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Stability of charge density waves in electron-phonon systems

Tadahiro Miyao

Abstract

We demonstrate that electron-phonon interactions enhance the stability of charge density waves in low-temperature phases of many-electron systems. Our proof method involves an appropriate application of the Pirogov--Sinai theory to electron-phonon systems. Combining our findings with existing results, we obtain rigorous information regarding the low-temperature phase diagram for half-filled electron-phonon systems.

Stability of charge density waves in electron-phonon systems

Abstract

We demonstrate that electron-phonon interactions enhance the stability of charge density waves in low-temperature phases of many-electron systems. Our proof method involves an appropriate application of the Pirogov--Sinai theory to electron-phonon systems. Combining our findings with existing results, we obtain rigorous information regarding the low-temperature phase diagram for half-filled electron-phonon systems.
Paper Structure (25 sections, 15 theorems, 217 equations, 10 figures)

This paper contains 25 sections, 15 theorems, 217 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of results
  4. Organization
  5. Basic setup and main theorem
  6. Fundamental properties of the Fock spaces
  7. Definition of the model
  8. Basic assumptions
  9. Boundary conditions
  10. Main theorem
  11. Example: Holstein--Hubbard Model
  12. Connection to contour models
  13. The strategy for the proof of Theorem \ref{['MainThm']}
  14. Contour representation of the partition function
  15. Factorization property of the contour activities
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $0<\varepsilon<W$ and $(U, \mu)\in S_{{\rm ep}, \varepsilon}$. There exist certain constantsThe constant $t_0$ depends on $d$ and $\beta_0$. Additionally, $\omega_*$ is chosen such that $\omega_* > \beta_0^{-1} \log 2$. For more details, refer to Remark GeInt and Subsection ExHH.$0<\bet

Figures (10)

  • Figure 1: Phase diagram of the ground state configurations for $H_{\rm e}^{(0)}$.
  • Figure 2:
  • Figure 3:
  • Figure 5: The colored boxes represent $\boldsymbol D$. The light gray region represents $\boldsymbol D_{c}$, and the dark gray represents $\boldsymbol D_q$. The red and green lattice points correspond to lattice points $[\boldsymbol D_{q}]$ and $[\boldsymbol D_c]$, respectively. The lattice points on the cross-section of $\boldsymbol D$ by the horizontal line represent $D^{(t)}$.
  • Figure 6: An example of $\mathbb{Y}=\{Y_1, Y_2, Y_3, Y_4\}$ in $1+1$-dimensional space-time. The gray regions represent $\mathrm{supp}Y_i\ (i=1, \dots, 4)$ in $1+1$-dimensional space-time. The red, blue, and green regions represent the ground state configurations of the classical part $H^{(0)}$. The narrower green region along with the red and blue regions represents $\mathrm{int} Y_i$. Each of the red, blue, and green regions is numbered as 1, 2, and 3, respectively. When the contour corresponding to the region floating in the blue region is denoted as $Y_1$, then $Y_2, Y_3, Y_4$ are all $3$-contours, and $\mathbb{Y}$ is a set of mutually compatible and matching contours.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2: Brief summary
  • proof
  • Remark 1.3
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • ...and 20 more