A Condensed-Matter Primer for Holographers

This page is a translation guide. It is written for readers who know AdS/CFT better than condensed matter, and for condensed-matter readers who want to see which pieces of their subject holography is trying to reorganize.

The goal is not to compress a full graduate course in condensed matter into one page. The goal is sharper: isolate the concepts that will recur throughout holographic quantum matter.

Those concepts are:

• quasiparticles and their failure;
• order parameters and the infrared meaning of symmetry breaking;
• quantum critical scaling;
• finite-density transport;
• the special role of momentum;
• Mott physics and strange metals;
• why linear-in-$T$ resistivity is a clue, not a theory.

The theme is that condensed matter is the art of finding simple infrared descriptions of complicated microscopic systems. Holography enters when the usual simple description — weakly interacting quasiparticles — is absent.

The bridge question

A condensed-matter problem becomes a holographic-quantum-matter problem when the infrared observables are organized by scaling, conservation laws, and dissipation, but not by long-lived quasiparticles.

Microscopic systems and infrared descriptions

A typical electronic model begins with a lattice Hamiltonian, for example the Hubbard model,

$$H = -t\sum_{\langle ij\rangle,\sigma} \left(c^\dagger_{i\sigma}c_{j\sigma}+c^\dagger_{j\sigma}c_{i\sigma}\right) +U\sum_i n_{i\uparrow}n_{i\downarrow} -\mu\sum_i n_i.$$

Here $t$ is the hopping amplitude, $U$ is the local repulsion, and $\mu$ fixes the density. This Hamiltonian is simple to write and hard to solve. The difficulty is not merely technical. At finite density, the Pauli principle, strong interactions, and many low-energy states combine into an enormous sign-sensitive many-body problem.

Condensed matter succeeds when it identifies an infrared description that is insensitive to most microscopic details. Examples include:

Microscopic input	Infrared description	Organizing data
weakly interacting electrons	Fermi liquid	Fermi surface, quasiparticle residue, Landau parameters
broken global symmetry	ordered phase	order parameter, Goldstone modes, defects
Cooper instability	superconductor or superfluid	condensate, phase stiffness, vortices, gap
tuned continuous transition	quantum critical point	scaling exponents, relevant operators
strong local repulsion at commensurate filling	Mott insulator	charge gap, spin sector, local constraints
strongly scattered finite-density metal	strange metal	scaling, short relaxation times, broad spectra

The same ultraviolet Hamiltonian can have several nearby infrared tendencies. A cuprate, heavy-fermion compound, pnictide, organic conductor, or cold-atom system is rarely “just” one of the entries in this table. The point of the table is to name the competing languages.

A schematic map of the condensed-matter ideas used in holographic quantum matter. The important message is not the literal geometry of the diagram, but the competition between quasiparticle phases, ordered phases, quantum critical scaling, Mott constraints, finite-density strange metals, and transport controlled by slow modes.

Quasiparticles: the conventional miracle

A quasiparticle is not merely an excitation. It is an excitation that is long-lived enough to behave as a particle over the time and length scales of interest.

For a fermionic operator $c_k$, the retarded Green function is

$$G^R(\omega,k) = -i\int dt\, e^{i\omega t}\theta(t) \langle [c_k(t),c_k^\dagger(0)]_+\rangle.$$

The spectral function measured by angle-resolved photoemission, in a simplified setting, is

$$A(\omega,k)=-2\operatorname{Im}G^R(\omega,k).$$

A quasiparticle appears when $G^R$ has a pole close to the real axis:

$$G^R(\omega,k) \simeq \frac{Z_k}{\omega-\epsilon_k+i\Gamma_k} +G^R_{\rm inc}(\omega,k),$$

with

$$\Gamma_k \ll |\epsilon_k|.$$

The residue $Z_k$ is the overlap between the microscopic operator and the emergent quasiparticle. The width $\Gamma_k$ is its decay rate. When $Z_k$ is finite and $\Gamma_k$ is small, a complicated interacting system can be treated as a gas of weakly interacting emergent particles.

