Holographic superconductivity of a critical Fermi surface

This paper derives a holographic formulation of triplet superconductivity in a compressible quantum critical metal by mapping the pairing problem from a large-$N$ Yukawa-SYK model onto a scalar field theory in an emergent AdS$_2 \otimes \mathbb{R}_2$ spacetime, where the superconducting instability corresponds to a Breitenlohner-Freedman instability equivalent to the linearized Eliashberg equations.

The Gist

Imagine you are trying to understand why a metal suddenly becomes a superconductor (a material that conducts electricity with zero resistance) when it is pushed to the very edge of a "quantum crisis."

In the world of physics, this crisis is called a Quantum Critical Point (QCP). It's like a tightrope walker teetering on the edge of a cliff. At this point, the electrons in the metal are behaving chaotically; they are losing their individual identities and becoming a "soup" of collective behavior. Usually, physicists use complex math to describe this soup, but it's so messy that it's hard to see the big picture.

This paper, by Veronika Stangier and Jörg Schmalian, does something brilliant: it translates this messy, chaotic quantum soup into a story about gravity and black holes.

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The Chaotic Dance of Electrons

Normally, electrons in a metal are like people walking through a crowded hallway. They bump into each other, but they mostly keep their own path.

• Near the Critical Point: The hallway starts shaking violently. The electrons stop walking and start dancing in a wild, synchronized, yet chaotic way. They are trying to pair up (like dance partners) to become superconductors, but the shaking makes it hard to see if they will succeed.
• The Old Way: Physicists usually try to calculate this by looking at every single electron's step. It's like trying to predict the weather by tracking every single water molecule. It's impossible to get a clear view.

2. The Solution: The "Holographic" Trick

The authors use a concept called Holography. Think of a hologram: a 2D image on a piece of plastic that, when you shine a light on it, creates a 3D image.

• The Trick: They realized that the messy, 2D "dance floor" of the electrons can be mapped onto a 3D curved space (like the inside of a black hole).
• The Extra Dimension: In this new 3D world, there is an extra dimension (let's call it "Depth"). In the real metal, this "Depth" doesn't exist physically. Instead, it represents time and the internal rhythm of the electron pairs.
  • Analogy: Imagine a 2D shadow of a spinning top. The shadow looks flat and confusing. But if you step back and look at the 3D top, you can see exactly how it spins. The "Depth" dimension in their math is like stepping back to see the 3D top.

3. The Discovery: The Black Hole Connection

When they mapped the electrons to this 3D space, something amazing happened. The math describing the electrons looked exactly like the math describing a Reissner-Nordström black hole.

• The Geometry: The space they found has a specific shape: AdS₂ ⊗ R².
  • AdS₂ (Anti-de Sitter Space): This is a curved, saddle-shaped space that acts like a "gravity well." It represents the time part of the electron dance.
  • R² (Flat Space): This is a flat sheet. It represents the space (left/right, up/down) of the metal.
• The Meaning: This tells us that while the electrons are chaotic in time (they are jittery and unpredictable), they are actually quite calm and local in space. They aren't tangled up across the whole metal; they are just jittering in place.

4. The "Instability" That Creates Superconductivity

In this new 3D gravity world, the electrons are represented by a field (a kind of wave).

• The Breitenlohner-Freedman Instability: In normal physics, if you drop a ball, it falls. But in this curved "gravity well" of the black hole, there is a rule: if the "weight" (mass) of the ball gets too light (or negative enough), it becomes unstable and starts to roll down the hill.
• The Result: When the electron pairs get "light" enough (due to the quantum critical fluctuations), they become unstable in this gravity world. They "roll down the hill" and condense into a superconducting state.
• The Translation: This "rolling down the hill" in the black hole math is exactly the same thing as the electrons deciding to pair up and become a superconductor in the real metal.

5. The Radon Transform: The Translator

How did they connect the messy electron math to the clean black hole math? They used a mathematical tool called a Radon Transform.

