Quasi-normal-mode analysis & phase diagram

In general the numerical solution to a system of coupled linear differential equation is found by setting the initial values and numerically integrating its evolution. Alternatively, boundary conditions can be set and thus one is sometimes forced to vary the boundary conditions on one

“side” of the integration domain to match the wanted values on the other “side”. This is usually done using a shooting method which additionally employs a root finding algorithm as explained in Section 4.1 for the s-wave superconductor. In our case we will follow a slightly different strategy. For2^ndorder phase transitions the critical temperature/chemical potential can be found by looking at the background solution with vanishing fields but non-vanishing fluctuations about this particular background solution, such that the overall value of the field isΦÝÑΦ`δφ“δφ.

The condensation of the scalar field is triggered by an instability which can be seen from the poles of the Green function being located in the upper complex ω plane. In particular, the pole is moving through the origin of the complexω-plane located at ω “0, exactly when the

temperature reaches the critical valueT_c. The appearance of a zero mode is indicating a global symmetry breaking that generates a massless mode according to the Goldstone’s Theorem on page46. Decreasing the temperature further leads to the breakdown of the effective field theory constructed by the saddle-point solution and thus to a negative mass squared of the fluctuations causing acausal behavior. In practice the instability and the critical temperature can be found by:

• Find solution to the linearized fluctuation equations with

ω “0(zero mode) andk“0(zero momentum). Finite momenta characterize transport processes beyond the thermodynamic validity,e.g. hydrodynamic properties of the system.

• Vary the chemical potential/temperature and look for poles in the Green function at the origin of the complexωplane for vanishing momenta.

In the case of ω “ 0 andk “ 0 we see that the linearized fluctuation equations (4.97) are identical to the scalar equation for the background field Φ (4.68) and the poles of the Green function correspond to solutions of the background equations with vanishing source (e.g. the leading term of the boundary expansion of the background field is zero which corresponds to the denominator of the fluctuation Green function). Generically, this is only true for a special choice of the background scalar field potential, namely a quadratic potential . Since we are probing with small fluctuations about theΦ“0solution up to quadratic order in the action, all that remains of the full-fledged potential is its leading quadratic term. After fixinguH, the only parameter left to vary is^T{^µ. For the fluctuation equations in the probe limit we haveµ¯as an external parameter coming from the solutions of the background fields ΦandA_t. For a second order differential equation inδφwe have two free parameters. These are fixed by the infalling boundary condition at the horizon for the fluctuations δφ and the overall normalization (which can be set to one since it is linear in all terms of the expansion at the horizon). Therefore the Green function forω “ 0 andk “ 0 only depends on the dimensionless parameter µ¯ “ µuH „ ^µ{^T. Taking the backreaction into account we get an additional external parameterαwhich determines the strength of the backreaction. Therefore we have to determine for each α the corresponding

¯

µc and take care of not running into temperatures that are non-positive since here the relation betweenµ,T andαis more complicated,e.g. see (4.52).

T

µ “d´^pd_d´^´21^q2µ¯²α²

4π¯µ ÝÑ T

µ ˇˇ

ˇα“0“ d 4π¯µ

ñ µ¯“

´2pd´1qπ^T_µ ` c

4pd´1q²π²´

T µ

¯2

`dpd´1qpd´2q²α²

pd´2q²α² “

cd´1 d´2

Q¯

α. (4.248) Settingα“0we recover the probe limit relation of^T{^µandµ, see (4.67). Including the backreac-¯ tion we use the holographic dissipation-fluctuation theorem in order to determine the complex-valued Green function of scalar fluctuations about the fixed scalar background in the normal phase. For convenience we state again the equation of motion for the scalar field fluctuations (4.97) in the normal phase

δφ²puq ` ˆf¹puq

fpuq ´d´1 u

˙

δφ¹puq `

„pω`A_tq² fpuq² ´ k²

fpuq´ L²m² u²fpuq



δφpuq “0, (4.249) with the Reissner-Nordström black brane blackening factor (4.49) and the background gauge field (4.48). After fixing the incoming wave condition at the horizon of the AdS-Reissner-Nordström black brane, we integrate out to the boundary of AdS-space and fit the numerical