That is the conventional miracle. Holographic quantum matter is built for cases where the miracle fails.

Operational test

Do not ask only whether a system is “strongly coupled.” Ask whether the low-energy spectral weight contains narrow poles whose lifetime is parametrically long compared with the microscopic or thermal time scale.

The Fermi gas and the Fermi surface

For noninteracting fermions with dispersion $\epsilon_k$, the ground state at zero temperature fills all states below the chemical potential:

$$\epsilon_k < \mu.$$

The Fermi surface is the locus

$$\epsilon_k=\mu.$$

For a rotationally invariant system in $d$ spatial dimensions,

$$\epsilon_k=\frac{k^2}{2m}, \qquad k_F=(2m\mu)^{1/2}.$$

The key fact is that low-energy excitations live near the Fermi surface, not throughout the full Brillouin zone. Write

$$k=k_F+\ell, \qquad |\ell|\ll k_F.$$

Then

$$\epsilon_k-\mu \simeq v_F\ell, \qquad v_F=\left.\frac{\partial \epsilon_k}{\partial k}\right|_{k_F}.$$

Thus each small patch of a smooth Fermi surface looks like a one-dimensional chiral problem in the direction normal to the surface. This is why the Fermi surface is more than a geometric surface: it is an infrared manifold of gapless modes.

The low-temperature thermodynamics follows immediately. Only states within energy $T$ of the Fermi surface are thermally active, so the entropy density and specific heat are linear in $T$:

$$s\sim N(0)T, \qquad c_V\sim N(0)T,$$

where $N(0)$ is the density of states at the Fermi surface.

This is already an important contrast with a relativistic CFT at zero density, where dimensional analysis gives

$$s\sim T^d.$$

A Fermi surface behaves as though only one dimension is thermally scaled: the radial direction perpendicular to the surface. This intuition will return when we discuss hyperscaling violation.

Landau Fermi liquid theory

Landau’s insight was that weak interactions are not necessary for quasiparticles to exist. A Fermi liquid is an interacting phase adiabatically connected to the free Fermi gas. It has the same low-energy kinematics — a Fermi surface and long-lived fermionic quasiparticles — but with renormalized parameters.

The energy functional for small changes $\delta n_k$ in the quasiparticle distribution is

$$\delta E = \sum_k \epsilon_k^*\,\delta n_k +\frac{1}{2}\sum_{k,k'} f_{k k'}\,\delta n_k\delta n_{k'}+ \cdots.$$

The $f_{k k'}$ are Landau interaction parameters. They do not destroy the quasiparticles; they tell us how one quasiparticle energy shifts in the presence of others.

The defining lifetime estimate in a clean Fermi liquid is

$$\Gamma(\omega,T) \sim \frac{\omega^2+(\pi T)^2}{E_F},$$

up to dimensionless angular factors and interaction strengths. Therefore

$$\frac{\Gamma}{T}\to 0 \qquad \text{as} \qquad T\to 0$$

for thermal excitations with $\omega\sim T$. Quasiparticles become sharper at low temperature.

This is the standard against which strange metals look strange. In a strange metal, the decay rate inferred from transport or spectroscopy is often of order $T$, not $T^2/E_F$. There is no parametrically long quasiparticle lifetime.

Drude theory and the first transport trap

The Drude model treats charge carriers as particles with density $n$, charge $q$, mass $m$, and relaxation time $\tau$. In an electric field,

$$m\frac{dv}{dt}=qE-\frac{m v}{\tau}.$$

For $E(t)=E e^{-i\omega t}$,

$$v(\omega)=\frac{q\tau}{m}\frac{1}{1-i\omega\tau}E(\omega),$$

so the current $J=nqv$ gives

$$\sigma(\omega) = \frac{n q^2\tau}{m}\frac{1}{1-i\omega\tau}.$$

This formula is useful, but it hides two different physical processes under the single symbol $\tau$.

