AdS2 Throats and Local Criticality

The main idea

The planar Reissner-Nordström-AdS black brane has a striking low-temperature feature. As $T/\mu\to0$, its horizon becomes extremal, the blackening factor develops a double zero, and the near-horizon geometry approaches

$$\mathrm{AdS}_2\times\mathbb R^{d-1}.$$

This is much more than a geometric curiosity. In the boundary theory, the AdS$_2$ factor controls the low-energy response of a finite-density state. The spatial directions $\mathbb R^{d-1}$ do not scale in the same way as time. Under the emergent IR scaling,

$$t\to \lambda^{-1}t, \qquad \rho\to \lambda\rho, \qquad \vec x\to \vec x,$$

where $\rho$ is an AdS$_2$ radial coordinate. Frequencies scale, but momenta are spectators. This is why the corresponding boundary behavior is often called local criticality or, more accurately, semi-local criticality:

• correlation functions show nontrivial scaling in time or frequency;
• spatial momentum labels different IR operators rather than scaling with frequency;
• equal-time spatial correlations are not forced to become scale invariant;
• the IR theory behaves roughly like a continuum of CFT$_1$ sectors, one for each momentum $k$.

An extremal charged AdS black brane develops a near-horizon throat $\simeq \mathrm{AdS}_2\times\mathbb R^{d-1}$. The AdS$_2$ factor rescales time and the throat coordinate, while the transverse $\vec x$ directions are spectators. Boundary momenta $k$ therefore label different IR scaling dimensions $\nu_k$ rather than scaling as part of a relativistic $d$-dimensional critical point.

The throat is one of the central mechanisms behind holographic quantum matter. It explains why many finite-density holographic systems have low-energy correlators of the form

$$\mathcal G_k^R(\omega) \propto (-i\omega)^{2\nu_k}$$

at zero temperature, and

$$\mathcal G_k^R(\omega,T) = T^{2\nu_k}\,\Phi_k\!\left(\frac{\omega}{T}\right)$$

at small nonzero temperature. The exponent $\nu_k$ is determined by an effective mass in the AdS$_2$ region. This is the technical origin of many non-Fermi-liquid, strange-metal, and quantum-critical phenomena in bottom-up holography.

There is also an important warning. The extremal RN-AdS throat has a finite horizon area at $T=0$ in the classical large-$N$ limit. Thus it carries a nonzero entropy density

$$s_0\neq0.$$

This is useful because it gives a solvable strongly coupled IR sector, but it is also suspicious as a literal ground state of a finite-$N$ theory. In many systems, the extremal throat is unstable to superconducting order, scalar condensation, electron-star formation, lattice effects, higher-derivative corrections, or other low-temperature physics. A good way to think about the RN-AdS throat is therefore:

$$\boxed{ \text{AdS}_2\times\mathbb R^{d-1} \text{ is often an intermediate or parent IR fixed point, not necessarily the final ground state.} }$$

Extremality and the double zero

Use the planar charged black-brane ansatz from the previous page,

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

with boundary at $z=0$ and horizon at $z=z_h$. For a simple Einstein-Maxwell theory in $d+1$ dimensions, one convenient dimensionless parametrization is

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

and

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

The formula for $A_t$ assumes the regular gauge $A_t(z_h)=0$. The temperature is

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

Extremality means $T=0$, so

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

At this value,

$$f'(z_h)=0.$$

The horizon is no longer a simple zero of $f(z)$; it is a double zero. Expanding near $z=z_h$ gives

$$f(z) = d(d-1) \left(1-\frac{z}{z_h}\right)^2 + O\!\left[\left(1-\frac{z}{z_h}\right)^3\right].$$

This double zero is the whole story. A simple zero gives Rindler space near the horizon, which is the usual finite-temperature nonextremal behavior. A double zero gives AdS$_2$.

Horizon type	Blackening factor	Near-horizon geometry	Boundary meaning
nonextremal	$f(z)\sim z_h-z$	Rindler	finite $T$ thermal dissipation
extremal	$f(z)\sim (z_h-z)^2$	AdS$_2$	emergent IR scaling

The extremal limit is therefore singular in a useful way. The geometry develops an infinitely long throat, and low-frequency probes spend a long radial time in this throat before escaping to the asymptotic AdS region.

