TELEFONÍA

theories which are invariant under the conformal groupSO_p2, d_qinddimensions. Since at the fixed point the couplings remain invariant under a change in the characteristic length scaleu, the geometry of scale invariant fixed point theories must be scale invariant underx ÝÑ c´1x

anduÝÑc´1u, which is realized by the AdS-space with metric34 ds2_“ L

2

u2

`

´dt2_{`</sub>dx2<sub>`}du2˘. (3.88) Note that the isometries of the AdS-spacetime ind`1 dimensionsSOp2, dqis exactly the con- formal group inddimensions. The crucial question arises, which fixed point is described by the AdS-spacetime. Clearly, we need to start with the microscopic theory in the UV in order to de- fine the couplingsgaof the underlying lattice theory in terms of physical microscopic interaction strengthsJa. Thus, we rediscover the holographic dictionary entry that the field living in the bulk geometry of the AdS-spacetime must correspond to the microscopic couplings in the UV,i.e.

gapxqˇˇˇ UV“J a_{pxq “ϕ}a_{px, uq}ˇˇ ˇ u“0“ϕ a_pxqˇˇ ˇ BpAdSq. (3.89)

Here we can make the relation between the scaling dimension and the nature of the associated operator clear. The coefficientϕd´∆of the leading term in the boundary expansion of the field scales withd´∆under scaling transformations, which is exactly the eigenvalueyJ of a scaling

field close to the fixed point (see (2.123)). The scaling dimension of the operator is related to the scaling of the source field byyJ “d´∆O. The behavior in the vicinity of the fixed point is thus controlled by the mass of the bulk field. According to Table3.3, for a scalar field with mass

mthe corresponding scaling field of the operator is relevant when yJ ą0 which corresponds

to ∆ ă dand ´d2_{4L2 ă m2 ă 0. For m “ 0 we find ∆ “ d and thus a marginal operator

yJ “0, whereas formą0,∆ądandyJ ă0the operator is irrelevant. Similarly, for all other

bulk fields listed in Table 3.3we can do the same analysis. Additionally, from (3.89) follows that the bulk fields in AdS-spacetime must carry the same quantum numbers and charges as the corresponding couplings. Since the effective action at an RG fixed point is a conformal field theory we can extend the dictionary by identifying the coupling with the sources of the effective CFT action (2.156) Ssource_“ ż ddx`ϕa_px_q

O

apxq `AaµJaµ`gµνTµν ˘ , (3.90)

which in turn correspond to the associated bulk fields via (3.89) with the same structure,i.e. the effective action of the bulk theory includes scalar fieldsϕpx, uqfor each scalar operator

O, vector

(gauge) fieldsAApx, uqfor each currentJµand a spin-two fieldgABfor the canonical energy mo-

mentum tensorTµν_{, arising due toPolyakov’s Theoremon page65. The existence of a spin-two}

field is the key to the gauge/gravity correspondence. According to the Weinberg-Witten theo- rem, and precursors [91,199], a Lorentz invariant spin-two field theory describes a topological theory which would not affect the couplings/sources of the QFT side or couple universally due to the equivalence principle which effectively describes gravity. Thus, the AdS-spacetime arises

34_{Strictly speaking we work in Euclidean signature which corresponds to a statistical system, but we like to replace the}

computation of physical quantities such as thermodynamic and transport properties by a gravitational computation with the gravity dual. As we will shortly see, we can extend the holographic dictionary to include calculations involving correlation functions by real-time calculations as is made explicit in the holographic fluctuation-dissipation theorem (3.102) discussed in Section3.5.1.

from a gravitational theory described by classical gravity with negative cosmological constant Λ“dpd´1q_{2L2 SAdS“ ₂1 κ2 ż dd`1 x?´g ´ R´2Λ`L_{matterrϕ, AAs}¯_{. (3.91)} Note that the energy-momentum tensor of the gravitational bulk theory is given by (3.3)

TAB“ ´ 2 ? ´g δp?´gL_matterq δgAB “ ´2 δL_matter δgAB `gABLmatter, (3.92) or35 TAB“ ?2 ´g δp?´gL_matterq δgAB , (3.93)

and must not be confused with the energy-momentum tensor of the boundary conformal field theory. In summary, the UV fixed point conformal quantum field theory in d dimensions can be viewed as the boundary of a d`1 dimensional gravitational theory described by an AdS- spacetime. The source of the conformal energy-momentum tensor is the boundary value of the spacetime metric and the matter fields in the bulk AdS-spacetime describe the dynamics of the couplings under the RG flow of the quantum field theory operators. The boundary values of these matter fields correspond to the UV fixed point couplings.