The first is local equilibration. Electron-electron scattering can rapidly redistribute energy and momentum among carriers. This can be very fast in a strongly coupled system.

The second is momentum relaxation. The total momentum of a perfectly translation-invariant system cannot decay. Momentum can relax through impurities, phonons, Umklapp scattering, explicit lattices, disorder, boundaries, or coupling to external baths.

At finite density, electric current often overlaps with momentum. If momentum is conserved, a DC electric field accelerates the fluid forever. The clean DC conductivity is then infinite even if the system equilibrates locally very rapidly.

A clean hydrodynamic metal has the schematic conductivity

$$\sigma(\omega) = \sigma_Q + \frac{\rho^2}{\chi_{PP}}\frac{i}{\omega},$$

where $\rho$ is the charge density and $\chi_{PP}$ is the momentum susceptibility. With weak momentum relaxation rate $\Gamma$,

$$\sigma(\omega) = \sigma_Q + \frac{\rho^2}{\chi_{PP}}\frac{1}{\Gamma-i\omega}.$$

This formula is one of the most important pieces of condensed-matter grammar for holography. A black hole horizon can produce strong dissipation, but finite DC resistivity at finite density also requires a channel that relaxes momentum, or an incoherent current that does not overlap with momentum.

Order parameters and broken symmetry

Many phases are simple because a symmetry is broken. A ferromagnet chooses a spin direction; a crystal chooses an origin and lattice; a superfluid chooses a phase; a charge-density wave chooses a spatial modulation.

An order parameter $\mathcal O$ is an operator whose expectation value distinguishes the phase:

$$\langle \mathcal O\rangle = 0 \quad\text{in the symmetric phase}, \qquad \langle \mathcal O\rangle \neq 0 \quad\text{in the ordered phase}.$$

A simple Ginzburg—Landau free energy is

$$F[\phi] = \int d^d x \left[ \frac{1}{2}(\nabla\phi)^2 +\frac{r}{2}\phi^2 +\frac{u}{4}\phi^4 \right], \qquad u>0.$$

For $r>0$, the minimum is $\phi=0$. For $r<0$, the minimum has

$$|\phi|^2=-\frac{r}{u}.$$

If the broken symmetry is continuous, there are low-energy Goldstone modes. If the broken symmetry is discrete, the ordered phase has domain walls but no Goldstone boson.

In holography, order usually appears as bulk hair. A boundary order parameter is dual to a bulk field. A source-free normalizable mode of that field is the gravitational signal of spontaneous symmetry breaking.

Superconductors and superfluids

A superconductor or superfluid breaks a global $U(1)$ symmetry. In a neutral superfluid the $U(1)$ is particle number. In a charged superconductor, electromagnetism is dynamical, so the would-be Goldstone mode is reorganized by the Anderson—Higgs mechanism.

The BCS mechanism begins with a Fermi surface. An attractive interaction in the Cooper channel makes the Fermi liquid unstable. The order parameter is schematically

$$\Delta \sim \langle c_{k\uparrow}c_{-k\downarrow}\rangle.$$

The low-energy quasiparticle spectrum becomes

$$E_k = \sqrt{\xi_k^2+|\Delta|^2}, \qquad \xi_k=\epsilon_k-\mu.$$

For an $s$-wave superconductor this opens a full gap. For a $d$-wave superconductor, nodes can remain, and the low-energy excitations are not fully gapped.

Holographic superconductors are not usually BCS superconductors. The simplest holographic mechanism is a charged scalar instability near a charged horizon. Nevertheless, the condensed-matter language remains the same: identify the broken $U(1)$, the condensate, the superfluid density, the optical conductivity, vortices, and collective modes.

