6.4 Probes as deformations and gapless excitations in complex systems

The dynamical regime in which brane effective actions hold is particularly suitable to describe physical systems made of several interacting subsystems in which one of them has a much smaller number of degrees of freedom. Assume the larger subsystems allow an approximate description in terms of a supergravity background. Then, focusing on the dynamics of this smaller subsector, while keeping the dynamics of the larger subsystems frozen, corresponds to probing the supergravity background with the effective action describing the smaller subsystem. This conceptual framework is illustrated in Figure 9View Image.
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Figure 9: Conceptual framework in which the probe approximation captures the dynamics of small subsystems interacting with larger ones that have reliable gravity duals.

This set-up occurs when the brane degrees of freedom are responsible for either breaking the symmetries of the larger system or describing an interesting isolated set of massless degrees of freedom whose interactions among themselves and with the background one is interested in studying. In the following, I very briefly describe how the first approach was used to introduce flavour in the AdS/CFT correspondence, and how the second one can be used to study physics reminiscent of certain phenomena in condensed-matter systems.

Probing deformations of the AdS/CFT:
Deforming the original AdS/CFT allows one to come up with set-ups with less or no supersymmetry. Whenever there is a small number of degrees of freedom responsible for the dynamics (typically D-branes), one may approximate the latter by the effective actions described in this review. This provides a reliable and analytical tool for describing the strongly-coupled behaviour of the deformed gauge theory.

As an example, consider the addition of flavour in the standard AdS/CFT. It was argued in [333] that this could be achieved by adding k D7-branes to a background of N D3-branes. The D7-branes give rise to k fundamental hypermultiplets arising from the lightest modes of the 3-7 and 7-3 strings, in the brane array

D3: 1 2 3 x x x x x x D7: 1 2 3 4 5 6 7 x x . (462 )
The mass of these dynamical quarks is given by mq = L ∕2πα ′, where L is the distance between the D3- and the D7-branes in the 89-plane. If gsN ≫ 1 the D3-branes may be replaced (in the appropriate decoupling limit) by an AdS5 × S5 geometry, as in the standard AdS/CFT argument, whereas if, in addition, N ≫ k then the back-reaction of the D7-branes on this geometry may be neglected. Thus, one is left, in the gravity description, with k D7-brane probes in AdS5 × S5. In the particular case of k = 1, one can use the effective action described before. This specific set-up was used in [348] to study the linearised fluctuation equations for the different excitations on the D7-probe describing different scalar and vector excitations to get analytical expressions for the spectrum of mesons in 𝒩 = 2 SYM, at strong coupling.

This logic can be extended to non-supersymmetric scenarios44. For example, using the string theory realisation of four-dimensional QCD with Nc colours and Nf ≪ Nc flavours discussed in [499Jump To The Next Citation Point]. The latter involves Nf D6-brane probes in the supergravity background dual to Nc D4-branes compactified on a circle with supersymmetry-breaking boundary conditions and in the limit in which all the resulting Kaluza–Klein modes decouple. For Nf = 1 and for massless quarks, spontaneous chiral symmetry breaking by a quark condensate was exhibited in [349] by working on the D6-brane effective action in the near horizon geometry of the Nc D4-branes.

Similar considerations apply at finite temperature by using appropriate black-hole backgrounds [499] in the relevant probe action calculations. This allows one to study phase transitions associated with the thermodynamic properties of the probe degrees of freedom as a function of the probe location. This can be done in different theories, with flavour [379], and for different ensembles [343, 378].

The amount of literature in this topic is enormous. I refer the reader to the reviews on the use of gauge-gravity duality to understand hot QCD and heavy ion collisions [137] and meson spectroscopy [207], and references therein. These explain the tools developed to apply the AdS/CFT correspondence in these set-ups.

Condensed matter and strange metallic behaviour:
There has been a lot of work in using the AdS/CFT framework in condensed matter applications. The reader is encouraged to read some of the excellent reviews on the subject [283, 296, 385, 284, 285], and references therein. My goal in these paragraphs is to emphasise the use of IR probe branes to extract dynamical information about certain observables in quantum field theories in a state of finite charge density at low temperatures. Before describing the string theory set-ups, it is worth attempting to explain why any AdS/CFT application may be able to capture any relevant physics for condensed matter systems. Consider the standard Fermi liquid theory, describing, among others, the conduction of electrons in regular metals. This theory is an example of an IR free fixed point, independent of the UV electron interactions, describing the lowest energy fermionic excitations taking place at the Fermi surface k = kf. Despite its success, there is experimental evidence for the existence of different “states of matter”, which are not described by this effective field theory. This could be explained by additional gapless bosonic excitations, perhaps arising as collective modes of the UV electrons. For them to be massless, the system must either be tuned to a quantum critical point or there must exist a kinematical constraint leading to a critical phase.

