PPreprint typeset in JHEP style - HYPER VERSION
CCTP-2014-21
Magnetic Critical Solutions in Holography
N. Angelinos
Crete Center for Theoretical Physics, Department of Physics, University ofCrete 71003 Heraklion, Greece
Abstract:
The AdS/CFT correspondence is a realization of the holographic princi-ple in the context of string theory. It is a map between a quantum field theory and astring theory living in one or more extra dimensions. Holography provides new toolsto study strongly-coupled quantum field theories. It has important applications inquantum chromodynamics (QCD) and condensed matter (CM) systems, which areusually complicated and strongly coupled. Quantum critical CM theories have scalingsymmetries and can be connected to higher-dimensional scale invariant space-times.The Effective Holographic Theory paradigm may be used to describe the low-energy(IR) holographic dynamics of quantum critical systems by the Einstein-Maxwell-Dilaton (EMD) theory. We find the magnetic critical scaling solutions of an EMDtheory containing an extra parity-odd term F ∧ F . Previous studies in the absenceof magnetic fields have shown the existence of quantum critical lines separated byquantum critical points. We find this is also true in the presence of a magnetic field.The critical solutions are characterized by the triplet of critical exponents ( θ, z, ζ ),the first two describing the geometry, while the latter describes the charge density. Keywords:
AdS/CFT, AdS/CMT, holography, quantum criticality, finite density,magnetic. a r X i v : . [ h e p - t h ] J a n ontents
1. Introduction 32. The AdS/CFT correspondence and Holography 3
AdS /CF T
3. Effective Holographic Theories 16
4. Setup and Equations of Motion 18
5. Constant Scalar Solutions 23 . Running Scalar Solutions 27
7. Conclusion 38Acknowledgements 38Appendix 39A. The anti-de Sitter space-time 39B. On Hyperscaling Violating metrics 40
B.1 Properties 41
C. Equations of motion and ansatz 42D. Null-Energy Condition 43E. Constant scalar solutions 44
E.1 Neutral solution 45E.2 Dyonic solution 45
F. Running scalar in the IR 47
F.1 Solutions without intrinsic P-violation 48F.1.1 Electric solutions 49F.1.2 Magnetic solutions 50F.1.3 Dyonic solutions 51F.2 Solutions with intrinsic P-violation 51
G. Solutions with subleading scalar potential ( V → ) 54H. Linear Perturbations 55 H.1 Neutral hyperscaling violating solution 56H.2 Electric/Magnetic hyperscaling violating solutions 57H.3 Neutral AdS × R
58– 2 –.5 Corrections due to the PQ term 59H.5.1 Neutral solution 59H.5.2 Magnetic solution at zero charge density 59H.5.3 Electric solution at finite charge density 60H.5.4 Magnetic solution at finite charge density 60
References 60
1. Introduction
The AdS/CFT correspondence, also known as gauge/gravity correspondence, sug-gests a duality between a string theory and a quantum field theory. In its earliestincarnation, [1] the gauge/gravity duality connected a 10-dimensional theory on anti-de Sitter space-time (AdS × S ) with a 4-dimensional conformal field theory (CFT)living on its boundary. It was discovered in the context of string theory by studyingfield theories on hypersurfaces (D-branes) embedded in a higher dimensional space-time, and the associated black holes.A strongly coupled field theory corresponds to a weakly coupled string theory,which is easier to handle. Because of that, holography provides an alternative way ofstudying strongly coupled QFTs. The AdS/CFT correspondence has a wide varietyof applications ranging from quantum chromodynamics (AdS/QCD) to condensedmatter physics (AdS/CMT) and to relativistic hydrodynamics.In this paper we focus on quantum critical theories at finite charge density. Suchtheories can be strongly coupled and hard to study in more than 1 dimensions.However, quantum critical points have special symmetries and the theory can beconnected to a simple gravitational dual. We study quantum critical solutions atfinite density in the presence of a magnetic field, using the holographic approach.This thesis is structured as follows. We first motivate and introduce the AdS/CFTcorrespondence, [1] and explain how it realizes holography. We then introduce theEffective Holographic Theories, [7] which are useful in the study of quantum criticalpoints. Finally we discuss our setup and solutions. Details about the calculationscan be found in the appendices.
2. The AdS/CFT correspondence and Holography
In this section we present the basic idea of the Holographic Principle. We also discussthe connection between large-N gauge theories with string theories. We explain howthe AdS/CFT correspondence arises when studying a system of D-branes inside a10-dimensional bulk space-time, [1] and we present some of its applications.– 3 – .1 What is Holography
The Holographic Principle states that a theory with gravity in a closed region ofspace-time is described by degrees of freedom that live on the boundary. The de-scription of the volume is completely encoded on a dual theory living on its boundary.Holography was inspired by Bekenstein’s bound, which states that in a theory ofgravity the maximal entropy in a region of space scales with its surface area, ratherthan its volume. This condition is saturated by black holes. Bekenstein argued thatif this bound was not satisfied by a system, then it would be possible to violate thesecond law of thermodynamics, [43]. In contrast, local quantum field theory predictsthat the number of degrees of freedom inside a region scales with its volume. For thatreason, Bekenstein’s bound has been controversial and is one of the points that makesQFT seemingly incompatible with gravity. It is believed, however, that a successfultheory of quantum gravity must satisfy the Holographic bound. Holography has beenembedded in the framework of string theory and is a widely studied subject.The most succesful realization of Holography is the Anti de-Sitter/ConformalField Theory (AdS/CFT) correspondence discovered by J. Maldacena in 1997, [1].A superstring theory on AdS × S is conjectured to be equivalent to 4-dimensional N = 4 super Yang-Mills. The latter is a conformal field theory (CFT) living on theboundary of the space-time, [2]. The theories are equivalent even though they livein a different number of dimensions. Every field in the AdS string theory can betranslated to an operator in the CFT and vice versa. It was first suggested by ’t Hooft that a strongly coupled gauge theory in the large-Nlimit can be described by an effective string theory at weak coupling, [42]. In thissection we introduce some basic ideas of string theory and present the stucture of itsperturbation theory, which turns out to be a topological expansion. We also presentthe perturbative structure of large-N Yang-Mills theory, which has a topologicalexpansion of identical structure if we identify the string coupling constant g s with1 /N , [6]. The conclusion is that in the large-N limit the effective string theorydescription is weakly coupled (small g s ). We start from the simple case of a relativistic particle. Its motion describes a curve inspacetime. This curve is called the world-line of the particle. The equation of motionof this particle can be derived by minimizing the length of its world-line between twopoints, [6]. S = − m (cid:90) ds (2.1)where m is the mass of the particle and ds the line element of its trajectory.– 4 –he action for a relativistic string is built following the same idea. A relativisticstring is a one-dimensional continuous object and its motion in space-time describesa two-dimensional surface (world-sheet). In analogy with the relativistic particle, theequations of motion for the string are derived by minimizing the surface of its world-sheet between two string configurations. In the case of closed strings the world-sheetis a tube, while for open strings it is a strip. The action describing the motion of thestring is the Nambu-Goto action, [49], [6]: S NG = − T (cid:90) dA (2.2)where T = (2 π(cid:96) s ) − is the tension of the string and dA is the surface element ofits worldsheet. The parameter (cid:96) s is the characteristic length of the theory (stringlength).We can take two coordinates ξ a ( a = 0 ,
1) to parametrize the world-sheet Σ( ξ , ξ ).The string moves in a space-time with metric G µν (target space). The target spaceinduces a metric on the world-sheet, [6]: ds = G µν ( X ) dX µ dX ν = G µν ∂X µ ∂ξ a ∂X ν ∂ξ b dξ a dξ b = ˜ G ab dξ a dξ b (2.3)where the induced metric is ˜ G ab = G µν ∂X µ ∂ξ a ∂X ν ∂ξ b (2.4)The Nambu-Goto action (2.2) can be written explicitly as, [6]: S NG = − T (cid:90) d ξ (cid:113) − det ˜ G ab (2.5)The equations of motion can be solved in flat target space-time for both Neumannand Dirichlet boundary conditions. The relativistic string can then be quantized byfollowing standard methods, like canonical quantization, [6]. Figure 1:
The topological expansion of closed string theory. This figure was taken from[66] .We are not interested in the details of string quantization for our purpose, butwe examine the perturbative structure of string theory. In closed string theory the– 5 –asic interaction is a string splitting into two (or the inverse), [6]. The world-sheet ofsuch an interaction can be obtained by thickening the lines of a triple vertex in QFT.A one-loop diagram can be obtained by combining two triple vertex diagrams andthe resulting world-sheet is a Riemann surface with one hole (genus 1). Repeatingthis process we obtain higher order diagrams of higher genus. Their world-sheet isa surface characterized by the topological number, h, which is the number of holes(see figure 1). Therefore the perturbation series in string theory is a topologicalexpansion. By weighing each triple vertex with the dimensionless string couplingconstant g s the string perturbative expansion is of the form, [6]: ∞ (cid:88) h =0 g h − s F h ( a (cid:48) ) (2.6)As we will see below the perturbative expansion of large-N gauge theories is identicalto (2.6). Consider a U(N) Yang-Mills theory: L = − g Y M
T r [ F µν F µν ] (2.7)The gauge field strength tensor is given by: F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] (2.8)The gauge field components A µ are N × N Hermitian matrices and the theory isnon-Abelian.In order to take the large-N limit, we first have to know how to scale the couplingconstant g Y M . In quantum field theory the beta function encodes the dependence ofthe coupling parameter g Y M on the energy scale µ . From the one-loop beta functionfor a non-Abelian U(N) gauge theory we obtain the RG flow equation, [6]: µ d g Y M d µ = − N g Y M (4 π ) + O ( g Y M ) (2.9)We can find the appropriate scaling by demanding that the leading terms are of thesame order. Therefore we can see that the combination λ = g Y M N (2.10)which is called the ’t Hooft coupling, [42], must be kept constant as N goes to infinity.The Lagrangian (2.7) can be rewritten as L = − Nλ T r [ F µν F µν ] (2.11)– 6 –he ’t Hooft expansion corresponds to the expansion of the amplitudes in powersof N while keeping λ constant. A convenient way to write the vacuum to vacuumdiagrams is the ’t Hooft double-line notation, [42], which substitutes each line in theFeynman diagrams with two lines of opposite orientations, [6]. Figure 2:
On the left: the zeroth order diagram can be drawn on a surface with zerohandles (sphere). On the right: the first order diagram cannot be drawn on a sphere, buthas to be drawn on a surface of genus 1 (torus). This figure was taken from [66] .Every propagator contributes a factor of λ/N and every vertex a factor of
N/λ .In addition every loop contributes a power of N (because of the summation over Ncolors). We can now find the factors associated with a diagram. A diagram with Epropagators (edges), V vertices and F loops (faces) has a coefficient proportional to[6]: (cid:18) λN (cid:19) E (cid:18) Nλ (cid:19) V N F = N χ λ E − V (2.12)where χ = V − E + F is the Euler number of the surface, [6]. For a closed compactsurface with h handles χ = 2 − h . Therefore such a diagram has a coefficient of order O ( N − h ). The ’t Hooft expansion organizes diagrams according to their topology,[42]. In the large-N limit the dominant diagrams are the ones with the minimumnumber of handles ( h = 0), which have the topology of a sphere. These diagramsare called planar because they can also be drawn on a plane. Non-planar diagramscorrespond to surfaces with handles and are suppressed in the large-N limit by anadditional factor of 1 /N h .The standard perturbative expansion for any correlator can be written at largeN as, [6]: ∞ (cid:88) h =0 N − h Z h ( λ ) = ∞ (cid:88) h =0 N − h ∞ (cid:88) i =0 c i,h λ i (2.13)– 7 –hich suggests a connection with the topological expansion of string theory (2.6) ifwe identify the string coupling constant g s with: g s ∼ /N (2.14)This connection indicates that in the large-N limit the effective string theory descrip-tion is weakly coupled. AdS /CF T String theory contains, besides strings, objects extended in more than one spatialdimensions, called branes, [6]. The most important ones for the AdS/CFT corre-spondence are the D p -branes, which are defined as (p+1)-dimensional hypersurfaceson which open strings can end with Dirichlet boundary conditions. On one handthe brane dynamics can be described pertubatively in terms of open strings. Onthe other hand, the D-branes interact gravitationally and are supergravity solutions.In this section we will review how the AdS/CFT correspondence was discovered byconjecturing a duality between these two different descriptions of the same systemof D-branes, [1]. D p -branes have two types of excitations, [66]. The first type is the motion anddeformation of their shapes which can be parametrized by their coordinates φ i inthe (9-p)-dimensional transverse space. These degrees of freedom are scalar fields onthe brane’s world-volume. The second type are internal excitations caused by thecharged end of a string. A charge is a source to a gauge field and the D p -brane hasan Abelian gauge field A µ , ( µ = 0 , , ..., p ) living on its world-volume. The actionthat describes these types of excitations is the Dirac-Born-Infeld (DBI) action, [66]: S DBI = − T D p (cid:90) d p +1 x (cid:113) − det( g µν + 2 π(cid:96) s F µν ) (2.15)where T D p = (2 π ) − p g − s (cid:96) − p − s is the tension of the brane, (cid:96) s the string length and g µν is the induced metric on the brane’s world-volume which depends on the scalar fields φ i . Consider now a D -brane in flat target space. We can write the induced metricon the brane using the 6 scalar fields φ i as g µν = η µν + (2 π(cid:96) s ) ∂ µ φ i ∂ ν φ i . The DBIaction (2.15) for a D -brane can be written as: S brane = − T D (cid:90) d x (cid:113) − det [ η µν + (2 π(cid:96) s ∂ µ φ i )(2 π(cid:96) s ∂ ν φ i ) + 2 π(cid:96) s F µν ] (2.16)We notice that every field φ, F is accompanied by a factor of 2 π(cid:96) s . We can nowexpand in powers of 2 π(cid:96) s (which is equivalent to expanding in powers of the gravi-tational constant κ and keeping the string coupling g s constant since κ ∼ g s (cid:96) s , [6],– 8 –sing the familiar identity from linear algebra: det( A ) = exp[ T r (log A )], where A isa square matrix. The leading order terms are (ignoring the constant zeroth orderterm): S brane = − πg s (cid:90) d x (cid:18) F µν F µν + 12 ∂ µ φ i ∂ ν φ i + . . . (cid:19) (2.17)where we used T D = ((2 π ) g s (cid:96) s ) − The rest of the terms are suppressed by additional factors of (cid:96) s . Therefore at thelow-energy limit we have a U(1) theory living on the world-volume of the D -brane.We now consider a system of N coincident parallel D -branes ( p = 3). Such a sys-tem generate a non-Abelian U(N) gauge theory. The branes live in a 10-dimensionalbulk space and are located at the same point of the transverse 6-dimensional space.Except for the open strings, which are excitations of the branes, the theory alsocontains closed strings, which are excitations of the bulk space. We can write thelow-energy action of this theory schematically: S = S bulk + S brane + S interactions (2.18)The first term S bulk describes the dynamics of the bulk space in terms of closed strings,which are described by IIB superstring theory. The second term S brane describe thedynamics of the branes in terms of open strings. The third term S interactions containsthe interactions between open and closed strings.At the low-energy limit, the superstring theory living on the bulk reduces tofree IIB supergravity. By expanding S bulk around the free point in powers of thegravitational constant κ = 8 πG N as g µν = η µν + κh µν we obtain, [6]: S bulk ∼ κ (cid:90) d x (cid:113) − det ( g µν ) R + · · · ∼ (cid:90) d x (cid:2) ( ∂h ) + κh ( ∂h ) + . . . (cid:3) (2.19)where we have not indicated explicitly all the bulk fields for simplicity. The importantresult is that all the interaction terms are proportional to positive powers of κ ,therefore at very low energies (small κ ) they become very weak compared to thekinetic term and can be ignored.The second term S brane governs the open degrees of freedom (open strings). Aswe indicated earlier, a single D -brane has a U(1) theory living on its world-volumecontaining a gauge field and 6 scalar fields. A system of N branes will generatea U(N) theory containing a vector boson and 6 scalars transforming as adjoints ofU(N). In particular such a system is equivalent to N = 4 , U ( N ) super Yang-Mills(SYM) theory in the low-energy limit, [6]: S brane ∼ − πg s (cid:90) d xT r (cid:20) F µν F µν (cid:21) + . . . (2.20)– 9 –here we kept only the gauge field terms for simplicity.The third term describing the interactions between open-closed degrees of free-dom is subleading in the low energy limit. We expand S interactions in powers of κ ,[6]: S interactions ∼ (cid:90) d x (cid:113) − det ( g µν ) T r (cid:2) F (cid:3) + · · · ∼ κ (cid:90) d xh µν T r (cid:20) F µν − δ µν F (cid:21) + . . . (2.21)where again the only terms indicated are the kinetic terms of the gauge field forsimplicity.We conclude that in the low-energy limit, this theory is described by free IIBsupergravity on the bulk and N = 4 , U ( N ) super Yang-Mills theory on the branesnot interacting with each other. We now consider the same system from a different point of view. The D -branes aresolutions of supergravity in 10-dimensions. The exact solution for N D -branes isgiven by, [6]: ds = H − / ( − dt + d(cid:126)x ) + H / ( dr + r d Ω ) (2.22)The 3 spatial coordinates (cid:126)x are parallel to the branes, while dr + r d Ω is the metricof the 6-dimensional transverse space. In particular r is the distance from the branes,and the metric changes as we move along r because of the warp factor: H = 1 + L r , L = 4 πg s l s N (2.23)Since the coefficient of dt in the metric depends on the distance from the branes, r , the energy measured also depends on r (due to gravitational redshift). If at somepoint r we measure energy E r , an observer at infinity would measure E ∞ = H − E r ,[6]. Therefore an object moving near the branes r → H − → r = 0), since any finite energy is redshiftedto zero. In the low-energy limit the two types of excitations decouple from eachother.Excitations of the first kind propagate away from the branes where the space isflat. For very large r, H ≈ r ,– 10 – / ≈ L /r and the metric (2.22) becomes: ds = r L ( − dt + d(cid:126)x ) + L r ( dr + r d Ω ) (2.24)Changing the radial coordinate to u = L /r we obtain: ds = L r ( du − dt + d(cid:126)x ) + L d Ω (2.25)which describes the product space AdS × S . Therefore the low-energy limit of thissystem is described by IIB free supergravity and IIB supergravity in AdS × S whichare not interacting with each other.In both descriptions we notice that the low-energy description of the system ofN D -branes reduces to the sum of two non-interacting theories. In both cases one ofthose theories is free IIB supergravity. We therefore expect that the two remainingtheories, gravity in AdS × S and N = 4 , U ( N ) SYM, are equivalent. The latteris a conformal field theory (CFT) in 4-dimensions. This is why this special case ofholography is called AdS /CF T correspondence. We now examine the region of validity of the two dual descriptions. We would firstlike to find the connection between the dimensionless parameters of the two theories.Starting from (2.20) and keeping only the gauge field terms: L Y M = 12 πg s T r (cid:20) F µν F µν (cid:21) = c πg s F aµν F a,µν , F µν = F aµν T a (2.26)where T a are the generators of the non-Abelian group and c is their normalizationconstant: T r (cid:2) T a T b (cid:3) = cδ ab . A popular choice that we are going to use is c = 1 / g Y M = 4 πg s (2.27)Combining (2.27) with the expression for the radius of the AdS space-time from(2.23): (cid:18) L(cid:96) s (cid:19) = N g Y M = λ (2.28)where λ is the ’t Hooft coupling defined in (2.10). This is one of the formulas wewere looking for.The 10-dimensional Newton constant is given by:16 πG = (2 π ) g s (cid:96) s (2.29)– 11 –ombining this with (2.27) and (2.10) we obtain: G (cid:96) s = π λ N (2.30)We can now find the region of validity of the two descriptions. From (2.30) we seethat G ∼ /N , which means that quantum effects are suppressed for large N. Ifthe CFT is strongly coupled ( λ (cid:29)
1) then, according to (2.28), L (cid:29) (cid:96) s , which meansthat the string theory is weakly curved and can be approximated by supergravity.Therefore the large-N limit of the sYM theory is described well by the two-derivativeaction of IIB supergravity in AdS × S . We will now briefly compare the symmetries of the two descriptions. We first considerthe conformal symmetry. The N = 4 SYM theory is a 4-dimensional CFT. Thereforeit is invariant under the conformal group SO(2,4). This is exactly the isometry groupof the AdS × S background of the dual string theory, [6].We now consider the sypersymmetries of the two descriptions. The N = 4 SYMtheory is maximally supersymmetric with 32 conserved fermionic supercharges. The AdS × S is also a maximally supersymmetric solution of 10-dimensional supergrav-ity with 32 Killing spinors, which correspond to the 32 supercharges of the dual gaugetheory, [6]. In addition N = 4 SYM is invariant under the R-symmetry group SO(6),which rotates the 6 scalar fields φ i into each other. This symmetry can be identi-fied with the rotational symmetry of the 5-sphere component of the dual AdS × S space-time, [6]. Consider the limit on the energy as we are taking (cid:96) s → E ∞ = H − E r ∼ E r r(cid:96) s (2.31)We must keep the energy in the near-horizon region fixed in string units (cid:96) s E r , [6], aswell as the energy in the near-boundary region E ∞ since this is the energy measuredin the CFT. From: E ∞ ∼ E r rl s = ( E r (cid:96) s ) r(cid:96) s (2.32)we see that U = r(cid:96) s must be kept fixed as we are taking the r → U = r(cid:96) s in the near-horizon metric,[1]: ds = (cid:96) s (cid:20) U √ πg s N ( − dt + d(cid:126)x ) + (cid:112) πg s N (cid:18) dU U + d Ω (cid:19)(cid:21) (2.33)– 12 –he coordinates t, (cid:126)x are the space-time coordinates of the CFT. The extra coordinate, U , of the AdS part behaves like the energy scale of the CFT.This argument involved the decoupling limit. There are, however, other argu-ments that indicate that the radial direction behaves like an energy scale for thegauge theory with the UV located near the boundary. We write the AdS metric inPoincar´e coordinates: ds = 1 u ( du − dt + d(cid:126)x ) (2.34)The metric is invariant under SO(1,1):( u, t, (cid:126)x ) → ( au, at, a(cid:126)x ) (2.35)The boundary is located at u = 0 in these coordinates. If we scale up the coordinatesof the gauge theory on the boundary ( t, (cid:126)x ), which means going down in energy, wemust also scale up u, which means moving away from the boundary at u = 0.Therefore the high-energy limit of the gauge theory (UV) corresponds to u = 0(the boundary) while the low-energy limit (IR) corresponds to u = ∞ (the Poincar´ehorizon), [6]. In this section we introduce a cutoff in the
AdS metric and calculate the degrees offreedom (entropy) of the theory on the boundary and the bulk theory.We begin by cutting out the 5-sphere part of the metric (2.25). It turns out thatby reducing the S part, every massless field in the original theory corresponds toan infinite tower of massive fields on AdS with ever increasing mass, [6], [66]. Theonly interesting detail for our purpose is the value of the 5-dimensional Newton’sconstant G after the reduction on S . The relationship between G and G can befound by considering the reduction of the Einstein-Hilbert term, [66]:116 πG (cid:90) d x d Ω (cid:112) − det( g ) R = V πG (cid:90) d x (cid:112) − det( g ) R (2.36)where V = π L is the volume of an S of radius L. It follows that G = G /V = (2 π ) l s g s (16 π ) π L = πL N (2.37)where we also used (2.23).Now that we removed the sphere we consider the AdS metric in global coordi-nates (more details can be found in appendix A): ds = L ( − cosh ( ρ ) dτ + dρ + sinh ( ρ ) d Ω ) (2.38)Changing the radial coordinate to u = tanh( ρ ), (2.38) becomes: ds = L ( − (cid:18) u − u (cid:19) dτ + 4(1 − u ) ( du + u d Ω )) (2.39)– 13 –n this coordinate system the boundary is located at u = 1 and the interior at0 ≤ u < u = 1 − (cid:15) with (cid:15) very small.This corresponds to a UV cutoff in the dual gauge theory according to the previoussection. The gauge theory lives on an S of radius L. The distance between twopoints on the cutoff sphere scales as log( | x − x | /(cid:15) ). Therefore we may view (cid:15) (cid:28) L(cid:15) (in units of length), there are 1 /(cid:15) fundamental cells of radius Lin the 3-sphere. Since the gauge theory has order N degrees of freedom and theentropy is proportional to the regulated volume of the boundary we obtain: S F T ∼ N (cid:15) (2.40)On the AdS side the area of the sphere at the regulated boundary can be readdirectly from the metric (2.39): A = 8 L u (1 − u ) (cid:12)(cid:12)(cid:12)(cid:12) u =1 − (cid:15) ∼ L (cid:15) (2.41)The gravitational entropy is given by the Bekenstein bound: S AdS ∼ AG ∼ N (cid:15) (2.42)where G is the AdS Newton constant we calculated earlier. The two descriptionshave the same degrees of freedom. Therefore the AdS/CFT correspondence success-fully realized holography in string theory.We can also calculate the volume of
AdS up to the shifted boundary u = 1 − (cid:15) : V = L (cid:90) u (1 − u ) d u d Ω = L Ω (cid:90) u =1 − (cid:15) u (1 − u ) d u ∼ L (cid:15) (2.43)where Ω is the volume of the 3-sphere. Comparing with (2.41) we notice that thevolume of AdS scales with the same power of (cid:15) as its area close to the boundary.From that aspect holography seems trivial. However the non-triviality of our previousanalysis stems from the fact that area and volume scale with different powers of L. The gauge/gravity correspondence has a wide range of applications, mostly becauseof the fact that it enables the study of a strongly coupled field theory in terms ofa weakly coupled theory with gravity. We briefly discuss the most important areaswhere the AdS/CFT correspondence is applied.– 14 –
AdS/QCD
Quantum chromodynamics, the theory of strong interactions, becomes stronglycoupled at low energies. Holography provides a new method to study the low-energy spectrum of the theory, [9, 14]. There are two approaches to AdS/QCD.The first is the top-down approach, in which a brane configuration in stringtheory is engineered so that its low-energy spectrum shows poperties similar toQCD, [15, 16]. The second is the bottom-up approach, where models are builtbased on phenomenology, [17, 18, 19, 20, 21, 22, 23, 24]. • AdS/CMT
Theories describing interesting states of matter such as superconductors, su-perfluids, Bose-Einstein condensates as well as theories describing transitionsat zero temperature (quantum critical points) can be strongly coupled. Mostof these phenomena are well studied by experiments, however they may be dif-ficult to explain theoretically using the usual techniques from QFT. In holog-raphy, superconductors and superfluids are studied at finite temperature interms of a black hole in a higher dimensional space-time. It has been dis-covered that near the horizon of an AdS black hole the Abelian symmetrycan be spontaneously broken [10], [4], which is analogous to what happenswhen a system transits to a superconducting phase. There has been some suc-cess in the holographic description of superconductors, superfluids and insula-tors, as well as superfluid/insulator and superconductor/insulator transitions[7, 13, 41, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. • Fluid/gravity correspondence
Hydrodynamics is an effective long-distance description of any QFT that islocally in thermodynamic equilibrium. Fluid dynamics is described by theNavier-Stokes equations, which are non-linear partial differential equationsand are difficult to solve. Fluid dynamics can be connected to a classicalgravitational theory, [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].The geometries dual to fluid dynamics are black hole space-times with regu-lar event horizons. It has also been discovered that the cosmological Einsteinequations reduce to the relativistic Navier-Stokes equations at the appropriatelong-wavelength limit and it has been shown that there is a one to one cor-respondence between dissipative Navier-Stokes equations Einstein equationscoupled to matter. Results are summarized in [29, 30, 45].– 15 – . Effective Holographic Theories
The concept of Effective Holographic Theories, [7], is very useful in the study ofholographic IR fixed points. They are developed in analogy with Effective FieldTheories (EFT) which are used to study the low-energy limit of a QFT. In analogywith EFTs, the strategy is to select a collection of fields that dominates the low-energy dynamics and build an action that governs their behavior. However a stringtheory, in general, has an infinite number of fields with non-zero vacuum expectationvalues. The central point of the approximation is to truncate string theory to a finitespectrum, keeping only a few fields dual to the most important QFT operators.
