Momentum space entanglement of four fermion field theory

TL;DR

The paper addresses momentum-space entanglement in a relativistic four-fermion quantum field theory by extending the Wilsonian effective-action framework and replica trick to fermions. It derives the low-energy action $S_{\mu}$, separates it into local and nonlocal temporal parts, and shows that perturbative entanglement between momentum shells arises solely from the nonlocal component $S_{\mu,\text{nl}}^{n\beta}$, which is captured diagrammatically by basketball-type Feynman diagrams with new fermionic rules. The authors apply the method to a $(\bar{\psi}\psi)^2$ theory, obtaining explicit expressions for six Rényi-entropy contributions $H_n^{2,1},\ldots,H_n^{6}$ and showing how the glued action decomposes into a local part equivalent to the original action at temperature $n\beta$ plus a nonlocal remainder. This framework provides a principled, perturbative route to compute fermionic momentum-space entanglement and suggests universal structural features of entanglement diagrams across bosonic and fermionic theories, with potential for systematic higher-order analyses and links to RG flow.

Abstract

Momentum space entanglement of four fermion field theory is calculated from the Wilsonian effective action pertubatively using replica trick, local terms in low energy effective action are proved to be non-relevant pertubatively and nonlocal terms are the only source of entanglement between different momentum modes. The final result again can be represented by a set of basketball feynmann diagrams with new feynmann rules proposed to inteprete them.

Figures (18)

• Figure 1: Order $g$ contribution to the low-energy effective action: momentum conservation prevents entanglement between high- and low-momentum modes, so this term is omitted.
• Figure 2: Order $g^2$ two-point contribution to the low-energy effective action. In fermionic systems, two types of contractions of the legs are possible, as shown in Fig. \ref{['fig:contractiontype2']}.
• Figure 3: Order $g^2$ four-point contribution to the low-energy effective action, with three possible fermionic leg contractions (see Fig. \ref{['fig:contraction4type3']}).
• Figure 4: Order $g^2$ six-point contribution to the low-energy effective action, denoted by $S_{\mu}^{\beta, (6)}$.
• Figure 5: Two contractions of the order $g^2$ two-point contribution to the low-energy effective action, distinguished by Dirac indices and denoted $S_{\mu}^{\beta, (2, 1)}$ and $S_{\mu}^{\beta, (2, 2)}$ respectively.