Deriving the AdS$_2\times\mathbb R^{d-1}$ throat

Define a near-horizon coordinate

$$\rho = \frac{d(d-1)}{z_h} \left(1-\frac{z}{z_h}\right).$$

As $z\to z_h$, keep $\rho$ fixed and use the double-zero expansion of $f(z)$. Since $z\simeq z_h$ in the overall warp factor,

$$\frac{L^2}{z^2}\simeq \frac{L^2}{z_h^2}.$$

The time component becomes

$$\frac{L^2}{z^2}f(z)dt^2 \simeq \frac{L^2}{z_h^2} \,d(d-1) \left(1-\frac{z}{z_h}\right)^2dt^2 = \frac{L^2}{d(d-1)}\rho^2dt^2.$$

The radial component becomes

$$\frac{L^2}{z^2}\frac{dz^2}{f(z)} \simeq \frac{L^2}{d(d-1)}\frac{d\rho^2}{\rho^2}.$$

Thus the near-horizon metric is

$$ds^2 \to L_2^2 \left( -\rho^2dt^2+ \frac{d\rho^2}{\rho^2} \right) + \frac{L^2}{z_h^2}d\vec x^{\,2},$$

where

$$\boxed{ L_2^2=\frac{L^2}{d(d-1)}. }$$

This is AdS$_2$ in Poincaré-like coordinates times a flat transverse space. If one defines $\zeta=1/\rho$, the AdS$_2$ part takes the more familiar form

$$ds_2^2 = \frac{L_2^2}{\zeta^2} \left(-dt^2+d\zeta^2\right).$$

The gauge field also has a universal near-horizon limit. Expanding

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

near $z=z_h$ gives

$$A_t \simeq \mu(d-2)\left(1-\frac{z}{z_h}\right) = e_d\rho,$$

with

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

Therefore the AdS$_2$ region is supported by a constant electric field,

$$F_{\rho t}=\partial_\rho A_t=e_d.$$

This electric field is not a small perturbation. It is the near-horizon remnant of the charge density of the boundary state. It is also what makes charged probes in the throat interesting: their effective IR dimensions are shifted by the electric field.

The near-extremal throat

At exactly $T=0$, the AdS$_2$ throat is infinitely long. At small but nonzero temperature, the throat is cut off by an AdS$_2$ black hole. The near-horizon geometry becomes

$$ds^2 \simeq L_2^2 \left[ -(\rho^2-\rho_0^2)dt^2 + \frac{d\rho^2}{\rho^2-\rho_0^2} \right] + \frac{L^2}{z_h^2}d\vec x^{\,2},$$

with AdS$_2$ temperature

$$T=\frac{\rho_0}{2\pi}$$

up to the time normalization already chosen above. The throat is no longer infinite; its proper length is cut off in the deep IR. Parametrically, for $T\ll\mu$, the throat length grows like

$$\ell_{\rm throat} \sim L_2\log\frac{\mu}{T}.$$

This logarithm is useful intuition. As the temperature decreases, there is a wider radial interval in which the geometry is well approximated by AdS$_2\times\mathbb R^{d-1}$. Low-frequency probes with

$$\omega\ll\mu$$

first see the AdS$_2$ throat. Probes with

$$\omega\lesssim T$$

are sensitive to the near-extremal AdS$_2$ black-hole horizon.

The entropy density of the planar black brane is still the horizon area density,

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

At extremality, $z_h$ remains finite because it is set by $\mu$ and the charge density. Thus

$$s(T\to0)=s_0 = \frac{1}{4G_{d+1}}\left(\frac{L}{z_*}\right)^{d-1},$$

where $z_*$ is the extremal horizon position. The low-temperature expansion of the classical RN-AdS saddle has the schematic form

$$s(T)=s_0+\gamma T+\cdots.$$

The linear-in-$T$ term is characteristic of the AdS$_2$ throat. The constant term is the famous zero-temperature entropy of the extremal black brane. It is both a calculational gift and a conceptual headache.

What does the boundary theory see?

The throat is a statement about the IR part of the bulk geometry. The boundary theory interpretation is an emergent low-energy sector controlled by the symmetries of AdS$_2$.