The main idea of EHTs is to select a collection of operators that are expected todominate the low-energy dynamics and build a general action containing their dualstring fields at the two-derivative level.We first consider the simplest case. A quantum field theory always has a stress-energy tensor, therefore the EHT must contain a spin-2 tensor, the metric, whichencodes the energy distribution of the dual field theory. The minimal theory contain-ing a metric with AdS solution is the Einstein gravity with a cosmological constant.In a system at finite charge density another important operator is the conservedcurrent J µ . In this case we include a massless gauge field A µ in the holographic theory,dual to the conserved current. The theory in this case is the Einstein-Maxwell theory.Using these two fields the basic solution is the AdS-Reissner-N¨ordstrom black hole,[47], [48] which has many interesting properties.The next step is to add a scalar field dual to the most important scalar operator.The resulting EHT is an Einstein-Maxwell-Dilaton (EMD) theory. This theory has3 fields, the metric g µν which controls the energy distribution of the field theory, thegauge field A µ which controls the charge density, and a scalar φ controlling a scalarcoupling constant and vacuum expectation value (vev). The most general action at the two-derivative level containing a scalar, a U(1) gaugefield and a metric is the EMD theory, which is given in (d+1)-dimensions by: S EMD = M d − (cid:90) d d +1 x (cid:20) √− g (cid:18) R −
12 ( ∂φ ) + V ( φ ) − Z ( φ ) F (cid:19)(cid:21) (3.1)In the above action d is the number of the space-time dimensions of the dual fieldtheory. The action contains the Ricci scalar, R , a scalar potential V and the gaugefield strength tensor F µν = ∂ µ A ν − ∂ ν A µ .The asymptotic behavior of the coupling functions when φ → ±∞ is motivatedby generic examples in string theory and is exponential. We parametrize it as: V ( φ ) = e − δφ , Z ( φ ) = e γφ (3.2)– 16 –ero temperature solutions to this theory have geometries with scaling symme-tries that should be dual to quantum critical theories. We will analyze this topic inthe next section.Zero temperature solutions, in general, have naked singularities. Although inclassical gravity such solutions are unacceptable, in holography they are not alwaysunphysical. Gubser has studied and presented the criterion for acceptable singulari-ties in holography [3]. We expect that singularities that satisfy the Gubser criterionare resolvable. This can happen by either embedding them in a higher-dimensionsolution (see for example [11]) or by stringy effects [12]. In this section we describe the most important results from previous studies of theEMD theory. The zero temperature scaling solutions of the EMD theory have ametric of the form: ds = r θ (cid:18) − dt r z + dr + dx + dx r (cid:19) (3.3)This metric has the following scaling symmetry: t → λ z t, r → λr, x i → λx i , ds → λ θ/ ds (3.4)This symmetry appears in CM theories near quantum critical points.The gauge field in general scales in the IR as A t = µ + Qr ζ − z (3.5)where µ, Q are identified as the chemical potential and charge density of the fieldtheory.A solution is characterized by the 3 critical exponents ( z, θ, ζ ). The exponent z is the dynamical critical exponent , or Lifshitz exponent and θ is the hyperscalingviolation exponent . CFTs have θ = 0 , z = 1, while Lifshitz theories have θ =0 , z (cid:54) = 1. The conductivity exponent , ζ , is in general independent of the other twoexponents and determines the behavior of the charge density in the IR.When θ (cid:54) = 0 the proper distance ds also has to be scaled. This indicates violationof hyperscaling in the boundary theory, [46]. In (d+1)-dimensional field theories withhyperscaling symmetry the free energy of the system scales by its naive dimension.In theories with Lorentz symmetry ( z = 1 , θ = 0) the entropy is proportional to ∼ T d , where d is the number of spatial dimensions of the field theory. In Lifshitztheories ( z (cid:54) = 1 , θ = 0) the entropy S depends on the temperature as S ∼ T d/z .When hyperscaling is violated ( z (cid:54) = 1 , θ (cid:54) = 0) the entropy scales as S ∼ T d − θz andthe field theory has an effective dimensionality of d eff = d − θ .The behavior of the exponents z, θ is closely related to the behavior of the chargedensity and scalar field respectively. The solutions are scale invariant ( z = 1) in the– 17 –bsence of charges, while inhomogeneous metrics ( z (cid:54) = 1) can be obtained whenthe charge density is finite to leading order. The hyperscaling violating exponent θ depends on the behavior of the scalar. We obtain hyperscaling violating geometries( θ (cid:54) = 0) by letting the scalar run logarithmically to ±∞ in the IR.Recent studies of EMD in the absence of magnetic fields, [5], have shown theexistence of hyperscaling violating quantum critical lines separated by hyperscalinginvariant ( θ = 0) quantum critical points. We find that this is also true in thepresence of magnetic fields.Magnetic critical solutions have been studied in the context of Einstein-Maxwell-Chern-Simons theory in [8]. The results include a magnetic zero charge densityAdS × R geometry and a magnetic solution at finite density in which the chargedensity depends on the magnetic field. Our results include a pure magnetic AdS × R solution and a magnetic solution at finite density in which the charge density dependson the magnetic field.Critical solutions at finite density in the presence of an external magnetic fieldhave been studied in [67]. Magnetic solutions of the Einstein-Maxwell-axion-dilatontheory have also been studied recently in [68].
4. Setup and Equations of Motion
We consider the EMDPQ class of Effective Holographic Theories (EHT) involvingthe metric g µν , a scalar field φ and a massless gauge field A µ in 3+1-dimensionalspace-time: S = M (cid:90) d x (cid:20) √− g (cid:18) R −
12 ( ∂φ ) + V ( φ ) − Z ( φ ) F − W ( φ ) F ∧ F (cid:19)(cid:21) (4.1)with F ∧ F = 12 √− g F µν (cid:15) µνρσ F ρσ The dual QFT in this case lives in 2+1 dimensions. The PQ (Peccei-Quinn)term, F ∧ F , is important when studying magnetic fields at finite density. When themagnetic field is zero it can be ignored. This term does not depend on the metricand therefore does not appear in the Einstein equations, but affects the geometryimplicitly via the two other fields φ, A µ .The equations of motion stemming from (4.1) by varying the metric, the gaugefield and scalar are respectively: R µν − g µν R = T µν (4.2a) ∇ µ ( Z ( φ ) F µν + W ( φ ) (cid:63) F µν ) = 0 (4.2b) (cid:3) φ + d V eff d φ = 0 (4.2c)– 18 –here the stress energy tensor in (4.2a) is given by: T µν = V ( φ )2 g µν + 12 ∂ µ φ∂ ν φ − g µν ∂φ ) + Z ( φ )2 (cid:104) F ρµ F νρ − g µν F (cid:105) (4.3)and the effective potential in (4.2c) is given by V eff ( φ ) = V ( φ ) − Z ( φ )4 F − W ( φ )4 F ∧ F (4.4)We use the standard radial ansatz for the metric and the scalar field: ds = − D ( r ) dt + B ( r ) dr + C ( r )( dx + dx ) , φ = φ ( r ) (4.5)The coordinates ( x , x , t ) are the space and time coordinates of the dual (2 + 1)-dimensional field theory, while r is the holographic coordinate. This metric ansatzcomes with a gauge freedom related to the freedom in reparametrizing the radialcoordinate.We also consider an electric field potential depending only on r, related to thecharge density of the boundary theory. In addition, we include a uniform magneticfield in the radial direction which is related to the magnetic field of the boundarytheory: A µ = ( A t ( r ) , , h x , − h x ) (4.6)The r-dependence of the magnetic field is forbidden by the Einstein equations, un-less the metric is non-diagonal. The gauge field equation (4.2b) can be integratedobtaining a conserved charge, q : q = √− gZ ( φ ) F − W ( φ ) h, F = A (cid:48) t BD , ∂ r q = 0 (4.7)The equations of motion can be written in the above coordinate system as: φ (cid:48) + 2 C (cid:48)(cid:48) C = (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) CBC (cid:18) ( q + hW ) Z + h (cid:19) Z = D (cid:48)(cid:48) D − C (cid:48)(cid:48) C + 12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) BV + 14 B (cid:48) B C (cid:48) C = 12 (cid:18) D (cid:48)(cid:48) D + C (cid:48)(cid:48) C (cid:19) − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D − C (cid:48) C (cid:19) φ (cid:48)(cid:48) + φ (cid:48) ( C (cid:48) C + D (cid:48) D − B (cid:48) B ) = B C ∂ φ (cid:18) ( q + hW ) Z + h Z − C V (cid:19) A (cid:48) t = ( q + hW ) Z √ DBC (4.8)The electric and magnetic fluxes are calculated respectively byΦ E = 14 π (cid:90) R ( Z ( φ ) (cid:63) F + W ( φ ) F ) = − V π (cid:18) CZA (cid:48) t √ BD − hW (cid:19) = − V π q (4.9)– 19 – B = 14 π (cid:90) R F = − V π h (4.10)The integrals are over the spatial coordinates ( x , x ) and V = (cid:82) R dx dx isthe area of the surface of integration. The value of the electric flux depends only onthe integration constant q (4.7). For the rest of this paper we will refer to q as theelectric flux and to h as the magnetic field. We characterize a solution as “electric” if the electric flux (4.9) is non-zero, whilethe magnetic flux (4.10) vanishes in the IR. A solution is named “magnetic” if theinverse is true. If both fluxes are non-zero we characterize the solution as “ dyonic”.We also distinguish between solutions with finite charge density ( A (cid:48) t (cid:54) = 0) and zerocharge density ( A (cid:48) t = 0).We split the solutions into two sections depending on the behavior of the scalarfield. From (4.2c) we distinguish 2 cases: • The scalar field settles to a finite constant φ (cid:63) in the IR, which ex-tremizes V eff (4.4) These solutions are studied in section 5. In the neutral case looking for asymp-totic solutions of the form ds = r θ − ( − dt + dr + dx i dx i ) (4.11)we obtain an AdS geometry. Including an electric or magnetic field the scalarequation (4.2c) is satisfied only if C ( r ) in (4.5) is constant. The solution in theIR is AdS × R . The solution can be generalized to an AdS black hole. Westudy these cases in subsection 5.2.1. • The scalar field runs to infinity in the IR
These solutions are studied in section 6. We obtain hyperscaling violatinggeometries by allowing the scalar to run logarithmically ( φ = φ + a log r )in the IR. In supergravity the coupling functions, in general, are given by acombination of exponentials of the scalar φ . When φ runs to ±∞ we can keeponly the exponential with the largest value. Therefore we adopt the followingasymptotic behavior for the scalar coupling functions V ( φ ) = V e − δφ , Z ( φ ) = Z e γφ , W ( φ ) = W e χφ (4.12)We look for asymptotic solutions in the deep IR, which is located either at r → r → ∞ . For that purpose a power ansatz is enough. In particular we– 20 –earch for hyperscaling violating solutions which feature both a hyperscaling( θ ) and Lifshitz ( z ) exponent: ds = r θ (cid:18) − dt r z + dr + dx + dx r (cid:19) (4.13)While for the gauge field we also have the conductivity exponent ζ : A t = µ + A r ζ − z (4.14)The constant A is fixed by the equations and can be identified with the chargedensity of the boundary theory. The constant µ is the chemical potential, whichcannot be fixed by the equations due to the gauge invariance of the vector field.The equations of motion using the above ansatz are:( θ − θ − z + 2) = a − z )( θ − z −
2) = B r − θ (cid:18) h Z + ( q + hW ) Z (cid:19) ( θ − z − θ − z −
