[Paper Review] Low-lying excitations in one-dimensional lattice electron systems

Hal Tasaki|arXiv (Cornell University)|Jul 23, 2004

Quantum and electron transport phenomena3 citations

TL;DR

This paper rigorously proves that in one-dimensional lattice electron systems with non-integer $P\nu$, the infinite-volume ground state is either degenerate (due to symmetry breaking) or has gapless low-energy excitations—ruling out a unique gapped ground state. The proof uses a novel method based on charge correlation functions and a unitary twist operator to construct localized trial states with energy close to the ground state, extending the Lieb-Schultz-Mattis theorem to interacting electron systems non-perturbatively.

ABSTRACT

We consider a general one-dimensional tight-binding electron model which has a period $P$. For any filling factor $ν$ such that $Pν$ is non-integral, we prove that the model in the infinite volume limit has either a symmetry breaking or a unique ground state with gapless excitations. The proof is based on the idea of Yamanaka, Oshikawa and Affleck (cond-mat/9701141), who extended the Lieb-Schultz-Mattis argument to electron systems.

Motivation & Objective

• To extend the Lieb-Schultz-Mattis theorem to strongly correlated one-dimensional electron systems in a fully non-perturbative way.
• To resolve the gap in earlier work by constructing localized excited states that remain orthogonal to the ground state in the infinite volume limit.
• To prove that for non-integer $P\nu$, a unique gapped ground state is impossible—only symmetry breaking or gapless excitations can occur.
• To develop a general method applicable beyond systems with high symmetry, using charge correlation functions to construct low-lying excited states.

Proposed method

• Introduce a unitary twist operator $U = \exp\{2\pi i \sum_{j=0}^{LP-1} ([j/P]+1)(\hat{n}_{j,\uparrow}/L)\}$ to generate a trial excited state $\Psi = U\Phi_0$.
• Define the density operator $\hat{\rho}_L = (LP)^{-1} \sum_{j=0}^{LP-1} \hat{n}_{j,\uparrow}$ and analyze its fluctuation $f(L) = \langle(\hat{\rho}_L - \nu)^2\rangle$.
• Use the Schwarz inequality to bound $|\langle\Phi_0, \Psi\rangle|^2 \leq \langle \alpha + \beta(\hat{\rho}_L - \nu)^2 \rangle \leq \alpha + \beta f(L)$, showing $|\varepsilon|^2 \leq 1 - \delta$ for large $L$.
• Decompose $\Psi = \sqrt{1 - |\varepsilon|^2} \Psi' + \varepsilon \Phi_0$ to extract an orthogonal excited state $\Psi'$ with energy $\langle \Psi', H \Psi' \rangle \leq E_{\rm GS} + \frac{\gamma}{\delta L}$.
• Leverage the bound on energy to show the existence of low-lying excitations with finite support $L$, and conclude that either symmetry breaking or gapless excitations must occur.

Experimental results

Research questions

• RQ1 Can the Lieb-Schultz-Mattis theorem be extended to interacting one-dimensional electron systems in a non-perturbative way?
• RQ2 Is it possible to construct a localized excited state that remains orthogonal to the ground state in the infinite volume limit for electron systems with non-integer $P\nu$?
• RQ3 What are the possible low-energy behaviors of one-dimensional electron systems when $P\nu$ is not an integer?
• RQ4 Can a unique, gapped ground state exist in such systems under general interactions and short-range hopping?

Key findings

• For any one-dimensional tight-binding electron system with period $P$ and non-integer $P\nu$, the infinite-volume ground state is either degenerate (symmetry breaking) or has gapless excitations.
• A unique, gapped ground state is rigorously ruled out when $P\nu$ is not an integer, regardless of interaction strength or correlation effects.
• The method constructs a trial excited state $\Psi'$ that differs from the ground state only in a finite region of length $L$, with energy $\langle \Psi', H \Psi' \rangle \leq E_{\rm GS} + \frac{\gamma}{\delta L}$.
• The bound $|\langle \Phi_0, \Psi \rangle|^2 \leq 1 - \delta$ for some $\delta > 0$ ensures the existence of a non-trivial orthogonal excited state, even without full orthogonality.
• The result holds for general short-ranged hopping and long-ranged interactions, provided the system has period $P$ and $P\nu \notin \mathbb{Z}$.