3.4.2. Finite temperature & density deformations

In order to understand the global RG flow diagram and critical phenomena, we need to deform the fixed point CFT by relevant deformations allowing us to flow to other fixed points. Gener- ically, the β-functions encoding the global RG flow are not accessible in complicated strongly coupled or strongly correlated systems. A holographic realization of fixed point deformation is realized by deforming the spacetime geometry in such a way that we recover the AdS-spacetime asymptotically. This amounts to a theory with a UV fixed point and a non-trivial IR behavior. Such a scenario is known from almost all condensed matter theories, where the short-range mi- croscopic theory is known, but the long-range/low-energy behavior emerges non-trivially from the microscopic degrees of freedom. There are many possibilities for non-trivial IR geometries, but the holographic principle provides us already with the most simple ones. In order to define a field theory with thermodynamic properties as temperature, entropy and a free energy, we may consider a black hole geometry with horizon at u“ uH which approaches AdS-spacetime asymptotically foruÑ0. Two well-known AdS-black hole solutions to Einstein’s equations with negative cosmological constant are given by the AdS-Schwarzschild and AdS-Reissner-Nordström black hole, extensively discussed in Chapter4and applied to render a holographic dual of su- perconductors and charged superfluids. Therefore, our holographic dictionary can be extended by including thermal field theories with finite temperature, set by the Hawking temperature (3.7) TH, finite entropy, defined by the black hole horizon or the Bekenstein-Hawking entropy

SBH (3.8) and a finite chemical potential related to the charge of the Reissner-Nordström black

35_{The minus sign arises from}

δ´gABgBC ¯ “0 _ñ δgAB_{“ ´}gACgBDδgCD and gACgBDTAB“TCD“ 2 ? ´gg AC_gBDδgCD δgAB looooooooomooooooooon “´1 δ_p?_´gL_matterq δgCD .

Boundary field theory inddimensions Bulk gravity ind`1dimensions Global current Jµ_{pxq é A} Apx, uq Gauge field Aµpxq é AApx, uq ˇ ˇ BpAdSq xJµ_{pxq y é ΠrAs}A_{px, uq}ˇ_ˇ BpAdSq Energy-momentum tensor Tµν_{pxq é g} abpx, uq spacetime metric gµν_{pxq é g} ABpx, uq ˇ ˇ BpAdSq xTµνpxq y é ΠrgsAB_{px, uq}ˇ_ˇ BpAdSq

Entropy S _é SBH _{Hawking entropy}Bekenstein-

Free energy F é IGravity Euclidean on-shell action

Temperature T _é TH Hawking Temperature

Chemical potential µ _é QBH Charge of black hole

Table 3.5. From the holographic RG flow viewpoint the couplings of the strongly coupled QFT correspond to the fields with the same symmetries, quantum numbers and tensorial structure in the gravitational theory. The UV fixed point couplings are the sources of the fixed point CFT operators that correspond to the boundary values of the fields in asymptotic AdS spacetime.

hole. According to Table2.1, once we have a thermal field theory, the thermodynamic potentials such as the free energy are determined by the logarithm of the partition function. In the case of gauge/gravity dualities we may employ the saddle-point approximation to the gravitational theory for strongly coupled field theories and thus the thermodynamic potentials reduce to the regularized Euclidean on-shell action. The extended holographic dictionary is listed in Table3.5. The non-trivial geometry arises from a matter Lagrangian36_L

matterdesigned in such a way that

the boundary values of the matter fields correspond to the sources of the strongly coupled QFT we want to describe holographically. In general, the so-called backreaction of the matter fields onto the simple AdS geometry generates the non-trivial IR geometries which arise as consistent solutions of Einstein’s equations with negative cosmological constant, c.f. Figure 3.13. Apart from the black hole solutions there are scaling solutions for non-trivial IR fixed points and there are other geometries like hard-wall solutions with a hard cut-off in the geometry introducing a mass-gap, or solitonic solutions connecting two AdS-spaces with different radii. There are also more exotic black hole solutions, such as the dionic black hole, including sources for magnetic fields. The main task to apply holography to physical systems is to identify the correct gravita- tional dual encoding the properties of the system and consistently solve the coupled equations of motion with two constraints. The first constraint arises from regularity in the IR,i.e. infalling boundary conditions at the black-hole horizon, that fixes one of the two solutions of the bulk equations of motion. The second constraint is that the geometry must be asymptotically AdS at the boundary,i.e. approach the UV fixed point CFT. In the next section, we will describe how

36_{In the following, we typically denote every Lagrangian including non-gravitational degrees of freedom as matter}

Figure 3.13.

