Holographic Quantum Matter

Holographic quantum matter is the use of gauge/gravity duality to study strongly coupled many-body systems, especially regimes where ordinary quasiparticle language is absent, incomplete, or actively misleading. The subject sits at the meeting point of quantum criticality, black-hole physics, hydrodynamics, finite-density field theory, condensed matter phenomenology, and string-theoretic model building.

The prerequisite is a working knowledge of the AdS/CFT dictionary: sources and expectation values, bulk fields and boundary operators, finite-temperature black branes, retarded Green’s functions, and infalling horizon boundary conditions. The pages below develop the quantum-matter-specific material from the beginning.

A roadmap for holographic quantum matter. The path begins with quantum critical matter and dissipative horizons, then moves through finite density, metallic transport, fermions, ordered phases, translation breaking, magnetic and anomalous response, information diagnostics, nonequilibrium dynamics, and experimental interpretation.

What the subject is really about

The shortest slogan is:

$$\text{strongly coupled quantum matter} \quad\longleftrightarrow\quad \text{classical gravitational dynamics with horizons}.$$

But the slogan hides three important qualifications.

First, holography is most useful when the boundary theory has many strongly interacting degrees of freedom. The classical gravity limit is not a generic property of all quantum systems. It is a controlled limit of special large-$N$ theories, and a useful modeling tool for broader universality questions.

Second, the most robust predictions usually concern universal structures: scaling, Ward identities, thermodynamics, hydrodynamic poles, horizon regularity, IR instabilities, conserved fluxes, entropy, and causal response. These are the places where gravitational language is most trustworthy.

Third, holographic models should be read with an epistemic label:

Label	Meaning	Typical examples
Exact duality	A precise string-theoretic or field-theoretic equivalence is known or strongly evidenced.	Canonical AdS/CFT examples and controlled descendants.
Top-down model	The bulk theory descends from string theory or supergravity, but the target condensed-matter system is idealized.	Probe branes, consistent truncations, D-brane intersections.
Bottom-up model	The bulk action is engineered to capture desired field-theory structures.	Einstein—Maxwell—Dilaton models, axion lattices, simple holographic superconductors.
Phenomenological analogy	Holography provides a mechanism or scaling template rather than a microscopic dual.	Strange-metal modeling, Planckian language, experimental comparisons.

A trustworthy treatment keeps these labels visible. It is fine for a bottom-up model to be useful. It is not fine to pretend that it is a derivation of a real material.

The central physical questions

The pages are organized around six questions.

What replaces quasiparticles? In a Fermi liquid, low-energy physics is built from long-lived excitations near a Fermi surface. In many strongly coupled systems, no such long-lived particle basis exists. Holography replaces quasiparticle poles with quasinormal modes of black branes: damped collective modes whose complex frequencies control relaxation.

Why are horizons dissipative? A black-hole horizon is an absorber. In Lorentzian AdS/CFT, retarded response is computed with infalling boundary conditions at the future horizon. Boundary dissipation becomes horizon absorption.

What does finite density mean holographically? A chemical potential is a boundary value of a bulk gauge field. Charge density is radial electric flux. Charged black branes provide the simplest compressible holographic phases, and their extremal near-horizon regions often control the low-energy response.

How can a metal conduct without quasiparticles? Clean finite-density systems have a momentum bottleneck: at nonzero density, electric current overlaps with conserved momentum, so the DC conductivity is singular unless translations are broken or current has an incoherent component. Holography gives calculable examples of both mechanisms.

How do ordered phases appear? Instabilities of black branes give superconducting, superfluid, striped, helical, crystalline, and other ordered phases. The diagnostic is not just a new bulk field; it is a source-free expectation value and a thermodynamically preferred nonlinear solution.

What can be compared to experiments? The honest answer is mechanisms, scaling structures, dimensionless ratios, response functions, and universality classes—not one-to-one material derivations. Holography is strongest when it clarifies how a mechanism works at strong coupling.

