[Paper Review] Two roads to hydrodynamic effective actions: a comparison

Felix M. Haehl, R. Loganayagam+1 more|ePrints Soton (University of Southampton)|Jan 26, 2017

Lattice Boltzmann Simulation StudiesEngineering43 references21 citations

TL;DR

This paper compares two modern approaches to constructing hydrodynamic effective field theories: one by Haehl, Loganayagam, and Rangamani (HLR), and another by Crossley, Glorioso, and Liu (CGL). Both frameworks describe hydrodynamics via a Schwinger-Keldysh-type doubled field theory, with a key result being the identification of the same dissipative effective action despite different derivations—highlighting a universal structure for non-dissipative and dissipative transport in hydrodynamics.

ABSTRACT

We make a detailed comparison between two attempts in recent years to construct hydrodynamic effective actions: we compare our work [1-7] with that of Crossley-Glorioso-Liu [8] and Glorioso-Liu [9]. The general philosophy espoused by the two approaches has a degree of overlap, despite various differences. We will try to outline the similarities to eke out the general lessons that have been uncovered, hoping that it will ease the access to the subject for interested readers.

Motivation & Objective

• To compare the distinct formalisms of HLR and CGL for constructing hydrodynamic effective field theories.
• To identify common structural features in their approaches despite different derivations and formalisms.
• To clarify the role of the $U(1)_{ m T}$ gauge symmetry and the $\mathbb{Z}_2$ symmetry in enforcing the second law of thermodynamics.
• To demonstrate that both frameworks yield the same effective action for hydrodynamics, particularly in the dissipative sector.
• To unify insights from anomalous transport, hydrostatic partition functions, and gravity duals into a coherent classification of hydrodynamic transport.

Proposed method

• Adopt a covariant superspace approach to construct the effective action, using $U(1)_{\rm T}$ gauge invariance to constrain couplings.
• Implement a Schwinger-Keldysh (SK) doubling of fields to describe real-time dynamics and dissipative processes.
• Identify the dissipative tensor $\bm{\eta}^{(ab)(cd)}$ as equivalent to $W^{\mu\nu,MN}$ in the CGL framework, linking it to entropy production.
• Use the $\mathbb{Z}_2$ combination of $U(1)_{\rm T}$ and CPT symmetry to probe time-reversal breaking and derive the second law.
• Match the HLR action (Eq. 4.4) with the CGL action (Eq. D24) by setting $ (i\mathring{\mathscr{F}}_{\theta\bar{\theta}}|, {\sf g}_{ab})_{\bm{\beta}} = \pounds_{\bm{\beta}}{\sf g}_{ab} $ and omitting the $U(1)_{\rm T}$ gauge field.
• Show that the $i$-factor in the second term ensures convergence of the path integral and positivity of entropy production.

Experimental results

Research questions

• RQ1 How do the HLR and CGL approaches to hydrodynamic effective actions compare in their underlying formalisms and assumptions?
• RQ2 What is the role of the $U(1)_{\rm T}$ gauge symmetry in both frameworks, and how does it relate to the second law of thermodynamics?
• RQ3 To what extent do the two approaches yield equivalent effective actions, particularly in the dissipative regime?
• RQ4 How is the $\mathbb{Z}_2$ symmetry (combination of $U(1)_{\rm T}$ and CPT) used to enforce time-reversal breaking and entropy production?
• RQ5 What is the physical significance of the SK doubling of fields in describing hydrodynamic transport, especially in the presence of anomalies?

Key findings

• The HLR and CGL frameworks yield the same effective action for hydrodynamics, with the key dissipative term matching exactly when the $U(1)_{\rm T}$ gauge field is omitted and the geometric condition $ (i\mathring{\mathscr{F}}_{\theta\bar{\theta}}|, {\sf g}_{ab})_{\bm{\beta}} = \pounds_{\bm{\beta}}{\sf g}_{ab} $ is imposed.
• The presence of the $U(1)_{\rm T}$ gauge field in HLR's formulation allows for a direct variational derivation of the dissipative entropy current.
• The $i$-factor in the second term of the HLR action ensures convergence of the path integral and enforces positivity of the dissipative tensor, linking to entropy production.
• The $\mathbb{Z}_2$ symmetry combining $U(1)_{\rm T}$ and CPT is equivalent to the dynamical KMS symmetry used by CGL to derive the second law.
• The eightfold classification of hydrodynamic transport (seven adiabatic, one dissipative) is consistent across both frameworks, with the dissipative class governed by the tensor $\bm{\eta}^{(ab)(cd)}$.
• The structure of couplings between Ret/Av and Adv/Dif fields in the SK-doubled theory is universal, revealing a deep commonality in the effective field theory description of hydrodynamics.