[Paper Review] Stability of the U(1) spin liquid with spinon Fermi surface in 2+1 dimensions

Sung-Sik Lee|arXiv (Cornell University)|Apr 23, 2008

Advanced Condensed Matter Physics4 citations

TL;DR

This paper demonstrates that the U(1) spin liquid with a spinon Fermi surface in 2+1 dimensions is stable against instanton proliferation for any nonzero number of spinon flavors. By mapping the spinon Fermi surface to an infinite set of 1+1D chiral fermions, it is shown that the instanton scaling dimension diverges due to the infinite sum of finite contributions, rendering instantons irrelevant in the low-energy limit and confirming the non-compact U(1) gauge theory as a valid low-energy description.

ABSTRACT

We study the stability of the 2+1 dimensional U(1) spin liquid state against proliferation of instantons in the presence of spinon Fermi surface. By mapping the spinon Fermi surface into an infinite set of 1+1 dimensional chiral fermions, it is argued that an instanton has an infinite scaling dimension for any nonzero number of spinon flavors. Therefore, the spin liquid phase is stable against instantons and the non-compact U(1) gauge theory is a good low energy description.

Motivation & Objective

• To determine the stability of the U(1) spin liquid phase with a spinon Fermi surface against instanton proliferation in 2+1 dimensions.
• To resolve the open question of whether fractionalized phases with non-relativistic spinons on a Fermi surface remain stable when instantons are present.
• To provide a non-perturbative analysis of instanton dynamics in the presence of a Fermi surface, overcoming limitations of prior RPA and perturbative approaches.
• To establish that the non-compact U(1) gauge theory remains a valid low-energy description of the spin liquid phase.

Proposed method

• Map the 2+1D spinon Fermi surface to an infinite set of 1+1D chiral fermions, each labeled by the direction of their velocity (angular momentum).
• Treat the instanton as a twist operator in each 1+1D chiral fermion sector, with the scaling dimension of the instanton operator being the sum of individual scaling dimensions.
• Use conformal field theory techniques to compute the scaling dimension of the twist operator in each 1+1D fermion, yielding a contribution of $ \frac{(2\eta - 1)^2}{8} $ per fermion.
• Introduce a large-$ N $ limit to control gauge field fluctuations and suppress vertex corrections, enabling a consistent scaling analysis.
• Derive the dressed gauge propagator and fermion self-energy using one-loop Feynman diagrams from the effective action, confirming the $ |\omega|^{2/3} $ self-energy behavior.
• Ensure gauge invariance by canceling the diamagnetic term $ K $ through regularization, fixing the gauge field propagator to $ \mathcal{D}^{-1} \propto \gamma |\nu|/|q| + \chi q^2 $.

Experimental results

Research questions

• RQ1 Is the U(1) spin liquid state with a spinon Fermi surface stable against instanton proliferation in 2+1 dimensions?
• RQ2 What is the scaling dimension of the instanton operator in the presence of a Fermi surface of spinons?
• RQ3 Does the infinite sum of contributions from 1+1D chiral fermions on the Fermi surface lead to an infinite scaling dimension for the instanton?
• RQ4 Can a non-perturbative analysis of instanton dynamics be performed when conformal symmetry is broken by the non-relativistic Fermi surface?
• RQ5 Does the non-compact U(1) gauge theory remain a valid low-energy description when spinons form a Fermi surface?

Key findings

• The instanton scaling dimension diverges to infinity due to the infinite sum of finite contributions from the 1+1D chiral fermion modes on the Fermi surface.
• For any nonzero number of spinon flavors $ N $, the instanton operator is irrelevant in the low-energy limit, implying it does not proliferate.
• The non-compact U(1) gauge theory remains a valid low-energy effective theory for the spin liquid phase with a spinon Fermi surface.
• The fermion self-energy acquires a non-analytic $ \Sigma_s(\omega,k,\theta) \propto i c \text{sgn}(\omega) |\omega|^{2/3} $ dependence, indicating strong coupling effects.
• The gauge field propagator is renormalized to $ \mathcal{D}^{-1} \propto \gamma |\nu|/|q| + \chi q^2 $, with the $ \gamma |\nu|/|q| $ term arising from screening by the Fermi surface.
• The analysis confirms that the spin liquid phase is stable even for small $ N $, resolving a long-standing open question in fractionalized quantum matter.