1) = B V r θ A ( ζ − z ) r ζ − = (cid:112) B ( q + hW ) Z (4.15)Every term in the equations of motion is a power of r. Since we are lookingfor solutions in the deep IR, which is located either at r → r → ∞ , wecan ignore the terms that are subleading in that limit. The strategy is to solvethe above equations to leading order by matching the powers and coefficientsof r on the left and right hand side. We distinguish cases depending on whichterms are leading. More details are presented in appendix F.The values of z, θ are restricted by the null-energy condition (appendix D) andthe allowed values are plotted in the same appendix. The solutions are alsoapproximately valid for small finite temperatures, at which the dual field theoryhas effectively d eff = 2 − θ dimensions and the thermal entropy scales as S ∼ T − θz (4.16)More details about hyperscaling violating metrics can be found in appendix B.Anisotropic scale invariant geometries ( z (cid:54) = 1) are obtained when the gaugefield participates to leading order in the equations. When z = 1 the chargedensity is always zero in the IR. In this subsection we discuss the electromagnetic duality of the EMD theory. Werewrite the equations of motion (4.8): – 21 – (cid:48) + 2 C (cid:48)(cid:48) C = (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) CBC (cid:18) ( q + hW ) Z + h Z (cid:19) = D (cid:48)(cid:48) D − C (cid:48)(cid:48) C + 12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) BV + 14 B (cid:48) B C (cid:48) C = 12 (cid:18) D (cid:48)(cid:48) D + C (cid:48)(cid:48) C (cid:19) − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D − C (cid:48) C (cid:19) φ (cid:48)(cid:48) + φ (cid:48) ( C (cid:48) C + D (cid:48) D − B (cid:48) B ) = B C ∂ φ (cid:18) ( q + hW ) Z + h Z − C V (cid:19) A (cid:48) t = q + hWZ √ DBC (4.17)Consider the Einstein and scalar equations and ignore the gauge equation fornow. The electric and magnetic fluxes q, h as well as the functions Z and W alwaysappear as the combination: ( q + hW ) Z + h Z When W is constant, if we swap ( q + hW ) with h and transform Z → /Z theequations do not change. The electromagnetic duality has the following form: q → h (1 − W ) − qW, h → q + hW, Z → /Z (4.18)When Z → ∞ the solution is purely magnetic, as the term ( q + hW ) Z is subleading.When Z → h Z is subleading and the solution is, in general, dyonic. Thistransformation, therefore, connects a purely magnetic solution to a dyonic one andvice versa.Now consider the gauge field equation. When Z → A (cid:48) t is, in general, non-zero. The electromagneticallydual solution is purely magnetic, with zero charge density A (cid:48) t = 0. Dyonic solutionsare connected to the magnetic solutions by the electromagnetic duality, with thedifference that the latter exist at zero charge density.In the special case W = 0 the electromagnetic duality takes the simpler form: q → h, h → q, Z → /Z (4.19)which maps a purely electric solution at finite charge density to a purely magneticsolution at zero charge density and vice versa. In the running scalar case we adoptthe behavior (4.12) for the coupling functions and the duality takes the form q /Z ↔ h Z , γ ↔ − γ . – 22 – . Constant Scalar Solutions In this section we study the solutions with the scalar settling to a finite value φ = φ (cid:63) ,that extremizes the effective potential:d V eff d φ (cid:12)(cid:12)(cid:12)(cid:12) φ = φ (cid:63) = 0 , V eff ( φ ) = V ( φ ) − Z ( φ )4 F − W ( φ )4 F ∧ F (5.1)The equations and details about the calculations are given in appendix E, whilethe perturbations around these solutions are presented in appendix H. There aremagnetic phases both at zero and non-zero charge density on AdS × R space-time.There are instabilities coming from the scalar field, depending on the values of thesecond derivatives of the coupling functions.For the rest of this section we use the following conventions: We define F (cid:63) = F ( φ (cid:63) ), where F ( φ ) is an arbitrary function of φ . We also define F (cid:48) (cid:63) = ∂∂φ F ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ (cid:63) . We first consider the simplest case, in which both the electric and magnetic fluxesare zero ( q = 0 = h ). In this case the scalar settles to a constant φ (cid:63) which extremizesthe value of the scalar potential V. V (cid:48) (cid:63) = 0 (5.2)This case has been recently studied in [5]. The leading order solution corresponds toa pure Einstein theory with a cosmological constant. Since there is no charge densitythe solution must be scale invariant ( z = 1). Indeed, when the value of V (cid:63) is non-zerowe have an AdS geometry: ds = L r ( − dt + dr + dx i dx i ) , L = 6 V (cid:63) (5.3)With the IR located at r → ∞ . Turning on the gauge field in this backgroundwe obtain A t = µ + A r (5.4)The second term creates a constant non-zero electric flux in the IR. It is a relevantdeformation. There is also a pair of modes coming from the scalar field perturbations(see appendix H). Both of them are relevant for V (cid:48)(cid:48) (cid:63) > V (cid:48)(cid:48) (cid:63) > L and thesolution is dynamically unstable. – 23 – .2 Electric/Magnetic solutions Including a magnetic or electric field the equations force the function C(r) (4.5) tobe a fixed constant (see appendix E). The geometry is the same for all the followingsolutions, but the value φ (cid:63) that the scalar assumes and the behavior of the chargedensity differ. The general solution for the metric has been found (in appendix E)and is studied in the next subsection 5.2.1. The general solution for the metric is the following: ds = L (cid:18) − f ( r ) r dt + dr f ( r ) r (cid:19) + C dx i dx i (5.5)where f ( r ) = 1 + K r + K r , C = 12 L E (cid:63) , L = 1 V (cid:63) , E = h Z + ( q + hW ) Z (5.6)The spatial coordinates of the field theory x i decouple from the radial and tem-poral coordinates. The coordinates x i enjoy rotational and translational symmetries,but not scaling symmetry.The UV is located at r → × R .Additionally, the metric is AdS × R for every value of r if K = K = 0. The lattercase has been studied in [5]: ds = L (cid:18) − dt + dr r (cid:19) + C dx i dx i (5.7)Time scales the same way as the radial coordinate, however the spatial part does notscale.The IR is located at r → ∞ where the behavior of the metric depends on thetwo integration constants K , K . The scalar curvature is independent of theintegration constants K , K and equal to − L . This means that, in principle, wecan transform the metric to AdS × R . However in some cases the transformation issingular and the metric contains a black hole. We consider different cases dependingon the behavior of the function f ( r ). The function f ( r ) is a quadratic polynomialwith discriminant ∆ = K − K . We distinguish the following cases regarding ∆and we focus on the AdS component of the metric: • ∆ > f ( r ) has two real roots r ± h = K ±√ ∆2 . In this case we have an AdS black hole with two horizons: the inner horizon is located at r − h and the outerat r + h : ds = L (cid:18) − K ( r − r − h )( r − r + h ) r dt + dr K ( r − r − h )( r − r + h ) r (cid:19) (5.8)– 24 – ∆ = 0: This is the extremal limit of the previous case, where the two horizonscoincide. We can get rid of the horizons with a simple redefinition of the radialcoordinate: ds = l u ( − d ˜ t + du ) (5.9)where u = r K r . The metric is AdS and has zero temperature. • ∆ <
0: In this case f ( r ) has no real roots. We can transform the metric toAdS (see appendix E): ds = ˜ L cos u ( − dτ + du ) (5.10) The simplest magnetic solution is obtained by setting q = 0 , W = 0. In this casewe have a magnetic field at zero charge density. The scalar field settles to a value φ (cid:63) that, instead of the scalar potential V , extremizes the effective potential (4.4) whichreads: V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) − V (cid:63) Z (cid:48) (cid:63) Z (cid:63) = 0 (5.11) Next, we study the magnetic solution at finite charge density which is obtained when q = 0 , W (cid:54) = 0. In this case the effective potential reads: V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) − V (cid:63) M (cid:48) (cid:63) M (cid:63) = 0 (5.12)Where M (cid:63) = Z (cid:63) + W (cid:63) Z (cid:63) (5.13)The value of the scalar, φ (cid:63) does not depend on the value of the magnetic field, h .When W is non-zero there is a finite IR charge density: A t = µ − Qr , µ = constant , Q = 2 W (cid:63) V (cid:63) Z (cid:63) ( W (cid:63) + Z (cid:63) ) (5.14)The charge density, Q, is fixed by the equations and is independent of the magneticfield. It depends only on the values of the coupling functions at φ (cid:63) . When W (cid:63) = 0the charge density is zero. – 25 – .2.4 Electric solution at finite charge density In this case we only have an electric field ( h = 0). This case is the electric dual ofthe solution 5.2.2 with W = 0 and has been studied in [5]. The scalar field settles toa value φ (cid:63) that extremizes the effective potential: V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) + V (cid:63) Z (cid:48) (cid:63) Z (cid:63) (5.15)The solution is again AdS × R with finite charge density: A t = µ − Qr , µ = constant , Q = 2 V (cid:63) Z (cid:63) (5.16)The IR charge density is again fixed by the equations and is independent of theelectric flux. It depends only on the values of the coupling functions at φ (cid:63) . In thispurely electric solution there is always a non-zero charge density, since Q cannot bezero. We now consider the most general solution with finite electric and magnetic fluxes( q (cid:54) = 0 , h (cid:54) = 0). This is the electromagnetic dual of case 5.2.2 with W (cid:54) = 0. In thiscase, the scalar field settles to a value φ (cid:63) that extremizes the effective potential: V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) − E (cid:48) (cid:63) E (cid:63) V (cid:63) (5.17)Where E (cid:63) = h Z (cid:63) + ( q + hW (cid:63) ) Z (cid:63) (5.18)The value φ (cid:63) that the scalar assumes now depends on the values of the electric ( q )and magnetic ( h ) fluxes.The charge density is given by: A t = µ − Qr , µ = constant , Q = V (cid:63) Z (cid:63) (cid:32) (cid:18) hZ (cid:63) q + hW (cid:63) (cid:19) (cid:33) (5.19)The charge density is fixed. In contrast to all the previous cases it now dependson the value of the magnetic field, h .Consider now the special case q + hW (cid:63) = 0. The effective potential is: V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) − Z (cid:48) (cid:63) Z (cid:63) V (cid:63) (5.20)and is independent of q, h . The charge density, Q , is now zero. This means thatwhen the magnetic field assumes the specific value h = − qW (cid:63) in terms of the electricflux, the charge density, Q , vanishes. This is similar to what is expected to happenin the quantum Hall effect. – 26 – . Running Scalar Solutions We now assume that the scalar field runs logarithmically: φ = φ + a log( r )to ±∞ in the IR and that asymptotically the coupling functions behave as: Z ( φ ) = Z e γφ , V ( φ ) = V e − δφ , W ( φ ) = e χφ (6.1)We first consider the pure EMD theory discarding the PQ term F ∧ F of theaction. This theory has already been studied in the past, [5], [7],[44]. Our ansatz isdifferent, as it also contains the magnetic field. The solutions come in pairs, due tothe electromagnetic duality of the EMD theory.We next consider the case with intrinsic parity violation, where we keep all theterms of the action ( W (cid:54) = 0). In general we expect new solutions.We look for metrics with a dynamical exponent z and a hyperscaling violatingexponent θ : ds = − r θ − z dt + B r θ − dr + r θ − ( dx + dx ) (6.2)while for the gauge field we assume the scaling form A t = µ + A r ζ − z (6.3)The triplet of critical exponents ( θ, z, ζ ) determines the geometry and the behaviorof the charge density and is, in turn, determined by the exponents γ, δ, χ that appearin the asymptotic behavior of the coupling functions. The constants µ, Q correspondto the chemical potential and charge density of the dual QFT respectively.The solutions with V subleading in the IR were also found. Such solutions seemproblematic because not only are there important parameters of the solution thatcannot be fixed, but also the IR is never well-defined for any values of the exponents γ, δ . It is not known if they are able to describe real physical systems, however wepresent them in appendix G for completeness. In this section we present the running scalar solutions of the EMD theory ignoringthe PQ term ( W = 0). The charge density in this case is closely related to theelectric flux. Solutions with an electric flux always have a finite charge density, whilesolutions without electric flux always have zero charge density. We find a neutral andan electric solution which have been recently studied in [5]. We also find the magneticdual of the electric solution. They are connected by γ → − γ, q /Z ↔ h Z . In thespecial case γ = 0 we have a dyonic solution which is electromagnetically self-dual.– 27 – .1.1 Neutral solution In this case the gauge field is zero and the Lorentz invariance of the field theory isrestored ( z = 1). This solution has been studied in the past, [5],[7],[11],[44]. ds = r θ (cid:18) − dt + B dr + dx i dx i r (cid:19) , e φ = e φ r √ θ ( θ − θ = 2 δ δ − , z = 1 , B = 2 e δφ V − δ ( δ − (6.4)The thermal entropy of the field theory for small temperatures scales as: S ∼ T − θz ∼ T − δ (6.5)When the exponent is negative the field theory has a mass gap and a discrete spec-trum, [7].Turning on the t-component of the gauge field in this background we find A t = µ + A r − γ √ θ ( θ − (6.6)with A ∼ q . The second term creates a constant finite electric flux in the IR when q (cid:54) = 0.The singularity is always located in the IR. However the location of the IR (aswell as the behavior of the entropy) depends on the value of δ : • δ <