The AdS$_2$ metric

$$ds_2^2 = L_2^2 \left(-\rho^2dt^2+\frac{d\rho^2}{\rho^2}\right)$$

is invariant under

$$t\to\lambda^{-1}t, \qquad \rho\to\lambda\rho.$$

But the transverse metric

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

does not participate in this scaling. Hence

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

This is very different from an ordinary relativistic critical point, where

$$t\to\lambda^{-1}t, \qquad \vec x\to\lambda^{-1}\vec x.$$

It is also different from a Lifshitz critical point with finite dynamical exponent $z$, where

$$t\to\lambda^{-z}t, \qquad \vec x\to\lambda^{-1}\vec x.$$

The AdS$_2$ throat corresponds heuristically to

$$z=\infty.$$

This does not mean the theory has no spatial structure. It means spatial momentum $k$ is an exactly good label of the IR problem, but it does not scale with frequency. A boundary operator at momentum $k$ maps in the throat to an AdS$_2$ field whose effective mass depends on $k$.

The practical dictionary is:

Bulk throat statement	Boundary finite-density statement
AdS$_2$ factor	emergent low-energy conformal symmetry in time
$\mathbb R^{d-1}$ factor	spatial directions are spectators of the IR scaling
finite electric field	finite charge density remains in the IR
AdS$_2$ black hole at $T>0$	universal $\omega/T$ scaling at low temperature
finite horizon area	nonzero large-$N$ entropy density at $T=0$
momentum-dependent AdS$_2$ mass	momentum-dependent IR dimension $\Delta_k$

This is why one often says that the extremal charged black brane is dual to a locally critical state. The word local refers to spatial locality: the critical scaling is local in space and extended in time.

Scaling dimensions in the throat

Consider a neutral scalar field in the full AdS$_{d+1}$ geometry,

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

Fourier decompose along the boundary directions:

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

In the AdS$_2\times\mathbb R^{d-1}$ region, the transverse momentum contributes to the effective AdS$_2$ mass. Since

$$g^{ij}_{\rm throat} = \frac{z_h^2}{L^2}\delta^{ij},$$

the effective mass is

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

The AdS$_2$ wave equation then gives the IR scaling dimension

$$\Delta_k^{\rm IR} = \frac12+\nu_k,$$

where

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

Thus every momentum $k$ has its own IR exponent. This is the simplest mathematical expression of local criticality.

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

$$\nu_k = \sqrt{ \frac14 +L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right) -q^2e_d^2 },$$

where $q$ is the bulk charge and $A_t\simeq e_d\rho$ in the throat. The precise coefficient multiplying $q^2e_d^2$ depends on gauge-field normalization, but the physical point is robust: the electric field can lower the effective AdS$_2$ mass of charged probes.

At zero temperature, the retarded Green function of the IR CFT has the form

$$\mathcal G_k^R(\omega) = C_k\,(-i\omega)^{2\nu_k},$$

for real $\nu_k$, up to normalization and contact-term conventions. At nonzero temperature,

$$\mathcal G_k^R(\omega,T) = T^{2\nu_k}\Phi_{\nu_k,q}\!\left(\frac{\omega}{T}\right),$$

where $\Phi_{\nu_k,q}$ is determined by solving the field equation in the AdS$_2$ black hole with electric field.

This is the origin of many unusual finite-density holographic response functions. In ordinary weakly coupled metals, low-energy response is often organized around quasiparticles near a Fermi surface. In the RN-AdS throat, the basic low-energy object is instead an emergent strongly coupled IR operator with momentum-dependent dimension.

Matching the throat to the UV

The AdS$_2$ region is not the whole spacetime. A full boundary correlator is obtained by matching two solutions:

1. an inner solution in the AdS$_2$ throat, valid at small $\omega$ and near the horizon;
2. an outer solution in the asymptotic AdS$_{d+1}$ region, valid at $\omega=0$ away from the horizon.

There is an overlap region when

$$\omega\ll \rho \ll \mu,$$

in appropriate throat units. The inner solution encodes the nonanalytic low-frequency dependence. The outer solution encodes UV data such as operator normalization, source/response mixing, and momentum-dependent coefficients.

For a bosonic field, the low-frequency Green function often takes the schematic form

$$G_R(\omega,k) = \frac{ b_+^{(0)}(k)+b_-^{(0)}(k)\mathcal G_k^R(\omega)+\cdots }{ a_+^{(0)}(k)+a_-^{(0)}(k)\mathcal G_k^R(\omega)+\cdots }.$$

The coefficients $a_\pm^{(0)}(k)$ and $b_\pm^{(0)}(k)$ come from the outer region at $\omega=0$. The function $\mathcal G_k^R(\omega)$ is the universal AdS$_2$ IR Green function. This formula separates universal IR scaling from UV matching data.

