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Chern-Simons-Fermion Vector Model with Chemical Potential

Shuichi Yokoyama

TL;DR

This work solves the SU(N) Chern-Simons–fermion vector model with a U(1) flavor chemical potential in the 't Hooft limit, revealing an exact solution for |λ|<1 with a thermal mass and a Fermi surface at $p_s = |μ|\, ext{sqrt}(1-λ^2)$. Finite-temperature results show a thermal mass parameter $c$ determined by $ ext{sqrt}(c) = |λ| \, ext{log}(2( ext{cosh} ext{(sqrt{p^2+c})} + ext{cosh} ilde μ))$, and a grand potential $G(T, μ)$ that reduces to known limits; the solution ceases to exist at |λ|=1, hinting at a phase transition. By introducing a diagonal U(1) holonomy in a U(N) setup, the authors discuss possible instabilities for |λ|>1, finding that the minimized free energy can indicate an unstable, non-conformal phase. The work highlights a puzzle regarding 3-point functions and bosonization in 3d and points to the need for 1/N corrections and operator content beyond conformal symmetry to fully understand the regime |λ|≥1.

Abstract

We study SU(N) level k Chern-Simons theories coupling to a fundamental fermion with chemical potential for U(1) flavor symmetry. We solve this system exactly in the 't Hooft limit, N,k\to\infty with λ= N/k fixed. The solution exists up to |λ|=1 for a fixed value of chemical potential. We study the system beyond |λ|=1 by considering U(N) gauge group and taking the diagonal U(1) holonomy into account. The result suggests the system becomes unstable and a phase transition happens at |λ|=1.

Chern-Simons-Fermion Vector Model with Chemical Potential

TL;DR

This work solves the SU(N) Chern-Simons–fermion vector model with a U(1) flavor chemical potential in the 't Hooft limit, revealing an exact solution for |λ|<1 with a thermal mass and a Fermi surface at . Finite-temperature results show a thermal mass parameter determined by , and a grand potential that reduces to known limits; the solution ceases to exist at |λ|=1, hinting at a phase transition. By introducing a diagonal U(1) holonomy in a U(N) setup, the authors discuss possible instabilities for |λ|>1, finding that the minimized free energy can indicate an unstable, non-conformal phase. The work highlights a puzzle regarding 3-point functions and bosonization in 3d and points to the need for 1/N corrections and operator content beyond conformal symmetry to fully understand the regime |λ|≥1.

Abstract

We study SU(N) level k Chern-Simons theories coupling to a fundamental fermion with chemical potential for U(1) flavor symmetry. We solve this system exactly in the 't Hooft limit, N,k\to\infty with λ= N/k fixed. The solution exists up to |λ|=1 for a fixed value of chemical potential. We study the system beyond |λ|=1 by considering U(N) gauge group and taking the diagonal U(1) holonomy into account. The result suggests the system becomes unstable and a phase transition happens at |λ|=1.

Paper Structure

This paper contains 6 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Note added
  3. Chern-Simons-fermion vector theory with chemical potential
  4. Finite temperature analysis
  5. Comment on instability in $|\lambda|>1$
  6. Summary and discussion

Figures (2)

  • Figure 1: The graphs of $\sqrt c$ and $-{G(T,\mu)\over NT^3}$ are shown as a function of $\lambda$ varying $\mu/T = 0,1,2,3,4$ from the bottom in Fig.\ref{['sqrtc']}, Fig.\ref{['grandpot']}, respectively. They are taking finite values in the region $0\leq\lambda <1$.
  • Figure 2: The U(1) flavor charge density, entropy density and heat capacitance normalized by $NT^2$ are plotted as a function of $\lambda$ varying $\mu/T = 0,1,2,3,4$ from the bottom in Fig.\ref{['charge']}, Fig.\ref{['entropy']}, Fig.\ref{['heatcap']}, respectively.