One interesting possibility involving this mechanism consists on the emergence of gauge fields (“photons”) at the onset of such critical phases. For example, 2 + 1 Maxwell theory in the presence of a Fermi surface (chemical potential μ)

1 ( ) ℒ = − --F2 + ψ¯Γ ⋅ (i∂ + A) + Γ 0 μ ψ , (463 ) 4
is supposed to describe at energies below μ, the interactions between gapless bosons (photons) with the fermionic excitations of the Fermi surfaces. The one-loop correction to the classical photon propagator at low energy ω and momenta k is
−1 -ω- 2 D (ω, k ) = γ|k| + |k | . (464 )
Due to the presence of the chemical potential, this result manifestly breaks Lorentz invariance, but there exists a non-trivial IR scaling symmetry (Lifshitz scale invariance)
t → λ3t, |x| → λ|x |, (465 )
with dynamical exponent z = 3, replacing the UV scaling {t, |x|} → λ {t, |x|}. Since these systems are believed to be strongly interacting, it is an extremely challenging theoretical task to provide a proper explanation for them. It is this strongly-coupled character and the knowledge of the relevant symmetris that suggest one search for similar behaviour in “holographic dual” descriptions.

The general set-up, based on the discussions appearing, among others, in [334Jump To The Next Citation Point, 398Jump To The Next Citation Point, 286Jump To The Next Citation Point], is as follows. One considers a small set of charged degrees of freedom, provided by the probe “flavour” brane, interacting among themselves and with a larger set of neutral quantum critical degrees of freedom having Lifshitz scale invariance with dynamical critical exponent z. As in previous applications, the latter is replaced by a gravitational holographic dual with Lifshitz asymptotics [324]

( ) 2 2 dt2 dv2- dx2 +-dy2- dsIR = L − v2z + v2 + v2 , (466 )
where v will play the role of the holographic radial direction. Turning on non-trivial temperature corresponds to considering black holes having the above asymptotics [162, 370, 102, 34]
( f (v)dt2 dv2 dx2 + dy2 ) ds2IR = L2 − --------+ -------+ ---------- , (467 ) v2z f(v)v2 v2
where the function f (v) depends on the specific solution and characterises the thermal nature of the system.

In practice, one embeds the probe “flavour” brane in the spacetime holographic dual, which may include some non-trivial cycle wrapping in internal dimensions when embedded in string theory, and turns on some non-trivial electric (Φ (v)) and magnetic fluxes (B ) on the brane

V = Φ(v)dt + Bxdy . (468 )
At low energies and in a quantum critical system, the only available scales are external, i.e., given by temperature T, electric and magnetic fields {E, B} and the density of charge carriers Jt. Solving the classical equations of motion for the world volume gauge field, allows one to integrate Φ(v), whose constant behaviour at infinity, i.e., at v → 0 in the above coordinate system, defines the chemical potential μ of the system. Working in an ensemble of fixed charge carrier density Jt, which is determined by computing the variation of the action with respect to δVt(0)= δμ, the free energy density f is given by
-F-- T-SDp- t f ≡ vol = vol + μJ , (469 ) 2 2
where vol2 stands for the volume of the non-compact 2-space spanned by {x, y } and SDp is the on-shell Dp-brane action. As in any thermodynamic system, observables such as specific heat or magnetic susceptibility can be computed from Eq. (469View Equation) by taking appropriate partial derivatives. Additionally, transport observables, such as DC, AC or DC Hall conductivities can also be computed and studied as a function of the background, probe embedding and the different constants controlling the world volume gauge field (468View Equation).

More than the specific physics, which is nicely described in [334, 398, 286], what is important to stress, once more, is that using the appropriate backgrounds, exciting the relevant degrees of freedom and considering the adequate boundary conditions make the methods described in this review an extremely powerful tool to learn about physics in regimes of parameters that would otherwise be very difficult to handle, both analytically and conceptually.

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