of instability which is driven by the charge term becomingsufficiently negative. Unlike in the case of the standard holographic superconductor [3], however,here the gauge field term (which approaches zero as r →
0) dominates not in the deep IR, but rathernear the r ∼ r tr transition region to AdS. Indeed, within the hyperscaling violating portion of thegeometry, Q r dn attains its largest value at r = r tr , and that is where we expect the superfluidinstability to be localized. As a result, a necessary condition for the formation of instabilities is˜ L Q r dntr > ˜ L B + (cid:16) m + 12 dn (cid:17)(cid:16) m + 12 dn − (cid:17) , (55)which can be satisfied by increasing the charge or alternatively pushing the transition region r tr closer and closer to the UV. Note that in these constructions r tr plays a crucial role in controlling theonset of the phase transition. The discussion above breaks down in the IR neutral background (25),for which Q = 0 while q (cid:54) = 0. The effective mass M eff is the same as that of the neutral case (47)but with z = 1, and therefore (51) is the appropriate criterion for the superfluid instability triggeredin the IR. We will not stress this special case in what follows.18inally, to describe AdS × R d with Q (cid:54) = 0, we set m = 1 , n = τ = 0 to find M eff = ˜ L B − ˜ L Q , (56)leading to the well-known AdS instability window M eff = ˜ L B − ˜ L Q < − . (57) B. Non-scaling case, τ (cid:54) = 2( m − (i) Parameter choices τ < m − and B < : The coupling B ( φ ) contribution to (43) approaches negative infinity as r →
0, while the remainingterms in (43) stay finite. Compared to the scaling case, the effective mass here is much morenegative along radial flow towards the IR, and thus instabilities are expected to be genericand form much more easily. Moreover, there should be unstable modes at arbitrarily smallvalues of the charge Q , associated with the deep IR portion of the geometry. As a consequence,we expect neutral scalars to condense generically, independently of how small or large B is(in contrast to the standard AdS case). We will return to this point in the next section, butanticipate to be able to find a superfluid phase transition at arbitrarily low temperature and charge.(ii) Parameter choices τ < m − and B > : On the other hand, in this case the contribution to (43) coming from the coupling B willapproach positive infinity as r →
0, preventing the formation of an unstable mode in the deep IR.Nevertheless, a sufficiently large value of the charge Q may trigger a superfluid instability nearthe scale r ∼ r tr , where the gauge field term ∼ Q r dn is largest. For this parameter range weexpect that a minimal charge will be needed in order for the charged condensate to form. We willexamine this point in detail in Section V.(iii) Parameter choices τ > m − : When τ > m − B r τ − m − and Q r dn in (43) both vanish at r →
0, and itis challenging to obtain a clean instability criterion. Whether an unstable mode will be presentdepends on whether the Q and B terms will compete against each other (when B >
0) orenhance each other (when B < Q min belowwhich no instabilities will form. It is difficult to be more quantitative at this stage, but we will19eturn to these two cases in more detail in V.(iv) Parameter choices τ − m −
1) = 2 dn : When τ = 2( m −
1) + 2 dn we see that the coupling and gauge field terms in (43) scale in thesame way, ∝ r dn (cid:2) B − Q (cid:3) . Thus, when B > Q the effective mass will never be negative inthe hyperscaling violating portion of the geometry, ensuring the absence of instabilities in thatregime. Interestingly, this is true even for very large charge. On the other hand, when B < Q the radially dependent part of M eff will be negative, but will approach zero towards the IR. Thus,we expect to have a condensate as long as the effective mass can become sufficiently negative near r tr . However, even in this case we will always have a minimal charge, since r dn → B = Q . We leave amore detailed treatment of this case to future work.We are now ready to compare the intuition developed here with what one can learn by recastingthe scalar equation in the form of a Schr¨odinger equation. Indeed, the presence of bound states inthe Schr¨odinger potential can also be taken as an indicator of instabilities, as we discuss next. V. EFFECTIVE SCHR ¨ODINGER POTENTIAL AND INSTABILITIES
By an appropriate combination of a change of coordinates and a field redefinition, the chargedscalar field equation of motion (12) can be rewritten in Schr¨odinger form – as done, for example,in holographic studies of the conductivity [47]. Inspecting the sign of the resulting Schr¨odingerpotential can then offer a window into the presence of instabilities in the system [3]. In particular,if the Schr¨odinger equation has a negative energy bound state, there will be unstable modes.Negative energy in this context corresponds to ω < i.e. imaginary frequencies and thereforesolutions which grow exponentially in time. Also, if for a certain range of parameters the effectivepotential remains positive everywhere in the hyperscaling violating portion of the geometry, we areguaranteed the absence of superfluid instabilities there.We turn on the charge of ψ and work with the parametrisation given by (22), taking the couplingto the neutral scalar to be B = B r τ . Recall that τ = 2( m −
1) is the case that preserves someof the scaling symmetry. We work at zero momentum and take ψ = e − iωt ψ ( r ). Introducing a newradial variable ξ and rescaling the charged scalar field, dξdr = ˜ Lr − m , ˜ ψ ( ξ ) = r dn/ ψ ( r ) , (58)20he equation for the perturbation takes the form of a Schr¨odinger equation − d dξ ˜ ψ + V Schr ˜ ψ = ω ˜ ψ , (59)with the effective Schr¨odinger potential given by˜ L V Schr ( r ) = r m − (cid:20) ˜ L (cid:16) B r τ − m − − Q r dn (cid:17) + dn ( dn + 4 m − (cid:21) , (60)where we used the original radial coordinate for simplicity and we recall that Q was defined in (42).Notice that the overall factor r m − → r → dn ( dn + 4 m −
