AdS2 Throats and Local Criticality

The charged black brane from the previous page has a special zero-temperature limit. At nonzero temperature its horizon is ordinary: the Euclidean time circle caps off smoothly, perturbations fall through the future horizon, and the state is a finite-density thermal fluid. At zero temperature something sharper happens. The horizon becomes extremal, the blackening factor develops a double zero, and the near-horizon region stretches into a long throat with geometry

$$AdS_2\times \mathbb R^{d_s}.$$

This throat is the first genuinely strange finite-density IR geometry in holographic quantum matter. Its scaling symmetry acts on time and the radial coordinate, but not on the boundary spatial coordinates:

$$t\to \lambda t, \qquad \zeta\to\lambda\zeta, \qquad \vec x\to \vec x.$$

In boundary language, the IR theory has critical temporal correlations but no ordinary spatial scaling. Equivalently, it has an infinite dynamical exponent,

$$z_{\rm dyn}=\infty.$$

This is called local or semi-local quantum criticality. It is local not because the theory has no spatial structure whatsoever, but because the deep IR scaling symmetry does not rescale space. Momentum $k$ becomes a label on a family of $0+1$-dimensional critical problems. The result is one of the signature formulas of holographic finite-density physics:

$$G^R_{\rm IR}(\omega,k)\sim \omega^{2\nu_k}.$$

The exponent depends on momentum. That is the odd little jewel of the $AdS_2$ throat: criticality in time, momentum-dependent scaling dimensions, and a finite density of low-energy spectral weight without quasiparticles.

Throughout this page, $d_s$ denotes the number of boundary spatial dimensions, $d=d_s+1$ is the boundary spacetime dimension, and the bulk dimension is $d+1$. The boundary is at $z=0$, the horizon is at $z=z_h$, and we set $\hbar=k_B=c=1$.

The extremal charged black brane

The planar Reissner—Nordström AdS black brane can be written as

$$ds^2 = \frac{L^2}{z^2} \left[-f(z)dt^2+d\vec x_{d_s}^{\,2}+\frac{dz^2}{f(z)}\right],$$

with

$$f(z) = 1-(1+Q^2)\left(\frac{z}{z_h}\right)^d +Q^2\left(\frac{z}{z_h}\right)^{2d-2}.$$

The gauge potential may be chosen regular at the horizon,

$$A=A_t(z)dt, \qquad A_t(z)=\mu\left[1-\left(\frac{z}{z_h}\right)^{d-2}\right], \qquad d>2.$$

The temperature is

$$T = \frac{1}{4\pi z_h}\left[d-(d-2)Q^2\right].$$

Thus extremality occurs when

$$Q^2=Q_{\rm ext}^2=\frac{d}{d-2}.$$

At this value,

$$f(z_h)=0, \qquad f'(z_h)=0.$$

The horizon is not merely a simple zero of $f$. It is a double zero. Expanding near $z=z_h$ gives

$$f(z) = d(d-1)\left(1-\frac{z}{z_h}\right)^2+\cdots.$$

This one line is the seed of the whole page. A simple zero gives Rindler space and a finite-temperature horizon. A double zero gives an $AdS_2$ throat and an emergent low-energy scaling region.

Extremal does not mean uncharged

The extremal brane has $T=0$, but it still has nonzero $\mu$, nonzero $\rho$, and a nonzero electric field in the bulk. The horizon is cold, not neutral.

Deriving the $AdS_2\times\mathbb R^{d_s}$ throat

Let

$$r=z_h-z.$$

Near the extremal horizon, $r\ll z_h$, and

$$f(z)\simeq d(d-1)\frac{r^2}{z_h^2}.$$

Substituting this into the metric, the time-radial part becomes

$$ds^2_{t,r} \simeq -\frac{L^2}{z_h^2}\,d(d-1)\frac{r^2}{z_h^2}dt^2 + \frac{L^2}{z_h^2}\frac{dr^2}{d(d-1)r^2/z_h^2}.$$

Define

$$L_2=\frac{L}{\sqrt{d(d-1)}}$$

and

$$R=\frac{d(d-1)}{z_h^2}r.$$

Then

$$ds^2_{t,R} = L_2^2\left(-R^2dt^2+\frac{dR^2}{R^2}\right).$$

This is $AdS_2$ in Poincare coordinates. Equivalently, with

$$\zeta=\frac1R=\frac{z_h^2}{d(d-1)(z_h-z)},$$

the throat metric is

$$\boxed{ ds^2 \to \frac{L_2^2}{\zeta^2}\left(-dt^2+d\zeta^2\right) + \frac{L^2}{z_h^2}d\vec x_{d_s}^{\,2}. }$$

