List of Figures
Low energy excitations of a stack of D3 branes at weak and strong ’t Hooft coupling. Weak coupling: N2 light string states connecting the N D3 branes. Strong coupling: classical gravitational excitations that are strongly red-shifted by an event horizon.
Essential dictionary: The boundary value h of a bulk field ϕ is a source for an operator 𝒪 in the dual QFT.
The radial direction: Events in the interior of the bulk capture long distance, low energy dynamics of the dual field theory. Events near the boundary of the bulk describe short distance, UV dynamics in the field theory dual. The simplest way to think about this is that the interior events are increasingly redshifted relative to the boundary energy scales (a similar logic was used in the decoupling argument of §1.4 to derive holographic duality, here we are discussing redshifts within the near horizon AdS region itself).
The Wilsonian cutoff: The cutoff is at r = 𝜖, while the UV fixed point theory is at r = 0. The partition function Z𝜖[ϕ𝜖] is over all modes at r > 𝜖, subject to the boundary condition ϕ(𝜖) = ϕ𝜖. The partition function 𝜖 and r = 0, with boundary conditions at both ends.
Minimal surface Γ in the bulk, whose area gives the entanglement entropy of the region A on the boundary.
MERA network and a network geodesic that cuts a minimal number of links allowing it to enclose a region of the physical lattice at the top of the network. Each line between two points can be thought of as a maximally entangled pair in the auxiliary Hilbert space, while the points themselves correspond to projections that glue the pairs together. At the top of the network, the projectors map the auxiliary Hilbert space into the physical Hilbert space. The entanglement entropy of the physical region is bounded by the number of links cut by the geodesic: SE ≤ ℓlogD.
Ground states of bosons on a square lattice with tunneling amplitude between the sites w, and on-site repulsive interactions U. Bosons are condensed in the zero momentum state in the superfluid, and so there are large number flucutations in a typical component of the wavefunction shown above. The Mott insulator is dominated by a configuration with exactly one particle on each site.
Phase diagram of S as a function of s and T. The quantum critical region is bounded by crossovers at T ∼ |s−sc|zν indicated by the dashed lines. Conventional quasiparticle or classical-wave dynamics applies in the non-quantum-critical regimes including a Kosterlitz-Thouless phase transition above which the superfluid density is zero. One of our aims in this review is to develop a theory of the non-quasiparticle dynamics within the quantum critical region.
Phase diagram of the Hubbard model for spin S = 1/2 fermions on the honeycomb lattice at a density of one fermion per site. The large U/w (s < sc) state breaks spin rotation symmetry, and is an insulating antiferromagnet with a gap to all charged excitations. The small U/w (s > sc) state is described at low energies by a CFT3 of free Dirac fermions.
RVB states and excitations. (a) Sketch of a component of a resonating valence bond wavefunction on the square lattice. (b) Excited state with neutral excitations carrying spin S = 1/2. In the field theory 𝒮z each excitation is a quantum of the zα particle, while the gauge field A represents the fluctuations of the valence bonds (see Figure 2.5).
Number operators, , counting the number of singlet valence bonds on each link of the square lattice; here we have numbered links by integers, but in the text we label them by the sites they connect. Modulo a phase factor, these operators realize the electric field operator of a compact U(1) lattice gauge theory.
Phases of a square lattice antiferromagnet. The vicinity of the critical point is described by the theory Sz in (2.14) (compare Figure 2.3). The phase with Néel order is the Higgs phase of the gauge theory, while the Coulomb phase of Sz is destabilized by monopoles, leading to valence bond solid order.
Cigar geometry. The r coordinate runs from 0 at the boundary to r+ at the horizon, where the Euclidean time circle shrinks to zero.
Thermal screening. The geodesic runs along the horizon over a distance x. This contribution to its length dominates the correlation function and leads to an exponentially decaying correlation (2.52) in space, with scale set by the horizon radius r+.
Schematic of the real part of the conductivity of a CFT3 of M Dirac fermions coupled to a U(1) gauge field in the large M limit. Similar features apply to other CFT3s in the vector large M limit. The peak at zero frequency is a remnant of the quasiparticles present at M = ∞, and the total area under this peak equals (T ln2)/2 as M → ∞.
Quasinormal poles as a function of z. Hollow dots denote quasinormal frequencies at z = 1. The arrows show the motion of the poles as z is increased. For z ≥ d the modes are all overdamped, lying along the negative imaginary axis. All modes have momentum k = 0. [Figure adapted with permission from ]
|σ(ω)| in the lower half plane, with γ = 1/12 (left) and γ = −1/12 (right). The dominant features are poles and zeros, which are shaded darker than the background. The plot has been clipped at e2|σ| = 2, and so poles are associated with the solid/sharply shaded regions.
Motion of poles and zeros as γ is decreased. Crosses are poles and circles are zeros. Pais of poles and zeros move to the imaginary frequency axis, and then move up and down the axis to annihilate with a zero or pole. In these plots w = ω/(4πT). Figure taken with permission from 
Fermionic excitations at the Ising-nematic critical point. We focus on an extended patch of the Fermi surface, and expand in momenta about the point 4.14). The coordinate y represents the d − 1 dimensions parallel to the Fermi surface.
Fermi surfaces of ψ1 and ψ2 fermions in the plane defined by the Fermi velocities . The gapless Fermi surfaces are one-dimensional, and are indicated by the full lines. The lines intersect at the hot spot, which is the filled circle at the origin.
Schematic of a component of a state obtained by doping the resonating valence bond state in Figure 2.4a by a density p of holes. The circles represent spinless ‘holons’ hq of electromagnetic charge +e, and emergent gauge charge q = ±1. The rectangular dimers, dα, are bound states of holons and the zα quanta: the dα are neutral under the emergent gauge field, but carry electromagnetic charge +e and spin S = 1/2. Finally, the resonance between all the dimers is captured by the emergent gauge field Aμ.