2) = ˜ L L (cid:104) ( z + d − θ ) − z (cid:105) = ν − B ˜ L − z ˜ L L > , (61)where ν was introduced in (49). This expression can be used to rewrite the potential in the followingsuggestive form, V Schr ( r ) = r m − (cid:20) B r τ − m − − Q r dn + 14 (cid:18) ν ˜ L − B − z L (cid:19)(cid:21) . (62)To recap, unstable modes will correspond to negative energy bound states for which ω < ω = 0. Thus, a necessary conditionfor the existence of an instability is that the potential V Schr develops at least one negative region inthe bulk. Inspecting (62) we see that the possible sources of instabilities are again transparent:the relative interplay between the charge, the coupling B and the value of the index ν . Recall thatin this paper we are only after unstable modes associated with the hyperscaling violating regionitself , for which 0 ≤ r ≤ r tr when the IR is at r = 0. As a consequence, we are only looking fornegative regions of (62), and not of the potential which determines the UV behavior of the theoryand the asymptotic AdS geometry. The statement can be made more precise if we know the profile of the potential (60) in the entire bulk region, fromthe IR to the UV. By using the WKB approximation, one can obtain a bound state at zero energy in a potentialwell for each integer k (cid:62) k − π = 2 (cid:90) dξ (cid:112) − V Schr ( ξ ) , (63)where the integral is carried out in the region of negative Schr¨odinger potential. We leave the study of thisinteresting feature to future work. The AdS UV geometry may have additional instabilities which are not captured by the behaviour of (60). However,those are already well understood via standard BF bound arguments, and will be ignored here. . Neutral Scalar Let’s focus on the neutral scalar case Q = 0 first, and consider different choices for the coupling:1. Scaling choice τ = 2( m − : When the effective mass term ∼ B ( φ ) ψ respects the scaling of the kinetic term, the potentialreduces to the simple expression V Schr ( r ) = r m − (cid:20) B + 14 L (cid:2) ( z + d − θ ) − z (cid:3)(cid:21) = r m − (cid:20) ν ˜ L − z L (cid:21) , (64)and is always positive everywhere in the hyperscaling violating part of the geometry if B > B < ν , the condition for V Schr < ν < z ˜ L /L . Notice however thatthe violation of the generalized BF bound (50) corresponds to a smaller window, ν < , (65)associated with the index becoming imaginary. Thus, we see an offset (by an amount ∝ z )between the violation of the generalized BF bound and the condition V Schr ≤
0. However,one should keep in mind that V Schr ≤ not a sufficient condition for instabilities, butonly a necessary one. In other words, the potential should be “negative enough” in order forbound states to form, and one should quantify how deep the potential well needs to be.One way to test whether in the additional window0 < ν ˜ L < z L , (66)the ψ scalar may condense (without a violation of the BF bound) is to examine the behaviorof its IR perturbations. In particular, in order for a condensate to form we must have atleast one irrelevant perturbation mode in the IR, without which a non-trivial scalar profilewould not be supported . Indeed, recall that in Section IV we found that in the IR thescalar had the form (48), with modes ψ ∼ r − d ( z + d − θ ) zd − θ ± ν , ν = 4 ˜ L B + d ( z + d − θ ) ( dz − θ ) . (67) An explanation for the origin of the shift ∝ z / If the perturbations of ψ were relevant, we would expect backreaction of the charged scalar on the background tobecome important, and to lead to a new geometry which would not be that of our simple { z, θ } solutions. Whilethis situation is clearly interesting, it is beyond the scope of our paper, and we will not consider it here. d ( z + d − θ ) zd − θ >
0, as seen from (30), and the IR corresponds to r = 0, one can easily checkthat the range (66) does not allow for irrelevant perturbations and hence is ruled out as apossible “condensation window”.The precise windows of instability in a given model can of course be tested numerically,using these analytical arguments as guidance. We will return to this issue in Section VI,but for now let’s summarize by pointing out that we have identified two mechanisms thatwill indicate the presence of a condensate. First, the violation of the analog of the AdS BFbound. Second, the presence of IR irrelevant modes, without which the boundary conditionswhich would allow for a condensate would not be satisfied. Thus, the absence of irrelevantmodes for the IR expansion of the neutral or charged scalar can be used as a criterionagainst condensation in certain regions of parameter space, especially in cases for which theSchr¨odinger potential analysis is not necessarily conclusive.2. Arbitrary scaling B = B r τ : The Schr¨odinger potential is now given by V Schr ( r ) = r m − (cid:20) B r τ − m − + 14 L (cid:2) ( z + d − θ ) − z (cid:3)(cid:21) . (68)Again, to trigger any instabilities one needs B < B < τ − m − >
0, so that the coupling B ( φ ) approaches zerotowards the IR. Then, in the hyperscaling violating portion of the geometry the term | B | r τ − m − is largest when r = r tr . This implies that we are guaranteed no insta-bilities when | B | r τ − m − tr < L (cid:2) ( z + d − θ ) − z (cid:3) , (69)since the potential is, again, everywhere positive in that case.(ii) On the other hand, when τ − m − < There are no IR irrelevant modes in the larger window 0 < ν < ˜ L L ( z + d − θ ) . Notice that z < ( z + d − θ ) inour parameter space (34). . Charged Scalar As can be easily seen from (60), since n > V Schr always decreases towards r = 0, and is therefore largest precisely near the transition region r ∼ r tr to AdS. As a result, we expect the bound states to be generically localized there and not in thedeep IR. This is in sharp contrast with the standard holographic superconductor setup with an AdS IR region, for which the charge contribution ∼ Q r dn is constant, as n = 0. Once again, weare going to examine the structure of the effective Schr¨odinger potential at zero temperature fordifferent choices of coupling B ( φ ), but this time with Q (cid:54) = 0:1. Scaling choice τ = 2( m − : In the presence of charge we have V Schr ( r ) = r m − (cid:20) − Q r dn + 14 (cid:18) ν ˜ L − z L (cid:19)(cid:21) . (70)We consider the following cases:(i) When ν ˜ L < z L the effective Schr¨odinger potential is negative everywhere independentlyof how large the charge is. Thus, we expect the scalar to be able to condense even when Q is very small . In particular, when the stricter condition ν < , (71)is satisfied, the condensation is triggered at zero charge, as anticipated by the neutral scalarfield analysis above. Note that this particular neutral scalar field instability – which isnothing but the violation of the generalized BF bound – originates from the far IR of thegeometry. It is visible both from the behavior of the effective mass as well as from theSchr¨odinger potential (70).On the other hand, when 0 < ν < z L L , even though the Schr¨odinger potential (70) developsa negative region as r →
0, we are not guaranteed the onset of a superfluid phase transitionin the far IR. Indeed, recall that in this range the IR perturbations of a neutral field areinconsistent with the formation of a condensate – there are no IR irrelevant perturbations.A similar perturbation analysis needs to be done for the charged scalar, to ensure that the This will be the case when the coupling B ( φ ) is of the scaling form. On the other hand, when B is chosen todiverge towards the IR, and B <
0, this story will change, as we will see. This was already anticipated by the neutral scalar analysis discussed above.
