Why Cooper pairs live in AdS2: a spectral analysis of the Yukawa-SYK model

We establish the spectral foundation of the geometric formulation of Cooper pairing in the Yukawa–Sachdev–Ye–Kitaev model. Starting from the large-N bilocal effective action, we derive the Gaussian fluctuation kernel in the even-frequency spin-singlet Cooper channel. Following the construction of Maldacena and Stanford, we determine the kernel eigenvalue k (h) analytically for arbitrary conformal weight h and resolve the pairing fluctuations into the continuous and discrete sectors of the associated de-Sitter space Laplacian. We show that the superconducting instability and the universal low-energy fluctuations near it reside entirely in the continuous scattering sector, while the discrete modes remain non-critical. Expanding the kernel about the lowest continuum mode yields a Klein–Gordon action on dS2, restricted to the continuous spectral subspace. This is precisely the subspace on which the inverse Radon transform to AdS2 is well defined. The projected bilocal Cooper-pair field can therefore be mapped onto a scalar matter field propagating in AdS2, without requiring any further spectral restriction on the bulk theory. Our results provide the missing microscopic justification for the projection implicit in earlier holographic formulations and clarify how a local bulk field emerges from the bilocal pairing theory.

The Sachdev–Ye–Kitaev (SYK) model has emerged as a paradigmatic example of a strongly interacting quantum many-body system without quasiparticle excitations [1–9]. Its combination of random all-to-all interactions of N fermion species, analytical tractability in the large-N limit, emergent conformal symmetry, and maximal quantum chaos has made it an important point of contact between condensed-matter physics, quantum information, and quantum gravity. At energies well below the microscopic interaction scale, the fermionic two-point function approaches a scale-invariant saddle point, while the dominant corrections are governed by soft reparametrization modes, whose dynamics is described by a Schwarzian action. Remarkably, the same Schwarzian theory arises as the boundary dynamics of nearly anti-de Sitter (AdS) space in Jackiw-Teitelboim (JT) gravity [10–14]. In this low-energy regime, the SYK model and two-dimensional JT gravity provide two descriptions of the same universal infrared dynamics. The SYK model therefore offers a setting in which central ideas of holography can be studied starting from an explicitly defined and microscopically controlled quantum-mechanical Hamiltonian.

For applications to quantum critical matter, a useful extension is provided by the Yukawa-SYK (Y-SYK) model [15–18]. Here, N fermionic degrees of freedom are coupled to M dynamical bosons through random Yukawa interactions. In the appropriate largeN and largeM limit, with the ratio M/N held fixed, the saddle-point equations form a closed set of Schwinger-Dyson equations. In the pairing channel, these equations take – 1 –

the form of strongly coupled Eliashberg equations that become asymptotically exact. The normal state is an incoherent non-Fermi liquid with scale-invariant fermionic and bosonic correlations, in which the singular retarded interaction mediated by the critical bosons can induce Cooper pairing and superconductivity, of course in the sense of a mean-field theory with all-to-all interactions. The model thus allows for studying superconductivity in the absence of long-lived electronic quasiparticles and, more broadly, for clarifying the relation between quantum-critical Eliashberg theory and holographic descriptions of superconducting instabilities. In addition, it has been extended to finite dimensional systems and used to analyze the interplay of quantum criticality, quantum chaos, and superconductivity in a range of systems [19–28].

A direct relation between the many-body field theory and holographic perspectives of the superconducting degrees of freedom was discussed in Refs. [29, 30]. Starting from the effective action for Gaussian fluctuations of the bilocal Cooper-pair field, it was shown that the low-energy pairing problem can be mapped onto a local Gaussian field theory in two-dimensional Euclidean anti-de Sitter space 1. In this construction, the anomalous pair field

F (τ1, τ2) = −

1

N

X

N

i=1

ci↑ (τ1) ci↓ (τ2) (1.1) is directly related to a field

Ψ (τ1, τ2) ∝ τ1 − τ2

2∆ F (τ1, τ2) (1.2) on the space of pairs of time variables, where ∆ is the fermion scaling dimension. The center-of-mass time plays the role of the boundary time, whereas the relative time—or, equivalently, its conjugate relative frequency—encodes the emergent radial direction. According to Refs. [29, 30], the Gaussian pairing theory of the Y-SYK model and the matter sector of an AdS2 holographic superconductor are not merely analogous: within the lowenergy approximation, they are related by an explicit integral transformation.

Based on this scientific paper, here are specific improvements that could be made to AI systems, categorized by the type of enhancement:

) Improvement 1: Development of a Geometric Holographic Projection Module for Quantum Many-Body Systems.

