Phase transition and critical behavior of thed=3 Gross-Neveu model

Abstract

A second order phase transition for the three dimensional Gross-Neveu model is established for one fermion species N = 1. This transition breaks a parity-like discrete symmetry. It constitutes its peculiar universality class with critical exponent ν = 0.63 and scalar and fermionic anomalous dimension ησ = 0.31 and η ψ = 0.11, respectively. We also compute critical exponents for other N . Our results are based on exact renormalization group equations.An understanding of systems with many fermionic degrees of freedom is one of the big challenges in statistical physics. Due to the anticommuting nature of the variables numerical simulations are not straightforward-analytical methods are crucially needed. One typically has to solve a functional integral for a d dimensional system with Grassmann variables. Approximate solutions for "test models" would be of great value. The Gross-Neveu (GN) model [1] is one of the simplest fermionic models. In three dimensions a discrete symmetry forbids a mass term unless it is spontaneously broken. In the symmetric phase the GN model is therefore a realization of a statistical system of gapless fermions. For a large number N of fermion species it is known [2-5] that a second order phase transition separates the symmetric phase from an ordered phase where the symmetry is spontaneously broken and the fermions become massive. Using methods based on an exact renormalization group equation [6] a second order transition for N ≥ 2 was confirmed. We know, however, of no previous work which clarifies the existence and nature of the phase transition in the simplest model with only one fermion species. The model with one fermion species is inaccessible to lattice simulations due to the fermion doubling problem and the 1/N expansion is not expected to give reasonable results for N = 1. The case N = 1 is also of special interest since an order parameter ψ j ψ j = 0 leads to a ground state which does not admit any discrete symmetry involving the reflection of all coordinates, in contrast to the models with N ≥ 2.In this letter we improve the exact renormalization group approach and establish a second order phase transition for N = 1. We also compute the critical exponents. This is important beyond a possible relevance for real physical systems: the GN model constitutes a peculiar universality class due to the presence of massless fermions at the critical point. Just as the O(N )-Heisenberg models for bosons, the GN model could in the future become a benchmark for our understanding of critical systems in presence of fermions.The GN model describes N fermionic fields with local quartic interaction. Here ψj, j = 1...N , are irreducible representations of the group O(d) including parity reflections, i. e. 2 d/2 component Dirac spinors for d even and 2 (d−1)/2 for d odd.The classical Euclidean actionis symmetric under a coordinate reflection ψ(x) → −ψ(−x) ,ψ(x) →ψ(−x). (We note that ψ andψ are independent variables in an Eucidean formulation.) A nonvanishing expectation value ofψj ψ j spontaneously bre...

Critical behavior of the (2+1)-dimensional Thirring model

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“…For reviews see [70][71][72][73][74][75][76] and [77][78][79] for a particular emphasis on fermionic systems. Prominent benchmark examples in three dimensions are bosonic O(N ) models [70,80,81] or the GrossNeveu model [55,82,83].…”

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