Fractionalized vs. cohesive charge. Left: The electric flux dual to the field theory charge density emanates from a charged event horizon. The charge is said to be ‘fractionalized’. Right: The electric flux is instead sourced by a condensate of charged matter in the bulk. The charge is said to be ‘cohesive’. Figures taken from  with permission.
Emergence of AdS2. A CFT, dually described by Einstein-Maxwell theory in the bulk, is placed at a nonzero chemical potential. This induces a renormalization group flow in the bulk, leading to the emergence of semi-locally critical AdS2 × ℝd in the far IR, near horizon region. Figure adapted with permission from .
The matching argument. When ω ≪ μ, then there is a parametrically large region of overlap between the range where the near-horizon solution (4.71) is valid, and the asymptotic range where the frequency may be set to zero. This allows the solution satisfying infalling boundary conditions to be extended to the boundary of the full spacetime.
Pair production in the near horizon region leads to a discharging of the black hole if electromagnetic screening overcomes gravitational clumping. Figure taken with permission from .
The electron star. A charged, gravitating fluid is present in the spacetime for ∞ < r < rs. All the charge is carried by the cohesive fermion fluid. Figure taken with permission from .
Phase diagram of Einstein-Maxwell-Chern-Simons theory as a function of temperature and magnetic field, at fixed nonzero charge density. The Chern-Simons coupling k > 3/4 (for 3/4 > k > 1/2 the temperature scaling of s in the critical region is different). Figure taken with permission from . This phase diagram ignores certain spatial modulation instabilities that will be discussed in Chapter 6.
A ‘phase space’ of metals. In this section we will discuss the coherent and incoherent non-quasiparticle metals that appear on the left column. Figure developed in discussion with Aharon Kapitulnik.
Relevant and irrelevant inhomogeneities. In the left plot the inhomogeneities grow towards the interior of the spacetime whereas in the right plot they decay towards the interior. In the irrelevant case one case use perturbation theory in the interior, almost homogeneous, geometry to describe momentum relaxation. In the relevant case one must find the new inhomogeneous interior spacetime, typically by solving PDEs.
Spectral weight transfer: Optical conductivity in the metallic (left) and insulating (right) phases of the model from . Curves with lower (higher) σdc are at higher temperature in the metallic (insulating) phase. Figure adapted with permission.
ac conductivity in an inhomogeneous background. Illustrative plot of the real and imaginary parts of σ(ω) in a holographic model deformed by the periodic chemical potential (5.204). The dashed lines show the corresponding conductivities without the lattice deformation. The lattice plot shows a Drude peak at small frequencies, a resonance due to scattering off the lattice with ω ∼ kL and the asymptotic behavior σ → 1. Figure taken from  with permission.
A chain of coupled SYK sites with complex fermions: each site contains N ≫ 1 fermions with on-site interactions as in (5.205). The coupling between nearest neighbor sites are four fermion interaction with two from each site. Figure adapted from  with permission.
The zero temperature endpoint of the instability: a charged bosonic condensate in the spacetime. Figure taken with permission from .
Critical temperature as a function of UV scaling dimension Δ and charge γq of the operator that condenses. Contours show values of γTc/μ. Figure adapted with permission from .
Optical conductivity in a holographic superconductor with broken translation invariance. Dashed line is above Tc, topmost solid line is at Tc, subsequent lines are at progressively lower temperatures. In the left plot, translations are weakly broken, and the sharp Drude peak in the normal state persists into the superconducting state. In the right plot, translations are strongly broken. The weak Drude peak in the normal state eventually disappears at low temperatures in the superconducting state. All plots exhibit the loss of spectral weight into the superconducting delta function as temperature is lowered. Figures taken with permission from .
Zero temperature fermionic spectral densities as a function of ω and k. Top left: Normal state extremal RN-AdS background, exhibiting gapless but strongly scattered fermions. Top right: Zero temperature superconducting state (with z = 1) with η5 = 0. Gapless fermions still exist but are now very weakly scattered within the z = 1 lightcone. Bottom left: Small nonzero η5 in the superconducting state. The fermions are now gapped, but still long lived within the lightcone. Bottom right: Larger nonzero η5. The fermions are both gapped and short lived. Figures taken with permission from .
The profile of the Ricci scalar (normalized to the AdS value) in the striped phase for T = 0.87Tc (left) and T = 0.04Tc (right). Figure taken from  with permission.
Triangular lattice. The expectation value of the current in the boundary theory, in a black hole with a triangular lattice ‘ground state’. The largest currents flow in circular patterns, and the smallest current is in the interstitial regions between them. Figure taken from  with permission.
A sketch of the dynamics of the order parameter in a holographic superfluid quench. Top row: ⟨𝒪⟩ as a function of t. Bottom row: location of the lowest lying QNMs, and their motion upon increasing λ. Columns left to right: 0 < λ < λ1; λ1 < λ < λ2; λ > λ2.
A quench connecting two heat baths. Left: two CFTs placed next to each other at different temperature. Right: a NESS forms as energy flows from the left to right bath.
List of Table
The canonical examples of holographic duality for field theories in 3+1, 2+1 and 1+1 spacetime dimensions. The first column shows the string theory setup of which one should consider the low energy excitations. In the limit of no gravitational backreaction the low energy degrees of freedom are described by a quantum field theory, given in the second column. In the limit of strong gravitational back reaction, the low energy degrees of freedom are described by classical gravitational dynamics about the backgrounds shown in the third column.