[Paper Review] Ginzburg-Landau Type Approach to the 1+1 Gross Neveu Model - Beyond Lowest Non-Trivial Order

Anees Ahmed|arXiv (Cornell University)|Feb 25, 2018

Theoretical and Computational Physics7 references4 citations

TL;DR

This paper investigates the convergence of Ginzburg-Landau (GL) expansions in the 1+1-dimensional Gross-Neveu model by increasing the order of the GL expansion beyond the lowest non-trivial order. It demonstrates that higher-order GL expansions yield phase diagrams increasingly resembling the exact phase diagram, with the crystal phase becoming progressively more accurate and approaching the exact critical chemical potential at zero temperature as the expansion order increases.

ABSTRACT

This paper presents a case study of the effects of increasing the order of a Ginzburg-Landau type expansion, by using the well known Gross-Neveu model in 1+1 dimensions as a test case. It is found that as the order of expansion increases, the predicted phase diagram increasingly resembles the known exact phase diagram. Finally, some properties of arbitrary large order phase diagrams are examined.

Motivation & Objective

• To examine the convergence of Ginzburg-Landau-type expansions in the 1+1D Gross-Neveu model as the expansion order increases.
• To understand how higher-order GL expansions affect the accuracy of the predicted phase diagram, particularly the crystal phase.
• To analyze the asymptotic behavior of the crystal phase in the large-order limit and its approach to the exact phase structure.
• To determine whether increasing the order of the GL expansion improves the description of the phase transition between massive and crystal phases.

Proposed method

• The study uses a Ginzburg-Landau expansion of the effective action in powers of the condensate field, up to order λ²ᴺ.
• The phase diagram is derived by minimizing the grand canonical potential Ψ with respect to the condensate parameters λ and ν, which parametrize the Jacobi elliptic function solution of the gap equation.
• The analysis focuses on the asymptotic behavior of the crystal phase by solving the GL expansion equations at high orders, particularly for α₂ₙ(T,μ)=0.
• The paper derives the leading-order asymptotic scaling of the critical chemical potential μ_c as a function of the expansion order N, using the relation ε_N(μ) ∼ (1/ln μ)^(1/(N−2)).
• It examines the large-N limit of the phase boundary, showing ε_N(μ)|_{ν=1} ∼ 1/N for μ > 1/2.
• The method relies on the known exact solution of the gap equation in terms of Jacobi elliptic functions and the asymptotic properties of the associated special functions (elliptic integrals, ψ functions).

Experimental results

Research questions

• RQ1 How does increasing the order of the Ginzburg-Landau expansion affect the accuracy of the predicted phase diagram in the 1+1D Gross-Neveu model?
• RQ2 What is the asymptotic behavior of the crystal phase boundary as the expansion order N increases?
• RQ3 Does the Ginzburg-Landau approach converge to the exact phase diagram in the limit of high-order expansions?
• RQ4 Can the critical chemical potential for the massive-to-crystal phase transition be accurately captured by high-order GL expansions?
• RQ5 What is the role of the UV cutoff and renormalization in the high-order GL expansion of the Gross-Neveu model?

Key findings

• The Ginzburg-Landau expansion at order λ²ᴺ predicts a crystal phase that becomes increasingly accurate as N increases, with the phase boundary approaching the exact phase diagram.
• The critical chemical potential for the phase transition scales asymptotically as μ_c ∼ 1 / (ln μ)^(1/(N−2)) in the large-μ limit, indicating convergence toward the exact result.
• For large N, the leading-order correction ε_N(μ)|_{ν=1} scales as 1/N, suggesting improved accuracy at high orders.
• The crystal phase never touches the μ-axis at T=0, as T=0 is not a solution of α₂ₙ(T,μ)=0 for any finite μ and N≥2.
• The lower edge of the crystal phase is confined to μ>1/2, consistent with the known exact critical value μ_c=2/π≈0.6366.
• The asymptotic expansion correctly captures the qualitative structure of the phase diagram, with increasing accuracy at higher orders.