T µ

α d“3

0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.2 0.4 0.6 0.8 1.0

m²L²“ ´2 m²L²“0 m²L²“4

T µ

α 0.0

0.00 0.05 0.10 0.15

0.2 0.4 0.6 0.8 1.0

d“4

m²L²“ ´4 m²L²“ ´3 m²L²“0 m²L²“5

Figure 4.8. Phase diagram of the holographic s-wave superconductor ford“4 andd “3as a function of the backreaction parameterα, depending on the scalar field mass.

Colored regions (ford“3are encoded asm²L² “4,m²L² “0,m²L²“ ´2and ford“4 we havem²L² “5,m²L² “0, m²L² “ ´3,m²L² “ ´4) show phases where the scalar field condenses yielding a superfluid phase. Due to the largeNc

limit, theColeman-Mermin-Wagner Theorem on page53 is evaded ind “ 2`1 dimensions allowing for a phase transition to happen. Quantum fluctuations arise only in the¹{^Nc corrections. On the gravity side the Anderson-Higgs mechanism for removing gauge symmetries is not subject to any constraint, formally the lower critical dimension is8due to Elitzur’s theorem.

solution to the boundary expansion given in (4.42). Finally, we read off the Green function and determine the critical values of^T{^µfor a given value of the strength of the backreactionαyielding critical curves as shown in Figure4.8. Additionally, we can also vary the mass of the scalar field related to the scaling dimension of the dual operator via

∆_˘“1 2

´ d˘a

d²`4L²m²¯

, m²L²“∆_˘p∆˘´dq. (4.250) Note that for positive masses,i.e.m²L²ą0the dual operator becomes relevant in the UV which would drastically alter the UV conformal theory. Since this operator is not sourced, the RG flow to the UV fixed point is not affected by the non-zero profile of the scalar field with positive masses in the bulk.

Analytic value ofα_c at zero temperature

In the case of vanishing temperature, we can find an analytic solution for the critical value of the strength of the backreaction. The charge of the extremal Reissner-Nordström black brane for T “0can be determined from (4.248) to be

Q¯²“ d

d´2 ñ T “0. (4.251)

with finite black brane horizonuHfollowing from (4.50), uH“

adpd´1q d´2

1

αµ. (4.252)

Curiously, the entropy density remains finite in this case. Taking the definition of the entropy given in (4.62) and inserting the zero temperature black brane horizonuHwe find

s“ S

Vd´1 “ 4π 2κ²

L^d´1 u^d´_H ¹ “ 4π

2κ²

L^d´1α^d´1µ^d´1

”dpd´1q pd´2q²

ı^pd´₂^1q “2π

„pd´2q² dpd´1q

^pd´1q{²

L^d´3α^d´3µ^d´1

e² (4.253)

where in the last equalityκ²has been replaced via the definition of the backreactionα²L²“^κ2{^e2. Ind“3andd“4dimensions the zero-temperature entropy density reduces to

sˇˇ

d“3“πµ²

3e³, sˇˇ

d“4“ 2π 3?

3 Lαµ³

e² . (4.254)

In the zero temperature limit, the near horizon limit of the blackening factor reads fpuq “1´

ˆ d d´2 `1

˙u^d u^d_H ` d

d´2 u^2pd´1q

u²_H^pd´1q « dpd´1q

u²_H pu´uHq²,

which gives rise to an AdS2metric ds²« L²

u²_H ˆ

´dpd´1q

u²_H pu´uHq²dt²` u²_H

dpd´1qpu´u²_Hqdpu´uHq²`dx²

˙

“ L² dpd´1q

ˆ

´d²pd´1q²

u⁴_H pu´uHq²dt²`dpu´uHq²

pu´uHq² `dpd´1q u²_H dx²

˙

. (4.255)

