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Quasi-local Frustration-Free Free Fermions

Shunsuke Sengoku, Hoi Chun Po, Haruki Watanabe

Abstract

Recent studies have revealed that frustration-free models, expressed as sums of finite-range interactions or hoppings, exhibit several properties markedly different from those of frustrated models. In this work, we demonstrate that, by relaxing the finite-range condition to allow for exponentially decaying hoppings, one can build gapped frustration-free systems that realize Chern insulators as well as quasi-degenerate ground states with finite-size splittings. Moreover, by permitting power-law decaying hoppings, we also construct a gapless band metal whose finite-size gap scales inversely with the system size $L$. These findings serve as an important step toward clarifying the general properties of frustration-free systems and those represented by tensor network states.

Quasi-local Frustration-Free Free Fermions

Abstract

Recent studies have revealed that frustration-free models, expressed as sums of finite-range interactions or hoppings, exhibit several properties markedly different from those of frustrated models. In this work, we demonstrate that, by relaxing the finite-range condition to allow for exponentially decaying hoppings, one can build gapped frustration-free systems that realize Chern insulators as well as quasi-degenerate ground states with finite-size splittings. Moreover, by permitting power-law decaying hoppings, we also construct a gapless band metal whose finite-size gap scales inversely with the system size . These findings serve as an important step toward clarifying the general properties of frustration-free systems and those represented by tensor network states.
Paper Structure (21 sections, 94 equations, 5 figures, 1 table)

This paper contains 21 sections, 94 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formulation
  3. Tight-binding model
  4. Frustration-free decomposition
  5. Exponential localization
  6. Gapless model with $O(1/L)$ finite-size gap
  7. One-band model
  8. Frustration-free decomposition
  9. Chern insulator
  10. Qi--Wu--Zhang model
  11. Frustration-free decomposition
  12. Finite-size splitting of quasi-degenerate ground states
  13. Su--Schrieffer--Heeger model
  14. Frustration-free decomposition
  15. Relation to alternative approach
  16. ...and 6 more sections

Figures (5)

  • Figure 1: Results for the one-band model in Eq. \ref{['ex1']} with $t=1$. (a) The band dispersion $\epsilon_k=-2\cos k$ for $L=42$. (b) The finite-size gap $\epsilon_{k_*}=2\sin(\pi/L)$ with $k_*\coloneqq\frac{\pi}{2}+\frac{\pi}{L}$. The solid line indicates $2\pi/L$. (c) The plot of $(\sqrt{- h^{(-)}})_{i,j}$ as a function of $i-j$ for $L=1002$. The solid curve represents $(4\pi)^{-1/2}|i-j|^{-3/2}$ for comparison.
  • Figure 2: Results on the Qi-Wu--Zhang model in Eq. \ref{['Chern']} with $t=1$. (a) The band gap $\Delta$ as a function of $m$. (b) $A$ in Eq. \ref{['Achern']} as a function of $m$. Orange dots represent values obtained by fitting data for $L=201$ and $0<m<3$. (c,d) The plot of $\|(\sqrt{h^{(+)}})_{\bm{0},\bm{R}}\|\coloneqq\sqrt{\sum_{\sigma,\sigma'}|(\sqrt{h^{(+)}})_{\bm{0}\sigma,\bm{R}\sigma'}|^2}$ as a function of $\bm{R}=(x,y)$ for $m=1$ and $L=101$. The line in (d) represents $e^{-A|\bm{R}|}$ with $A=\log2$.
  • Figure 3: Results for the Su--Schrieffer--Heeger model in Eq. \ref{['SSH']} with $t=2$ and $\mu=1$ (a,b) Illustration of the solutions of Eq. \ref{['eqkn']} and Eq. \ref{['eqlambda']} for $L=6$. (c,d) The plot of $\|(\sqrt{h^{(+)}})_{\bm{0},\bm{R}}\|\coloneqq\sqrt{\sum_{\sigma,\sigma'}|(\sqrt{h^{(+)}})_{i\sigma,j\sigma'}|^2}$ [$(i,\sigma)=(1,1)$ for (c) and $(i,\sigma)=(50,2)$ for $(d)$] as a function of $j$ for $L=101$. The blue (orange) dots correspond to $\sigma'=1$ ($\sigma'=2$). The slopes of the solid lines are $\pm\log(t/\mu)$.
  • Figure 4: Eigenvalues of $\hat{H}_i$ (gray) and $\hat{\tilde{H}}_i$ (blue) for $L=12$ obtained by exact diagonalization. Panel (a) shows results for $B=2$, while panel (b) corresponds to $B=0.5$. The inset plots the minimum eigenvalue of $\hat{\tilde{H}}_i$ as a function of $L$.
  • Figure 5: The minimum eigenvalue of $\hat{\tilde{H}}_i$ at $B=2$ obtained by mapping to free fermions by the Jordan-Wigner transformation. Orange (blue) points correspond to even (odd) fermion parity. The inset is for larger values of $L$.