1: The IR is located at r → ∞ . The entropy scales with a positive powerof T and vanishes at zero temperature. In the special case δ = 0 we arrive atthe AdS solution with constant scalar (in section 5.1). According to [7] thespectrum of the theory is continuous without a mass gap. • δ = 1: The geometry is AdS with constant scalar (same as section 5.1). Thethermal entropy scales as ∼ T . • < δ <
3: The IR is located at r →
0. The thermal entropy scales with anegative power of temperature and becomes very large for zero temperature.The boundary theory has a discrete spectrum with a mass gap, [44], [7]. • δ = 3: This value can only be reached by allowing V to be subleading. Thesolution is presented in appendix G. • δ >
3: Such values are unacceptable since the r-coordinate becomes time-like.They violate the Gubser bound [3].– 28 – .1.2 Magnetic solution at zero charge density
In this case the magnetic field is finite and the electric flux is zero. The Lifshitzexponent now depends on the parameters γ, δ . ds = − r θ − z dt + r θ − (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r θ/δ θ = 4 δδ − γ , z = 3 δ − γ + 2 γδ − δ − γ B = (2 + z − θ ) (cid:18) z − θV (cid:19) γγ + δ (cid:18) z − h Z (cid:19) δγ + δ , e ( γ + δ ) φ = 2 z −
11 + z − θ V h Z (6.7)The thermal entropy of the field theory scales at low temperatures as S ∼ T − θz ∼ T γ + δ )2 γ − γδ − δ (6.8)According to [7] a negative exponent indicates an unstable black hole, as the entropybecomes very large at extremality. It also indicates that the field theory may have adiscrete spectrum with a mass gap, [22]. There is a finite entropy at extremality inthe special case γ = − δ , which corresponds to an AdS × R space-time according to[5]. The parameter values for which the exponent is negative are plotted in figure 3.When the magnetic flux becomes zero ( z = 1 ⇒ γδ = 2 − δ ) we arrive at theprevious case 6.1.1. In the special case δ = 0 the metric becomes Lifshitz and theexponent of the entropy is positive for any γ . We can also arrive at a conformallyRindler metric when z = 0. This corresponds to the curves ( γ − δ ) = 4( δ −
1) onthe δ − γ plane. These curves separate the thermodynamically stable and unstableregions on the δ − γ plane. The behavior of the entropy in these cases is not knownand needs to be studied further. What can be done in this case is discussed in [22],[77].Turning on the t-component of the gauge field in this background we obtain: A t = µ + A r − z − θ γδ (6.9)The amplitude is proportional to A ∼ q . The region of the parameter space where A (cid:48) t is relevant is shown in figure 3. – 29 – igure 3: The region of validity of the magnetic solution 6.1.2 is shown above. Withinthe light blue area r −−→ IR r −−→ IR ∞ . The solution is validfor r → ∞ only if the electric flux is zero to all orders (see appendix F). The diagonals γ = − δ, γ = δ correspond to AdS × R and conformally AdS × R geometries respectively.The singularity is in most cases located in the IR, except for the values within the purplearea. In the light red area the entropy scales with a negative power of temperature, whichindicates that the field theory may have a mass gap. A (cid:48) t is relevant within the green areaand irrelevant everywhere else. We now consider the cases δ = ± γ , [5], which make the exponents θ, z blow up: • γ = − δ : In this case z → ∞ while θ remains finite. Changing the radialcoordinate r z = ρ we obtain the AdS × R geometry with constant scalar. • γ = δ : In this case z → ∞ , θ → ∞ , while their ratio θz = λ is finite. Afterchanging the radial coordinate r z = ρ the metric becomes: ds = ρ λ ( dρ − dt ρ + dx i dx i ) (6.10)which is conformally AdS × R . This solution has some interesting propertiesand for this it has been named semi-locally critical [5].– 30 – .1.3 Electric solution at finite charge density In this case the electric flux is finite and the magnetic field is zero. It has beenstudied in the past in [5],[7],[44]. This is the electric dual of case 6.1.2 with W = 0,which means that the geometry and scalar of this solution can be obtained with theduality transformation γ → − γ, h Z → q /Z . The analysis of this solution isanalogous to 6.1.2. The only difference is that there is no magnetic field and thecharge density is now non-zero. A t = µ + q γδ − γ (cid:118)(cid:117)(cid:117)(cid:116) (2 z − γ − δγ − δ (1 + z − θ ) γδ − γ Z δγ − δ V γγ − δ (2 + z − θ ) r θ − − z θ = 4 δδ + γ , z = 3 δ − γ − γδ − δ − γ (6.11) Figure 4:
The region of validity of the electric solution 6.1.3 is shown above. Within thelight blue area r −−→ IR r −−→ IR ∞ . The solution is valid in the darkblue area only if the magnetic field is zero to all orders (see appendix F). In the light redarea the IR is still at r −−→ IR
0, but the entropy scales with a negative power of temperature,which means that the boundary theory has a mass gap. – 31 –he charge density assumes a value proportional to a power of the electric flux: A ∼ q γδ − γ ∼ q − θ θ − (6.12)The entropy of the field theory scales at low temperatures as S ∼ T − θz ∼ T γ − δ )2 γ γδ − δ (6.13)The sign of the exponent is plotted in figure 4. In the leading order solution both electric and magnetic fluxes are finite if Z isconstant, which means that γ = 0, if we require the scalar field to run to infinity.This solution is electromagnetically self-dual. ds = − r − z dt + r (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r /δ , A t = µ + (cid:115) z − Z (2 + z − θ ) r ζ − z θ = 4 , z = 3 δ − δ , ζ = 2 B = (2 + z − θ )( z −
1) 2 Z q + h Z , e δφ = 2 z −
11 + z − θ V Z q + h Z (6.14)The charge density in this case is not affected by the existence of the magneticfield. However when the charge density is zero ( z = 1) both the electric flux and themagnetic field must be zero.The IR is located at r → δ <
2. Values 2 < δ < δ > r → S ∼ T − θz ∼ T δ − δ (6.15)For δ < the entropy scales with a positive power of temperature. When < δ < δ = 0 in which z → ∞ . Redefining the radial coordinateas: r z = ρ we obtain an AdS × R space-time with constant scalar: ds = 1 ρ (cid:0) − dt + B (cid:63) dρ (cid:1) + dx i dx i , B (cid:63) = lim z →∞ B z = 1 V (6.16)– 32 – .2 Scaling solutions with intrinsic P-violation In this section we study solutions with W (cid:54) = 0. When W is subleading we obtain thesame leading order solutions studied in the previous section. We correct these casesto first order of W by including a subleading term in appendix H.5. In the presentsection we study the solutions when W participates in the equations to leading order. There is a dyonic solution when W is constant and leading in the IR (this impliesthat χ = 0). This is the electromagnetic dual of solution 6.1.2 with W (cid:54) = 0. ds = − r θ − z dt + r θ − (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r θ/δ , A t = µ + A r ζ − z θ = 4 δδ + γ , z = 3 δ − γ − γδ − δ − γ , ζ = θ − B = (2 + z − θ ) (cid:18) z − θV (cid:19) γγ − δ (cid:18) z − Z ( q + hW ) (cid:19) δδ − γ , e ( δ − γ ) φ = 2 z −
11 + z − θ V Z ( q + hW ) A = ( q + hW ) γδ − γ (cid:118)(cid:117)(cid:117)(cid:116) (2 z − γ − δγ − δ (1 + z − θ ) γδ − γ Z δγ − δ V γγ − δ (2 + z − θ ) (6.17)The values of the scalar and the charge density both depend on the electric andmagnetic fluxes. The values of the exponents θ, z, ζ are the same as in case 6.1.3.The analysis of this solution is identical to case 6.1.3, however this solution shows aninteresting behavior when z = 1 which we will now consider. Setting z = 1 (whichis equivalent to setting γ = δ − /δ ) in the above equations we obtain: ds = r θ (cid:18) − dt + B dr + dx i dx i r (cid:19) , e φ = e φ r θ/δ θ = 2 δ δ − , z = 1 , B = 2 e δφ V − δ ( δ − , q + hW = 0 (6.18)This is the generalization of the neutral solution studied in section 6.1.1 when W (cid:54) = 0. The important difference is that instead of q = h = 0 we have the less strictrequirement q + hW = 0. This is satisfied, of course, when both q, h are zero, butit is also satisfied for non-zero values of q, h . The important conclusion is that whenthe magnetic field assumes a specific value in terms of the electric flux h = − qW thecharge density vanishes. This behavior also appears in the constant scalar case 5.2.5on an AdS × R background. – 33 – .2.2 Magnetic solution at finite charge density In this case W participates in the leading order solution and χ is, in general, non-zero.This is a stable magnetic quantum critical line, characterized by 3 exponents ( γ, δ, χ ).In previous cases the conductivity exponent, ζ was dependent on the hyperscalingviolating exponent θ . In this case the 3 exponents ( z, θ, ζ ) are completely independentfrom each other. ds = − r θ − z dt + r θ − (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r θδ , A t = µ + A r ζ − z θ = 4 δγ + δ − χ , z = 3 δ − ( γ − χ ) − δ ( γ − χ ) − δ − ( γ − χ ) , ζ = 2 δ − γγ + δ − χe ( δ +2 χ − γ ) φ = 2 z −
11 + z − θ Z V h W , B = (cid:18) z − θV (cid:19) χ − γδ +2 χ − γ (2 + z − θ ) (cid:18) z − Z h W (cid:19) δδ +2 χ − γ A = ( hW ) γδ +2 χ − γ ζ − z (cid:115) (2 z − γ − δ − χγ − δ − χ (2 + z − θ ) (cid:18) V z − θ (cid:19) γγ − δ − χ Z δ +2 χγ − δ − χ (6.19)The entropy at low temperatures scales as: S ∼ T − θz ∼ T χ − γ + δ )24+4 χ γ γδ − δ − χ ( γ + δ ) (6.20)For values of χ : γ + δ −√ δ − < χ < γ + δ + √ δ − A ∼ h γδ +2 χ − γ ∼ h θ − ζ θ − (6.21)which shows a similarity to the results in [8].The parameter χ must obey a constraint for this solution to be valid in the IR.This constraint comes from the requirement that W/Z → ∞ in the IR (for detailssee appendix F) and is the following:( γ + δ − χ )( χ − γ ) > , r −→ IR ∞ (6.22a)( γ + δ − χ )( χ − γ ) < , r −→ IR W ∼ r aχ in the IR depends on the value of χ relative to theother two exponents γ, δ . • χ ( χ − γ + δ ) <
0: Then W → ∞ if r −→ IR ∞ and W → r −→ IR • χ ( χ − γ + δ ) >