24R mode expansion is compatible with the presence of a condensate. Indeed, from (53), wecan obtain the asymptotic behavior in the far IR, ψ = r − d ( z + d − θ ) zd − θ − ν ( c + Q r dn + O ( Q r dn )) + r − d ( z + d − θ ) zd − θ + ν ( c + Q r dn + O ( Q r dn )) , (72)with ν = 4 ˜ L B + d ( z + d − θ ) ( dz − θ ) . Notice that the contribution from U (1) gauge field onlyappears as subleading corrections. Once again, one can easily see that the range 0 < ν
0, whilethe charge contribution will become weaker. Thus, the potential is guaranteed tobe positive along the entire region 0 < r < r tr . This tells us that there will be aminimal charge below which the condensate will not form, set by r = r tr , Q min = r dntr (cid:20) B r τ − m − tr + 14 L (cid:2) ( z + d − θ ) − z (cid:3)(cid:21) . (76)26e can adjust Q min by varying the size r tr of the hyperscaling violating regime(the larger the region, the smaller the minimal charge), as well as by increasing n = d − θzd − θ .(b) The situation for τ > m −
1) is more complicated, and one has to take intoaccount the relative scaling between the charge and the coupling terms to identifywhat sets the value of Q min . The existence of a minimal charge is still genericbecause, as the IR is approached, at some point the positive constant term willdominate the potential, unless the charge is increased above some critical value.The special value τ = 2( m −
1) + 2 dn discussed in case (iv) of Section IV B naivelyfalls within this category, but needs to be treated separately. Indeed, notice thatwhen B > Q the potential is always positive, no matter how large the charge is.Thus, a condensate will not form.(ii) When B <
0, we can rewrite the potential suggestively as V Schr ( r ) = r m − (cid:18) L (cid:2) ( z + d − θ ) − z (cid:3) − (cid:104) | B | r τ − m − + Q r dn (cid:105)(cid:19) . (77)Again the behavior depends on the range of τ :(a) When τ > m − Q and B are taken to be, which is differentfrom the scaling choice, in which one can simply tune B to be large enough totrigger the instability in the deep IR. Thus, to find V Schr < | B | r τ − m − + Q r dn can be is set by the transition scale to AdS, r = r tr .Thus, superfluid instabilities will not develop as long as | B | r τ − m − tr + Q r dntr < ( z + d − θ ) − z . (78)This again sets a minimal charge, as in the previous cases. The new feature com-pared to the standard holographic superconductor [3] is that the minimal chargeis present independently of how negative B is tuned to be. The special choice τ = 2( m −
1) + 2 dn discussed in case (iv) of Section IV B falls within this category.(b) On the other hand, when τ < m −
1) the contribution from the coupling B becomes infinitely negative in the deep IR. As a result, we expect to have a charged27calar condensate (this time localized in the far IR) for any value of the charge, nomatter how small it is, without having to tune B to be large. This is in contrast tothe AdS case, for which the instability is associated with the mass term m beingvery large and negative. This is a new feature, due entirely to having allowed foran arbitrary scaling for B .To summarize, the simple analytical arguments we have formulated can be used to highlightthe competition between different sources of instabilities – in particular, the interplay betweenthe coupling B ( φ ) and the charge term – and the criteria under which they are triggered orsuppressed. Although the analysis was performed using extremal solutions, it provides guidanceto detailed numerical studies of instability windows, and analytical intuition for when a minimalcharge should exist. Next, we will examine our estimates numerically. VI. NUMERICS
So far our discussion has been restricted to zero temperature solutions, but in what follows wewill switch on a finite temperature, and examine these instability windows in the background ofhyperscaling violating black branes that are asymptotic to AdS. For simplicity, we are going to workwith an analytical solution which arises from the supergravity setup of [48], and is characterizedby z, θ → ∞ with the ratio θ/z held fixed. We will examine the condensation of the chargedscalar on top of this analytical background numerically, in a number of examples which will lendevidence to the simple estimates of the last two sections. Although the latter apply only to extremalsolutions, they provide a guide towards a classification of superfluid transitions at finite (but low)temperature. Finally, even though the z, θ → ∞ limit is rather special, we believe that it capturesall the essential features of our analysis, and postpone a more thorough look at black brane solutionswith finite z and θ to future work.Working with d = 2 and choosing the scalar potential and gauge kinetic function in (5) to be Z ( φ ) = e φ/ √ , V ( φ ) = − φ/ √ , (79)we obtain the three-equal-charge black brane solution of [48], ds = − f ( r ) dt + 1 f ( r ) dr + h ( r )( dx + dy ) ,f = r / ( r + Q ) / (cid:18) − ( r h + Q ) ( r + Q ) (cid:19) , h = r / ( r + Q ) / ,A t = (cid:112) Q ( r h + Q ) (cid:18) − r h + Qr + Q (cid:19) , φ = √
32 ln(1 +