Lecture-note sequence

Page	Topic	Main role
01	What Is Holographic Quantum Matter?	Conceptual entry point: quasiparticles, horizons, large $N$, and model status.
02	Quantum Critical Matter and the Holographic Dictionary	Scaling, fixed points, neutral black branes, and the zero-density starting point.
03	Horizons, Dissipation, and Quantum Critical Transport	Retarded response, infalling conditions, spectral weight, QNMs, diffusion, and viscosity.
04	Finite Density and Charged Black Branes	Chemical potential, radial flux, RN-AdS black branes, extremal throats, and fractionalized charge.
05	Einstein—Maxwell—Dilaton Backgrounds and IR Scaling Geometries	Running couplings, Lifshitz scaling, hyperscaling violation, and IR domain walls.
06	Metallic Transport without Quasiparticles	Momentum bottlenecks, incoherent conductivity, thermoelectric response, and Drude-like poles.
07	Momentum Relaxation, Lattices, Q-Lattices, and Disorder	Explicit translation breaking, memory matrix logic, axion models, Q-lattices, and horizon DC formulas.
08	Strange Metals and Non-Fermi-Liquid Transport	Linear-$T$ resistivity, Planckian language, coherent/incoherent mechanisms, and scaling caveats.
09	Holographic Fermi Surfaces and Fermionic Response	Bulk spinors, spectral functions, IR matching, $k_F$, and non-Fermi-liquid self-energies.
10	Electron Stars, Dirac Hair, and Fractionalized Charge	Backreacted fermion matter, cohesive charge, horizon charge, and Luttinger-count deficits.
11	Probe Flavor at Finite Density and DBI Transport	Flavor branes, DBI electric displacement, embeddings, zero sound, and probe-sector conductivity.
12	Holographic Superconductors and Superfluids	Charged scalar instabilities, hairy black branes, condensates, optical conductivity, and superfluid stiffness.
13	Competing Orders, Stripes, and Spatially Modulated Phases	Finite-momentum instabilities, source-free modulated vevs, phonons, pinning, and phase relaxation.
14	Magnetic Field, Hall Transport, and Topological Response	Conductivity tensors, Hall angle, magnetization currents, dyonic branes, and contact terms.
15	Anomalies, Weyl Semimetals, and Chiral Transport	Chiral anomaly, Chern—Simons terms, CME/CVE, Weyl semimetals, and consistent vs covariant currents.
16	Entanglement, Chaos, and Information Diagnostics of Quantum Matter	Mutual information, hyperscaling-violating entanglement, butterfly velocity, chaos, and diffusion diagnostics.
17	Nonequilibrium Holographic Quantum Matter	Quenches, thermalization, hydrodynamization, Kibble—Zurek dynamics, turbulence, and time-dependent order.
18	Experimental Connections and Epistemic Limits	How to compare holographic mechanisms to graphene, cuprates, pnictides, heavy fermions, and Weyl semimetals.

Suggested reading routes

For a complete route, read pages 01—18 in order. The sequence is designed so that each new page introduces one genuinely new physical layer.

For transport, read pages 01—08, then 14—15. This route covers quantum critical response, finite density, metallic transport, momentum relaxation, strange metals, Hall response, and anomalous transport.

For fermions and compressible phases, read pages 01—05, then 09—11. This route covers charged black branes, IR scaling, probe fermions, electron stars, and probe flavor.

For ordered phases, read pages 01—07, then 12—13. This route covers the general transport background, homogeneous condensates, spatial modulation, pinning, and phase relaxation.

For diagnostics and experiments, read pages 01—08, then 16—18. This route emphasizes how to interpret holographic mechanisms rather than just how to solve bulk equations.

Common notation

Symbol	Meaning
$d$	number of boundary spatial dimensions
$D=d+1$	boundary spacetime dimension
$D+1=d+2$	bulk spacetime dimension
$L$	AdS radius
$r$ or $u$	radial coordinate; convention stated page by page
$T$	temperature
$s$	entropy density
$\mu$	chemical potential
$\rho$	charge density
$\epsilon$	energy density
$p$ or $P$	pressure; momentum is written explicitly when ambiguity matters
$J^i$	electric current
$Q^i=T^{ti}-\mu J^i$	heat current, in the common relativistic convention
$\sigma,\alpha,\bar\kappa$	electric, thermoelectric, and open-circuit thermal response coefficients
$\kappa$	thermal conductivity at zero electric current
$\Gamma$	momentum relaxation rate
$z$	dynamical critical exponent
$\theta$	hyperscaling violation exponent
$G_R$	retarded Green’s function
$\omega_*$	quasinormal-mode frequency

Radial-coordinate conventions vary across the literature. Whenever the distinction matters, the page will state whether the boundary is at $r\to\infty$, $r=0$, or $u=0$.

A few equations to keep in mind

The finite-temperature neutral CFT entropy density scales as

$$s\sim T^d .$$

For a hyperscaling-violating IR geometry with spatial dimension $d$, dynamical exponent $z$, and hyperscaling exponent $\theta$,

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

At finite density with slow momentum relaxation, the electrical conductivity often has the schematic low-frequency form

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

The first term is the incoherent conductivity. The second term is the broadened momentum pole. Confusing these two mechanisms is one of the most common mistakes in interpreting holographic metals.

The Lyapunov time of maximally chaotic holographic horizons is

$$\tau_L=\frac{1}{2\pi T},$$

in units where $\hbar=k_B=1$. This is a statement about many-body chaos, not by itself a formula for electrical resistivity.