Redefining the coordinate̺“ pu´uHq^´1and rescalingtandxby a constant accordingly, (4.255) looks like an AdS2ˆR^d´1metric

ds²“L²_AdS2

̺²

´d˜t²`d̺²¯

`d˜x<sup>2</sup>“ds<sup>2</sup><sub>AdS2</sub>`dE^d´1, (4.256) where the AdS2radius is is related to theAdSd`1radius by

L²_AdS2 “ L²

dpd´1q. (4.257)

The gauge field/chemical potential for vanishing temperature is given by

A“

adpd´1q pd´2q

1 uHα

˜

1´u^d´2 u^d´_H ²

¸

dt, (4.258)

and the near horizon expansion ofAtpuq²up to leading order reads A_tpuq²« dpd´1q

u⁴_Hα² pu´uHq². (4.259)

For the near horizon expansion of (4.249) we need to insert the near horizon expansions offpuq andA_tpuq²which up to leading order yields

δφ²pu´uHq ` ˆ 2

u´uH´d´1 uH

˙

δφ¹pu´uHq

`

„ dpd´1q

d²pd´1q²α²pu´u²_Hq´ L²m² dpd´1qpu´uHq²



δφpu´uHq “0,

δφ²pu´uHq ` ˆ 2

u´uH

˙

δφ¹pu´uHq ´ 1 dpd´1q

ˆ

L²m²´ 1 α²

˙δφpu´uHq

pu´uHq² “0. (4.260) The effective mass term in (4.260) can be read off by comparing to the scalar field equation in AdS224

δφ²prq `2

rδφ¹prq ´L²_AdS2m²

r² δφprq “0, (4.261)

and compare it to the Breitenlohner-Freedman stability bound in one dimension since we are in AdSd`1“2-space

L²_AdS2m²_eff“ 1 dpd´1q

ˆ

L²m²´ 1 α²

˙

ďL²_AdS2m²_BF“ ´1

4, (4.262)

which indicates, that the stability bound for the effective scalar field massmeffis lowered by the strength of the backreactionα. This is true in general since the minimal coupling of the scalar field to theUp1qgauge field gives rise to an effective mass term. Looking at (4.8) we see

∇a∇^aΦ`i∇aA<sup>a</sup>Φ´`

m²`AaA^a˘

Φ“0. (4.263)

Of course, there is still the possibility for more complicated potentials which may lead to more complicated mass terms in the first place. Assuming that we only have a non-zero time compo-nent of the gauge field dependent on the radial coordinate, (4.263) can be rewritten as

∇_a∇^aΦ´`

m²`g^ttA_tA_t˘

Φ“0. (4.264)

Thus, there could be a critical value ofαwhere the scalar field condenses in the near horizon region. Solving (4.262) forαgives rise to the condition that scalar field condensation occurs

1

α² ě dpd´1q

4 `L²m², or α²ď 1

dpd´1q

4 `L²m². (4.265) In the case ofd“3,4we find the critical value to bec.f. Figure4.9

α²_c ˇˇ

ˇd“3“ 1

3

2`L²m², α²_c ˇˇ

ˇd“4“ 1

3`L²m². (4.266) We see if the mass is already below the AdS2Breitenlohner-Freedman bound, there is no critical value forαand hence no quantum critical point or phase transition at zero temperature between the condensed phase and the normal phase. The different masses used in the numerical calcula-tion and the corresponding values for the scaling and the critical backreaccalcula-tion strength are listed in Table4.5. This is true for allm²L²we have chosen ford“3,4, (m²L²“ ´2,´3,´4ă ´¹{⁴)

24It is easier to work in the coordinatesr“ u´u_Hwith the AdS2 metricds²_AdS

2 “L²`

r²d˜t`^dr2{^r2˘

in order to compare with the original AdSd`1 Schwarzschild metric inucoordinates since the double zeros of the blackening factor give rise to the metric structure.

