Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Frustration-free free fermions and beyond

Rintaro Masaoka, Seishiro Ono, Hoi Chun Po, Haruki Watanabe

Abstract

Frustration-free Hamiltonians provide pivotal models for understanding quantum many-body systems. In this paper, we establish a general framework for frustration-free fermionic systems. First, we derive a necessary and sufficient condition for a free fermion model to be frustration-free. In the case of translation-invariant, noninteracting systems, we show that any band touching between the valence and conduction bands is at least quadratic. Furthermore, by extending the Gosset-Huang inequality to fermionic systems, we demonstrate that even in interacting and non-translation-invariant cases, the finite-size gap of gapless excitations scales as $O((\log L)^2/L^2)$, provided the ground-state correlation function exhibits a power-law decay. Our results provide a foundation for studying frustration-free fermionic systems, including flat-band ferromagnetism and $η$-pairing states.

Frustration-free free fermions and beyond

Abstract

Frustration-free Hamiltonians provide pivotal models for understanding quantum many-body systems. In this paper, we establish a general framework for frustration-free fermionic systems. First, we derive a necessary and sufficient condition for a free fermion model to be frustration-free. In the case of translation-invariant, noninteracting systems, we show that any band touching between the valence and conduction bands is at least quadratic. Furthermore, by extending the Gosset-Huang inequality to fermionic systems, we demonstrate that even in interacting and non-translation-invariant cases, the finite-size gap of gapless excitations scales as , provided the ground-state correlation function exhibits a power-law decay. Our results provide a foundation for studying frustration-free fermionic systems, including flat-band ferromagnetism and -pairing states.

Paper Structure

This paper contains 47 sections, 1 theorem, 191 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. U(1)-symmetric free fermionic systems
  3. Setting
  4. Condition for frustration-freeness
  5. Translation invariant models
  6. Flat bands
  7. Frustration-freeness in momentum space
  8. Examples
  9. Ladder
  10. Checkerboard lattice
  11. Kagome lattice
  12. Pyrochlore lattice
  13. U(1) breaking models
  14. Derivation of BdG form
  15. Frustration-free condition
  16. ...and 32 more sections

Key Result

Theorem 1

Let $\hat{H}$ be a frustration-free Hamiltonian with the decomposition $\hat{H} = \sum_{\bm{R} \in \Lambda} \hat{H}_{\bm{R}}$ where each $\hat{H}_{\bm{R}}$ is a positive semidefinite operator Let $\hat{G}$ be the orthogonal projector onto the ground space and $\epsilon$ be the spectral gap. For oper The definitions of the maximum degree $g$, the chromatic number $c$, and the distance $D(\hat{\math

Figures (6)

  • Figure 1: (a) Lattice structure of the ladder. (b--e) Local orbitals and the band structure for the TB models in Eq. \ref{['Ladder1']} [(b)], Eq. \ref{['Ladder2']} [(c)], Eq. \ref{['Ladder3']} [(d)], and Eq. \ref{['Ladder4']} [(f)]. White (gray) dots represent positive (negative) coefficients.
  • Figure 2: (a) The checkerboard lattice. (b) Local orbitals $\hat{\psi}_{\bm{R}}$ in Eq. \ref{['CBpsi']} and $\hat{w}_{\bm{R}}$ in Eq. \ref{['CBw']}. (c) The dispersion relation with the additional term in Eq. \ref{['CBaddition']} for $t=1$ and $\tilde{\nu}=2$.
  • Figure 3: (a) The Kagome lattice. (b) The local orbitals $\hat{\psi}_{\bm{R}1}$ and $\hat{\psi}_{\bm{R}2}$ in Eqs. \ref{['Kagomepsi1']} and \ref{['Kagomepsi2']}. (c) The hexagonal state $\hat{w}_{\bm{R}}$ in Eq. \ref{['Kagomew']}. (d) The band dispersion for $t=\tilde{\nu}=1$.
  • Figure 4: The pyrochlore lattice.
  • Figure 5: (a) Quasi-orbitals for conduction bands ($\rho^{(1,2,3)}\coloneqq\sum_{\bm{k}}\rho_{\bm{k}}^{(1,2,3)}e^{i\bm{k}\cdot\bm{r}}$) and for valence bands ($\sigma^{(\pm)}\coloneqq\sum_{\bm{k}}\sigma_{\bm{k}}^{(\pm)}e^{i\bm{k}\cdot\bm{r}}$). (b) The band dispersion of $H_{\bm{k}}=H_{\bm{k}}^{(+)}+H_{\bm{k}}^{(-)}$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1: Gosset and Huang gossetCorrelationLengthGap2016