0: Then W → r −→ IR ∞ and W → ∞ if r −→ IR q (4.7) is non-zero we must require W → ∞ in the IRand we have an additional constraint. In this case q does not appear in the leadingorder solution. When q = 0 this constraint is absent.When χ assumes special values we arrive at some interesting cases: • χ = γ, W = Z : We obtain the magnetic solution of section 6.1.2. • χ = 0 , hW → q : In this case we obtain the electric solution of section 6.1.3. • χ = γδ − δ δ In this case ( h = 0 = A → z = 1) we obtain the neutral solutionof section 6.1.1. • χ → ∞ : In this case we arrive at the neutral AdS solution with constantscalar of section 5.1. • χ = γ − δ : In this case z → ∞ while θ is finite. Redefining the radial coordinate r z = ρ the geometry becomes AdS × R with constant scalar. Also the entropy(6.20) is constant and independent of the temperature. We have a finite entropyat extremality. • χ = γ + δ : In this case θ, z → ∞ , while their ratio is constant. Redefining theradial coordinate r z = ρ the geometry becomes conformally AdS × R . ds = ρ λ ( dρ − dt ρ + dx i dx i ) , λ = θz = 2 δ δ − • χ = γ + δ ± √− δ : In these cases z = 0 and the space-time is conformallyRindler. The curves corresponding to this case separate the thermodynamicallystable and unstable regions in the δ − γ plane. The formula for the entropy(6.20) no longer holds in this case and further studies are required, [22], [77].Studying the linear perturbations around this solution we find two pairs of modessumming to 2+ z − θ . The first pair corresponds to the finite temperature perturbationand the marginal perturbation, the latter of which can be absorbed by rescaling thecoordinates. The other two modes b ± are non-universal and also sum to 2 + z − θ .Their forms are not very enlightening and are presented in section H.2 of appendixH. One of these modes is always positive, while the other is always negative. Thesolution is, therefore, RG stable. The values of z, θ which make b ± complex areforbidden by combining the null-energy condition D.2 and the conditions for a well-defined IR (see appendix B), which shows that this solution is also dynamicallystable. – 35 –inally, in figure 5 we present the region of the γ − δ plane where this solution isvalid as well as the location of the IR for various values of χ . The constraints usedare the following:I Null-energy condition : The null-energy condition D is satisfied for (2 − z + θ )( θ − ≥ , ( z − z − θ ) ≥ Well-defined IR : The IR is well-defined (see appendix B) when ( θ − θ − z ) > r → z − θ < r → ∞ if 2 + z − θ > W/Z → ∞ : This combination must be leading in the IR in order to arrive atthis solution. The constraint is given by (6.22).IV
Space-like radial coordinate (Gubser bound) : B must be positive whichmeans ( θ − z − θ − z − > Real magnetic field : We require h > z − z − θ ) > Real charge density A : This requirement gives ( z − z − θ ) > igure 5: The region of validity of the solution 6.2.2 is shown above in the δ − γ planefor various values of χ . Within the light blue area the IR is located at r −−→ IR r −−→ IR ∞ . For values within the light red area the fieldtheory is expected to have a finite mass gap (as the entropy scales with a negative powerof temperature, [22]). – 37 – . Conclusion In the constant scalar case in the absence of magnetic fields we confirmed previousstudies, [5], which include a neutral phase on AdS and a charged phase on AdS × R .Turning on the magnetic field we obtain an AdS × R geometry both at zeroand finite charge density, which may contain a black hole with a pair of horizons.In both of these cases there are RG instabilities originating in the scalar and gaugefields. We also find a dyonic solution in which the charge density depends on themagnetic field. In this case the charge density vanishes for a special value of themagnetic field.In the running scalar case, in the absence of the PQ term ( W = 0), we found theusual neutral and electric solutions studied in the past [5],[7],[11],[44] along with theirmagnetic duals. They are connected by the transformation γ → − γ, q /Z ↔ h Z .We also found a dyonic solution which is electromagnetically self-dual. In this casethe magnetic field does not affect the value of the charge density.When W is constant and leading, we found a stable dyonic critical line in whichthe value of the charge density is affected by the presence of the magnetic field. Thecharge density vanishes when the magnetic field assumes a specific value in terms ofthe electric flux.We also found a stable magnetic critical line at finite density characterized bythree exponents ( γ, δ, χ ). In this case the charge density is proportional to a powerof the magnetic field, which depends on the values of two critical exponents θ, ζ . Acknowledgements
I would like to thank Elias Kiritsis for suggesting the topic and for helpful conversa-tions, comments and suggestions during the course of this work. I would also like toacknowledge helpful conversations with Chris Rosen.– 38 – ppendixA. The anti-de Sitter space-time
The anti-de Sitter space is the maximally symmetric solution of the Einstein equa-tions with a negative cosmological constant. Consider the Einstein-Hilbert actionwith a cosmological constant Λ in d+1 dimensions: S = 116 πG N (cid:90) d d +1 x (cid:112) − det g ( R − R µν − g µν R = − Λ g µν (A.2)Taking the trace we obtain the scalar curvature: R = 2 d + 1 d − S d +1 , the one withnegative Λ is the hyperbolic space H d +1 , while the one with Λ = 0 is the flat space R d +1 . In Minkowskian signature, the maximally symmetric solution with positive Λis called the de-Sitter (dS d +1 ) spacetime, the one with negative Λ is the anti-de-Sitter( AdS d +1 ) and the one with Λ = 0 is the Minkowski space.The spaces above can be defined from a d + 2 dimensional flat space via aquadratic constraint. We are interested in the AdS case, which can be defined from R ,d space [6] ds = − dX − dX d +1 + d (cid:88) i =1 dX i (A.4)as a hyperboloid of radius L via the condition: X + X d +1 − d (cid:88) i =1 X i = L (A.5)It is obvious from this condition that AdS d +1 has isometry group SO(2,d) and itis homogeneous and isotropic. The constraint (A.5) can be solved using the followingparametrization, [6]: X = Lcosh ( ρ ) cos ( τ ) , X p +1 = Lcosh ( ρ ) sin ( τ ) X i = Lsinh ( ρ )Ω i , (cid:0) i = 1 , , ..., d, Σ di =1 Ω i = 1 (cid:1) (A.6)– 39 –ubstituting (A.6) into (A.4) we obtain the AdS d +1 metric: ds = L ( − cosh ( ρ ) dτ + dρ + sinh ( ρ ) d Ω p − ) (A.7)where d Ω p − is the line element of the sphere S p − . Taking ρ ∈ (cid:60) + , τ ∈ [0 , π ) wecover the hyperboloid exactly once. Because of that these coordinates are calledglobal. Since τ is periodic we have closed time-like curves. We can obtain a causalspace-time by taking the universal cover, which means τ ∈ (cid:60) .There is another useful set of coordinates defined as, [1],[6]: X = u (cid:20) u ( L + (cid:126)x − t ) (cid:21) , X i = Lx i uX d = u (cid:20) − u ( L − (cid:126)x + t ) (cid:21) , X d +1 = Ltu (A.8)which brings the metric to the form ds = L u ( − dt + du + d(cid:126)x ) (A.9)These coordinates are called Poincar´e coordinates and they cover half the hyper-boloid. The Poincar´e symmetry of t, x i coordinates is now obvious, as well as thescale invariance: t → λt, u → λu, x i → λx i . In this coordinate system there is aboundary at u = 0. This is the coordinate system we use in this thesis.Finally there is another metric, also called Poincar´e metric, related to the pre-vious one by the transformation r = L /u : ds = L (cid:20) dr r + r ( − dt + d(cid:126)x ) (cid:21) (A.10)The boundary is located at r = ∞ . B. On Hyperscaling Violating metrics
Holography can be generalized to geometries which are not asymptotically AdS inthe IR. Such a generalization is very useful in Condensed Matter (CM) physics. Thecategory of Lifshitz theories is prime example. Such theories have a homogeneousscale invariant space characterized by the Lifshitz exponent, z: t → λ z t, x i → λx i (B.1)where t is the time component and x i are the spatial components of the space-time.The gravitational dual of a (d+1)-dimensional Lifshitz field theory can be definedon a metric with the same symmetry: ds d +2 = − dt r z + dr + dR d r (B.2)– 40 –here dR d = Σ di =1 dx i is the d-dimensional Euclidean metric.The characteristic of this space-time is the anisotropy between time and space.This geometry is not a solution to a pure cosmological Einstein gravity, becausethere is nothing to produce such an anisotropy. One needs to couple gravity to otherfields in order to obtain a Lifshitz metric. The Lifshitz geometry arises from anEinstein-Maxwell theory, where gravity is coupled to a gauge field.There is a more general class of geometries which can be reached by also includinga scalar field: they are solutions of an Einstein-Maxwell-Dilaton (EMD) theory.These metrics are conformal to Lifshitz and feature an extra parameter θ , called the”Hyperscaling Violation” exponent: ds d +2 = r θd ( − dt r z + dr + dR d r ) (B.3)This is the most general metric with homogeneous space and is invariant under thetransformation: t → λ z t, r → λr, x i → λx i , ds d +2 → λ θd ds d +2 (B.4)Obviously, when θ is non-zero the proper distance is not invariant under the scal-ing, which indicates violation of hyperscaling in the dual (d+1)-dimensional theory,[46]. In a field theory dual to the geometry (B.3) ( θ = 0) the thermal entropy scalesat low temperatures as T dz but in a field theory dual to (B.4) ( θ (cid:54) = 0) the entropyscales as T d − θz . Therefore a theory with non-zero θ has effectively d eff = d − θ dimensions at low temperature. B.1 Properties
The hyperscaling violating metric is the most general metric with homogeneous spaceand is extremal (zero temperature), ds d +2 = r θd ( − dt r z + dr + dR d r ) (B.5)The IR is located either at r = 0 or r = ∞ , depending on the values of θ , z, d.According to appendix B of [5] for the IR to be well defined we require:( θ − d )( θ − dz ) > θ < d, θ < dz (B.7)and is located at r = 0 if: θ > d, θ > dz (B.8)– 41 –n addition perturbations (appendix H) with a mode d + z − θ in general have finitetemperature according to appendix B of [5]. These modes should vanish in the UV,therefore in the IR we also require: d + z − θ > , r −→ IR ∞ (B.9a) d + z − θ < , r −→ IR θ determines the location of the singularity. The Ricci and Kretschmannscalars are: R ∝ r − θd , R µνρσ R µνρσ ∝ r − θd (B.10)Therefore if θ = 0 there are no singularities, if θ > r = 0 and if θ < C. Equations of motion and ansatz
We start with the action: S = M (cid:90) d x (cid:20) √− g (cid:18) R −
12 ( ∂φ ) + V ( φ ) − Z ( φ ) F (cid:19) − W ( φ ) F µν ˜ F µν (cid:21) , ˜ F µν = 12 (cid:15) µνρσ F ρσ (C.1)By varying the metric g µν , the gauge field A µ and the scalar field φ we obtainrespectively: R µν − R + V ( φ )2 g µν = 12 ∂ µ φ∂ ν φ − g µν ∂φ ) + Z ( φ )2 (cid:104) F ρµ F νρ − g µν F (cid:105) (C.2a)0 = ∇ µ (cid:18) Z ( φ ) F µν + W ( φ ) √− g ˜ F µν (cid:19) , (C.2b) (cid:3) φ = Z (cid:48) ( φ )4 F − V (cid:48) ( φ ) + W (cid:48) ( φ )4 √− g ˜ F , (C.2c)where ˜ F = F µν ˜ F µν (C.3)We will use the following radial ansatz for the metric and the scalar fieldd s = − D ( r )d t + B ( r )d r + C ( r ) (cid:0) dx + dx (cid:1) , φ ( r ) (C.4)while for the gauge field components A t ( r ) , A r = 0 , A = h x , A = − h x , h = constant (C.5)Substituting (C.4),(C.5) into (C.2) we obtain 4 Einstein equations with only 3linearly independent (the last 2 equations are identical):– 42 – (cid:48) C (cid:18) B (cid:48) B + C (cid:48) C (cid:19) = − BV + h Z BC + ZD A (cid:48) t + φ (cid:48) + 4 C (cid:48)(cid:48) Cφ (cid:48) + 2 BV = h Z BC + ZD A (cid:48) t + C (cid:48) C (cid:18) C (cid:48) C + 2 D (cid:48) D (cid:19) C (cid:48)(cid:48) C + 2 D (cid:48)(cid:48) D + φ (cid:48) = 2 BV + h Z BC + ZD A (cid:48) t + C (cid:48) C (cid:18) B (cid:48) B + C (cid:48) C − D (cid:48) D (cid:19) + D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D (cid:19) C (cid:48)(cid:48) C + 2 D (cid:48)(cid:48) D + φ (cid:48) = 2 BV + h Z BC + ZD A (cid:48) t + C (cid:48) C (cid:18) B (cid:48) B + C (cid:48) C − D (cid:48) D (cid:19) + D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D (cid:19) (C.6)as well as the gauge field equation (which has been integrated): A (cid:48) t = ( q + hW ) Z √ DBC , q, h = constant (C.7)and finally the scalar field equation: φ (cid:48)(cid:48) + φ (cid:48) ( C (cid:48) C + D (cid:48) D − B (cid:48) B ) = − B∂ φ V + h √ BDA (cid:48) t CD ∂ φ W + (cid:18) h B C − A (cid:48) t D (cid:19) ∂ φ Z (C.8)We can substitute A (cid:48) t from (C.7) into the Einstein and scalar equations of motion. φ (cid:48) + 2 C (cid:48)(cid:48) C = (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) C (C.9a) BC (cid:18) ( q + hW ) Z + h (cid:19) Z = D (cid:48)(cid:48) D − C (cid:48)(cid:48) C + 12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) (C.9b) BV + 14 B (cid:48) B C (cid:48) C = 12 (cid:18) D (cid:48)(cid:48) D + C (cid:48)(cid:48) C (cid:19) − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D − C (cid:48) C (cid:19) (C.9c) φ (cid:48)(cid:48) + φ (cid:48) ( C (cid:48) C + D (cid:48) D − B (cid:48) B ) = B C ∂ φ (cid:18) ( q + hW ) Z + h Z − C V (cid:19) (C.10)In the running scalar case we are interested in hyperscaling violating solutions, there-fore we use the following ansatz: D ( r ) = r θ − z , B ( r ) = B r θ − , C ( r ) = r θ − , A t = A r ζ − z (C.11) D. Null-Energy Condition
