[Paper Review] Controlled non-Fermi liquids from spacetime dependent couplings

Xi Dong, Samuel McCandlish+2 more|arXiv (Cornell University)|Feb 24, 2014

Physics of Superconductivity and Magnetism26 references3 citations

TL;DR

This paper proposes a perturbatively controlled non-Fermi liquid in 3+1 dimensions by introducing spacetime-dependent Yukawa couplings of the form $ g(x) \propto |x|^\kappa $ with small $ \kappa \ll 1 $. The mechanism balances classical gradient effects against quantum corrections from fermion-boson interactions, yielding anisotropic Fermi surface deformations and a stable non-Fermi liquid phase over an exponentially large scale window, with critical behavior controlled in $ \kappa $.

ABSTRACT

We construct perturbatively controlled non-Fermi liquids in 3+1 spacetime dimensions, using mild power-law translation breaking interactions. Our mechanism balances the leading tree level effects from such gradients against quantum effects from the interaction between the Fermi surface and a critical boson. We exhibit this in a model where finite density fermions interact with a scalar field via a Yukawa coupling of the form $g(x)\propto |x|^κ$. The approximate non-Fermi liquid behavior arises in the limit of small $κ$ and persists over an exponentially large window of scales, being cut off by the regime where the coupling becomes large, or by superconducting instabilities. The translation breaking coupling introduces anisotropic deformations of the Fermi surface depending on the direction of the gradient. An extension of this mechanism to 2+1 dimensions could provide a strongly translation-breaking, but weakly coupled non-fermi liquid, something we leave for further work.

Motivation & Objective

• To develop a perturbatively controlled non-Fermi liquid phase in finite-density quantum field theories.
• To address the challenge of strong coupling in non-Fermi liquids by introducing a physical control parameter via spacetime-dependent couplings.
• To extend the epsilon expansion and large-N methods by using mild translation-breaking via $ g(x) \propto |x|^\kappa $.
• To demonstrate that such a coupling stabilizes a non-Fermi liquid with anisotropic Fermi surface deformations.
• To provide a framework for weakly coupled, strongly translation-breaking non-Fermi liquids in 2+1D, left for future work.

Proposed method

• Introduce a Yukawa coupling $ g(x) = g_0 |x|^\kappa $ with small $ \kappa $ to break translation invariance mildly.
• Use classical scaling analysis to balance the $ |x|^\kappa $ gradient term against quantum corrections from the fermion-boson interaction.
• Analyze the fermionic self-energy and bosonic self-energy using one-loop diagrams in dimensional regularization.
• Compute the fermion self-energy $ \Pi(p) $ using contour integration, yielding a non-analytic dispersion relation with $ \Pi \sim 1 - \frac{p_0}{v|\vec{p}|} \tan^{-1}(v|\vec{p}|/p_0) $.
• Evaluate the $ \phi^4 $ vertex correction via four-fermion loop diagrams, confirming vanishing correction at zero momentum.
• Use the $ SU(N) $ global symmetry to ensure stability and control the large-N limit in the analysis.

Experimental results

Research questions

• RQ1 Can a non-Fermi liquid phase be constructed in 3+1 dimensions with perturbative control?
• RQ2 How does a spacetime-dependent coupling $ g(x) \propto |x|^\kappa $ with small $ \kappa $ stabilize a non-Fermi liquid?
• RQ3 What is the role of quantum corrections from the critical boson in balancing classical gradient effects?
• RQ4 How do anisotropic Fermi surface deformations arise from directional gradients in $ g(x) $?
• RQ5 Can this mechanism be extended to 2+1 dimensions to yield a weakly coupled, strongly translation-breaking non-Fermi liquid?

Key findings

• The non-Fermi liquid behavior emerges in the limit of small $ \kappa $, with an exponentially large scale window before breakdown due to strong coupling or superconducting instabilities.
• The fermion self-energy exhibits a non-analytic form $ \Pi(p_0, \vec{p}) = \frac{g^2 k_F^2}{2\pi^2 v} \left(1 - \frac{p_0}{v|\vec{p}|} \tan^{-1}\left(\frac{v|\vec{p}|}{p_0}\right)\right) $, signaling non-Fermi liquid behavior.
• The $ \phi^4 $ coupling correction vanishes at zero external momentum, confirming no unexpected backreaction on the bosonic sector.
• Anisotropic deformations of the Fermi surface arise due to directional dependence of the gradient in $ g(x) $.
• The model exhibits a perturbative fixed point controlled in $ \kappa $, with no need for large-N or $ \epsilon $-expansion.
• The mechanism provides a physically realizable alternative to large-N or $ \epsilon $-expansion for studying non-Fermi liquids in integer dimensions.