# Holographic Condensed Matter: Superconductors, Quantum Criticality, and Model Honesty

## Overview

RU is a certificate engine for universality and obstructions in constrained nonnegative propagation. Host structures supply finite inputs. Outputs are audited certificates or finite obstruction witnesses.

## RU fit

- **Fit level:** Low

## RU instantiation hooks

- Modeling context; RU enters only where the model reduces to finite positive propagation with explicit margins.
- Use with the moduli-chamber step when parameters vary.

## Purpose

This document clarifies what “string theory applied to condensed matter” actually means in practice.

The central claim is not that string theory has produced a catalogue of laboratory predictions. The central claim is that holography provides a **controlled sandbox** for non-quasiparticle physics and strongly coupled transport, with explicit models that can be studied end-to-end.

## The problem condensed matter brings to the table

Many experimentally relevant regimes are difficult for conventional weak-coupling tools:

- Quantum critical regions without long-lived quasiparticles.
- Strongly correlated transport where Boltzmann or Fermi-liquid intuition breaks.
- Dynamics of symmetry breaking at strong coupling.

Holography contributes a calculational framework precisely in the regimes where these features are dominant.

## What holography contributes, in one sentence

Holography replaces certain strongly coupled many-body problems with classical gravitational problems in one extra dimension, allowing one to compute:

- Finite-temperature correlators
- Conductivities and diffusion
- Phase transitions with symmetry breaking

using the same real-time prescription used in high-energy applications.

## Holographic superconductors: the example example

A minimal “holographic superconductor” model couples AdS gravity to:

- A Maxwell field (a conserved U(1) current on the boundary)
- A charged scalar (an order parameter candidate)

Below a critical temperature, the scalar condenses in the bulk, corresponding to spontaneous symmetry breaking on the boundary.

This construction and its properties are reviewed in Hartnoll–Herzog–Horowitz.

### What is genuinely reducible into RU certificates here

Even when the model is not derived from a specific material, it provides:

- A fully solvable example of a strongly coupled superconducting-like phase transition.
- A clean demonstration of how linear response and conductivity emerge from bulk equations.
- A laboratory for testing general ideas about universality, scaling, and transport bounds.

### What is not honest to claim

- That the simplest holographic superconductor model predicts a particular material’s critical temperature or microscopic mechanism.
- That the dual gravity is uniquely determined for a real substance.

The right claim is methodological:

- Holography supplies solvable, strongly coupled examples that are hard to construct otherwise.

## Quantum critical transport and “holographic quantum matter”

Beyond superconductors, holography is used to model “incoherent” transport and quantum critical dynamics.

A modern broad review is Hartnoll–Lucas–Sachdev, which connects holographic transport to hydrodynamics and memory-matrix methods and emphasizes regimes without quasiparticles.

This is one of the best places to see how the holographic toolbox interfaces with mainstream condensed matter language.

## A practical model taxonomy: top-down vs bottom-up

Holographic condensed matter splits naturally into two styles.

### Top-down (string-derived) models

- Start from a specific string or supergravity construction.
- Boundary theory is identified precisely.
- Control is better, but matching a specific material is rare.

### Bottom-up (effective holography)

- Write the simplest bulk theory consistent with desired symmetries.
- Tune couplings and potentials to realize phases of interest.
- Great for exploring mechanisms, less direct as “prediction.”

Both styles are legitimate. The crucial point is to label them correctly.

## What counts as “universal host suitability for RU instantiation” here

The host suitability for RU instantiation is universal in the same sense as in finite-temperature holography:

- The same dictionary and boundary-condition prescription repeatedly produces computable, testable response functions.

That universality is methodological. It does not require that every model maps to a laboratory sample.

## Use-case cards

| Question in many-body physics | Holographic move | Output | What to be careful about |
|---|---|---|---|
| What does conductivity look like without quasiparticles? | Compute retarded current correlator from bulk gauge field | σ(ω), scaling, sum-rule behavior | Model dependence (bulk action choices) |
| How can superconducting order arise at strong coupling? | Charged scalar condenses in black brane background | Phase transition, gap formation, σ(ω) features | “Mechanism” is effective, not necessarily microscopic |
| How does quantum criticality affect transport? | Use scale-invariant geometries; compute transport by horizon methods | Temperature scaling exponents, diffusion, relaxation | Matching exponents to a given system is nontrivial |

## How this connects back to the supplied lecture notes

The lecture notes motivate AdS/CFT as an equivalence between a gravitational bulk theory and a boundary gauge theory, emphasizing the generating-functional dictionary.

Holographic condensed matter is, structurally, the same idea with different boundary questions:

- Instead of “compute a correlator to learn about a CFT,” one asks “compute a correlator to understand transport and phases.”

## References to start with

- Hartnoll, Herzog, Horowitz, “Holographic Superconductors,” arXiv:0810.1563.
- Herzog, “Lectures on Holographic Superfluidity and Superconductivity,” arXiv:0904.1975.
- Horowitz, “Introduction to Holographic Superconductors,” arXiv:1002.1722.
- Hartnoll, Lucas, Sachdev, “Holographic quantum matter,” arXiv:1612.07324.