Quantum phase transitions

A classical thermal phase transition is driven by temperature. A quantum phase transition occurs at zero temperature as a nonthermal control parameter $g$ is tuned:

$$g=g_c.$$

Near a continuous transition, the correlation length diverges:

$$\xi\sim |g-g_c|^{-\nu}.$$

The correlation time diverges as

$$\xi_t\sim \xi^z,$$

where $z$ is the dynamical critical exponent. At finite temperature, imaginary time is compact with size

$$\beta=\frac{1}{T},$$

so temperature cuts off the quantum critical scaling. The thermal correlation length is

$$\xi_T\sim T^{-1/z}.$$

The simplest scaling form for the singular part of the free-energy density is

$$f_{\rm sing}(T,g) = T^{(d+z)/z}\,\Phi\left(\frac{g-g_c}{T^{1/(\nu z)}}\right),$$

assuming no hyperscaling violation. Therefore the entropy density at criticality scales as

$$s = -\frac{\partial f}{\partial T} \sim T^{d/z}.$$

For a relativistic CFT, $z=1$, and so

$$s\sim T^d.$$

Holographic black branes reproduce this scaling geometrically: the horizon radius sets the thermal scale. Later we will generalize this to geometries with hyperscaling violation, where

$$s\sim T^{(d-\theta)/z}.$$

Why quantum critical transport is hard

Thermodynamic scaling is only the beginning. Transport is harder because it depends on currents, conserved quantities, and relaxation mechanisms.

The conductivity is determined by a Kubo formula:

$$\sigma(\omega) = \frac{1}{i\omega} G^R_{J_xJ_x}(\omega,k=0),$$

up to possible contact terms fixed by conventions.

A critical system has no long-lived quasiparticles, but it still has conservation laws. Charge, energy, and momentum can dominate the response. In a relativistic quantum critical system at zero density, charge transport can be finite because charge current need not overlap with momentum. At finite density, charge current generally does overlap with momentum, producing the momentum bottleneck described above.

This is why holographic transport is not just “compute a black hole conductivity.” A serious calculation must say:

1. Which symmetries are present?
2. Which quantities are conserved?
3. Is momentum relaxed, and how?
4. Is the measured current coherent, incoherent, or a mixture?
5. Is the result controlled by horizon data, UV data, or both?

Mott physics: not just a large effective mass

A band insulator is insulating because all available bands are either full or empty. A Mott insulator can occur even when band theory predicts a partially filled band. The obstruction is local repulsion.

At half filling in the Hubbard model, large $U/t$ makes double occupancy expensive. Charge motion is frozen, but spin degrees of freedom can remain active. The low-energy spin exchange scale is

$$J\sim \frac{4t^2}{U}.$$

Thus a Mott insulator is not simply a bad metal with a heavy mass. It is a phase in which local constraints reorganize the Hilbert space. Doping a Mott insulator introduces mobile charge into a strongly constrained background, and this is one route to strange-metal and high-temperature-superconducting phenomenology.

Holography does not literally start from a Hubbard model. Its ultraviolet theories are usually large-$N$ gauge theories, often with matrix degrees of freedom. This mismatch matters. A responsible holographic comparison to Mott-based materials should therefore focus on robust infrared observables, not microscopic identification.

Strange metals

“Strange metal” is not a single microscopic theory. It is a name for metallic behavior that is hard to reconcile with conventional quasiparticle transport.

Common clues include:

• resistivity approximately linear in temperature over a wide range;
• broad single-particle spectra;
• absence of a clear quasiparticle peak;
• anomalous optical conductivity;
• unusual thermoelectric response;
• proximity to superconductivity, density waves, magnetism, or Mott physics;
• relaxation rates of order $k_B T/\hbar$.

A compact phenomenological statement is

$$\tau^{-1}\sim \frac{k_B T}{\hbar}.$$

This is often called Planckian dissipation. But the phrase must be used carefully. A Planckian time scale by itself does not explain a resistivity. To get resistivity, one must also know what current is being measured, what relaxes it, and how charge density, entropy density, susceptibilities, and irrelevant deformations enter.