The IR geometry is deformed by a relevant opera- tor which corresponds to a AdS-black-hole geom- etry, where the boundary is still asymptotic AdS- space corresponding to a UV fixed point CFT. More accurately, the black hole is a spatially infinite black brane extending across the flat spatial direction of the field theory. The black brane horizon sets the temperature of the deformed UV fixed point CFT, but the matter content on the gravity side is intro- duced at zero temperature. As explained in Section 3.5, fluctuations about the background solution to the full Einstein equations are related to dissipative effects described by the infalling bulk field fluctua- tions. For charged black branes, the electric flux em- anating from the black brane horizon sets the charge density on the boundary field theory.

u

ddimensional CFT onR1,d´1

d´1dimensional black brane with

temperatureTH

to retrieve physical properties in terms of response functions by applying linear response theory from Section2.2to our holographic setup.

3.5. Linear Response & Holography

Overview

• Transport processes of strongly correlated systems can be computed via the holographic fluctuations-dissipation theorem.

• Response functions are related to correlators of fluctuations about background solu- tions of Einstein’s equation.

• Finite temperature relaxation can be traced back to fluctuations being “swallowed” by black holes/branes.

As we have seen in the previous section, thermodynamic properties of physical systems are holo- graphically described by asymptotically AdS-black-hole geometries. In order to probe the systems beyond thermodynamics to determine their transport behavior and to extend the holographic dictionary to general response functions, we need to reformulate the Fluctuation-Dissipation Theoremon page30in terms of bulk field fluctuations.

3.5.1. Holographic fluctuation-dissipation theorem

In Section3.4.1we already derived correlation functions which can be reduced easily to linear response Green functions. The linear response Green function is defined as the ratio of the response of the systemx

O

yto an infinitesimal external sourceJ. This allows us to reduce the functional derivative to a simple division due to the linear dependence of the expectation value to the source. According to the holographic dictionary listed in Table3.4and3.5, the vacuum expectation value of an operator is given by the subleading term in the boundary expansionϕ∆

and the corresponding source by the leading termϕd´∆, respectively. Applying the holographic

dictionary and inserting (3.82) we find

G_pk_{q “} x

O

pkq y

J_pk_q “ulimÑ0u

2p∆´dqΠpk, uq

Φpk, u_q. (3.94)

However, to compute the Green function of real-time processes in this manner poses a serious problem. First of all, the Euclidean Green function should by analytically continued, which is only possible if an analytic solution relatingϕ∆toϕd´∆is known. As we already hinted at in the last paragraph of Section3.4.2for general bulk fields this solution cannot be obtained in closed analytical form and we need to resort to asymptotic expansions or even numerical solutions. Secondly, a direct calculation in real time is not feasible because we would need to modify the holographic dictionary concerning the equivalence of the partition functions of the field theory and the gravity side to _A

eişdd_{x J} pxqOpxq E QFT“ e iSGR ˇ ˇ ˇ ϕBpAdSq“J . (3.95)

As discussed in [78,200] for real-time Minkowski spacetime the regularity condition at the black hole horizon is insufficient to obtain a unique solution. This corresponds to the existence of multiple real-time Green functions, the retarded GR, the advanced GA and the time-ordered Green functionGT _{(2.74) in contrast to the unique imaginary time Green function}

Gτ _(2.75).

Luckily, there exists a powerful method, known as the Keldysh formalism, that circumvents vari- ous problems arising in treatments of complex systems37_{i.a. equilibrium and close-to-equilibrium} problems where the analytical continuation from Matsubara frequencies fails. Let us postpone the discussion of the Keldysh formalism to Section 3.5.2 and give the correct prescription to compute response functions in holography via the fluctuation-dissipation theorem:

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