The horizon is at

$$\zeta\to\infty,$$

while the boundary of the $AdS_2$ throat, where it glues onto the asymptotic $AdS_{d+1}$ region, is at

$$\zeta\to0.$$

The spatial directions are spectators. They do not form part of the $AdS_2$ scaling geometry; they sit as a flat $\mathbb R^{d_s}$ of fixed proper size

$$\ell_x=\frac{L}{z_h}.$$

The extremal charged black brane develops an $AdS_2\times\mathbb R^{d_s}$ throat. The UV region is asymptotically $AdS_{d+1}$ and encodes the microscopic CFT deformed by a chemical potential. The deep IR is controlled by the near-horizon $AdS_2$ region, where $t$ and the $AdS_2$ radial coordinate scale but the boundary spatial coordinates do not. At small nonzero temperature the throat is cut off at an energy scale $E\sim T$.

The radial direction is the holographic version of scale. Moving from the boundary toward the horizon means flowing from the UV to the IR. The charged black brane therefore says:

$$\text{UV CFT at finite }\mu \quad\longrightarrow\quad \text{IR }AdS_2\times\mathbb R^{d_s}\text{ throat}.$$

This is a geometric RG flow. The chemical potential $\mu$ introduces a scale. Above that scale the system remembers the original relativistic CFT. Below that scale the extremal horizon controls the IR.

The electric field in the throat

The gauge potential also has a simple near-horizon limit. Since

$$A_t(z)=\mu\left[1-\left(\frac{z}{z_h}\right)^{d-2}\right],$$

near $z=z_h$ we find

$$A_t\simeq \mu(d-2)\frac{z_h-z}{z_h}.$$

Using the $AdS_2$ coordinate $\zeta$, this becomes

$$A_t\simeq \frac{e_d}{\zeta}, \qquad e_d=\frac{\mu(d-2)z_h}{d(d-1)}.$$

The constant $e_d$ depends on the normalization of the bulk Maxwell field and on the convention used for $\mu$, but the structure

$$\boxed{A_t\propto \frac1\zeta}$$

is universal for the RN-AdS throat. The corresponding electric field is nonzero in the $AdS_2$ region:

$$F_{\zeta t}=\partial_\zeta A_t\propto -\frac1{\zeta^2}.$$

This field is compatible with the $AdS_2$ scaling symmetry. Under

$$t\to\lambda t, \qquad \zeta\to\lambda\zeta,$$

the one-form

$$A=A_tdt\sim \frac{dt}{\zeta}$$

is invariant.

The electric field is physically important. It is the near-horizon remnant of the finite charge density. It also lowers the effective mass of charged fields and can drive IR instabilities, including the holographic superconductor instability.

Why the throat is infinitely long

For a nonextremal black brane, the near-horizon geometry is Rindler-like. The proper radial distance from a point outside the horizon to the horizon is finite. For an extremal horizon, the double zero changes the story.

At fixed time, the radial proper distance near the extremal horizon is

$$ds_{\rm radial} \simeq L_2\frac{dr}{r}.$$

Therefore the distance from $r=r_{\rm UV}$ to $r=r_{\rm IR}$ is

$$\Delta s \simeq L_2\log\frac{r_{\rm UV}}{r_{\rm IR}}.$$

As $r_{\rm IR}\to0$, this diverges. The extremal horizon is down an infinitely long throat.

This logarithmic length is why the $AdS_2$ region can dominate low-energy physics. A perturbation sent from the boundary must propagate through a long scaling region before reaching the horizon. The throat is not a small correction to the UV geometry; it is the IR fixed point.

At small but nonzero temperature, the throat is cut off. The near-horizon region is better described by an $AdS_2$ black hole,

$$ds^2_2 = \frac{L_2^2}{\zeta^2} \left[-\left(1-\frac{\zeta^2}{\zeta_T^2}\right)dt^2 +\frac{d\zeta^2}{1-\zeta^2/\zeta_T^2}\right],$$

with

$$T=\frac{1}{2\pi\zeta_T}.$$

The throat length is then roughly

$$\Delta s\sim L_2\log\frac{\mu}{T}.$$

So the low-temperature limit is a long-throat limit. This is a useful mental picture: finite temperature caps off the $AdS_2$ throat, while $T\to0$ makes it infinitely long.