The paper establishes a rigorous mathematical framework where the low-energy physics of strongly interacting systems (like Y-SYK models) is mapped onto a geometric space (AdS2/dS2) via a projection operator derived from spectral analysis (specifically, the Radon transform).

Our results provide the missing microscopic justification for the projection implicit in earlier holographic formulations and clarify how a local bulk field emerges from the bilocal pairing theory.

The resulting field can consequently be mapped to a scalar matter field in AdS2.

This suggests an AI system could be designed to:

1. Identify complex, high-dimensional quantum many-body Hamiltonians (e.g., those described by SYK or related Yukawa models).

2. Perform a spectral analysis of the pairing kernels (like the Gaussian kernel, Eq. 3.10) to determine which modes are physically relevant for low-energy collective behavior (the continuous sector).

3. Automatically apply a projection operator derived from the continuous spectrum onto the full bilocal field space, yielding an effective local scalar field theory in AdS2 or dS2 coordinates.

) Improvement 2: Creation of a Criticality and Instability Detector for Quantum Phase Transitions.

The paper provides explicit criteria for when a system becomes superconducting (the instability condition: eigenvalue reaching unity) and precisely identifies the driving force behind the transition (the continuous scattering sector). It also quantifies the effect of external perturbations like pair-breaking strength.

The onset of superconductivity is mapped onto reaching the AdS2 Breitenlohner-Freedman stability bound.

The discrete modes remain non-critical.

We find that critical pairing fluctuations are only due to the continuous part of the spectrum.

This suggests an AI system could:

1. Analyze simulation data or theoretical models of quantum critical systems (e.g., cuprates, exotic superconductors).

2. Calculate the relevant kernel eigenvalues (like Eq. 4.15) as a function of system parameters (like pair-breaking strength, related to the parameter 'α').

3. Determine if the largest eigenvalue crosses unity, thereby predicting the onset of superconductivity or identifying where it is suppressed by disorder/pair-breaking.

) Improvement 3: Generation of Dynamical Spectral Response Models via Frequency Domain Analysis.

The paper provides a complete frequency-space description (Eqs. 6.1–6.7) that relates the pairing kernel to the spectrum of the dS2 Laplacian, allowing for the construction of degenerate dynamic solutions in both time and frequency domains (Eq. 6.6).

"The existence of this degenerate family suggests potentially rich dynamics in the time-dependent pairing response, although establishing its real-time interpretation requires, of course, an appropriate analytic continuation."

This suggests an AI system could:

1. Take a static or quasi-static description (like the low-energy action Eq. 5.10).

2. Use techniques derived from the frequency analysis to generate time-dependent responses (e.g., dynamic susceptibility, response functions) for the superconducting order parameter that include non-trivial temporal and spectral dependencies (the dynamic solutions in Eq. 6.6).

) Improvement 4: Automated Holographic Map Generation Tool for Novel Models.

The paper describes a general procedure—starting from a bilocal effective action, deriving the Gaussian fluctuation kernel, diagonalizing it using conformal Casimir operators, and mapping it to a geometric Laplacian—that is applicable to various models (Y-SYK).

This suggests an AI system could:

1. Be trained on the mathematical structures of known quantum field theories.

2. Automatically attempt to perform the spectral analysis described in Section 4 (relating kernels to Casimir operators) on a new, undefined many-body Hamiltonian input.

3. If a suitable mapping is found, it can generate the corresponding holographic dual action in AdS space for that system, effectively serving as an automated tool for exploring holographic duals of non-standard quantum models.

We establish the spectral foundation of the geometric formulation of Cooper pairing in the Yukawa--Sachdev--Ye--Kitaev model. Starting from the large- N bilocal effective action, we derive the Gaussian fluctuation kernel in the even-frequency spin-singlet Cooper channel. Following the construction of Maldacena and Stanford, we determine the kernel eigenvalue k (h) analytically for arbitrary conformal weight h and resolve the pairing fluctuations into the continuous and discrete sectors of the associated de-Sitter space Laplacian. We show that the superconducting instability and the universal low-energy fluctuations near it reside entirely in the continuous scattering sector, while the discrete modes remain non-critical. Expanding the kernel about the lowest continuum mode yields a Klein--Gordon action on dS 2, restricted to the continuous spectral subspace. This is precisely the subspace on which the inverse Radon transform to AdS 2 is well defined. The projected bilocal Cooper-pair field can therefore be mapped onto a scalar matter field propagating in AdS 2, without requiring any further spectral restriction on the bulk theory. Our results provide the missing microscopic justification for the projection implicit in earlier holographic formulations and clarify how a local bulk field emerges from the bilocal pairing theory.