0.5

0.5 0.0

´1.0

´1.5

´2.0

´2.5

´3.0

0.4 0.6 0.8 1.0 1.2

α L2 AdS2m2 eff

d“3

L²m²“ ´2 L²m²“0 L²m²“4

0.0

´0.5

´1.0

´1.5

0.4 0.6 0.8 1.0 1.2

α L2 AdS2m2 eff

d“4

L²m²“ ´4 L²m²“ ´3 L²m²“0 L²m²“5

Figure 4.9. The left plot shows the effective AdS2 mass in d “ 3 for different masses of the scalar field. The black constant line denotes the Breitenlohner-Freedman bound in one dimension. In the case ofm²L²“0,4, the mass of the scalar field is above the BF bound where the intersection determinesα_c. Ind “4 dimensions, shown on the right, we see that only form²L²“0,5the masses are above the Breitenlohner-Freedman bound in the IR and so there is a critical valueα_c.

so the superfluid phase is extended along the T “ 0 axis for all values ofα. Finally, we can calculate the corresponding critical value ofµ¯via (4.50) to be

¯ µ_c“

cdpd´1q d´2

1

α_c ÝÑ µ_c ˇˇ ˇd“3“

?6

α_c, and µ_c ˇˇ ˇd“4“

?3

α_c, (4.267) which upon insertion into^T{^µgiven in (4.248) will give zero. Our results as displayed in Figure 4.8show that the critical temperature decreases with increasing backreaction strengthα. More-over, if the scalar mass is larger than a critical value, the critical temperature goes to zero for a finite value ofα. This is the case most reminiscent of a real highT_c superconductor, when the dome in Figure4.8has similarities with the right hand side of the dome in the phase diagram of a highT_csuperconductor.

Physical interpretation of the holographic superconductor

The physical interpretation ofαis that it corresponds to the ratio of the number ofSUp2qcharged degrees of freedom over all degrees of freedom [219]. The phase diagrams above indicate that an increase ofαreduces the numbers of degrees of freedom which can participate in pair for-mation and condensation, such that T_c is lowered. A similar mechanism also seems to be at work when adding a double trace deformation to the holographic superconductor [240], and has been discussed within condensed matter physics using a BCS approach in [241]. For holo-graphic superconductors, this mechanism is most clearly visible for the top-down holoholo-graphic superconductors involving D7 brane probes [215,217] in which the dual field theory Lagrangian and thus the field content of the condensing operator is known. In these models, there is aUp2q symmetry which factorizes into anSUp2qIˆUp1qB,i.e. into an isospin and a baryonic symmetry.

A chemical potential is switched on for theSUp2qI isospin symmetry and the condensate is of ρ-meson form,

J_x¹“ψσ¯ ¹γ_xψ`φσ¯ <sup>1</sup>Bxφ“ψ¯<sub>Ò</sub>γ<sub>x</sub>ψ<sub>Ó</sub>`ψ¯_Óγ_xψ_Ò`bosons, (4.268) whereψ“ pψ_Ò, ψ_Óqandφ“ pφ_Ò, φ_Óqare the quark and squark doublets, respectively, which in-volve up and down flavors, withσⁱthe Pauli matrices in isospin space andγ_µthe Dirac matrices.

d“3 d“4

m²L² 4 0 ´2 5 0 ´3 ´4

∆_´ ´1 0 1 ´1 0 1 2

∆_` 4 3 2 5 4 3 2

α²_c 2

11

2

3 ´2 1

8

1

3 8 ´1

Table 4.5. List of the critical values for the strength of the backreactionαfor different masses in three and four dimensions. Note that the instability condition is satisfied for α ă α_c. In particular, for the negative values we do not have a critical value of αPR, so in this case forT “0we always find a condensed/superfluid phase. ∆_˘ describes the scaling of the dual operator according to the boundary expansionΦ« Φsourceu^∆´`@

O

∆_`

Du^∆`. Ford “3 andm²L² “ ´2 an alternative quantization scheme yields the operator

O

∆_´. Ford“4with saturated Breitenlohner-Freedman bound we need to introduce an additional implicit UV cut-offlnp^u{^δq.

