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A two dimensional analogue of the Luttinger model

Edwin Langmann

TL;DR

This work introduces a 2D analogue of the Luttinger model by deriving a partial continuum limit of a 2D spinless lattice fermion system with local interactions away from half filling. It separates fermions into nodal and antinodal sectors, achieving an exact bosonization treatment for the nodal part while handling the antinodal sector with mean-field methods, and identifies a finite-doping regime where the nodal-antinodal structure yields an analytically tractable, potentially exactly solvable truncation. The key result is a six-flavor nodal-antinodal model with explicit commutation relations for nodal densities and a bosonized nodal Hamiltonian that can be diagonalized; integrating out the nodal bosons produces an effective antinodal theory with a possible CDW gap. The framework connects to ARPES-inspired phenomenology and suggests a controlled route to study 2D interacting fermions beyond weak coupling, with potential extensions to more general band structures and spinful cases.

Abstract

We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.

A two dimensional analogue of the Luttinger model

TL;DR

This work introduces a 2D analogue of the Luttinger model by deriving a partial continuum limit of a 2D spinless lattice fermion system with local interactions away from half filling. It separates fermions into nodal and antinodal sectors, achieving an exact bosonization treatment for the nodal part while handling the antinodal sector with mean-field methods, and identifies a finite-doping regime where the nodal-antinodal structure yields an analytically tractable, potentially exactly solvable truncation. The key result is a six-flavor nodal-antinodal model with explicit commutation relations for nodal densities and a bosonized nodal Hamiltonian that can be diagonalized; integrating out the nodal bosons produces an effective antinodal theory with a possible CDW gap. The framework connects to ARPES-inspired phenomenology and suggests a controlled route to study 2D interacting fermions beyond weak coupling, with potential extensions to more general band structures and spinful cases.

Abstract

We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.

Paper Structure

This paper contains 9 sections, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Two dimensional $t$-$V$ model
  3. Notation
  4. Definitions
  5. Summary of results
  6. Physics motivation
  7. Derivation of the model
  8. Bosonization and partial exact solution
  9. Final remarks

Figures (1)

  • Figure 1: Division of the Brillouin zone in six regions $\Lambda^*_{r,s}$, $r=\pm$, $s=0,\pm$. The six dots mark the points ${\bf Q}_{r,s}/a$ in (\ref{['vQ']}) for filling $\nu=0.55$. The regions $\Lambda^*_{r,s}$ are equal to the six-sided polygons (for $s=\pm$) and small diamonds (for $s=0$) shifted such that the corresponding points ${\bf Q}_{r,s}/a$ coincide with the origin of the coordinate system. Shown are also the half-filled square surface (large diamond) and the assumed interacting Fermi surface for the nodal fermions (dashed lines).