Q/r ) , (80)28here r h denotes the horizon. The corresponding temperature and chemical potential are given by T = 3 (cid:112) r h ( r h + Q )4 π , µ = (cid:112) Q ( r h + Q ) . (81)The extreme limit T /µ → r h /Q = 0, and the corresponding IR geometryhas the hyperscaling violating form (22) with exponents m = 3 / n = 1 /
4. Note that thissolution is conformal to AdS × R . More precisely, if we introduce ρ = (cid:113) Q r , then the extreme IRlimit of (80) can be written as ds = Q √ ρ (cid:20) − dt ρ + 43 Q dρ ρ + dx + dy (cid:21) ,A t = Q √ ρ , φ = √ √ ρ ) , (82)with the IR now located at ρ → ∞ . This kind of geometry can be obtained from (18) by consideringthe limit z, θ → ∞ with θ/z = − ψ = e − iωt ψ ( r ) of the charged scalar, whose linearized equation of motion reads ψ (cid:48)(cid:48) ( r ) + (cid:18) f (cid:48) f + h (cid:48) h (cid:19) ψ (cid:48) ( r ) − f (cid:18) B ( φ ) − q A t f (cid:19) ψ ( r ) = − ω f ψ ( r ) . (83)As in the zero temperature case, (83) can be written in Schr¨odinger form, − d dξ ˜ ψ + V Schr ˜ ψ = ω ˜ ψ , (84)after a change of variable and field redefinition given by dξdr = 1 f , ˜ ψ = √ h ψ . (85)The corresponding effective Schr¨odinger potential is then V Schr = f h (cid:48)(cid:48) h − f h (cid:48) h + f f (cid:48) h (cid:48) h + B ( φ ) f − q A t . (86)We emphasize once again that the background geometry will be unstable if the Schr¨odinger equa-tion (84) has a negative energy bound state ω <
0, corresponding to a solution which growsexponentially in time. Furthermore, if there is an unstable mode Im ( ω ) >
0, then at the onset ofthe instability one should expect to find a zero mode with ω = 0. Clearly, its profile will dependon the entire geometry, from the IR to the UV. This case is known as semi-local criticality and can give rise to interesting behavior [49–52]. ψ (cid:48)(cid:48) ( r ) + (cid:18) f (cid:48) f + h (cid:48) h (cid:19) ψ (cid:48) ( r ) − f (cid:18) B ( φ ) − q A t f (cid:19) ψ ( r ) = 0 , (87)which therefore determines the zero modes. After specifying the coupling B ( φ ), the critical temper-ature as a function of charge q can be determined by solving (87) numerically. The two boundaryconditions needed to fully specify the solution will be chosen as follows. First, we will imposeregularity at the horizon . The second boundary condition will come from specifying the UVasymptotics. Indeed, as is well known, there are two modes in the UV AdS region – one is inter-preted as the source of the dual scalar operator, while the other as its expectation value. Here weadopt the standard quantization, i.e. choose the faster falloff to describe the expectation value,and the leading term to be the source. Moreover, we will set the latter to zero, so that the U (1)symmetry is broken spontaneously. For a given temperature T , we expect such normalizable zeromodes to appear at a special value of q . Finally, we will work in the grand canonical ensemble byfixing the chemical potential – in particular, we will set it to µ = 1.For concreteness in our numerics we choose the coupling to be B ( φ ) = M cosh(ˆ τ φ ) , (88)with M and ˆ τ constant. At the asymptotic boundary, where r → ∞ , it behaves as B ( φ ) ∼ M (1 + ˆ τ φ + · · · ) , as φ → , (89)and the leading terms in the UV expansion of ψ are ψ ( r ) = ψ ( − ) r ∆ − (1 + · · · ) + ψ (+) r ∆ + (1 + · · · ) , ∆ ± = 3 ± √ + 4 M . (90)Since we are not allowing for a source term, we set ψ ( − ) = 0 in the expansion above.On the other hand, in the extreme IR with φ ∼ ln(1 /r ) → ∞ , B ( φ ) takes the form we haveassumed in the previous sections, B ( φ ) = M e ˆ τφ , as φ → ∞ . (91)Note that to obtain this relation we have assumed ˆ τ >
0. It is helpful to point out that in thepresent case ˆ τ is related to τ of (39) by ˆ τ = − √ τ . (92) Note that ψ (cid:48) ( r h ) is fully determined by ψ ( r h ), which can be set to unity due to the linearity of (87). Since the coupling (88) is an even function of φ , ˆ τ < τ = 0 describes the case in which B ( φ ) = M is a constant. Finally, the casein which the mass term B ( φ ) ψ scales in the same way as the kinetic term ( ∂ψ ) is obtainedfrom (40) by choosing m = 3 / τ = 2( m −
1) = − ⇒ ˆ τ = 1 √ . (93)One of the questions we are interested in is whether the superfluid instability in these modelscan appear at arbitrary small values of the charge q . Guided by our analytical estimates – in termsof the effective mass in Section IV or the Schr¨odinger potential in Section V – we will considerexamples that address the following scenarios:1. Scaling case τ = 2( m − AdS case. If the bound is violated, the superfluid instability can betriggered for arbitrarily small charges, while if the bound is unbroken a minimal charge isrequired. The latter can be tuned by changing the location of the transition ∼ r tr to the UVAdS geometry.2. Non-scaling case with τ < m −
1) and B <
0: As we discussed in case (i) of Section IV B,the effective mass (43) approaches negative infinity as r →
0, and therefore the superfluidinstability is expected to appear even at zero charge, i.e. there is no Q min . From theSchr¨odinger potential (60) standpoint, we see a large negative well as r →
0. Therefore,the corresponding superfluid instability is expected to be associated with the far IR of thehyperscaling violating region.3. Remaining parameter ranges: A minimal charge is generically required in order to triggera superfluid instability. For case (ii) of Section IV B, the effective mass (43) approachespositive infinity as r →