How to read the model labels

A page may use a bottom-up action such as

$$S=\int d^{d+2}x\sqrt{-g}\left[ R-\frac12(\partial\phi)^2-V(\phi)-\frac14 Z(\phi)F^2 \right].$$

This kind of model is powerful because it isolates mechanisms: running couplings, charged horizons, IR scaling, translation breaking, or instabilities. But a useful bottom-up model is not automatically a microscopic derivation of a material.

Likewise, a top-down probe-brane system may have a precise string-theoretic origin, but still describe a highly idealized large-$N$ quantum system rather than electrons in a crystal. The right question is usually not “is this material literally dual to this black brane?” but “which observable structure is controlled by the same robust mechanism?”

How the pages are written

Each page is meant to function as a lecture note. The usual rhythm is:

1. physical motivation,
2. boundary formulation,
3. bulk model,
4. dictionary and observables,
5. controlled limits and caveats,
6. worked example,
7. common pitfalls,
8. exercises with solutions,
9. further reading.

The exercises are meant to test understanding, not just algebra. Many ask you to distinguish similar-looking mechanisms: Drude transport versus incoherent transport, explicit versus spontaneous translation breaking, superfluid stiffness versus a momentum-induced delta function, and Hall response versus anomaly-induced response.

Common pitfalls

Do not equate “black brane” with “real material.” A black brane is a calculational description of a large-$N$, strongly coupled state. Material relevance requires additional arguments.

Do not read linear-$T$ resistivity as a mechanism. It is a scaling behavior. Different mechanisms can produce it.

Do not call every non-Fermi-liquid spectral function a strange metal. Spectral response and transport are related but distinct.

Do not confuse finite density with finite DC conductivity. Exact translations at finite density generally produce a singular DC response.

Do not treat an IR scaling metric as a complete state. It must be embedded into a full geometry with UV boundary conditions and a sensible thermodynamic ensemble.

Do not forget magnetization currents. In magnetic field, local currents are not automatically transport currents.

Exercises

Exercise 1: Why horizons are central

Explain why a future horizon naturally computes retarded rather than advanced response.

Solution

A retarded Green’s function describes causal response: the system responds after a source is applied. In the bulk, this is implemented by allowing perturbations to fall through the future horizon. Nothing classically emerges from the future horizon. An outgoing condition would instead correspond to radiation coming out of the horizon and would not describe ordinary causal dissipation in the thermal state.

Exercise 2: Momentum bottleneck

Assume a finite-density system with exact translations. Explain why the DC conductivity can be infinite even if the system has no quasiparticles.

Solution

At finite density, the electric current overlaps with total momentum. If translations are exact, total momentum is conserved. A homogeneous electric field injects momentum into the system, and there is no mechanism to relax it. The resulting persistent acceleration gives a zero-frequency delta function in $\mathrm{Re}\,\sigma(\omega)$ and an $i/\omega$ pole in $\mathrm{Im}\,\sigma(\omega)$. This argument uses conservation laws, not quasiparticles.

Exercise 3: Scaling of entropy

For an IR geometry with entropy density $s\sim T^{(d-\theta)/z}$, what value of $\theta$ gives $s\sim T$ when $d=2$ and $z=1$?

Solution

Set

$$\frac{d-\theta}{z}=1.$$

For $d=2$ and $z=1$, this gives $2-\theta=1$, hence

$$\theta=1.$$

This value is also special because $\theta=d-1$ is associated with logarithmic violations of the entanglement area law, often interpreted as Fermi-surface-like behavior.

Exercise 4: What counts as evidence?

A holographic model gives $\rho_{\rm DC}\sim T$ and an optical conductivity with no narrow Drude peak. Is that enough to identify the model with the cuprate strange metal?

Solution

No. Those are suggestive features, not a microscopic identification. One should compare a network of observables: thermodynamics, charge susceptibility, Hall response, magnetoresistance, optical spectral weight, momentum relaxation, scaling regimes, superconducting instabilities, and material-specific symmetries. A holographic model may provide a mechanism for non-quasiparticle transport, but matching one or two exponents is not enough to identify a real material.

Further reading

Good external references include:

• Sean A. Hartnoll, Andrew Lucas, and Subir Sachdev, Holographic Quantum Matter.
• Jan Zaanen, Yan Liu, Ya-Wen Sun, and Koenraad Schalm, Holographic Duality in Condensed Matter Physics.
• Martin Ammon and Johanna Erdmenger, Gauge/Gravity Duality: Foundations and Applications.
• Makoto Natsuume, AdS/CFT Duality User Guide.
• Subir Sachdev, Quantum Phase Transitions.

The reading order above is designed so that these references become progressively easier to parse.