Pole-skipping without master variable and holographic superfluids

The pole-skipping is a universal property of Green’s functions at strong coupling found by the AdS/CFT duality. There is a conventional formalism of the pole-skipping, but it relies on the existence of a “master variable.” Namely, it is applicable to a system with a single field. We propose an alternative formalism that does not rely on a master variable. As an example, we study the pole-skipping of holographic superfluids. A “hydrodynamic” pole such as the diffusion pole is usually regarded as a pole-skipping point. But we point out that not all hydrodynamic poles are pole-skipping points.

💡 Research Summary

The paper addresses a limitation of the existing pole‑skipping analysis in holographic duality, namely its reliance on a single “master variable” that reduces a set of coupled bulk equations to one second‑order differential equation. While this works for simple single‑field systems, many physically relevant holographic models—such as holographic superfluids, superconductors, or any theory with several interacting bulk fields—do not admit a convenient master variable, or the choice of master variable may miss some pole‑skipping points.

To overcome this, the authors develop a general matrix formalism. Starting from (m) bulk fields each obeying a second‑order equation, they rewrite the system as (2m) first‑order equations and collect the fields and their radial derivatives into a vector (\mathbf{X}). The dynamics become (\mathbf{X}’ = M(u),\mathbf{X}), where (M) is a (2m\times 2m) matrix. Near the black‑hole horizon ((u\to1)) the matrix is expanded as
(M = M_{-1}(u-1)^{-1}+M_0+M_1(u-1)+\dots).
A Frobenius series (\mathbf{X}=(u-1)^{\lambda}\sum_{n=0}^{\infty}\mathbf{x}n (u-1)^n) is inserted. The indicial equation ((\lambda-M{-1})\mathbf{x}0=0) yields eigenvalues (\lambda=\pm i\omega/(2\pi T)) and corresponding eigenvectors; the negative sign selects the incoming mode. The recursion relation
((\lambda+n-M{-1})\mathbf{x}n = \sum{k=0}^{n-1} M_{n-1-k}\mathbf{x}_k)
generates higher‑order coefficients. A pole‑skipping point occurs when, for a particular complex frequency (\omega) and momentum (q), the coefficient vector (\mathbf{x}_n) becomes ambiguous (all components simultaneously acquire a (0/0) form). This signals that the bulk solution is not uniquely fixed by the incoming boundary condition, and consequently the dual retarded Green’s function is not uniquely defined.

The authors illustrate the method with three concrete examples.

1. Scalar field: For a minimally coupled scalar with (m^2=-4) in the 5‑dimensional Schwarzschild‑AdS background, the matrix formalism reproduces the known pole‑skipping at (\omega = -i,2\pi T) (the first “chaotic” point). The ambiguity appears already at (\mathbf{x}_1) when ((\omega,q^2)=(-i,-1/2)). Higher‑order coefficients generate the infinite tower (\omega = -i n,2\pi T).

2. Maxwell‑scalar (diffusive) mode: The coupled system of the temporal component (a_t), radial component (a_u), and a complex scalar is considered. Using the traditional master variable (a_t) yields both the hydrodynamic pole‑skipping at (\omega=0) (diffusion pole) and the chaotic point at (\omega=-i,2\pi T). Choosing instead (a_u) as the master variable reproduces only the chaotic point, missing the hydrodynamic one. This demonstrates that the master‑variable approach can overlook pole‑skipping points depending on the choice.

3. Matrix approach without a master variable: By taking (\mathbf{X}=(a_t, f a_u)^T) the authors treat both fields simultaneously. The recursion produces ambiguities at both ((\omega,q)=(0,0)) and ((\omega,q)=(-i,1/2)), confirming that the hydrodynamic pole‑skipping is present even when no single master equation exists.

Having validated the method on simple systems, the authors apply it to a holographic superfluid model (Einstein‑Maxwell‑complex scalar). In the high‑temperature (normal) phase the scalar decouples from the gauge field, and the analysis reduces to the Maxwell case, reproducing the known pole‑skipping points. In the low‑temperature (superfluid) phase the scalar and gauge perturbations are coupled. Using the matrix formalism, the authors find that the massless order‑parameter mode does not generate a new hydrodynamic pole‑skipping point at (\omega\to0). Only the universal points (\omega=0) (diffusion) and (\omega=-i,2\pi T) (chaotic) remain. This clarifies a misconception in the literature that every hydrodynamic pole automatically corresponds to a pole‑skipping point.

A crucial new step in the matrix formalism is the handling of multiple incoming eigenvectors. For an (m)-field system there are (m) independent incoming eigenvectors (\mathbf{x}{0,\alpha}). The general incoming solution is a linear combination (\mathbf{y}0=\sum{\alpha}C\alpha\mathbf{x}_{0,\alpha}). At a pole‑skipping point all higher‑order coefficients (\mathbf{y}n) must have vanishing residues, which imposes constraints on the coefficients (C\alpha) and on the momentum (q). This extra algebraic step replaces the simplicity of a single master variable and is essential for correctly identifying pole‑skipping points in multi‑field systems.

In summary, the paper provides a systematic, master‑variable‑independent framework for locating pole‑skipping points in holographic theories with multiple coupled bulk fields. It demonstrates that hydrodynamic poles are not automatically pole‑skipping points and that the new formalism can uncover the full set of pole‑skipping locations, including both chaotic and hydrodynamic ones, in complex models such as holographic superfluids. The method is expected to be broadly applicable to other condensed‑matter‑inspired holographic setups, including superconductors, anisotropic backgrounds, and models with higher‑derivative interactions.