In a simple Drude estimate,

$$\rho_{\rm dc} =\frac{m}{nq^2\tau}.$$

If $m$, $n$, and $q$ are treated as constants and $\tau^{-1}\sim T$, then $\rho\sim T$. But in a strongly coupled quantum critical metal, the Drude variables may not be meaningful. Linear resistivity can arise from several distinct mechanisms: momentum relaxation by critical modes, incoherent transport, umklapp, disorder, fluctuating order, semi-holographic baths, or horizon dynamics.

No one-line explanation

“$\rho\sim T$ because $\tau\sim \hbar/(k_B T)$” is a dimensional estimate, not a theory. A theory must identify the slow modes and the mechanism that relaxes the electric current.

Compressibility and finite density

A metal is compressible: changing the chemical potential changes the charge density. The charge susceptibility is

$$\chi = \frac{\partial \rho}{\partial \mu}.$$

Compressibility is one reason finite-density matter is more difficult than zero-density critical matter. At finite density, a system can carry current by dragging momentum. It can have Fermi surfaces, emergent gauge fields, charge-density waves, superconducting instabilities, and collective sound modes.

In holography, finite density is introduced by a bulk gauge field:

$$A_t(r\to \infty)=\mu+\cdots,$$

and the charge density is encoded in radial electric flux. Charged black branes are therefore natural gravitational models of compressible quantum matter. They are not automatically models of ordinary metals; they are controlled large-$N$ examples of finite-density states whose infrared may have no quasiparticle description.

What condensed matter teaches holography

Condensed matter gives holography discipline. It insists that a model be judged by observables, not by slogans.

Condensed-matter question	Holographic translation
Is the system compressible?	Does the bulk have electric flux or charged matter?
Are there quasiparticles?	Are there long-lived poles near the real axis, or only broad QNMs?
Is DC conductivity finite?	Is momentum relaxed, or is an incoherent current isolated?
What symmetry is broken?	Which bulk field condenses without a source?
What are the slow modes?	Which bulk fluctuations become hydrodynamic or pseudo-hydrodynamic?
What is the IR fixed point?	What is the near-horizon geometry?
What is the model status?	Is this top-down, a consistent truncation, bottom-up, or phenomenological?

A holographic model becomes useful when it answers these questions cleanly.

Common pitfalls

Pitfall 1: confusing strong scattering with finite resistivity. Strong electron-electron scattering can equilibrate a fluid without relaxing total momentum. At finite density, this gives a clean metal with infinite DC conductivity unless momentum is relaxed.

Pitfall 2: treating all linear-in-$T$ resistivity as the same phenomenon. The same exponent can emerge from different mechanisms. Exponents are clues, not fingerprints.

Pitfall 3: forgetting the lattice. Real solids are not translation invariant continua. A lattice allows Umklapp, changes momentum conservation, introduces Brillouin zones, and can stabilize commensurate order.

Pitfall 4: identifying a holographic black brane with a specific material too quickly. Holographic models are often large-$N$ quantum systems with different microscopic degrees of freedom from electrons in a solid. The safest comparisons use robust infrared structures and multiple observables.

Pitfall 5: assuming every ordered phase is weak coupling. Symmetry breaking can happen out of a strongly coupled normal state. Holographic superconductors are the canonical example.

Exercises

Exercise 1 — Drude conductivity

Starting from

$$m\frac{dv}{dt}=qE-\frac{m v}{\tau},$$

derive the optical conductivity for $E(t)=E e^{-i\omega t}$.

Solution

Use $v(t)=v(\omega)e^{-i\omega t}$. The equation becomes

$$-i\omega m v=qE-\frac{m}{\tau}v.$$

Therefore

$$\left(\frac{1}{\tau}-i\omega\right)v=\frac{q}{m}E,$$

so

$$v=\frac{q\tau}{m}\frac{1}{1-i\omega\tau}E.$$

Since $J=nqv$,

$$\sigma(\omega) =\frac{J}{E} =\frac{nq^2\tau}{m}\frac{1}{1-i\omega\tau}.$$

Exercise 2 — Why a clean finite-density metal has infinite DC conductivity

Assume a translation-invariant system at finite charge density $\rho$. Explain why exact momentum conservation generically produces an infinite DC conductivity.

