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Ground state of one-dimensional fermion-phonon systems

Tadahiro Miyao, Seigo Okida, Hayato Tominaga

Abstract

A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion-phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range.

Ground state of one-dimensional fermion-phonon systems

Abstract

A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion-phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range.

Paper Structure

This paper contains 49 sections, 50 theorems, 260 equations, 5 figures.

Table of Contents

  1. Introduction and summary of the results derived from the theory constructed in this paper
  2. Overview
  3. Interacting fermion-phonon system
  4. Principal questions
  5. Background positivity: $\mathscr{P}_{\mathrm{b}, \Lambda}$
  6. Reflection positivity: $\mathscr{P}_{\mathrm{r}, \Lambda}$
  7. Properties of the semigroup $e^{-\beta H_{\Lambda}}$
  8. The semigroup $e^{-\beta H_{\Lambda}}$ preserves reflection positivity
  9. Ergodicity of $e^{-\beta H_{\Lambda}^{\rm F}}$
  10. Infinite chain
  11. Properties of the ground state
  12. Long-range order
  13. Remark on the generalization of the interaction term
  14. Organization
  15. Order preserving operator inequalities
  16. ...and 34 more sections

Key Result

Proposition 1.4

(Perron--Frobenius--Faris) Let $A$ be a self-adjoint positive operator on $\mathfrak{X}$. Suppose that $0\unlhd e^{-\beta A}$ w.r.t. $\mathscr{P}$ for all $\beta \ge 0$, and that $E=\inf \mathrm{spec}(A)$ is an eigenvalue. Then, the following are equivalent:

Figures (5)

  • Figure 1: The mapping $r$ represents a reflection about $j=-1/2$.
  • Figure 2: The fermions are distributed across all sites with odd indices.
  • Figure 3: Every configuration $I_e\cup I_o$ can be partitioned into clusters.
  • Figure 4: The fermions occupying ${\sf C}$ are successively annihilated in pairs. This process is repeated iteratively until no fermions remain.
  • Figure 5: The fermions present in $\sf C$ are eliminated in pairs. This process is iterated until only a single fermion remains at the site $j+r$.

Theorems & Definitions (100)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Remark 1.7
  • Corollary 1.8
  • ...and 90 more