There are constraints on the exponents z, θ stemming from the null-energy condition.This condition states that the contraction of the stress-energy tensor, T with a null-vector, N (a vector with zero length, N µ N µ = 0), must be non-negative. We can use G µν = T µν , where G is the Einstein tensor to obtain the easier-to-handle expression: T µν N µ N ν ≥ ⇒ G µν N µ N ν ≥ − z + θ )( θ − ≥ , ( z − z − θ ) ≥ z, θ are given in the left figure of 6. Adding the conditions fora well-defined IR (see appendix B) we obtain the right figure of 6. Figure 6:
Left figure: Imposing only the null-energy condition
D.2 the allowed values of z, θ are within the black area. Right figure: Additionally we impose the conditions forwell-defined IR. Within the gray area the IR is located at r −−→ IR ∞ while in the black areathe IR is located at r −−→ IR E. Constant scalar solutions
Setting φ = φ (cid:63) in (C.9) we obtain:2 C (cid:48)(cid:48) C = (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) C (E.1a) BC (cid:18) ( q + hW (cid:63) ) Z (cid:63) + h (cid:19) Z (cid:63) = D (cid:48)(cid:48) D − C (cid:48)(cid:48) C + 12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) (E.1b) BV (cid:63) + 14 B (cid:48) B C (cid:48) C = 12 (cid:18) D (cid:48)(cid:48) D + C (cid:48)(cid:48) C (cid:19) − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D − C (cid:48) C (cid:19) } (E.1c)– 44 –hile from the scalar equation (C.10) we have ∂ φ (cid:18) ( q + hW ) Z + h Z − C V (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ (cid:63) = 0 (E.2)Where V (cid:63) , Z (cid:63) , W (cid:63) is the value of V, Z, W at φ = φ (cid:63) respectively. E.1 Neutral solution
We will look for solutions with q = 0 = h . This case is equivalent to a pure Einstein-Dilaton theory and has already been studied in the past [5] . Setting q = 0 = h and φ = φ (cid:63) in (E.2) we obtain: ∂ φ V ( φ (cid:63) ) = 0 (E.3)Therefore the scalar settles to a constant φ (cid:63) which extremizes the value of the scalarpotential V.Setting q = 0 = h and φ = φ (cid:63) in (C.9), the Einstein equations become:2 C (cid:48)(cid:48) C = (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) C (E.4a) C (cid:48)(cid:48) C − D (cid:48)(cid:48) D = 12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) (E.4b) BV (cid:63) + 14 B (cid:48) B C (cid:48) C = 12 (cid:18) D (cid:48)(cid:48) D + C (cid:48)(cid:48) C (cid:19) − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D − C (cid:48) C (cid:19) (E.4c)Looking for solutions of the form: ds = L r θ − ( − dt + dr + dx i dx i ) (E.5)we obtain θ ( θ −
2) = 0 , L V (cid:63) r θ = ( θ − θ −
3) (E.6)from which we obtain θ = 0 , L = 6 V (cid:63) , V (cid:63) (cid:54) = 0 (E.7) θ = 2 , V (cid:63) = 0 (E.8) E.2 Dyonic solution
We look for solutions with qh (cid:54) = 0 in general. We rewrite (E.2) (cid:18) D (cid:48) D + B (cid:48) B + C (cid:48) C (cid:19) C (cid:48) C = 2 C (cid:48)(cid:48) C (E.9a)12 (cid:18) C (cid:48) C − D (cid:48) D (cid:19) (cid:18) B (cid:48) B + D (cid:48) D (cid:19) = BC E (cid:63) + C (cid:48)(cid:48) C − D (cid:48)(cid:48) D (E.9b) C (cid:48) C (cid:18) C (cid:48) C + D (cid:48) D (cid:19) + B C E (cid:63) = BV (cid:63) (E.9c)– 45 –here E (cid:63) = h Z (cid:63) + ( q + hW (cid:63) ) Z (cid:63) (E.10)From the equation (E.2) we obtain C = E (cid:48) (cid:63) V (cid:48) (cid:63) = constant (E.11)Setting C (cid:48) = C (cid:48)(cid:48) = 0 in (E.9) we are left with D (cid:48)(cid:48) D − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D (cid:19) = BC E (cid:63) (E.12a)2 C V (cid:63) = E (cid:63) (E.12b)From (E.11) and (E.12b) we obtain V (cid:48) eff ( φ (cid:63) ) = V (cid:48) (cid:63) − E (cid:48) (cid:63) E (cid:63) V (cid:63) (E.13)The only equation that remains is (E.12a) which using (E.12b) can be written as D (cid:48)(cid:48) D − D (cid:48) D (cid:18) B (cid:48) B + D (cid:48) D (cid:19) = 2 V (cid:63) B (E.14)We use the gauge freedom stemming from the radial ansatz to set: D = L r f ( r ) , B = L r f ( r ) , L = 1 V ∗ (E.15)The equation becomes r f (cid:48)(cid:48) − rf (cid:48) + 2 f − f ( r ) = 1 + K r + K r (E.16)The metric is therefore: ds = L (cid:18) − f ( r ) r dt + dr f ( r ) r (cid:19) + C dx i dx i (E.17)With the transformation ρ = 2 arctan f (cid:48) ( r ) √− ∆ √− ∆ (E.18)the metric becomes: ds = L G ( ρ )( − dt + dρ ) , G ( ρ ) = − ∆16 cos ( √− ∆ ρ + arctan( √− ∆ K )) (E.19)– 46 –e transform again the radial coordinate and rescale the time coordinate as: u = 12 √− ∆ ρ + arctan( √− ∆ K ) , t = 2 τ √− ∆ , ˜ L = L/ √ ds = ˜ L cos u ( − dτ + du ) (E.21)which is AdS . Since the transformation involved √− ∆ it becomes singular when∆ > A t = µ − Qr , Q − = V (cid:63) Z (cid:63) (cid:32) (cid:18) hZ (cid:63) q + hW (cid:63) (cid:19) (cid:33) , µ = constant (E.22) F. Running scalar in the IR
We assume that the scalar field runs logarithmically in the IR and the couplingfunctions scale as: V ( φ ) = V e − δφ , Z ( φ ) = Z e γφ , W ( φ ) = W e χφ , e φ = e φ r a (F.1)Using the ansatz (C.11) with the Einstein equations (C.9) we obtain: k = a (F.2a) k = B r − θ (cid:18) h Z + ( q + hW ) Z (cid:19) (F.2b) k = B V r θ (F.2c)where k = ( θ − θ − z + 2) , k = 2(1 − z )( θ − z − , k = ( θ − z − θ − z −
1) (F.3)also from the gauge field equation (C.7) we obtain: A ( ζ − z ) r ζ − = (cid:112) B ( q + hW ) Z (F.4)We ignore the scalar equation because it always satisfied if the above equationsare satisfied. – 47 – .1 Solutions without intrinsic P-violation In this section we ignore the parity violating term F ∧ F of the action. We rewritethe above equations setting W = 0.The Einstein equations are the following: k = a (F.5a) k = B r − θ (cid:18) h Z + q Z (cid:19) (F.5b) k = B V r θ (F.5c)and the gauge field equation is: A ( ζ − z ) r ζ − = (cid:112) B qZ (F.6)In equation (F.5b) there are two terms on the right hand side which dependon different powers of r (because of the implicit r-dependence of Z). We have toconsider cases depending on which one of those terms is leading in the IR limit. Forthat reason we distinguish 3 cases regarding Z. From (F.1) we see that Z scales as Z ∼ r aγ . Depending on the sign of aγ and whether the IR is located at 0 or ∞ wehave the following cases:I Z →
0: Then aγ > aγ <
0) for r → r → ∞ ). In this case in leading order h Z + q Z → q Z .This is the electric solution. This case can also be reached by setting h = 0 andthe constraint on the sign of aγ is evaded.II Z → ∞ : Then aγ < aγ >
0) for r → r → ∞ ). In this case in leadingorder h Z + q Z → h Z This is the magnetic solution. This case can also be reached by setting q = 0and the constraint on the sign of aγ is evaded.III Z → Z e γφ : Then aγ = 0. In this case both terms are of the same order h Z + q Z → h Z e γφ + q Z e − γφ This case contains both electric and magnetic fields, however because the twoterms are of the same order only if they are both constant we have the constraint aγ = 0.Next we notice that in (F.5b), (F.5c) the left hand side is a constant and theright hand side is a function of r. The right hand sides can be leading or subleadingin the IR. We have to further distinguish 4 subcases regarding the constants k , k a k k (cid:54) = 0:In this subcase all the terms are leading.– 48 – k (cid:54) = 0 , k = 0:In this subcase the right hand side of (F.5c) is subleading. This corresponds tosolutions with subleading scalar potential ( V → k = 0 , k (cid:54) = 0:In this subcase the right hand side of (F.5b) is subleading. This corresponds toneutral solutions.d k = 0 = k :In this subcase both right hand sides of (F.5c) and (F.5b) are subleading. Thiscorresponds to neutral solutions with subleading scalar potential ( V = 0). Thegeometry in this case does not depend on the parameters δ, γ of the theory. F.1.1 Electric solutions a k k (cid:54) = 0: a = 4 γ + δ , θ = 4 δγ + δ , z = 3 δ − γ − γδ − δ − γ , ζ = θ − B = k V e δφ , q = 2 2 + δ ( γ − δ )2 + γ ( γ − δ ) Z V e ( γ − δ ) φ , A = q √ B Z ( θ − z − e − γφ (F.8) aγ < , r −→ IR ∞ aγ > , r −→ IR k (cid:54) = 0 , k = 0: a = 2 γ γ , θ = 2 + 21 + γ , z = θ − , ζ = θ − B q = 4 Z e γφ γ , A = − e − γφ (cid:112) Z (1 + γ ) (F.10) aγ < , θ < aδ, r −→ IR ∞ aγ > , θ > aδ, r −→ IR k = 0 , k (cid:54) = 0: a = 2 δδ − , θ = 2 δ δ − , z = 1 (F.11)– 49 – = 2 e δφ V − δ ( δ − (F.12) aγ < , θ + aγ > , r −→ IR ∞ aγ > , θ + aγ < , r −→ IR ∞ d k = 0 = k : a = 3 , z = 1 , θ = 3 (F.13) aγ < , θ < aδ, θ + aγ > , r −→ IR ∞ aγ > , θ > aδ, θ + aγ < , r −→ IR F.1.2 Magnetic solutions a k k (cid:54) = 0:The same as Ia with q → h Z , γ → − γaγ > , r −→ IR ∞ aγ < , r −→ IR k (cid:54) = 0 , k = 0:The same as Ib with q → h Z , γ → − γaγ > , θ < aδ, r −→ IR ∞ aγ < , θ > aδ, r −→ IR k = 0 , k (cid:54) = 0:The same as Ic with γ → − γaγ > , θ > aγ, r −→ IR ∞ aγ < , θ < aγ, r −→ IR k = 0 = k :The same as Id with γ → − γ – 50 – a = z (4 − z ) , θ = 2 + z, aγ = 0 aγ < , θ < aδ, θ > aγ, r −→ IR ∞ aγ > , θ > aδ, θ < aγ, r −→ IR F.1.3 Dyonic solutions a k k (cid:54) = 0:Same geometry as Ia with q → q + h Z , γ = 0Same gauge field as Ia with γ = 0 γ = 0b k (cid:54) = 0 , k = 0:Same geometry as Ib with q → q + h Z , γ = 0Same gauge field as Ia with γ = 0 γ = 0 , θ < aδ (F.14)c k = 0 , k (cid:54) = 0:Same geometry as Ic with a = 0d k = 0 = k :Same as Id aγ = 0 , θ < aδ, θ > F.2 Solutions with intrinsic P-violation
We distinguish 3 cases for W in the IR limit: • lim W = 0 Then aχ > aχ <
0) for r → r → ∞ ). In the leading ordersolution q + hWZ → qZ We obtain the same leading order solutions as the previous section. • lim W = W Then aχ = 0. In the leading order solution q + hWZ → q + hW Z This case contains both an electric and a magnetic field. We obtain the sameleading order solutions as the previous case with the difference q → q + hW .– 51 – lim W = ∞ Then aχ < aχ >
0) for r → r → ∞ ). In the leading ordersolution q + hWZ → hWZ We can also arrive at this case by setting q = 0 to avoid the constraints on thesign of aχ .For W → ∞ we rewrite the equations: k = a (F.16a) k = B h r − θ (cid:18) Z + W Z (cid:19) (F.16b) k = B V r θ (F.16c)and the gauge field equation: A ( ζ − z ) r ζ − = (cid:112) B h WZ (F.17)We distinguish 3 cases regarding W/Z :I WZ → a ( χ − γ ) > a ( χ − γ ) <
0) for r → r → ∞ ). In the leadingorder solution Z + W Z → Z .In this case we obtain again the magnetic solutions of the previous section.II WZ → W Z Then a ( χ − γ ) = 0. In the leading order solution Z + W Z → Z (1 + W Z )The same the previous case, but with Z → Z + W Z , aχ = aγ III WZ → ∞ Then a ( χ − γ ) < a ( χ − γ ) >