For holographic fermions, the same logic leads to a famous expression near a Fermi momentum $k_F$:

$$G_R(\omega,k) \simeq \frac{h_1}{ k_\perp-v_F^{-1}\omega-h_2 e^{i\gamma}\omega^{2\nu_{k_F}} },$$

where

$$k_\perp=k-k_F.$$

Depending on $\nu_{k_F}$, this can describe a Fermi liquid-like quasiparticle, a non-Fermi liquid, or a marginal-Fermi-liquid-like response. The next page will develop this fermionic application more carefully.

The general moral is simple:

$$\boxed{ \text{AdS$_2$ gives the nonanalytic IR function; the UV geometry tells that function how to enter the boundary correlator.} }$$

Local criticality versus ordinary quantum criticality

It is helpful to compare several scaling structures.

IR geometry	Scaling	Boundary interpretation
AdS$_{d+1}$	$t\to\lambda^{-1}t$, $\vec x\to\lambda^{-1}\vec x$	relativistic CFT
Lifshitz	$t\to\lambda^{-z}t$, $\vec x\to\lambda^{-1}\vec x$	finite-$z$ quantum criticality
hyperscaling-violating Lifshitz	finite-$z$ plus hyperscaling violation	nontrivial entropy and scaling dimensions
AdS$_2\times\mathbb R^{d-1}$	$t\to\lambda^{-1}t$, $\vec x\to\vec x$	local or semi-local criticality

For AdS$_2\times\mathbb R^{d-1}$, the dispersion relation is not determined by scale invariance in the usual way. Since $k$ does not scale, the IR scaling form is not

$$G_R(\omega,k)=k^{2\Delta-d-z}\,F\!\left(\frac{\omega}{k^z}\right).$$

Instead, one has a family of frequency scalings,

$$G_R(\omega,k) \sim A(k)+B(k)(-i\omega)^{2\nu_k}+\cdots.$$

This is why the term semi-local is useful. The state has local critical dynamics in time but retains nontrivial momentum dependence through $\nu_k$, $A(k)$, $B(k)$, and possible poles from the UV matching problem.

A common condensed-matter slogan is that AdS$_2$ describes a critical state with infinite dynamical exponent,

$$z=\infty.$$

This slogan is helpful, but it hides two subtleties:

1. The IR still knows about momentum through effective AdS$_2$ masses.
2. A genuine finite-density holographic ground state may replace the AdS$_2$ throat with another IR geometry at sufficiently low energy.

Thus AdS$_2$ is best treated as an exact classical saddle of the minimal model and as a robust organizing principle for intermediate low-energy physics.

The zero-temperature entropy puzzle

The extremal black brane has a horizon with nonzero area density. Therefore the classical entropy density remains finite at $T=0$:

$$s_0 = \frac{1}{4G_{d+1}}\left(\frac{L}{z_*}\right)^{d-1}.$$

From the boundary viewpoint, this is an enormous ground-state degeneracy of order $N^2$ or $C_T$. It is not what one expects from a generic isolated finite-density quantum system at finite $N$.

There are several possible interpretations, none of which should be confused with one another.

Viewpoint	Meaning
Classical gravity	the extremal RN-AdS black brane is a legitimate saddle with finite horizon area
Large-$N$ limit	exponentially many states may remain degenerate at leading order in $N$
Finite $N$ expectation	degeneracy may be lifted by quantum or stringy corrections
Effective-model viewpoint	the throat is a parent IR fixed point, often unstable to lower-temperature order
Top-down viewpoint	the final answer depends on the full string compactification and allowed charged fields

This finite entropy is not a reason to discard the throat. Instead, it is a sign that one should ask a sharper question:

Is the extremal RN-AdS saddle stable in the theory under consideration, and down to what energy scale does it control the physics?

Often the answer is: it controls an important intermediate regime, but not the ultimate ground state.

Instabilities of the AdS$_2$ throat

The AdS$_2$ throat is especially sensitive to charged or light fields. The simplest diagnostic is the AdS$_2$ Breitenlohner-Freedman bound. For a neutral scalar in AdS$_2$,

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

In the charged case, the electric field lowers the effective mass. Schematically the stability condition becomes

$$L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right)-q^2e_d^2 \ge -\frac14.$$

If this inequality is violated for some mode, the AdS$_2$ region is unstable even if the scalar is perfectly stable in the UV AdS$_{d+1}$ region. This is the mechanism behind many holographic superconducting instabilities: the finite-density electric field makes a charged scalar effectively tachyonic in the near-horizon throat.