0. Similarly, the Schr¨odinger potential (60) is positive in the IR. Thesuperfluid instability, if it is triggered, will be associated with the effective UV geometry ofthe hyperscaling violating regime, and not with its IR regime. In case (iii) of Section IV B,the potential (60) becomes and stays positive as one gets sufficiently close to r = 0, no matterhow large the values of B and Q are – even when B <
0. Unlike the case we have justdiscussed, however, the potential remains finite and instabilities can still be triggered, butare associated with the r ∼ r tr transition region. In these cases a minimal charge is neededto ensure that V Schr has a sufficiently negative region.Below we will provide concrete examples realizing each of these scenarios. In particular, we will31nvestigate (87) numerically in order to determine the critical temperature associated with the zeromode solutions as a function of the charge q . A. Scaling case
The scaling case corresponds to ˆ τ = 1 / √
3, so that our mass coupling is given by B ( φ ) = M cosh( φ/ √ . (94)The equation of motion for ψ on the background geometry (82) then becomes ψ (cid:48)(cid:48) ( ρ ) − ρ ψ (cid:48) ( ρ ) − M ρ − q )9 ρ ψ ( ρ ) = 0 , (95)and is solved by ψ ( ρ ) = ρq (cid:20) c Γ(1 − ν ) J − ν (cid:18) q ρ (cid:19) + c Γ(1 + ν ) J ν (cid:18) q ρ (cid:19)(cid:21) , (97)where the index ν = (cid:112) M /
3. The instability associated with the index becoming imaginary,when M < − /
2, is equivalent to the violation of the BF bound (50), with the parameter choice m = 3 / n = 1 /
4. Notice that in this case ν does not contain any charge dependence, unlikethe standard AdS case (57).We will consider two qualitatively different cases, by choosing first M = −
2, for which ν = (cid:112) − / M = − /
4, i.e. ν = (cid:112) / q for M = − T c decreases as we lower the charge q , but the zero mode survives even when the charge iszero. In that case the instability is due to the breaking of the local BF bound in the far IR of thehyperscaling violating geometry, where the charge term is negligible. Thus, here we see a modelwhich gives rise to a superfluid condensate at arbitrarily small values of the charge, in accordancewith the analogous AdS result.In figure 2 we show the critical temperature as a function of q for M = − /
4. In this case theindex ν is real and the corresponding BF bound is unbroken. Just as expected, there is a minimalcharge at which the background will become unstable to developing non-trivial scalar hair. We We point out that (97) holds when ν is not an integer. However, when M = 3( k − / k an integer, thesolution is given by ψ ( ρ ) = ρq (cid:20) c J k (cid:18) q ρ (cid:19) + c Y k (cid:18) q ρ (cid:19)(cid:21) . (96) .0 0.5 1.0 1.5 2.0 2.50.000.020.040.060.080.10 q T c FIG. 1: Critical temperature as a function of charge q for the scaling case with M = − τ = 1 / √ q = 0 is T c ≈ . µ = 1. log (cid:64) q (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) T c (cid:68) FIG. 2: Critical temperature versus charge q for the scaling case with M = − / τ = 1 / √
3. We see aminimal charge q ≈ .
327 below which the zero mode for superfluid instability does not exist. We work inunits in which the chemical potential is µ = 1. note that although the existence of a minimal charge is analogous to what would occur in AdS, thebehavior of the charge term in the hyperscaling violating geometries is not – it is most importantnear r tr and negligible in the far IR. B. Non-scaling case with infinitely negative effective mass
Here we are considering the scenario discussed in case (i) of Section IV B. The effective mass (43)approaches negative infinity as r →
0. Thus, the expectation is that the zero mode should survive33 q T c FIG. 3: Critical temperature as a function of charge q for the non-scaling case with M = − τ = √ q = 0 is T c ≈ . µ = 1. at arbitrarily small values of the charge. We consider the following coupling B ( φ ) = − √ φ ) , (98)which is obtained from (88) by choosing M = − τ = √ q . One can clearly see thatthere is a phase transition even in the limit of zero charge. It is helpful to compare this case to thescaling one with M = −
2, as they both share the same UV mass. Since the effective mass in thefar IR goes to negative infinity, in the present case (with ˆ τ = √
3) instabilities should be triggeredmuch more easily than in the scaling one (with ˆ τ = 1 / √ B ∼ M is, a feature due entirely to the non-trivial coupling B ( φ ), which is absent inthe standard holographic superconductor scenario. Thus, by appropriately choosing the functionaldependence of the coupling, we can facilitate the phase transition and increase T c . C. Remaining cases
In the remaining cases we discussed in Section IV B, we don’t expect the zero mode to exist atarbitrarily small values of q . Let’s focus on the choice B ( φ ) = M , (99)34 .2 0.4 0.6 0.8 1.0 1.2 1.4 log (cid:64) q (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) T c (cid:68) FIG. 4: Log-Log plot of the critical temperature versus charge q for the non-scaling case with M = − τ = 0. The critical temperature goes to zero at q ≈ .