• Analogy: Imagine you have a blurry photo of a crowd (the electrons). You want to know the shape of a specific person in the crowd. A Radon transform is like taking a series of X-ray slices through the crowd from different angles and reconstructing the person's shape from those slices.
• In this paper, the "slices" are the different frequencies of the electron vibrations. The transform takes those slices and builds the smooth, 3D "holographic" picture.

Why Does This Matter?

1. It's Not Just Magic: For a long time, using black hole math to explain superconductors was seen as a cool trick, but nobody knew why it worked. This paper proves exactly why. It shows that the "extra dimension" in the black hole math is actually just a clever way of organizing the internal time-dynamics of electron pairs.
2. A New Map: It gives physicists a new "map" to navigate the most difficult problems in condensed matter physics. Instead of getting lost in the chaotic math of electrons, they can now look at the simpler, geometric rules of gravity.
3. Universal Truth: It suggests that the behavior of electrons in a metal at a critical point is fundamentally the same as the physics near a black hole's event horizon. The universe seems to use the same geometric rules for both the very small (electrons) and the very massive (black holes).

In a nutshell: The authors took a messy, chaotic problem of electrons trying to pair up in a metal and showed that if you look at it through the "lens" of a black hole's gravity, the chaos turns into a beautiful, predictable geometric instability. They proved that the "extra dimension" in holographic theories isn't magic—it's just a way to see the hidden rhythm of time in the quantum world.

Technical Summary

1. Problem Statement

The paper addresses the fundamental challenge of understanding superconductivity near a Quantum Critical Point (QCP) in compressible metals (systems with a Fermi surface).

• The Context: Near a QCP, strong fluctuations destroy well-defined quasiparticles while generating singular interactions. This regime, known as "quantum-critical pairing," is typically described by Eliashberg theory.
• The Gap: While holographic duality (AdS/CFT) has been successfully applied to zero-dimensional models (like the SYK model) to describe holographic superconductivity, extending this to systems with finite spatial dimensions and a Fermi surface has remained elusive.
• The Goal: The authors aim to derive a microscopic holographic formulation for superconductivity in a 2D metal at a ferromagnetic QCP. They seek to establish a direct mapping between the standard many-body Eliashberg theory and a gravitational theory in an emergent curved spacetime, clarifying the physical meaning of the extra holographic dimension.

2. Methodology

The authors employ a multi-step theoretical approach combining many-body physics with mathematical tools from integral geometry:

1. Model Definition:

   • They start with a large-$N$ Yukawa-Sachdev-Ye-Kitaev (Yukawa-SYK) model.
   • The system consists of 2D compressible fermions coupled to quantum-critical Ising-ferromagnetic bosonic fluctuations.
   • The interaction involves random all-to-all couplings in flavor space, allowing for a controlled large-$N$ expansion.

2. Bilocal Collective Fields:

   • Instead of working with fundamental fermions and bosons, the authors reformulate the problem using bilocal collective fields (Green's functions $G$ and pairing fields $F$).
   • This allows them to integrate out the microscopic degrees of freedom, resulting in an effective action solely in terms of these collective fields.

3. Gaussian Fluctuation Analysis:

   • They analyze the system around the saddle point (the normal state at the QCP).
   • They focus on Gaussian fluctuations of the pairing field ($F$) and the anomalous self-energy ($\Phi$).
   • Crucially, they expand the pairing functions in terms of angular harmonics on the Fermi surface, freezing the radial momentum dependence (fixed by the Fermi surface) and retaining only the angular and frequency dependencies.

4. Mapping to Kinematic Space:

   • The authors identify that the pairing susceptibility depends on 4 coordinates: 2 spatial momenta, 1 center-of-mass frequency, and 1 relative frequency.
   • They utilize the Radon transform, a non-local integral transform, to map the fluctuating correlation function (defined on the space of geodesics, or "kinematic space") to a scalar field in Euclidean Anti-de Sitter space ($AdS_2$).
   • They exploit the mathematical relationship between the Laplacian on hyperbolic space ($H_2 \cong AdS_2$) and its space of geodesics ($G_2$).