The IR scaling symmetry

The $AdS_2$ metric

$$ds^2_2 = \frac{L_2^2}{\zeta^2}\left(-dt^2+d\zeta^2\right)$$

is invariant under

$$t\to\lambda t, \qquad \zeta\to\lambda\zeta.$$

The full near-horizon geometry is

$$AdS_2\times\mathbb R^{d_s},$$

so the spatial coordinates do not participate in the scaling:

$$\vec x\to\vec x.$$

This is unlike a relativistic CFT, where

$$t\to\lambda t, \qquad \vec x\to\lambda \vec x.$$

It is also unlike a Lifshitz fixed point with finite dynamical exponent $z$, where

$$t\to\lambda^z t, \qquad \vec x\to\lambda\vec x.$$

The $AdS_2\times\mathbb R^{d_s}$ throat is the limiting case

$$z=\infty.$$

Energy scales, but momentum does not scale. Frequency is an IR scaling variable; momentum is an external label.

This is the cleanest way to understand local criticality:

$$\boxed{ \omega\text{ scales, but }k\text{ does not.} }$$

That one sentence prevents a lot of confusion.

Local does not mean spatially featureless

The IR scaling symmetry does not rescale space, but correlators can still depend strongly on momentum. In fact, the hallmark of the $AdS_2$ throat is that the IR exponent $\nu_k$ depends on $k$.

Probe fields and momentum-dependent IR dimensions

Consider a neutral scalar bulk field $\phi$ of mass $m$ dual to a boundary operator $O$. In the throat, Fourier decompose

$$\phi(t,\zeta,\vec x)=e^{-i\omega t+i\vec k\cdot\vec x}\phi_{\omega,k}(\zeta).$$

Because the spatial metric in the throat is fixed,

$$ds_x^2=\frac{L^2}{z_h^2}d\vec x^{\,2},$$

the momentum term acts like an additional contribution to the mass in $AdS_2$:

$$\boxed{ m_k^2 = m^2+\frac{z_h^2}{L^2}k^2. }$$

The scalar wave equation near the $AdS_2$ boundary has two independent asymptotic behaviors,

$$\phi_{\omega,k}(\zeta) \sim A\,\zeta^{\frac12-\nu_k} + B\,\zeta^{\frac12+\nu_k},$$

where

$$\boxed{ \nu_k = \sqrt{\frac14+L_2^2m_k^2} }$$

for a neutral scalar.

The corresponding IR operator dimension is

$$\delta_k=\frac12+\nu_k.$$

This is the central $AdS_2$ fact: each momentum $k$ gives a different scaling dimension in the emergent $0+1$-dimensional theory.

For a charged scalar, the near-horizon electric field shifts the effective exponent. Schematically,

$$\boxed{ \nu_k^2 = \frac14 +L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right) -q_*^2, }$$

where

$$q_*\sim q e_d$$

is the dimensionless coupling of the charged field to the $AdS_2$ electric field. The precise normalization depends on the Maxwell convention. The sign is robust: the electric field lowers $\nu_k^2$ for charged fields.

For spinors the formula is modified by spin connection and gamma-matrix factors, but the physical structure is the same:

$$\nu_k=\text{a momentum-dependent IR exponent}.$$

Later, in the holographic fermion page, this $\nu_k$ will control whether a boundary fermionic excitation is Fermi-liquid-like, non-Fermi-liquid-like, marginal, or deeply incoherent.

IR Green’s functions

In a $0+1$-dimensional scale-invariant theory, an operator of dimension $\delta_k$ has a time-domain two-point function

$$G_{\rm IR}(t,k)\sim \frac1{|t|^{2\delta_k}}.$$

Fourier transforming gives, up to contact terms and phase conventions,

$$G^R_{\rm IR}(\omega,k) \sim \omega^{2\delta_k-1} = \omega^{2\nu_k}.$$

Thus

$$\boxed{ G^R_{\rm IR}(\omega,k) = \mathcal C_k\,e^{-i\pi\nu_k}\,\omega^{2\nu_k} + \cdots }$$

for real positive $\omega$ and real $\nu_k$, with $\mathcal C_k$ depending on conventions and on the normalization of the IR field.