0) for r → r → ∞ ). In the leadingorder solution Z + W Z → W Z .In this case we have new solutions.We further distinguish 4 subcases regarding the constants k , k a k k (cid:54) = 0:This subcase gives electric/magnetic solutions.b k (cid:54) = 0 , k = 0:This subcase corresponds to solutions with subleading scalar potential ( V = 0).c k = 0 , k (cid:54) = 0:This subcase corresponds to neutral solutions.– 52 – k = 0 = k :This subcase corresponds to neutral solutions with subleading scalar potential( V = 0).a k k (cid:54) = 0: a = 4 γ + δ − χ , θ = 4 δγ + δ − χ , z = 3 δ − ( γ − χ ) − δ ( γ − χ ) − δ − ( γ − χ ) , ζ = 2 δ − γγ + δ − χ (F.18) B = k V e δφ , h = k k Z V W e ( γ − δ − χ ) φ , A = (cid:112) B hW ( ζ − z ) Z e ( χ − γ ) φ (F.19) aχ > aγ, r −→ IR ∞ aχ < aγ, r −→ IR k (cid:54) = 0 , k = 0: a = 2( γ − χ )1 + ( γ − χ ) , θ = 2+ 21 + ( γ − χ ) , z = θ − , ζ = 2( χ − γ ) 1 + 2( γ − χ ) γ − χ ) (F.20) B h = 2( θ − Z W e ( γ − χ ) φ , A = (cid:112) B hW ( ζ − z ) Z e ( χ − γ ) φ (F.21) aχ > aγ, θ < aδ, r −→ IR ∞ aχ < aγ, θ > aδ, r −→ IR k = 0 , k (cid:54) = 0:The geometry is the same as Ic of the parity preserving case. aχ > aγ, θ > a (2 χ − γ ) , r −→ IR ∞ aχ < aγ, θ < a (2 χ − γ ) , r −→ IR k = 0 = k :The geometry is the same as Id of the parity preserving case. aχ > aγ, θ < aδ, θ > a (2 χ − γ ) , r −→ IR ∞ aχ < aγ, θ > aδ, θ < a (2 χ − γ ) , r −→ IR
0– 53 – . Solutions with subleading scalar potential ( V → ) In this section we present the solutions in which the scalar potential V is subleadingin the IR. The solutions are problematic because the IR is usually not well-definedand important parameters, such as B , cannot be fixed.I Neutral solutionThis case corresponds to an Einstein-Dilaton theory with V = 0. The solutionis conformally flat: ds = r (cid:0) − dt + B dr + dx i dx i (cid:1) , e φ = e φ r a (G.1)with B undetermined and a = 3. The IR located at r = 0.II Magnetic solution at zero charge densityThe IR is not well-defined and the solution is valid only for r → ds = − r − θ dt + r θ − (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r a (G.2) a = − γ γ , θ = 2 + 21 + γ , z = θ − , ζ = θ − B Z h = 4 e − γφ γ (G.4)Since θ is always positive, the singularity is located at r = 0. For γ → ∞ wearrive at a flat, neutral geometry.III Electric solution at finite charge densityThe IR is not well-defined and the solution is valid only for r →
0. This is theelectric dual of II. There is also a charge density: A t = µ + A r , A = − e − γφ (cid:112) Z (1 + γ ) (G.5)IV Dyonic solution at finite charge densityIn the leading order solution there is both a finite charge density and a non-trivial magnetic field only if Z is constant. In this case the scalar field must beconstant a = 0, and thus γ is free.– 54 – s = − dt r + r (cid:0) B dr + dx i dx i (cid:1) , A t = µ + A r , A = − q √ B Z e γφ (G.6) a = 0 , θ = 4 , z = 3 , ζ = 2 (G.7) B (cid:18) e γφ h Z + q Z e − γφ (cid:19) = 4 (G.8)The IR is not well-defined and the solution is valid only for r → W is participating in the leading order solution and V is subleading.The solution is valid only for r → χ → ∞ we obtain a flat neutral geometry with constant scalar, while for χ = γ/ ds = − r − θ dt + r θ − (cid:0) B dr + dx i dx i (cid:1) , e φ = e φ r a , A t = µ + A r ζ − z (G.9) a = 2( γ − χ )1 + ( γ − χ ) , θ = 2+ 21 + ( γ − χ ) , z = θ − , ζ = 2( χ − γ ) 1 + 2( γ − χ ) γ − χ ) (G.10) B h = 2( θ − Z W e ( γ − χ ) φ , A = 1 ζ − z (cid:115) θ − Z e γφ (G.11) H. Linear Perturbations
We consider two kinds of perturbations. The first is an expansion in amplitudes ofthe perturbations. The second is an expansion in powers of r, where we calculatethe first order corrections due to the subleading terms in the leading order solu-tions of appendix F. We are mostly interested in perturbations that maintain zerotemperature. In the second case we have an inhomogeneous linear 5x5 system ofequations with the amplitudes as the unknown variables. The inhomogeinity comesfrom the subleading term and the amplitudes can be fixed in terms of the parametersof the leading order solution, after choosing a gauge. In the first case the system ishomogeneous and has a non-trivial solution only if its determinant vanishes.We perturb around the running scalar solutions using the following ansatz:– 55 – ( r ) = B r θ − (1 + B r b ) , C ( r ) = r θ − (1 + C r c ) , D ( r ) = r θ − z (1 + D r d ) (H.1) φ = φ + a log( r ) + Φ r φ , A t = µ + r f ( A + A r f ) (H.2)We plug this perturbation ansatz into the equations of motion and set c = d = b = a = b . There is a residual gauge freedom stemming from the radial ansatz.Choosing a gauge practically means fixing one of the perturbation amplitudes. Wewill not choose a gauge, as the calculations are very simple even without a fixed gauge.We have a 5x5 homogeneous system with the perturbation amplitudes as unknownvariables. There are non-trivial solutions only when the determinant vanishes. Thedeterminant can be factored into: b ( b − − z + θ )( b − b + )( b − b − )( f + f ) (H.3)where b + + b − = 2 + z − θ, b + b − = − z − z − θ )(2 + z − θ )2 z − − θ (H.4)There are two pairs of conjugate modes each adding up to 2 + z − θ , whichshould be the case in this coordinate system. There is always a marginal mode b = 0 which only gives a shift of the constants of the leading order solution. Theseshifts can always be absorbed by rescaling t, x i and choosing a gauge. The mode b = 2 + z − θ corresponds to a finite temperature perturbation, however with somegauge choices it can maintain zero temperature according to appendix B of [5]. Inthe gauge C = 0 this mode always gives the emblackening factor and has finitetemperature: f ( r ) = 1 + D r z − θ ds = − r θ − z f ( r ) dt + r θ B dr r f ( r ) + r θ − dx i dx i (H.5)For the non-universal modes b ± we can see that for the neutral solutions ( z = 1)they are identical to the universal modes: b + = 2 + z − θ, b − = 0. However in allother cases they maintain zero temperature and one of them is relevant while theother is irrelevant, since always b + b − < H.1 Neutral hyperscaling violating solution
We perturb around the solution 6.1.1. • b = 0: Marginal mode B = δ Φ (H.6) • b = 2 + z − θ : Relevant mode with finite temperature B = − D − C + 2 δ Φ (H.7)– 56 – .2 Electric/Magnetic hyperscaling violating solutions We perturb around the solution 6.2.2, the perturbations for 6.1.2, 6.1.3 and 6.1.4 canbe obtained by setting χ = γ , χ = 0 and χ = 0 = γ respectively. • b = 0: Marginal mode B = δ Φ , C = Φ χ − γ + δ • b = 2 + z − θ : Relevant mode with finite temperature B = − D + ( δ + γ − χ )Φ , C = 12 ( δ − γ + 2 χ )Φ (H.9) • b = b ± : One relevant and the other irrelevant.The expressions for the amplitudes are too messy and we will not present them.This pair of modes shows a dynamical instability ( b ± become complex) when(2 + z − θ )(2 z − − θ )( −
20 + 2 z + 18 z + 16 θ − zθ + θ ) < H.3 Neutral AdS We perturb around the neutral constant scalar solution (5.1) using the ansatz (theperturbations around this solution have also been studied in [5]) B ( r ) = L r (1 + B r b ) , C ( r ) = L r (1 + C r c ) , D ( r ) = L r (1 + D r d ) (H.10) φ = φ (cid:63) + Φ r a , A t = µ + A r f (H.11)The determinant factors into c d f ( f − a − a + L V (cid:48)(cid:48) (cid:63) )[ c (6 − d + b ) + 3 d + b (2 d − f = 0 , f = 1corresponding to chemical potential and charge density. The latter mode creates aconstant finite flux.Next, coming from the scalar field, there are 2 modes given by: a − a + L V (cid:48)(cid:48) (cid:63) = 0 → a ± = 12 (cid:104) ± (cid:112) − L V (cid:48)(cid:48) (cid:63) (cid:105) (H.13)The mode a +1 is always relevant, while a − is relevant when V (cid:48)(cid:48) (cid:63) >
0. The solution isdynamically unstable when 4 L V (cid:48)(cid:48) (cid:63) > C = 0 we obtain d = b = 3 with amplitudes B = − D , C = Φ = A = 0. This is the finitetemperature perturbation. – 57 – .4 Electromagnetic AdS × R We perturb around the constant scalar
AdS × R solutions (5.2.4), (5.2.3), (5.2.5)using the ansatz B ( r ) = L r (1 + B r b ) , C ( r ) = C (1 + C r c ) , D ( r ) = L r (1 + D r d ) (H.14) φ = φ (cid:63) + Φ r φ , A t = µ + r − ( A + A r f ) (H.15)In all 3 cases the perturbation modes are similar. By setting h = 0 or q = 0 oneobtains the perturbation for the pure electric and pure magnetic case respectively.The determinant factors into: c d (1 + c )( d − f − P ( a ) (H.16)where P ( a ) = a − a + L V (cid:48)(cid:48) (cid:63) − E (cid:48)(cid:48) (cid:63) E (cid:63) → a ± = 12 ± (cid:114) − L V (cid:48)(cid:48) (cid:63) + E (cid:48)(cid:48) (cid:63) E (cid:63) (H.17)where E (cid:63) = ( q + hW (cid:63) ) /Z (cid:63) + h Z (cid:63) .There are six modes b = ( − , , , , a + , a − ) which pairwise sum to 1 as required.The amplitudes for each case are given below. • b = −
1: Irrelevant mode B = 13 (4 C + 3 D − L Φ V (cid:48) (cid:63) ) , C = Φ ( E (cid:48)(cid:48) (cid:63) E (cid:48) (cid:63) − V (cid:48)(cid:48) (cid:63) − V (cid:63) V (cid:48) (cid:63) ) A = q + hW (cid:63) LZV (cid:48) (cid:63) √ E (cid:63) (cid:20)(cid:18) V (cid:48) (cid:63) V (cid:63) − V (cid:48) (cid:63) V (cid:63) ( Z (cid:48) (cid:63) Z (cid:63) − hW (cid:48) (cid:63) q + hW (cid:63) ) (cid:19) Φ + 3 V (cid:48) (cid:63) V (cid:63) D (cid:21) • b = 0: Marginal mode B = − Φ L V (cid:48) (cid:63) , C = 0 , A = L ( q + hW (cid:63) )2 CZ (cid:63) (cid:32) − D + Φ ∂ φ log( V Z ( q + hW ) ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ (cid:63) (cid:33) (H.19) • b = 1: Relevant mode, finite temperature, shift of chemical potential B = − D − L V (cid:48) (cid:63) (H.20)We have additionally Φ = 0 in all cases except 5.2.4.– 58 – b = 2: Relevant mode C = Φ = 0 , A = ( B + D ) L ( q + hW (cid:63) ) Z (cid:63) √ E (cid:63) (H.21) • b = a ± : B = − a ± D , C = Φ = 0 , A = − D ( q + hW (cid:63) ) LZ (cid:63) √ E (cid:63) (H.22)These two modes come from the scalar field. The mode a +1 is always relevant,while a − is irrelevant if L V (cid:48)(cid:48) (cid:63) < E (cid:48)(cid:48) (cid:63) E (cid:63) . There is a dynamical instability if L V (cid:48)(cid:48) (cid:63) > + E (cid:48)(cid:48) (cid:63) E (cid:63) . H.5 Corrections due to the PQ term
When W is subleading in the IR, the leading order solutions are identical to thesolutions presented in section 6.1. In this section we present these solution alongwith the first order corrections coming from the PQ term. H.5.1 Neutral solution
The leading order solution in this case is described by 6.1.1. ds = − D ( r ) dt + B ( r ) dr + C ( r ) dx i dx i , e φ = e φ +Φ r b r a , A t = µ + A r β D ( r ) = r θ − z (1 + D r b ) , B ( r ) = B r θ − (1 + B r b ) , C ( r ) = r θ − (1 + C r b ) a = 2 δδ − , θ = 2 δ δ − , z = 1 , b = 4 + (2 χ − γ − δ ) a, β = 1 + ( χ − γ ) aB = 2 e δφ V − δ ( δ − (H.23)All the amplitudes D , B , C , Φ are fixed (after choosing a gauge) and areproportional to h W Z , while A is proportional to h W Z . The region of validity of thissolution is smaller than that of 6.1.1 as we also require that b, β be subleading in theIR. H.5.2 Magnetic solution at zero charge density
The leading order solution in this case is the same as 6.1.2. ds = − D ( r ) dt + B ( r ) dr + C ( r ) dx i dx i , e φ = e φ +Φ r b r a , A t = A t = µ + A r β D ( r ) = r θ − z (1 + D r b ) , B ( r ) = B r θ − (1 + B r b ) , C ( r ) = r θ − (1 + C r b ) a = 4 δ − γ , θ = 4 δδ − γ , z = 3 δ − γ + 2 γδ − δ − γ , b = 2 a ( χ − γ ) , β = 2 − z + a ( χ − γ ) B = ( θ − z − θ − z − V e δφ , h = 2 z −
11 + z − θ V Z e − ( γ + δ ) φ (H.24)– 59 –he amplitudes D , B , C , Φ are fixed (after choosing a gauge) and are pro-portional to h W Z , while A is proportional to h W Z . The region of validity of thissolution is smaller than that of 6.1.2 as we also require that b, β be subleading in theIR. H.5.3 Electric solution at finite charge density
The leading order solution in this case is described by 6.1.3. ds = − D ( r ) dt + B ( r ) dr + C ( r ) dx i dx i , e φ = e φ +Φ r b r a , A t = µ + A r θ − − z (1 + A r β ) D ( r ) = r θ − z (1 + D r b ) , B ( r ) = B r θ − (1 + B r b ) , C ( r ) = r θ − (1 + C r b ) a = 4 δ + γ , θ = 4 δδ + γ , z = 3 δ − γ − γδ − δ − γ , b = β = aχB = ( θ − z − θ − z − V e δφ , q = 2 z −
11 + z − θ V Z e ( γ − δ ) φ , A = (cid:115) z − Z e γφ (2 + z − θ )(H.25)All the amplitudes D , B , C , Φ , A are proportional to hW /Z . The region ofvalidity of this solution is smaller than that of 6.1.3 as we also require that b, β besubleading in the IR. H.5.4 Magnetic solution at finite charge density
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