[Paper Review] Shear Viscosity of a Non-Relativistic Conformal Gas in Two Dimensions

Jiunn-Wei Chen, Wen-Yu Wen|arXiv (Cornell University)|Aug 4, 2008

Black Holes and Theoretical Physics13 references12 citations

TL;DR

This paper investigates the shear viscosity-to-entropy density ratio (η/s) of a two-dimensional non-relativistic conformal Fermi gas. Using effective field theory and scattering amplitude analysis, it demonstrates that η/s diverges as the two-body binding energy B → 0, indicating the system becomes effectively free. This implies the 2D unitary Fermi gas lacks a weakly coupled gravity dual, contradicting predictions from non-relativistic AdS/CFT.

ABSTRACT

The shear viscosity, eta, of a fermi gas with non-relativistic conformal symmetry in two spatial dimensions is investigated. We find that eta/s, s being the entropy density, diverges as a gas of free particles in this system. It is in contrast to the eta/s=1/(4 pi) found using non-relativistic AdS/CFT correspondence, which requires a strongly interacting CFT. It implies the unitary fermi gas in two spatial dimensions is not likely to have a weakly interacting gravity dual.

Motivation & Objective

• To analyze the shear viscosity η of a 2D non-relativistic conformal Fermi gas with tunable two-body interactions.
• To determine whether the unitary limit (B = 0) in two spatial dimensions supports a weakly coupled gravity dual as predicted by non-relativistic AdS/CFT.
• To clarify the role of conformal symmetry and scattering behavior in low-dimensional Fermi systems.
• To resolve the apparent contradiction between the η/s = 1/4π prediction from NR AdS/CFT and the free-particle-like behavior observed in 2D.
• To identify conditions under which strongly interacting non-relativistic conformal field theories may exist in 2D.

Proposed method

• Employing an effective field theory (EFT) Lagrangian with a contact interaction term $ C_0(ar{ ho})^2 $, where $ ho = ar{ ho} $ is the density operator.
• Computing the two-body scattering amplitude via one-loop bubble diagrams in dimensional regularization, with the loop integral $ I $ evaluated in $ D $-dimensional momentum space.
• Analyzing the scattering amplitude $ \mathcal{A} $ in the center-of-mass frame and identifying poles corresponding to bound states at $ E = -B $.
• Evaluating the limit $ B \to 0 $ in $ d=2 $ spatial dimensions and showing that $ \mathcal{A} \to 0 $ for $ E > 0 $, indicating non-interacting behavior.
• Using the operator-field correspondence in non-relativistic AdS/CFT to examine scalar field solutions in the bulk, focusing on the $ \nu = 0 $ case for $ d=2 $.
• Demonstrating that only the $ z^2 $-behavior solution survives in $ d=2 $, corresponding to free fermions, and ruling out the unitary limit as a distinct conformal field theory.

Experimental results

Research questions

• RQ1 What is the behavior of the shear viscosity-to-entropy density ratio $ \eta/s $ in a 2D non-relativistic Fermi gas at unitarity (B = 0)?
• RQ2 How does the scattering amplitude behave in 2D when the two-body binding energy vanishes?
• RQ3 Why does the non-relativistic AdS/CFT correspondence predict $ \eta/s = 1/4\pi $, while the actual system behaves like a free gas?
• RQ4 Can a 2D non-relativistic conformal field theory with strong interactions exist that supports a weakly coupled gravity dual?
• RQ5 What is the role of the $ \nu = 0 $ mode in the bulk scalar field solution for $ d=2 $, and how does it affect the boundary CFT?

Key findings

• In two spatial dimensions, the shear viscosity-to-entropy density ratio $ \eta/s \to \infty $ as the two-body binding energy $ B \to 0 $, indicating a free-particle-like limit.
• The scattering amplitude $ \mathcal{A} \to 0 $ for $ E > 0 $ in the $ B \to 0 $ limit, confirming the absence of interactions in the 2D unitary limit.
• The divergence of $ \eta/s $ arises because the system fails to relax to equilibrium under perturbations due to lack of momentum exchange in scattering.
• In the gravity dual, the $ \nu = 0 $ case for $ d=2 $ allows only the $ z^2 $-behavior solution, corresponding to free fermions, and excludes the unitary CFT state.
• The $ B=0 $ limit in 2D is not a distinct strongly interacting CFT but degenerates into the free fermion theory, contradicting the $ \eta/s = 1/4\pi $ prediction of NR AdS/CFT.
• Thus, the 2D unitary Fermi gas is not expected to have a weakly coupled gravity dual, as the required strongly interacting CFT does not exist in this dimension.