Shear subdiffusion in non-relativistic holography

We study shear fluctuations in non-relativistic holographic systems coupled to torsional Newton-Cartan geometry, using asymptotically Lifshitz spacetimes in Einstein-Maxwell-dilaton gravity. We identify a universal subdiffusive shear mode characterized by the quartic dispersion relation $ω=-iD_4 k^4$, in sharp contrast to the conventional hydrodynamic diffusion. We derive this result analytically through a systematic higher-order matched asymptotic expansion connecting near-horizon and far-region solutions, and we verify it with direct numerical quasinormal mode calculations. Our numerics demonstrate that the first non-hydrodynamic mode is purely imaginary and gapped, following the dispersion relation $ω=-iω_0-i D k^2$, and that both the hydrodynamic and the first non-hydrodynamic modes pass through pole-skipping points. These results highlight Lifshitz holography as an efficient framework for anomalous transport in strongly coupled non-relativistic quantum matter.

💡 Research Summary

This paper investigates shear (transverse momentum) transport in strongly coupled non‑relativistic quantum field theories that are holographically dual to Lifshitz black holes in Einstein‑Maxwell‑dilaton (EMD) gravity. The bulk theory admits asymptotically Lifshitz spacetimes characterized by a dynamical exponent z > 1 and spatial dimension d = 2. The authors couple the bulk to torsional Newton‑Cartan (TNC) geometry on the boundary, which provides the appropriate non‑relativistic sources (time one‑form nμ, spatial metric σμν, and U(1) gauge field aμ) and defines the energy‑momentum complex (energy density E, energy flux Ei, momentum density Pi, stress tensor πij).

In the shear channel they consider parity‑odd linear perturbations of the metric (h_ty, h_xy) and the bulk gauge field (a_y). The coupled equations of motion are second‑order ordinary differential equations in the radial coordinate r. To isolate physical degrees of freedom they construct gauge‑invariant combinations. The standard AdS‑inspired invariants Z₁ = ω h_xy + k h_ty and Z₂ = a_y become linearly dependent for generic Lifshitz exponent z > 1, reflecting the fact that energy flux and momentum density are independent in a non‑relativistic theory. The authors therefore introduce a new invariant Z₃ = ω h_xy + k h_ty + (k s_z+2)/(2(z−1)) r^{z−2} a_y, which together with Z₂ provides a complete set of independent variables.

The analytical part employs a matched asymptotic expansion. Near the horizon they impose infalling boundary conditions and expand the solutions in powers of (r−r_h). Near the boundary they solve the equations in the Lifshitz background, identifying leading terms as sources (Z₃^(0), Z₂^(0)) and subleading terms as expectation values (Z₃^(1), Z₂^(1)). In an overlapping intermediate region the two expansions are matched, yielding a dispersion relation for the lowest hydrodynamic pole. Remarkably, the shear mode obeys a quartic (subdiffusive) dispersion relation

  ω = −i D₄ k⁴,

instead of the usual Fickian diffusion ω = −i D k². The coefficient D₄ depends on temperature, the dynamical exponent z, and the bulk parameters, but its existence is universal for any Lifshitz holographic model with z > 1. In the relativistic limit z → 1 the quartic term disappears and the standard diffusion is recovered.

To confirm the analytic result, the authors compute quasinormal modes (QNMs) numerically by solving the full radial ODEs with infalling conditions at the horizon and Dirichlet conditions at the UV boundary. The lowest QNM is purely imaginary and follows

  ω = −i ω₀ −i D k²,

representing a gapped non‑hydrodynamic mode. The next mode is the subdiffusive shear pole with the quartic dispersion. Both modes intersect special points in the complex (ω, k) plane where the retarded Green’s function becomes indeterminate – the so‑called pole‑skipping points. This phenomenon, previously observed in relativistic AdS/CFT, is shown to persist in the non‑relativistic Lifshitz setting.

The paper concludes that Lifshitz holography provides a natural and efficient framework for describing anomalous transport such as shear subdiffusion in strongly coupled non‑relativistic matter. The universal quartic shear mode arises from the interplay of broken Lorentz boost symmetry, torsional Newton‑Cartan structure, and the presence of a conserved dipole‑like quantity (momentum) that is not directly tied to energy flux. The authors suggest several extensions: exploring other values of d and z, incorporating higher‑derivative bulk terms, studying nonlinear response, and seeking experimental signatures in systems where momentum relaxation is strongly suppressed (e.g., cold atoms in optical lattices with synthetic gauge fields). Overall, the work bridges holographic techniques with contemporary condensed‑matter interest in subdiffusive dynamics, offering both analytic insight and robust numerical verification.