At nonzero temperature, scaling gives the finite-temperature form

$$G^R_{\rm IR}(\omega,k;T) = T^{2\nu_k}\, \mathcal F_k\left(\frac{\omega}{T}\right),$$

where $\mathcal F_k$ is an $AdS_2$ black-hole response function. Its explicit expression is a ratio of Gamma functions, but the scaling form is usually the key point. The only low-energy scale in the IR throat is $T$.

The spectral density behaves as

$$\rho_O(\omega,k) = -2\operatorname{Im}G^R(\omega,k) \propto \omega^{2\nu_k}$$

at zero temperature when the IR throat controls the response. This is low-energy continuum spectral weight, not a quasiparticle pole.

Matching the throat to the UV

The $AdS_2$ Green’s function is not automatically the full boundary Green’s function. The full geometry has two regions:

1. the UV and matching region, where the spacetime is not $AdS_2\times\mathbb R^{d_s}$;
2. the IR throat, where $AdS_2$ scaling controls low-frequency dynamics.

A standard matched-asymptotic calculation gives a structure of the form

$$\boxed{ G^R(\omega,k) = \frac{b_+(\omega,k)+b_-(\omega,k)\,\mathcal G_k(\omega)} {a_+(\omega,k)+a_-(\omega,k)\,\mathcal G_k(\omega)}. }$$

Here

$$\mathcal G_k(\omega)\equiv G^R_{\rm IR}(\omega,k)$$

is the $AdS_2$ response, while $a_\pm$ and $b_\pm$ are determined by solving the zero-frequency wave equation through the UV geometry. They are analytic in $\omega$ at small frequency.

Generically, if $a_+(0,k)\neq0$, expanding at small $\omega$ gives

$$G^R(\omega,k) = G^R(0,k)+c(k)\omega^{2\nu_k}+\cdots.$$

So the $AdS_2$ throat contributes the leading nonanalytic frequency dependence.

A special case occurs when

$$a_+(0,k_F)=0.$$

Then the denominator of $G^R$ is small at $k=k_F$, and the same $AdS_2$ response becomes a self-energy for a sharp or semi-sharp excitation. This is the origin of the famous holographic fermion formula

$$G^R(\omega,k) \sim \frac{1}{k-k_F-\omega/v_F-c\,\omega^{2\nu_{k_F}}}.$$

We will not develop that story here, but this page contains its IR engine.

What local criticality means physically

The $AdS_2$ throat gives a controlled large-$N$ model of a finite-density state with no ordinary quasiparticle description. The most important physical features are these.

First, the IR theory has a continuum of low-energy excitations. The horizon absorbs perturbations, and the $AdS_2$ region produces branch cuts rather than isolated stable quasiparticle poles.

Second, time and space behave asymmetrically. Time correlations are power-law critical. Spatial correlations are not governed by a scale transformation. Instead, spatial dependence enters through momentum-dependent exponents $\nu_k$.

Third, the charge is fractionalized in the pure RN-AdS solution. The electric flux continues into the horizon, so the boundary charge is carried by the deconfined large-$N$ sector rather than by visible gauge-invariant charged particles outside the horizon.

Fourth, the extremal entropy density is nonzero:

$$s_0 = \frac{1}{4G_{d+1}}\left(\frac{L}{z_h}\right)^{d_s}.$$

This makes the RN-AdS throat powerful but also suspicious. A finite ground-state entropy density is not expected in a generic stable ordinary material. In many holographic models the RN-AdS throat is not the final ground state; it is an intermediate scaling regime or the parent state that becomes unstable to another phase.

The useful attitude

The $AdS_2$ throat is not a claim that real strange metals literally have extremal black-hole horizons. It is a solvable mechanism for finite-density quantum criticality without quasiparticles. Its predictions should be treated as sharp theoretical structures to compare, deform, and sometimes falsify.

BF bounds and IR instabilities

Anti-de Sitter space can tolerate a negative mass squared, but only down to the Breitenlohner—Freedman bound. In $AdS_{p+2}$, a neutral scalar is stable if

$$m^2L_{p+2}^2\ge -\frac{(p+1)^2}{4}.$$

For $AdS_2$, this becomes

$$m_{\rm eff}^2L_2^2\ge -\frac14.$$

In the notation above, this is simply

$$\nu_k^2\ge0.$$

If

$$\nu_k^2<0,$$

then

$$\nu_k=i\lambda_k, \qquad \lambda_k>0,$$

and the IR scaling dimension is complex:

$$\delta_k=\frac12+i\lambda_k.$$

The response behaves schematically as

$$\omega^{2i\lambda_k} = \exp\left(2i\lambda_k\log\omega\right),$$

which is log-periodic in frequency. For bosonic modes, a complex IR scaling dimension is usually an instability of the $AdS_2$ throat. The system wants to condense the unstable mode and replace the RN-AdS interior with a new geometry.