The same logic also explains more exotic transitions. If the AdS$_2$ scaling exponent

$$\nu_k$$

becomes imaginary, the two IR fixed points associated with the two AdS$_2$ quantizations collide and move into the complex plane. This produces oscillatory behavior in frequency and can lead to Berezinskii-Kosterlitz-Thouless-like scaling in certain holographic quantum phase transitions.

Common low-temperature replacements of the RN-AdS throat include:

• charged scalar hair, giving holographic superconductors or superfluids;
• neutral scalar hair, giving density-wave, antiferromagnetic, or other ordered phases in suitable models;
• electron-star geometries, where charged fermion matter carries part of the charge outside the horizon;
• Lifshitz or hyperscaling-violating IR geometries;
• spatially modulated phases;
• stringy or quantum corrections that modify the extremal horizon.

The RN-AdS throat is therefore not a universal final answer. It is a universal starting point in the minimal finite-density Einstein-Maxwell model.

Relation to entropy, fractionalization, and charge behind the horizon

A charged black brane carries charge in the bulk electric field, and much of the charge is associated with the horizon in the classical geometry. In the boundary theory, this is often described as a fractionalized phase: the charge is carried by deconfined degrees of freedom not visible as gauge-invariant quasiparticles outside the horizon.

This language is model-dependent, but it is useful. Compare three possibilities:

Bulk charge distribution	Boundary intuition
electric flux enters a horizon	fractionalized charge behind the horizon
charged matter outside the horizon	cohesive charge carried by gauge-invariant or quasiparticle-like matter
mixed configuration	partially fractionalized phase

The extremal RN-AdS throat belongs to the first category. It has an electric field in the throat and a charged horizon. Holographic superconductors and electron stars move some charge outside the horizon, changing the deep IR physics.

This distinction is especially important for interpreting Fermi surfaces. A boundary theory can have finite charge density without an obvious gauge-invariant Fermi surface if the charge is hidden behind a horizon. Conversely, holographic fermion probes can show sharp spectral features even when the background charge remains fractionalized. One must carefully distinguish the charge carried by the background geometry from spectral features of a probe operator.

A minimal computational workflow

When using an AdS$_2$ throat to analyze a holographic observable, the workflow is usually:

1. Find the extremal background. Determine $z_*$, $L_2$, and the near-horizon electric field.
2. Choose the perturbation. Identify the bulk field dual to the boundary operator.
3. Reduce to AdS$_2$. Fourier transform along $\vec x$ so that $k$ becomes an effective mass parameter.
4. Compute the IR dimension. Extract $\nu_k$ and $\Delta_k^{\rm IR}=1/2+\nu_k$.
5. Solve the inner problem. Obtain $\mathcal G_k^R(\omega)$ in AdS$_2$ or the AdS$_2$ black hole.
6. Solve the outer problem. Match to the asymptotic AdS$_{d+1}$ region at $\omega=0$.
7. Assemble the boundary Green function. Combine UV coefficients with the IR Green function.
8. Check for instabilities. Look for imaginary $\nu_k$, poles at $\omega=0$, or modes violating an IR BF bound.

This workflow appears repeatedly in finite-density holography. It is the backbone of holographic non-Fermi liquids, holographic superconducting instabilities, semi-local critical response, and many quantum phase transitions.

Common mistakes

Mistake 1: Saying the whole bulk becomes AdS$_2$.

Only the near-horizon region becomes AdS$_2\times\mathbb R^{d-1}$. The full spacetime still interpolates to AdS$_{d+1}$ near the boundary. Boundary correlators require matching the throat to the UV region.

Mistake 2: Forgetting the transverse $\mathbb R^{d-1}$.

The transverse space is not decorative. It is why the IR dimensions depend on momentum $k$ and why the criticality is semi-local rather than an ordinary CFT$_1$ completely decoupled from space.

Mistake 3: Treating $s_0\neq0$ as automatically physical at finite $N$.

The finite entropy is a classical large-$N$ result. It often signals degeneracy, instability, or missing corrections. It is still an extremely useful leading saddle.

Mistake 4: Confusing local criticality with locality of the microscopic Hamiltonian.