As an additional control parameter, a chemical potentialµ_B for the baryonicUp1qB symmetry may be turned on. This leads to a decrease ofT_c[222], see also [242], which may be understood as follows: Under theUp1qB symmetry,ψ andψ¯ have opposite charge. The same applies toφ andφ. Turning on¯ µ_B leads to an excess ofψ over ψ¯ degrees of freedom, which implies that less degrees of freedom are available to form the pairs (4.268). The same applies toφandφ¯ degrees of freedom. Thus in this case, charge carriers in the normal state are also present in the superconducting phase, leading to the formation of a pseudo-gap.

Let us comment on the RG picture of the holographic superconductors, pictorially shown in Fig-ure4.10. The conformal UV fixed point is deformed by a relevant operatorJ^twhich scaling field is associated with the boundary value ofAtB“µ. According to Table3.3, a massless vector field withp“1andm“0is always a relevant scaling field withyµ “d´∆J ą0. More generally, the UV relevant operatorJ^t, driving the system out of the conformal UV fixed point, generates a RG flow where theUp1qgauge symmetry is removed and Lorentz invariance is broken. The deep interior IR geometry is stabilized by the scalar field condensate. In the zero temperature case discussed above, the emergent AdS2ˆR^d´2of the extremal AdS-Reissner-Nordström black brane allows for a stable scale invariant solution of the scalar field by choosing a suitable IR potential such that the ground state yields a irrelevant scalar deformation [245]. For the operatorJ^tthere are two possibilities, eitherJ^tbecomes irrelevant in the IR, restoring Lorentz symmetry which gives rise to a AdS4spacetime [246–248], or the operator’s anomalous dimension allows a more intricate IR fixed point geometry. In general, these solutions involve so-called Lifshitz geome-tries [158] with arbitrary dynamical scaling exponentz, defined in (2.159). In the casez“1we find a relativistic AdS4 geometry, whereas the extremal AdS-Reissner-Nordström black brane is recovered in the limitzÑ 8. Lifshitz solution with finitezdescribe completely discharged black brane geometries, where the charged scalar condensate sources the boundary field theory charge densityxJ^ty. A nice overview of the low-temperature behavior of the holographic condensed phases are given in [249]. More details of the related fermionic story described by electron stars can be found in [177]. Let us conclude with two open questions:

• What is the origin of the AdS2ˆR^d´2instability close to the quantum critical point atαc?

Φ Eu

u

u ^IR

IR UV UV

xJ^ty ‰0 x

O

y ‰0

Tc{^{µc T}{^µ

Figure 4.10. In the normal phase^T{^µą^Tc{^µcthe charge density of the boundary theoryxJ^ty is sourced completely by the electric flux through the black brane horizon con-nected to the electric fieldEH “ A¹_tHpuq. A charged pair producing instability characterizes the onset of the superconducting phase below ^Tc{^µc generating a charged condensate x

O

y that contributes to the boundary field theory charge density. For extremely low temperaturesT ! µthe IR geometry becomes more intricate. As we already discussed in the main text at zero temperature, the ex-tremal Reissner-Nordström black brane gives rise to an emergent AdS2ˆR^d´2IR geometry. More “exotic” IR fixed point geometries are possible which allow for a completely discharged and thus vanishing black brane, where the charge density is sourced entirely by the electric flux generated by the charged condensate. In a wider sense, geometries with charges “hidden” behind the black brane horizon are called fully fractionalized, whereas charge distributions inside and outside of the black brane are known as partially fractionalized. An electric flux sourced completely by charges in the bulk are called cohesive; well-known examples are bosonic holographic superconductors and fermionic electron stars [243,244].

• How do quantum corrections in¹{^Ncaffect the largeNc holographic picture? Clearly, on the field theory side we expect an instability due to the diverging entropy (4.254) allowing for a high ground state degeneracy. More importantly, ind“3dimensions we expect that the phase of the condensate is totally randomized due to quantum fluctuations, restoring theUp1qas explained in Section2.3.4.

Parts of these questions are tried to answered in a zero temperature finite density top-down setup discussed in Chapter5.

Im Dokument Gauge/Gravity duality (Seite 188-195)