This is the mechanism behind several important ordered phases.

Charged scalar instability

A charged scalar has

$$\nu_k^2 = \frac14 +L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right) -q_*^2.$$

The electric field can make $\nu_k^2$ negative even if the scalar is perfectly stable in the UV $AdS_{d+1}$ region. This means:

$$\text{UV theory is stable} \quad\text{but}\quad \text{finite-density IR throat is unstable}.$$

That is exactly what a phase transition should look like holographically. The UV theory is well-defined. The finite-density state is not the true IR endpoint.

For a homogeneous superfluid or superconductor instability, the first unstable mode is often at

$$k=0.$$

For spatially modulated phases, an instability may first appear at

$$k=k_\star\neq0.$$

Then the ordered phase breaks translations.

Neutral scalar instability

Even a neutral scalar can violate the $AdS_2$ BF bound if its UV mass lies in the window where it is stable in $AdS_{d+1}$ but unstable in the smaller $AdS_2$ throat. This is possible because the stability bounds are different in the two geometries.

For example, in $AdS_4$ the UV BF bound is

$$m^2L^2\ge -\frac94.$$

For an extremal RN-AdS$_4$ throat,

$$L_2^2=\frac{L^2}{6},$$

so the $AdS_2$ BF bound is

$$m^2L_2^2\ge -\frac14 \quad\Longleftrightarrow\quad m^2L^2\ge -\frac32.$$

Thus a scalar with

$$-\frac94<m^2L^2<-\frac32$$

is allowed in the UV $AdS_4$ theory but unstable in the $AdS_2$ throat.

This is a beautiful and useful distinction. A negative mass squared is not automatically bad; what matters is the geometry that controls the IR.

Local criticality versus ordinary quantum criticality

It is worth comparing the $AdS_2$ throat with an ordinary relativistic quantum critical point.

Feature	Relativistic CFT	$AdS_2\times\mathbb R^{d_s}$ throat
scaling	$t\to\lambda t$, $\vec x\to\lambda\vec x$	$t\to\lambda t$, $\vec x\to\vec x$
dynamical exponent	$z=1$	$z=\infty$
operator dimension	fixed by operator	depends on $k$ in the IR
low-energy correlator	$G(\omega,k)$ scales in $\omega/k$	$G(\omega,k)\sim\omega^{2\nu_k}$
spatial correlations	scale invariant	finite spatial scale set by $\mu$ in simple RN-AdS
entropy at $T=0$	usually zero density of entropy	finite $s_0$ in classical RN-AdS

The $AdS_2$ throat is therefore not just another CFT. It is a semi-local critical sector attached to every point in momentum space.

This makes it both powerful and peculiar. It gives simple analytic control over frequency dependence, but it does not by itself produce all the spatial structure of a real metal. Spatial coherence, Fermi surfaces, lattices, density waves, and disorder require additional ingredients or deformations.

The residual entropy problem

The extremal RN-AdS black brane has

$$s(T\to0)=s_0\neq0.$$

At the level of classical gravity this is not a contradiction. The entropy is an area density in Planck units, and the large-$N$ limit makes it order $N^2$. But as a model of quantum matter it raises a sharp question: what microscopic degrees of freedom account for a macroscopic ground-state entropy?

There are several common responses.

One response is that the RN-AdS throat is an intermediate scaling regime. Before reaching the true $T=0$ limit, another instability may intervene: superconductivity, electron-star formation, scalar condensation, a spatially modulated phase, or a dilaton-driven scaling geometry.

Another response is that the large-$N$ limit can hide low-temperature splittings. Effects that are subleading in $1/N$ may lift the degeneracy at parametrically low scales.

A third response is to replace the RN-AdS throat by a more general IR geometry, such as an Einstein—Maxwell—dilaton solution with Lifshitz or hyperscaling-violating scaling. That is the next page.