3. Key Contributions and Results

A. Emergent Geometry: $AdS_2 \otimes \mathbb{R}^2$

The most significant result is the derivation of an emergent spacetime geometry for the superconducting order parameter:
$$ds^2 = d\zeta^2 + \frac{d\tau^2}{\zeta^2} + k_F^2 d\mathbf{x}^2$$

• Structure: The geometry factorizes into a curved temporal sector ($AdS_2$) and a flat spatial sector ($\mathbb{R}^2$).
• Physical Interpretation:
  • The flat spatial sector reflects the fact that fermionic excitations near a metallic QCP are local in space (momentum-independent self-energy) but singular in time.
  • The curved $AdS_2$ sector encodes the critical temporal dynamics.
  • This geometry corresponds to the near-horizon limit of a Reissner-Nordström black hole, distinguishing it from Lifshitz geometries often associated with other critical systems.

B. The Holographic Map (Radon Transform)

The authors derive an explicit, non-local mapping between the microscopic Gor'kov function $F(\mathbf{k}, \omega, \epsilon)$ and the holographic scalar field $\psi(\mathbf{x}, t, \zeta)$:
$$F(\mathbf{k}, \omega, \epsilon) \propto \int_0^{c/|\epsilon|} d\zeta \, \cos\left(\omega \sqrt{(c/\epsilon)^2 - \zeta^2}\right) \psi(\mathbf{k}, \omega, \zeta)$$

• The extra holographic coordinate $\zeta$ is related to the inverse of the relative frequency ($\zeta \propto 1/|\epsilon|$).
• This coordinate encodes the internal dynamics of the Cooper pair (the relative time between electrons).

C. Equivalence of Instabilities

The paper demonstrates a rigorous equivalence between two seemingly different approaches:

1. Eliashberg Theory: The onset of superconductivity is determined by the linearized gap equation, which yields a critical pair-breaking parameter $\alpha^*$.
2. Holographic Theory: The onset of superconductivity corresponds to the Breitenlohner-Freedman (BF) instability of the scalar field in $AdS_2$.
   • The scalar field mass squared $m^2$ drops below the BF bound ($m_{BF}^2 = -1/4$).
   • The authors prove that the condition $m^2 = m_{BF}^2$ derived from the holographic mass formula is identical to the condition $\alpha = \alpha^*$ derived from the Eliashberg equations.

D. Cancellation of Singular Terms

A critical technical finding is the exact cancellation of more singular momentum-dependent terms in the pairing susceptibility.

• Naive power counting suggests a Lifshitz geometry (where space and time scale differently, $z_{ds} \neq 1$).
• However, due to the momentum-independence of the fermionic self-energy, the leading gradient terms cancel out.
• This cancellation forces the geometry to be $AdS_2 \otimes \mathbb{R}^2$ rather than a Lifshitz geometry, explaining why the spatial sector remains flat.

4. Significance

• Microscopic Foundation: The work provides the first rigorous microscopic derivation of holographic superconductivity for a system with a Fermi surface, moving beyond phenomenological models.
• Unification: It bridges the gap between Quantum Critical Eliashberg theory (condensed matter) and Holographic Duality (high-energy physics), showing they describe the same physics from different geometric perspectives.
• Physical Meaning of Holography: It clarifies that the "extra dimension" in holographic descriptions of metals is not arbitrary but encodes the internal temporal dynamics (relative time) of fluctuating Cooper pairs.
• Predictive Power: The framework allows for the calculation of holographic parameters (like the scalar mass) directly from microscopic material parameters (coupling constants, density of states), enabling controlled applications of holographic methods to real-world strongly correlated materials.

In summary, the paper establishes that the complex, non-Fermi liquid pairing physics near a ferromagnetic QCP can be exactly mapped to a scalar field theory in a factorized $AdS_2 \otimes \mathbb{R}^2$ spacetime, with the superconducting transition identified as a geometric instability (BF bound) in the emergent gravity dual.