88. We work in units with µ = 1. which corresponds to ˆ τ = 0 (or equivalently τ = 0) and falls under the category (iii) of Section IV B.Before discussing the numerics, we stress that in these cases it’s hard to identify sharp analyticalinstability criteria in the scaling regime. The equation for the ψ perturbation now reads ψ (cid:48)(cid:48) ( ρ ) − ρ ψ (cid:48) ( ρ ) − √ M ρ − q )9 ρ ψ ( ρ ) = 0 , (100)and the solution in general is given by ψ ( ρ ) = e iq ρ (cid:20) c a F (cid:18) − i M √ q , , iq ρ (cid:19) + c b U (cid:18) − i M √ q , , iq ρ (cid:19)(cid:21) , (101)where F ( a, b, x ) is the Kummer confluent hypergeometric function and U ( a, b, x ) is a confluenthypergeometric function. The special case with q = 0 needs to be treated separately, and is ψ ( ρ ) = ρ (cid:34) c a I (cid:32) / (cid:115) M ρ (cid:33) + c b K (cid:32) / (cid:115) M ρ (cid:33)(cid:35) . (102)In contrast to the scaling case (97), it is not immediately apparent how to extract information aboutpotential instabilities from the structure of the solutions. In particular, there is no simple analogof the generalized BF bound, illustrating the challenge of obtaining generic analytical conditionsfor the onset of the phase transition.The behavior of the critical temperature as a function of charge q for the choice M = − M = − / .2 0.4 0.6 0.8 1.0 1.2 1.4 log (cid:64) q (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) T c (cid:68) FIG. 5: Log-Log plot of the critical temperature as a function of charge q for the non-scaling case with M = − / τ = 0. The critical temperature goes to zero at q ≈ .
6. We work in units with µ = 1. However, we note that as we increase the size of M , a bigger minimal charge is required in orderto trigger the superfluid instability. This point can be understood qualitatively by comparing theSchr¨odinger potential (86) for different values of the mass (99) but keeping q and T fixed, as isdone in figure 6. Recall that we are working in the grand canonical ensemble and have thereforefixed the chemical potential to µ = 1.We choose parameters such that the thick magenta line in figure 6 corresponds to the zero modesolution of (87) for M = − / , ˆ τ = 0 at q ≈ . T ≈ . × − . From (63), this casegives the smallest negative potential well which supports a zero mode bound state for the chosenvalues of q and T (e.g. for k = 1 of (63)). One can in principle change M to obtain a much largernegative potential region such that (63) can be satisfied for k (cid:62)
2. However, it is easy to see fromfigure 6 that the range in which the Schr¨odinger potential is negative as well as its depth becomessmaller and smaller as one increases M . Therefore, in order to support a zero mode, a biggervalue of q is required to compensate for the increase in the mass parameter M . In addition, wenote that there is a positive potential region in the deep IR, which is too small to see from figure 6.Before closing this Section, we would like to point out one final feature visible from the numerics.As one can see from inspecting figures 1 to 5, when q is large the value of T c increases linearly with q . This behavior can be understood as follows [53]. Taking q → ∞ while keeping q Ψ and qA µ finite,we arrive at the probe limit in which the gauge field and the charged scalar do not backreact onthe background geometry. In order to compare our results with those in the probe limit, we haveto perform the scaling transformation Ψ → q Ψ and A µ → qA µ . After taking these rescalings intoaccount, the physical dimensionless temperature becomes T c /qµ . Since we are working with µ = 136 .0 0.2 0.4 0.6 0.8 1.0 (cid:45) s V Schr M (cid:61)(cid:45) (cid:144) (cid:61)(cid:45) (cid:144) (cid:61)(cid:45) (cid:61)(cid:45) (cid:144) (cid:61)(cid:45) (cid:144) (cid:61)(cid:45) (cid:144) FIG. 6: Schr¨odinger potential (86) as a function of radial coordinate s = r h + Qr + Q for the non-scaling caseˆ τ = 0. The different curves have the same values of charge q ≈ . T ≈ . × − but different values of M . The horizon is located at s = 1 and the UV AdS boundary at s = 0. We chooseparameters such that the thick magenta line corresponds to the zero mode solution for this particular choiceof q and T . We work in units of µ = 1. (recall that we are in the grand canonical ensemble), this tells us that T c ∝ q , which is preciselywhat is observed from the numerics in the large charge limit. The backreacton of the U (1) fieldand charged scalar on the geometry becomes smaller and smaller as q is increased, explaining againwhy we observe a linear behavior for T c when the charge is large. We confirm this in figure 7, whichhas the same choice of parameters as figure 1, but reaches higher values of q . It is clear that thelarge q behavior can be well approximated by the linear function T c = γq with γ a constant, asexpected from the probe limit argument. For small charges we deviate from the linear relationship,as clearly visible from figure 1 as well as figure 7. Acknowledgments