The practical lesson is simple:

$$\boxed{ AdS_2\text{ is an excellent IR saddle, but not always the final ground state.} }$$

Common pitfalls

Pitfall 1: thinking $AdS_2$ means a literal spatially zero-dimensional boundary theory. The full boundary theory still lives in $d_s$ spatial dimensions. The $AdS_2$ throat means the deep IR scaling problem is $0+1$-dimensional for each momentum $k$.

Pitfall 2: confusing local criticality with momentum independence. The exponent $\nu_k$ depends on momentum. The critical scaling is local in the sense that $k$ does not scale.

Pitfall 3: treating the residual entropy as harmless. It is harmless as a controlled classical saddle. It is not harmless as a claim about generic stable matter. Always ask what resolves or replaces the extremal entropy.

Pitfall 4: using the UV BF bound to decide IR stability. A field can be stable near the $AdS_{d+1}$ boundary and unstable in the $AdS_2$ throat. The IR geometry has its own stability bound.

Pitfall 5: forgetting the electric field. The RN-AdS throat is not just neutral $AdS_2\times\mathbb R^{d_s}$. It is supported by a near-horizon electric field, and charged probes feel that field.

Pitfall 6: expecting quasiparticles. The generic $AdS_2$ response is a branch cut, not a pole. Quasiparticle-like poles can appear after matching to the UV, but they are special and can decay into the IR bath.

Exercises

Exercise 1: The double zero produces $AdS_2$

At extremality,

$$f(z)\simeq d(d-1)\left(1-\frac{z}{z_h}\right)^2.$$

Starting from

$$ds^2=\frac{L^2}{z^2}\left[-f(z)dt^2+d\vec x^{\,2}+\frac{dz^2}{f(z)}\right],$$

show that the near-horizon metric is

$$ds^2 = \frac{L_2^2}{\zeta^2}(-dt^2+d\zeta^2) + \frac{L^2}{z_h^2}d\vec x^{\,2}, \qquad L_2=\frac{L}{\sqrt{d(d-1)}}.$$

Solution

Let

$$r=z_h-z.$$

Near the horizon,

$$1-\frac{z}{z_h}=\frac{r}{z_h}, \qquad f(z)\simeq d(d-1)\frac{r^2}{z_h^2}.$$

Also $z\simeq z_h$ in the overall prefactor. The time-radial metric becomes

$$ds^2_{t,r} \simeq -\frac{L^2}{z_h^2}d(d-1)\frac{r^2}{z_h^2}dt^2 + \frac{L^2}{z_h^2}\frac{dr^2}{d(d-1)r^2/z_h^2}.$$

Define

$$R=\frac{d(d-1)}{z_h^2}r, \qquad L_2^2=\frac{L^2}{d(d-1)}.$$

Then

$$ds^2_{t,R} = L_2^2\left(-R^2dt^2+\frac{dR^2}{R^2}\right).$$

Finally set

$$\zeta=\frac1R.$$

Since

$$\frac{dR^2}{R^2}=\frac{d\zeta^2}{\zeta^2}, \qquad R^2=\frac1{\zeta^2},$$

we obtain

$$ds^2_{t,\zeta} = \frac{L_2^2}{\zeta^2}(-dt^2+d\zeta^2).$$

The spatial part is constant at the horizon:

$$\frac{L^2}{z^2}d\vec x^{\,2}\to \frac{L^2}{z_h^2}d\vec x^{\,2}.$$

Thus the near-horizon geometry is $AdS_2\times\mathbb R^{d_s}$.

Exercise 2: The near-horizon gauge potential

Use

$$A_t(z)=\mu\left[1-\left(\frac{z}{z_h}\right)^{d-2}\right]$$

and

$$\zeta=\frac{z_h^2}{d(d-1)(z_h-z)}$$

to show that

$$A_t\simeq \frac{e_d}{\zeta}$$

near the extremal horizon. Find $e_d$.

Solution

Let

$$r=z_h-z, \qquad \frac{z}{z_h}=1-\frac{r}{z_h}.$$

For small $r/z_h$,

$$\left(1-\frac{r}{z_h}\right)^{d-2} = 1-(d-2)\frac{r}{z_h}+\cdots.$$

Therefore

$$A_t(z) \simeq \mu(d-2)\frac{r}{z_h}.$$

From the definition of $\zeta$,

$$r=\frac{z_h^2}{d(d-1)}\frac1\zeta.$$

Thus

$$A_t\simeq \mu(d-2)\frac1{z_h} \frac{z_h^2}{d(d-1)}\frac1\zeta = \frac{\mu(d-2)z_h}{d(d-1)}\frac1\zeta.$$

Hence

$$e_d=\frac{\mu(d-2)z_h}{d(d-1)}.$$

The normalization of $e_d$ can change if the Maxwell field is rescaled, but the $1/\zeta$ form is the invariant near-horizon statement.