The boundary theory can be a perfectly local quantum field theory. “Local criticality” means that the emergent IR scaling is local in space: time scales, but $\vec x$ does not.

Mistake 5: Assuming AdS$_2$ guarantees a stable strange metal.

The throat gives a powerful low-energy sector, but additional fields may condense, translation symmetry may break, fermions may backreact, and the deep IR may be replaced by another geometry.

Mistake 6: Ignoring gauge-field normalization.

The schematic exponent

$$\nu_k^2 = \frac14+m_k^2L_2^2-q^2e_d^2$$

is convention-dependent in the last term. Physical conclusions are invariant, but numerical formulas require a precise normalization of the Maxwell action and charge $q$.

Exercises

Exercise 1: The double zero of the extremal blackening factor

For

$$f(u)=1+Q^2u^{2d-2}-(1+Q^2)u^d, \qquad u=\frac{z}{z_h},$$

show that extremality corresponds to

$$Q^2=\frac{d}{d-2},$$

and that near $u=1$,

$$f(u)=d(d-1)(1-u)^2+O((1-u)^3).$$

Solution

The horizon is at $u=1$, and indeed

$$f(1)=1+Q^2-(1+Q^2)=0.$$

The derivative is

$$f'(u) = (2d-2)Q^2u^{2d-3} -d(1+Q^2)u^{d-1}.$$

At the horizon,

$$f'(1) = (2d-2)Q^2-d(1+Q^2) =(d-2)Q^2-d.$$

Extremality means the horizon is a double zero, so $f'(1)=0$. Therefore

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

Now compute

$$f''(u) = (2d-2)(2d-3)Q^2u^{2d-4} -d(d-1)(1+Q^2)u^{d-2}.$$

At extremality,

$$Q^2=\frac{d}{d-2}, \qquad 1+Q^2=\frac{2(d-1)}{d-2}.$$

Thus

$$f''(1) = \frac{d}{d-2}(2d-2)(2d-3) - \frac{2d(d-1)^2}{d-2} =2d(d-1).$$

Since $f(1)=f'(1)=0$,

$$f(u) = \frac12 f''(1)(u-1)^2+\cdots =d(d-1)(1-u)^2+\cdots.$$

Exercise 2: Derive the AdS$_2$ radius

Starting from

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

and the extremal near-horizon expansion

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

show that the near-horizon metric is AdS$_2\times\mathbb R^{d-1}$ with

$$L_2^2=\frac{L^2}{d(d-1)}.$$

Solution

Define

$$\rho = \frac{d(d-1)}{z_h} \left(1-\frac{z}{z_h}\right).$$

Then

$$1-\frac{z}{z_h} = \frac{z_h\rho}{d(d-1)}, \qquad dz=-\frac{z_h^2}{d(d-1)}d\rho.$$

Near the horizon $z\simeq z_h$. The time component becomes

$$-\frac{L^2}{z_h^2}d(d-1) \left(1-\frac{z}{z_h}\right)^2dt^2 = -\frac{L^2}{d(d-1)}\rho^2dt^2.$$

The radial component becomes

$$\frac{L^2}{z_h^2}\frac{dz^2}{d(d-1)(1-z/z_h)^2} = \frac{L^2}{d(d-1)}\frac{d\rho^2}{\rho^2}.$$

Therefore

$$ds^2 \to \frac{L^2}{d(d-1)} \left(-\rho^2dt^2+\frac{d\rho^2}{\rho^2}\right) + \frac{L^2}{z_h^2}d\vec x^{\,2}.$$

The AdS$_2$ radius is

$$L_2^2=\frac{L^2}{d(d-1)}.$$

Exercise 3: Momentum-dependent IR dimension

Consider a neutral scalar of bulk mass $m$ in the AdS$_2\times\mathbb R^{d-1}$ throat. Show that a mode with boundary spatial momentum $k$ has AdS$_2$ effective mass

