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The Second Sound of SU(2)

Christopher P. Herzog, Silviu S. Pufu

TL;DR

The paper develops a holographic SU(2) model in AdS5 under the probe limit to study a second-order superfluid phase transition. By exploiting an analytic zero mode at μ_c=4, the authors construct background solutions and derive the critical behavior, including the speed of second sound and the directional transport coefficients. Through systematic fluctuations and Green's-function calculations in the hydrodynamic limit, they uncover propagating second-sound modes, diffusive modes, and a London-type response that signals superconducting-like behavior, with clear Ward-identity consistency. The results are analytic and shed light on the universality of non-Abelian holographic superfluids, offering connections to vector-order systems and potential extensions to magnetic fields and broader correlators.

Abstract

Using the AdS/CFT correspondence, we calculate the transport coefficients of a strongly interacting system with a non-abelian SU(2) global symmetry near a second order phase transition. From the behavior of the poles in the Green's functions near the phase transition, we determine analytically the speed of second sound, the conductivity, and diffusion constants. We discuss similarities and differences between this and other systems with vector order parameters such as p-wave superconductors and liquid helium-3.

The Second Sound of SU(2)

TL;DR

The paper develops a holographic SU(2) model in AdS5 under the probe limit to study a second-order superfluid phase transition. By exploiting an analytic zero mode at μ_c=4, the authors construct background solutions and derive the critical behavior, including the speed of second sound and the directional transport coefficients. Through systematic fluctuations and Green's-function calculations in the hydrodynamic limit, they uncover propagating second-sound modes, diffusive modes, and a London-type response that signals superconducting-like behavior, with clear Ward-identity consistency. The results are analytic and shed light on the universality of non-Abelian holographic superfluids, offering connections to vector-order systems and potential extensions to magnetic fields and broader correlators.

Abstract

Using the AdS/CFT correspondence, we calculate the transport coefficients of a strongly interacting system with a non-abelian SU(2) global symmetry near a second order phase transition. From the behavior of the poles in the Green's functions near the phase transition, we determine analytically the speed of second sound, the conductivity, and diffusion constants. We discuss similarities and differences between this and other systems with vector order parameters such as p-wave superconductors and liquid helium-3.

Paper Structure

This paper contains 18 sections, 112 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. Critical Behavior
  4. The Background
  5. Superfluid flow
  6. The Free Energy
  7. Formal Remarks about Green's Functions
  8. Computation of two-point functions
  9. Fluctuations
  10. Pure transverse mode
  11. Transverse sound fluctuations
  12. Longitudinal sound fluctuations
  13. Transverse diffusive mode
  14. Longitudinal diffusive mode
  15. Conductivity and London Equations
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Plots of numerical results for ${g^2 \over T} {\rm Res}_{\omega = 0} {\rm Im} \sigma_{xx}$ and ${g^2 \over T} {\rm Res}_{\omega = 0} {\rm Im} \sigma_{xx}$ as functions of temperature (solid lines), as well as analytical approximations at small $\epsilon$ (dotted lines) given by equations \ref{['GotResidues']}.
  • Figure 2: The squared speeds of second sound $c_\parallel^2$ and $c_\perp^2$ as functions of the reduced temperature $T/T_c$ (solid lines) as well as analytical approximations given by eqs. \ref{['speedsoundperp']} and \ref{['speedsoundpar']} close to $T_c$ (dotted lines).