Exercise 3: Momentum as an effective mass in $AdS_2$

Consider a neutral scalar of mass $m$ in the throat metric

$$ds^2 = \frac{L_2^2}{\zeta^2}(-dt^2+d\zeta^2) +\ell_x^2d\vec x^{\,2}, \qquad \ell_x=\frac{L}{z_h}.$$

For a mode $e^{-i\omega t+i\vec k\cdot\vec x}\phi(\zeta)$, show that the spatial momentum contributes an effective $AdS_2$ mass

$$m_k^2=m^2+\frac{k^2}{\ell_x^2} = m^2+\frac{z_h^2}{L^2}k^2.$$

Solution

The scalar equation is

$$(\Box-m^2)\phi=0.$$

The Laplacian contains a spatial term

$$g^{ij}\partial_i\partial_j\phi.$$

In the throat,

$$g_{ij}=\ell_x^2\delta_{ij}, \qquad g^{ij}=\frac{1}{\ell_x^2}\delta^{ij}.$$

For a Fourier mode,

$$\partial_i\partial_i\phi=-k^2\phi.$$

Thus the spatial derivative term contributes

$$g^{ij}\partial_i\partial_j\phi = -\frac{k^2}{\ell_x^2}\phi.$$

In the $AdS_2$ radial equation this appears in the same place as $-m^2\phi$, so the effective $AdS_2$ mass is

$$m_k^2=m^2+\frac{k^2}{\ell_x^2}.$$

Since $\ell_x=L/z_h$,

$$m_k^2=m^2+\frac{z_h^2}{L^2}k^2.$$

Exercise 4: The $AdS_2$ scaling exponent

For a neutral scalar in $AdS_2$ with effective mass $m_k$, show that the near-boundary behavior is

$$\phi\sim A\zeta^{1/2-\nu_k}+B\zeta^{1/2+\nu_k},$$

with

$$\nu_k=\sqrt{\frac14+L_2^2m_k^2}.$$

Solution

Near the $AdS_2$ boundary, the frequency term is subleading for determining the indicial exponents. The scalar equation reduces to the standard $AdS_2$ mass equation. Try a power-law solution

$$\phi\sim \zeta^\alpha.$$

For $AdS_{n+1}$, the relation between mass and scaling exponent is

$$\Delta(\Delta-n)=m^2L_{n+1}^2.$$

Here $n=1$, so

$$\Delta(\Delta-1)=m_k^2L_2^2.$$

Solving gives

$$\Delta=\frac12\pm\sqrt{\frac14+m_k^2L_2^2}.$$

Thus

$$\nu_k=\sqrt{\frac14+m_k^2L_2^2},$$

and the two independent behaviors are

$$\phi\sim A\zeta^{1/2-\nu_k}+B\zeta^{1/2+\nu_k}.$$

The IR operator dimension in standard quantization is

$$\delta_k=\frac12+\nu_k.$$

Exercise 5: From $0+1$-dimensional scaling to $G(\omega)\sim\omega^{2\nu}$

In a scale-invariant $0+1$-dimensional theory, an operator of dimension $\delta$ has

$$G(t)\sim \frac1{|t|^{2\delta}}.$$

Use dimensional analysis to show that the frequency-space retarded function scales as

$$G^R(\omega)\sim \omega^{2\delta-1}.$$

Then set $\delta_k=1/2+\nu_k$.

Solution

Under a time rescaling

$$t\to\lambda t,$$

the correlator transforms as

$$G(t)\to \lambda^{-2\delta}G(t).$$

The Fourier transform is schematically

$$G(\omega)=\int dt\,e^{i\omega t}G(t).$$

The measure contributes one power of time, so the frequency-space object has scaling dimension

$$[G(\omega)]\sim [\omega]^{2\delta-1}.$$

Therefore

$$G^R(\omega)\sim \omega^{2\delta-1}.$$

For the $AdS_2$ throat,

$$\delta_k=\frac12+\nu_k,$$

so

$$G^R_{\rm IR}(\omega,k)\sim\omega^{2\nu_k}.$$

Exercise 6: The $AdS_2$ BF instability window in RN-AdS$_4$

In RN-AdS$_4$, the UV geometry is $AdS_4$ with radius $L$, while the extremal throat has

$$L_2^2=\frac{L^2}{6}.$$

For a neutral scalar at $k=0$, find the mass window in which the scalar is stable in the UV $AdS_4$ region but unstable in the $AdS_2$ throat.

