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Generalized global symmetries and dissipative magnetohydrodynamics

Sašo Grozdanov, Diego M. Hofman, Nabil Iqbal

Abstract

The conserved magnetic flux of U(1) electrodynamics coupled to matter in four dimensions is associated with a generalized global symmetry. We study the realization of such a symmetry at finite temperature and develop the hydrodynamic theory describing fluctuations of a conserved 2-form current around thermal equilibrium. This can be thought of as a systematic derivation of relativistic magnetohydrodynamics, constrained only by symmetries and effective field theory. We construct the entropy current and show that at first order in derivatives, there are seven dissipative transport coefficients. We present a universal definition of resistivity in a theory of dynamical electromagnetism and derive a direct Kubo formula for the resistivity in terms of correlation functions of the electric field operator. We also study fluctuations and collective modes, deriving novel expressions for the dissipative widths of magnetosonic and Alfven modes. Finally, we demonstrate that a non-trivial truncation of the theory can be performed at low temperatures compared to the magnetic field: this theory has an emergent Lorentz invariance along magnetic field lines, and hydrodynamic fluctuations are now parametrized by a fluid tensor rather than a fluid velocity. Throughout, no assumption is made of weak electromagnetic coupling. Thus, our theory may have phenomenological relevance for dense electromagnetic plasmas.

Generalized global symmetries and dissipative magnetohydrodynamics

Abstract

The conserved magnetic flux of U(1) electrodynamics coupled to matter in four dimensions is associated with a generalized global symmetry. We study the realization of such a symmetry at finite temperature and develop the hydrodynamic theory describing fluctuations of a conserved 2-form current around thermal equilibrium. This can be thought of as a systematic derivation of relativistic magnetohydrodynamics, constrained only by symmetries and effective field theory. We construct the entropy current and show that at first order in derivatives, there are seven dissipative transport coefficients. We present a universal definition of resistivity in a theory of dynamical electromagnetism and derive a direct Kubo formula for the resistivity in terms of correlation functions of the electric field operator. We also study fluctuations and collective modes, deriving novel expressions for the dissipative widths of magnetosonic and Alfven modes. Finally, we demonstrate that a non-trivial truncation of the theory can be performed at low temperatures compared to the magnetic field: this theory has an emergent Lorentz invariance along magnetic field lines, and hydrodynamic fluctuations are now parametrized by a fluid tensor rather than a fluid velocity. Throughout, no assumption is made of weak electromagnetic coupling. Thus, our theory may have phenomenological relevance for dense electromagnetic plasmas.

Paper Structure

This paper contains 20 sections, 122 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ideal magnetohydrodynamics
  3. Coupling external sources
  4. Hydrodynamic stress-energy tensor and current
  5. First-order hydrodynamics
  6. Transport coefficients
  7. Kubo formulae
  8. Application: Dissipative Alfvén and magnetosonic waves
  9. Magnetohydrodynamics at weak field
  10. Magnetohydrodynamics at strong field
  11. Cyclotron mode
  12. Magnetohydrodynamics with a baryon number
  13. Relativistic MHD at zero temperature
  14. Effective degrees of freedom
  15. Beyond ideal hydrodynamics
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Integration over a codimension-$2$ surface $S$ counts the number of strings that cross it at a given time.
  • Figure 2: Sum over current-current insertions to compute electrical resistivity.
  • Figure 3: Example of new diagram that contributes to electrical resistivity.
  • Figure 4: Sample plots illustrating velocity squared of Alfven wave $v_A^2$ (solid black line), fast magnetosonic wave $(v_M^2)_{\mathrm{fast}}$ (dotted line), and slow magnetosonic wave $(v_M^2)_{\mathrm{slow}}$ (dashed line) as a function of angle $\theta$ between the momentum of the wave and the background magnetic field.