We are grateful to Blaise Gout´ e raux, Elias Kiritsis and Jim Liu for many insightful comments onthe draft. We would also like to thank Tom Hartman for useful conversations. The work of LL wassupported in part by European Union’s Seventh Framework Programme under grant agreements37 q T c FIG. 7: The red solid line is the critical temperature as a function of charge q for the scaling case with M = − τ = 1 / √
3. The dashed black line corresponds to the probe limit result with T c /q ≈ . µ = 1. (FP7-REGPOT-2012-2013-1) no 316165 and the Advanced ERC grant SM-grav 669288. [1] S. S. Gubser, “Breaking an Abelian gauge symmetry near a black hole horizon,” Phys. Rev. D ,065034 (2008) doi:10.1103/PhysRevD.78.065034 [arXiv:0801.2977 [hep-th]].[2] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a Holographic Superconductor,” Phys.Rev. Lett. , 031601 (2008) doi:10.1103/PhysRevLett.101.031601 [arXiv:0803.3295 [hep-th]].[3] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Holographic Superconductors,” JHEP , 015(2008) doi:10.1088/1126-6708/2008/12/015 [arXiv:0810.1563 [hep-th]].[4] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class. Quant. Grav. , 224002 (2009) doi:10.1088/0264-9381/26/22/224002 [arXiv:0903.3246 [hep-th]].[5] C. P. Herzog, “Lectures on Holographic Superfluidity and Superconductivity,” J. Phys. A , 343001(2009) doi:10.1088/1751-8113/42/34/343001 [arXiv:0904.1975 [hep-th]].[6] G. T. Horowitz, “Introduction to Holographic Superconductors,” Lect. Notes Phys. , 313 (2011)[arXiv:1002.1722 [hep-th]].[7] R. G. Cai, L. Li, L. F. Li and R. Q. Yang, “Introduction to Holographic Superconductor Models,” Sci.China Phys. Mech. Astron. , no. 6, 060401 (2015) doi:10.1007/s11433-015-5676-5 [arXiv:1502.00437[hep-th]].[8] S. K. Domokos and J. A. Harvey, “Baryon number-induced Chern-Simons couplings of vec-tor and axial-vector mesons in holographic QCD,” Phys. Rev. Lett. , 141602 (2007)doi:10.1103/PhysRevLett.99.141602 [arXiv:0704.1604 [hep-ph]].[9] S. Nakamura, H. Ooguri and C. S. Park, “Gravity Dual of Spatially Modulated Phase,” Phys. Rev. D , 044018 (2010) doi:10.1103/PhysRevD.81.044018 [arXiv:0911.0679 [hep-th]].[10] H. Ooguri and C. S. Park, “Spatially Modulated Phase in Holographic Quark-Gluon Plasma,” Phys.Rev. Lett. , 061601 (2011) doi:10.1103/PhysRevLett.106.061601 [arXiv:1011.4144 [hep-th]].[11] A. Donos and J. P. Gauntlett, “Holographic striped phases,” JHEP , 140 (2011)doi:10.1007/JHEP08(2011)140 [arXiv:1106.2004 [hep-th]].[12] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, “Striped instability of a holographic Fermi-likeliquid,” JHEP , 034 (2011) doi:10.1007/JHEP10(2011)034 [arXiv:1106.3883 [hep-th]].[13] S. A. Hartnoll and D. M. Hofman, “Locally Critical Resistivities from Umklapp Scattering,” Phys. Rev.Lett. , 241601 (2012) doi:10.1103/PhysRevLett.108.241601 [arXiv:1201.3917 [hep-th]].[14] G. T. Horowitz, J. E. Santos and D. Tong, “Optical Conductivity with Holographic Lattices,” JHEP , 168 (2012) doi:10.1007/JHEP07(2012)168 [arXiv:1204.0519 [hep-th]].[15] A. Donos and S. A. Hartnoll, “Interaction-driven localization in holography,” Nature Phys. , 649(2013) doi:10.1038/nphys2701 [arXiv:1212.2998].[16] D. Vegh, “Holography without translational symmetry,” arXiv:1301.0537 [hep-th].[17] M. Blake and D. Tong, “Universal Resistivity from Holographic Massive Gravity,” Phys. Rev. D ,no. 10, 106004 (2013) doi:10.1103/PhysRevD.88.106004 [arXiv:1308.4970 [hep-th]].[18] A. Donos and J. P. Gauntlett, “Holographic Q-lattices,” JHEP , 040 (2014)doi:10.1007/JHEP04(2014)040 [arXiv:1311.3292 [hep-th]].[19] T. Andrade and B. Withers, “A simple holographic model of momentum relaxation,” JHEP , 101(2014) doi:10.1007/JHEP05(2014)101 [arXiv:1311.5157 [hep-th]].[20] B. Gout´ e raux, “Charge transport in holography with momentum dissipation,” JHEP , 181 (2014)doi:10.1007/JHEP04(2014)181 [arXiv:1401.5436 [hep-th]].[21] A. Donos, B. Goutraux and E. Kiritsis, “Holographic Metals and Insulators with Helical Symmetry,”JHEP , 038 (2014) doi:10.1007/JHEP09(2014)038 [arXiv:1406.6351 [hep-th]].[22] A. Salvio, “Transitions in Dilaton Holography with Global or Local Symmetries,” JHEP , 136(2013) doi:10.1007/JHEP03(2013)136 [arXiv:1302.4898 [hep-th]].[23] Z. Fan, “Holographic superconductors with hyperscaling violation,” JHEP , 048 (2013)doi:10.1007/JHEP09(2013)048 [arXiv:1305.2000 [hep-th]].[24] A. Lucas and S. Sachdev, “Conductivity of weakly disordered strange metals: from conformal tohyperscaling-violating regimes,” Nucl. Phys. B , 239 (2015) doi:10.1016/j.nuclphysb.2015.01.017[arXiv:1411.3331 [hep-th]].[25] E. Kiritsis and L. Li, “Holographic Competition of Phases and Superconductivity,” JHEP , 147(2016) doi:10.1007/JHEP01(2016)147 [arXiv:1510.00020 [cond-mat.str-el]].[26] H. Kodama and A. Ishibashi, “Master equations for perturbations of generalized static black holeswith charge in higher dimensions,” Prog. Theor. Phys. , 29 (2004) doi:10.1143/PTP.111.29 [hep-th/0308128].