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arXiv:2607.20944

Gaplessness indicator by topologically trivial twisting operators

Y Yao, Linhao Li, Feng-Feng Song

Published Jul 23, 2026• 0 citations
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Abstract

We propose several general necessary conditions for quantum many-body system in one dimension respecting U(1) symmetry to be gapped. We show that the ground-state expectation value of topologically trivial twisting operators must approach unity in the thermodynamic limit with a certain finite-size scaling. Equivalently, its violation can indicate gaplessness of U(1)-symmetric Hamiltonians. The topological triviality of such a twisting operator enables us to derive infinitely many other gaplessness indicators by static structure factor to any order in real experiments, which are impossible to obtain by earlier topologically nontrivial twisting operators. We also apply analytic and numerical calculations to test the efficiency and consistency of our results.

DOI
10.48550/arxiv.2607.20944

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We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.
The expectation value of the twist operator has been widely used as a polarization-based index for gapped and gapless phases in interacting quantum many-body systems. Although numerous studies support this usage in specific settings and rigorous results have established the validity of the criterion in important settings, the precise assumptions required for it to sharply distinguish gapped and gapless phases under more general conditions have not been fully clarified. In this work, we clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions. We identify the role of ground-state degeneracy in the statements for gapped systems and clarify the distinct assumptions required to exclude gapless scenarios that could otherwise mimic gapped behavior in the thermodynamic limit. Building on this controlled framework, we construct a meaningful extension beyond one dimension, emphasizing that such an extension is nontrivial and cannot be obtained by a straightforward generalization of the one-dimensional twist operator. Our results delineate the regime in which polarization-based quantities are justified as sharply defined many-body indices.
The equivalence of topological indices for U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​-symmetric quantum spin chains
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We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.