## Journal of Physical Mathematics

- J. Phys. Math.
- Volume 3 (2011), 1-12.

### The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model

Paulo A. Faria da Veiga and Petrus H. R. dos Anjos

#### Abstract

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or Gross-Neveu model in $d+1$ space-time dimensions ($d=1,2,3$) and with $N$-component fermions. Let $0<\kappa\ll 0$ be the hopping parameter, $\lambda>0$ the four-fermion coupling, $m>0$ the bare fermion mass and take $s\times s$ spin matrices ($s=2,4$). Our analysis of the one and the two-particle spectrum is based on spectral representation for suitable two- and four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by $\frac{sN}{2}$ isolated and identical dispersion curves, and the mass of particles has asymptotic value order $-\ln\kappa$. The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination does hold or does not, respectively. Two-particle bound states emerge from solutions to a lattice Bethe-Salpeter equation, in a ladder approximation. Within this approximation, the $(\frac{sN}{2}-1)\frac{sN}{4}$ identical bound states have $\mathcal{O}(\kappa^0)$ binding energies at zero system momentum and their masses are all equal, with value $\approx -2 \ln \kappa$. Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties.

#### Article information

**Source**

J. Phys. Math., Volume 3 (2011), 1-12.

**Dates**

First available in Project Euclid: 29 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jpm/1359468400

**Digital Object Identifier**

doi:10.4303/jpm/P110901

**Zentralblatt MATH identifier**

1264.81221

**Subjects**

Primary: 81Qxx: General mathematical topics and methods in quantum theory 81Txx: Quantum field theory; related classical field theories [See also 70Sxx] 81Vxx: Applications to specific physical systems

**Keywords**

Quantum theory Quantum field theory

#### Citation

dos Anjos, Petrus H. R.; da Veiga, Paulo A. Faria. The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model. J. Phys. Math. 3 (2011), 1--12. doi:10.4303/jpm/P110901. https://projecteuclid.org/euclid.jpm/1359468400