[27] C. Keeler, G. Knodel and J. T. Liu, “What do non-relativistic CFTs tell us about Lifshitz spacetimes?,” HEP , 062 (2014) doi:10.1007/JHEP01(2014)062 [arXiv:1308.5689 [hep-th]].[28] N. Iqbal, H. Liu, M. Mezei and Q. Si, “Quantum phase transitions in holographic models of mag-netism and superconductors,” Phys. Rev. D , 045002 (2010) doi:10.1103/PhysRevD.82.045002[arXiv:1003.0010 [hep-th]].[29] M. ˇCubrovi´c, “Confinement/deconfinement transition from symmetry breaking in gauge/gravity dual-ity,” arXiv:1605.07849 [hep-th].[30] C. Charmousis, B. Gout´ e raux, B. S. Kim, E. Kiritsis and R. Meyer, “Effective Holographic Theories forlow-temperature condensed matter systems,” JHEP , 151 (2010) doi:10.1007/JHEP11(2010)151[arXiv:1005.4690 [hep-th]].[31] N. Iizuka, N. Kundu, P. Narayan and S. P. Trivedi, “Holographic Fermi and Non-Fermi Liq-uids with Transitions in Dilaton Gravity,” JHEP , 094 (2012) doi:10.1007/JHEP01(2012)094[arXiv:1105.1162 [hep-th]].[32] B. Gouteraux and E. Kiritsis, “Generalized Holographic Quantum Criticality at Finite Density,” JHEP , 036 (2011) doi:10.1007/JHEP12(2011)036 [arXiv:1107.2116 [hep-th]].[33] L. Huijse, S. Sachdev and B. Swingle, “Hidden Fermi surfaces in compressible states of gauge-gravityduality,” Phys. Rev. B , 035121 (2012) doi:10.1103/PhysRevB.85.035121 [arXiv:1112.0573 [cond-mat.str-el]].[34] S. Harrison, S. Kachru and H. Wang, “Resolving Lifshitz Horizons,” JHEP , 085 (2014)doi:10.1007/JHEP02(2014)085 [arXiv:1202.6635 [hep-th]].[35] J. Bhattacharya, S. Cremonini and A. Sinkovics, “On the IR completion of geometries with hyperscalingviolation,” JHEP , 147 (2013) doi:10.1007/JHEP02(2013)147 [arXiv:1208.1752 [hep-th]].[36] N. Kundu, P. Narayan, N. Sircar and S. P. Trivedi, “Entangled Dilaton Dyons,” JHEP , 155(2013) doi:10.1007/JHEP03(2013)155 [arXiv:1208.2008 [hep-th]].[37] S. Cremonini and A. Sinkovics, “Spatially Modulated Instabilities of Geometries with Hyperscal-ing Violation,” JHEP , 099 (2014) doi:10.1007/JHEP01(2014)099 [arXiv:1212.4172 [hep-th],arXiv:1212.4172].[38] N. Iizuka and K. Maeda, “Stripe Instabilities of Geometries with Hyperscaling Violation,” Phys. Rev.D , no. 12, 126006 (2013) doi:10.1103/PhysRevD.87.126006 [arXiv:1301.5677 [hep-th]].[39] G. Knodel and J. T. Liu, “Higher derivative corrections to Lifshitz backgrounds,” JHEP , 002(2013) doi:10.1007/JHEP10(2013)002 [arXiv:1305.3279 [hep-th]].[40] S. Cremonini, “Spatially Modulated Instabilities for Scaling Solutions at Finite Charge Density,”arXiv:1310.3279 [hep-th].[41] S. Barisch-Dick, G. L. Cardoso, M. Haack and . Vliz-Osorio, “Quantum corrections to extremal blackbrane solutions,” JHEP , 105 (2014) doi:10.1007/JHEP02(2014)105 [arXiv:1311.3136 [hep-th]].[42] D. K. O’Keeffe and A. W. Peet, “Electric hyperscaling violating solutions in Einstein-Maxwell-dilatongravity with R corrections,” Phys. Rev. D , no. 2, 026004 (2014) doi:10.1103/PhysRevD.90.026004[arXiv:1312.2261 [hep-th]].
43] J. Bhattacharya, S. Cremonini and B. Goutraux, “Intermediate scalings in holographic RG flows andconductivities,” JHEP , 035 (2015) doi:10.1007/JHEP02(2015)035 [arXiv:1409.4797 [hep-th]].[44] S. S. Gubser, “Curvature singularities: The Good, the bad, and the naked,” Adv. Theor. Math. Phys. , 679 (2000) [hep-th/0002160].[45] A. Lucas, S. Sachdev and K. Schalm, “Scale-invariant hyperscaling-violating holographic theories andthe resistivity of strange metals with random-field disorder,” Phys. Rev. D , no. 6, 066018 (2014)doi:10.1103/PhysRevD.89.066018 [arXiv:1401.7993 [hep-th]].[46] J. Gath, J. Hartong, R. Monteiro and N. A. Obers, “Holographic Models for Theories with HyperscalingViolation,” JHEP , 159 (2013) doi:10.1007/JHEP04(2013)159 [arXiv:1212.3263 [hep-th]].[47] G. T. Horowitz and M. M. Roberts, “Zero Temperature Limit of Holographic Superconductors,” JHEP , 015 (2009) doi:10.1088/1126-6708/2009/11/015 [arXiv:0908.3677 [hep-th]].[48] S. S. Gubser and F. D. Rocha, “Peculiar properties of a charged dilatonic black hole in AdS ,” Phys.Rev. D , 046001 (2010) doi:10.1103/PhysRevD.81.046001 [arXiv:0911.2898 [hep-th]].[49] S. A. Hartnoll and E. Shaghoulian, “Spectral weight in holographic scaling geometries,” JHEP ,078 (2012) doi:10.1007/JHEP07(2012)078 [arXiv:1203.4236 [hep-th]].[50] A. Donos and S. A. Hartnoll, “Universal linear in temperature resistivity from black hole superradi-ance,” Phys. Rev. D , 124046 (2012) doi:10.1103/PhysRevD.86.124046 [arXiv:1208.4102 [hep-th]].[51] R. J. Anantua, S. A. Hartnoll, V. L. Martin and D. M. Ramirez, “The Pauli exclusion prin-ciple at strong coupling: Holographic matter and momentum space,” JHEP , 104 (2013)doi:10.1007/JHEP03(2013)104 [arXiv:1210.1590 [hep-th]].[52] A. Donos, J. P. Gauntlett and C. Pantelidou, “Semi-local quantum criticality in string/M-theory,”JHEP , 103 (2013) doi:10.1007/JHEP03(2013)103 [arXiv:1212.1462 [hep-th]].[53] R. G. Cai, L. Li and L. F. Li, “A Holographic P-wave Superconductor Model,” JHEP , 032 (2014)doi:10.1007/JHEP01(2014)032 [arXiv:1309.4877 [hep-th]]., 032 (2014)doi:10.1007/JHEP01(2014)032 [arXiv:1309.4877 [hep-th]].