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Holography and the Coleman-Mermin-Wagner theorem

Dionysios Anninos, Sean A. Hartnoll, Nabil Iqbal

TL;DR

The paper shows that in a holographic superconductor on a planar AdS4 black hole, infrared fluctuations of a hydrodynamic second-sound mode erase the classical order parameter phase at long distances, consistent with the Coleman-Mermin-Wagner mechanism in the holographic setting. Using a one-loop bulk calculation in the probe limit, it identifies a pole in the Green's function at ω^2 = c_s^2 k^2 and derives an infrared divergence in the order-parameter correction that leaves the magnitude finite but generates algebraic long-range order with a calculable exponent. The work develops a general Green's function formalism for coupled bulk fluctuations and demonstrates that IR quantum fluctuations in curved backgrounds can play a central role in determining the phase structure of 3+1D theories with gauged symmetries. These results suggest broad implications for IR physics in asymptotically AdS spacetimes and point toward potential BKT-like behavior near Tc and the importance of backreaction in future studies.

Abstract

In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of global symmetries is precluded by large thermal fluctuations of the order parameter. The holographic correspondence implies that analogous effects must also occur in 3+1 dimensional theories with gauged symmetries in certain curved spacetimes with horizon. By performing a one loop computation in the background of a holographic superconductor, we show that bulk quantum fluctuations wash out the classical order parameter at sufficiently large distance scales. The low temperature phase is seen to exhibit algebraic long range order. Beyond the specific example we study, holography suggests that IR singular quantum fluctuations of the fields and geometry will play an interesting role for many 3+1 dimensional asymptotically AdS spacetimes with planar horizon.

Holography and the Coleman-Mermin-Wagner theorem

TL;DR

The paper shows that in a holographic superconductor on a planar AdS4 black hole, infrared fluctuations of a hydrodynamic second-sound mode erase the classical order parameter phase at long distances, consistent with the Coleman-Mermin-Wagner mechanism in the holographic setting. Using a one-loop bulk calculation in the probe limit, it identifies a pole in the Green's function at ω^2 = c_s^2 k^2 and derives an infrared divergence in the order-parameter correction that leaves the magnitude finite but generates algebraic long-range order with a calculable exponent. The work develops a general Green's function formalism for coupled bulk fluctuations and demonstrates that IR quantum fluctuations in curved backgrounds can play a central role in determining the phase structure of 3+1D theories with gauged symmetries. These results suggest broad implications for IR physics in asymptotically AdS spacetimes and point toward potential BKT-like behavior near Tc and the importance of backreaction in future studies.

Abstract

In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of global symmetries is precluded by large thermal fluctuations of the order parameter. The holographic correspondence implies that analogous effects must also occur in 3+1 dimensional theories with gauged symmetries in certain curved spacetimes with horizon. By performing a one loop computation in the background of a holographic superconductor, we show that bulk quantum fluctuations wash out the classical order parameter at sufficiently large distance scales. The low temperature phase is seen to exhibit algebraic long range order. Beyond the specific example we study, holography suggests that IR singular quantum fluctuations of the fields and geometry will play an interesting role for many 3+1 dimensional asymptotically AdS spacetimes with planar horizon.

Paper Structure

This paper contains 14 sections, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical symmetry breaking solutions
  3. Green's functions for multiple fields
  4. Fluctuation equations in Feynman gauge
  5. The retarded Green's function
  6. Second sound mode and pole in Green's function
  7. Low frequency expansion in the far region and dispersion relation
  8. Green's function in the vicinity of the pole
  9. Washing out the order parameter
  10. General formalism for one loop corrections
  11. The IR divergence
  12. Algebraic long range order
  13. Discussion
  14. Green's function for one dimensional boundary value problem with multiple fields

Figures (1)

  • Figure 1: The classical background is a planar AdS$_4$ black hole, carrying charge and surrounded by symmetry breaking scalar 'hair'. We exhibit the divergent tadpole contribution of a hydrodynamic sound mode that randomizes the phase of the order parameter.