Solution

In a finite-density system, the electric current generally has an overlap with the total momentum. A uniform electric field exerts a force on the charge density, so in linear response

$$\dot P_x=\rho E_x.$$

If $P_x$ is exactly conserved in the absence of the external field, there is no internal mechanism that can dissipate the momentum injected by the field. The current component proportional to momentum therefore grows without bound in time. In frequency space, this appears as

$$\sigma(\omega) \supset \frac{\rho^2}{\chi_{PP}}\frac{i}{\omega}.$$

By the Kramers—Kronig relation, the imaginary pole corresponds to a delta function in the real part:

$$\operatorname{Re}\sigma(\omega) \supset \pi\frac{\rho^2}{\chi_{PP}}\delta(\omega).$$

Thus the DC conductivity is infinite. Strong interactions can make local equilibration fast, but they cannot relax a conserved total momentum.

Exercise 3 — Fermi-liquid entropy scaling

Use the fact that only fermions within energy $T$ of the Fermi surface are thermally excited to estimate the low-temperature entropy density of a Fermi liquid.

Solution

Let $N(0)$ be the density of states per unit volume at the Fermi surface. The number of thermally active states is of order

$$N(0)T.$$

Each active state contributes an entropy of order one in units where $k_B=1$. Hence

$$s\sim N(0)T.$$

More carefully, for a conventional Fermi liquid,

$$c_V=\gamma T,$$

with $\gamma$ proportional to the renormalized density of states at the Fermi surface. Since $c_V=T\partial s/\partial T$, this also gives $s\propto T$.

Exercise 4 — Quantum critical entropy

Assume a scale-invariant quantum critical point in $d$ spatial dimensions with dynamical exponent $z$ and no hyperscaling violation. Use dimensional analysis to derive

$$s\sim T^{d/z}.$$

Solution

At the critical point there is no intrinsic length scale. Temperature sets the thermal correlation length

$$\xi_T\sim T^{-1/z}.$$

A correlated thermal volume has size

$$\xi_T^d\sim T^{-d/z}.$$

The entropy per correlated volume is order one, so the entropy density scales as the inverse correlated volume:

$$s\sim \xi_T^{-d}\sim T^{d/z}.$$

Equivalently, the free-energy density has dimension energy per volume, so

$$f\sim \xi_T^{-(d+z)}\sim T^{(d+z)/z},$$

and therefore

$$s=-\frac{\partial f}{\partial T}\sim T^{d/z}.$$

Exercise 5 — Mott insulator versus band insulator

A half-filled single band would be metallic in noninteracting band theory. Explain how strong local repulsion can nevertheless make it insulating, and identify the low-energy scale for spin exchange in the large-$U$ Hubbard model.

Solution

At half filling, every lattice site has on average one electron. In the large-$U$ Hubbard model, moving an electron to a neighboring occupied site creates a doubly occupied site and an empty site, costing energy $U$. Charge motion is therefore suppressed even though band theory would predict a metal.

The remaining low-energy degrees of freedom are spins. Virtual hopping processes are still possible: an electron can hop to a neighbor and back, temporarily paying the energy cost $U$. Second-order perturbation theory in $t/U$ gives the antiferromagnetic exchange scale

$$J\sim \frac{4t^2}{U}.$$

Thus the Mott insulator has a charge gap controlled by $U$ but spin dynamics controlled by the much smaller scale $J$.

Further reading

For a condensed-matter-facing introduction to holographic quantum matter, see Zaanen, Liu, Sun, and Schalm, Holographic Duality in Condensed Matter Physics. For the transport and non-quasiparticle perspective used throughout this group, see Hartnoll, Lucas, and Sachdev, Holographic Quantum Matter. For standard condensed-matter background, the natural companions are Landau Fermi-liquid theory, Ginzburg—Landau theory, BCS theory, and quantum critical scaling.