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

and IR exponent

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

Solution

In the throat,

$$ds^2 = ds_2^2+\frac{L^2}{z_h^2}d\vec x^{\,2}.$$

Thus the inverse transverse metric is

$$g^{ij}=\frac{z_h^2}{L^2}\delta^{ij}.$$

For a Fourier mode

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

the transverse Laplacian contributes

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

In the AdS$_2$ wave equation, this is equivalent to shifting the mass to

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

For a scalar in AdS$_2$, the dimension satisfies

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

Writing

$$\Delta_k^{\rm IR}=\frac12+\nu_k$$

gives

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

Exercise 4: Why the criticality is semi-local

Use the AdS$_2\times\mathbb R^{d-1}$ metric to explain why $\omega$ scales but $k$ does not. Then explain why the low-frequency response can have the form

$$\mathcal G_k^R(\omega)\sim (-i\omega)^{2\nu_k}.$$

Solution

The AdS$_2$ part of the metric is invariant under

$$t\to\lambda^{-1}t, \qquad \rho\to\lambda\rho.$$

Since frequency is conjugate to time,

$$\omega\to\lambda\omega.$$

The transverse part of the metric is

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

which is invariant only if

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

Therefore the conjugate momentum $k$ also does not scale. Instead, $k$ enters the AdS$_2$ equation as a parameter in the effective mass. The AdS$_2$ field has dimension

$$\Delta_k^{\rm IR}=\frac12+\nu_k,$$

so its retarded Green function scales as

$$\mathcal G_k^R(\omega)\sim (-i\omega)^{2\Delta_k^{\rm IR}-1} =(-i\omega)^{2\nu_k}.$$

This is semi-local criticality: critical in frequency, with momentum-dependent exponents rather than ordinary momentum scaling.

Exercise 5: The AdS$_2$ BF instability criterion

A charged scalar in the throat has schematic exponent

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

What condition signals an instability of the AdS$_2$ throat?

Solution

The AdS$_2$ BF bound requires the effective mass to obey

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

Equivalently, the exponent $\nu_k$ should be real:

$$\nu_k^2\ge0.$$

Using the given expression, stability requires

$$\frac14+L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right)-q^2e_d^2\ge0.$$

An instability occurs when

$$L_2^2\left(m^2+\frac{z_h^2}{L^2}k^2\right)-q^2e_d^2 <-\frac14.$$

Physically, the electric field lowers the effective mass of a charged scalar in the throat. If the mass falls below the AdS$_2$ BF bound, the extremal RN-AdS throat is unstable, often toward a charged condensate.

Exercise 6: The matching formula

Suppose the full UV Green function has the schematic low-frequency form

$$G_R(\omega,k) = \frac{ b_+^{(0)}(k)+b_-^{(0)}(k)\mathcal G_k^R(\omega) }{ a_+^{(0)}(k)+a_-^{(0)}(k)\mathcal G_k^R(\omega) }.$$

Which part of this expression is universal IR data, and which part depends on the UV completion of the geometry?

Solution

The function

$$\mathcal G_k^R(\omega)$$

is the IR Green function computed in the AdS$_2$ throat. Its nonanalytic frequency dependence, such as

$$\mathcal G_k^R(\omega)\sim(-i\omega)^{2\nu_k},$$

is universal for a field with a given IR dimension.

The coefficients

$$a_+^{(0)}(k),\quad a_-^{(0)}(k),\quad b_+^{(0)}(k),\quad b_-^{(0)}(k)$$

come from solving the outer-region problem in the full asymptotically AdS$_{d+1}$ geometry at $\omega=0$. They depend on the UV theory, the operator normalization, the bulk mass and couplings outside the throat, and the boundary conditions.

Thus the formula separates universal IR scaling from UV matching data.

Further reading

• Thomas Faulkner, Hong Liu, John McGreevy, and David Vegh, “Emergent quantum criticality, Fermi surfaces, and AdS$_2$”. The essential paper explaining how the AdS$_2$ throat controls finite-density low-energy response and holographic Fermi surfaces.
• Sean A. Hartnoll, “Lectures on holographic methods for condensed matter physics”. A standard entry point to RN-AdS black holes, finite-density holography, and holographic superconductors.
• John McGreevy, “Holographic duality with a view toward many-body physics”. A broad and physically motivated discussion of finite-density holographic states.
• Nabil Iqbal, Hong Liu, and Mark Mezei, “Semi-local quantum liquids”. A focused discussion of semi-local criticality, finite spatial correlation length, infinite correlation time, and the interpretation of AdS$_2$ throats.
• Nabil Iqbal, Hong Liu, and Mark Mezei, “Quantum phase transitions in semi-local quantum liquids”. A useful source for bifurcating, hybridized, and marginal critical points controlled by AdS$_2$ physics.
• Subir Sachdev, “Holographic metals and the fractionalized Fermi liquid”. Helpful for the fractionalization interpretation of horizon charge and finite-density holographic phases.