Solution

The $AdS_4$ BF bound is

$$m^2L^2\ge -\frac94.$$

The $AdS_2$ BF bound is

$$m^2L_2^2\ge -\frac14.$$

Using

$$L_2^2=\frac{L^2}{6},$$

the $AdS_2$ bound becomes

$$m^2\frac{L^2}{6}\ge -\frac14,$$

or

$$m^2L^2\ge -\frac32.$$

Stable in the UV but unstable in the IR means

$$m^2L^2\ge -\frac94$$

and

$$m^2L^2< -\frac32.$$

Therefore the window is

$$\boxed{ -\frac94\le m^2L^2< -\frac32. }$$

A scalar in this window is allowed in the UV theory but condenses in the extremal near-horizon region.

Exercise 7: Momentum can stabilize an IR instability

For a charged scalar, suppose

$$\nu_k^2=\nu_0^2+L_2^2\frac{z_h^2}{L^2}k^2,$$

with $\nu_0^2<0$. Find the maximum momentum $k_c$ for which the $AdS_2$ mode is unstable.

Solution

The mode is unstable when

$$\nu_k^2<0.$$

Using the given expression,

$$\nu_0^2+L_2^2\frac{z_h^2}{L^2}k^2<0.$$

Thus

$$k^2<\frac{L^2}{L_2^2z_h^2}(-\nu_0^2).$$

The critical momentum is

$$\boxed{ k_c=\frac{L}{L_2z_h}\sqrt{-\nu_0^2}. }$$

Modes with $k<k_c$ are unstable, while sufficiently large momentum increases the effective $AdS_2$ mass and stabilizes the mode.

Exercise 8: Matching and the leading nonanalytic term

Suppose the full boundary Green’s function has the low-frequency matching form

$$G^R(\omega,k) = \frac{b_0(k)+b_1(k)\mathcal G_k(\omega)} {a_0(k)+a_1(k)\mathcal G_k(\omega)},$$

where $a_0(k)\neq0$ and

$$\mathcal G_k(\omega)=C_k\omega^{2\nu_k}+\cdots.$$

Show that the leading nonanalytic term in $G^R$ is proportional to $\omega^{2\nu_k}$.

Solution

Factor out $a_0$ from the denominator:

$$G^R = \frac{b_0+b_1\mathcal G}{a_0\left(1+\frac{a_1}{a_0}\mathcal G\right)}.$$

For small $\mathcal G$,

$$\frac{1}{1+\frac{a_1}{a_0}\mathcal G} = 1-\frac{a_1}{a_0}\mathcal G+\cdots.$$

Therefore

$$G^R = \frac{b_0}{a_0} + \left(\frac{b_1}{a_0}-\frac{b_0a_1}{a_0^2}\right)\mathcal G_k(\omega) + \cdots.$$

Since

$$\mathcal G_k(\omega)=C_k\omega^{2\nu_k}+\cdots,$$

the leading nonanalytic term is

$$G^R(\omega,k) = G^R(0,k)+\tilde C_k\omega^{2\nu_k}+\cdots,$$

where

$$\tilde C_k= \left(\frac{b_1}{a_0}-\frac{b_0a_1}{a_0^2}\right)C_k.$$

Further reading

For the $AdS_2\times\mathbb R^d$ throat of extremal Einstein—Maxwell theory, charged horizons, and semi-local critical response, see Hartnoll, Lucas, and Sachdev, Holographic Quantum Matter, especially the sections on compressible quantum matter. For a condensed-matter-facing treatment of the RN strange metal, local quantum criticality, and momentum-dependent $AdS_2$ exponents, see Zaanen, Liu, Sun, and Schalm, Holographic Duality in Condensed Matter Physics, chapters 8 and 9. For textbook derivations of the extremal near-horizon geometry and the BF-bound instability mechanism, see Ammon and Erdmenger, Gauge/Gravity Duality, section 15.2, and Natsuume, AdS/CFT Duality User Guide, section 14.3.