Polarization-based indices in quantum many-body systems: validity and extension beyond one dimension

Abstract

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.

10 pages, 7 figures

References in corpus (37)

• The ITensor Software Library for Tensor Network Calculations
• Lieb-Schultz-Mattis in Higher Dimensions
• Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
• The Insulating State of Matter: A Geometrical Theory
• Variational description of Mott insulators
• Electron localization in the insulating state: application to crystalline semiconductors
• Many-Body Electric Multipole Operators in Extended Systems
• Many-Body Order Parameters for Multipoles in Solids
• Order Parameter to Characterize Valence-Bond-Solid States in Quantum Spin Chains
• mVMC - Open-source software for many-variable variational Monte Carlo method
• Strong electronic correlation in the Hydrogen chain: a variational Monte Carlo study
• Ground-state properties of the hydrogen chain: insulator-to-metal transition, dimerization, and magnetic phases
• A Multi-Dimensional Lieb-Schultz-Mattis Theorem
• Difficulties in operator-based formulation of the bulk quadrupole moment
• Lattice Twisting Operators and Vertex Operators in Sine-Gordon Theory in One Dimension
• A many-body index for quantum charge transport
• Electric polarization as a nonquantized topological response and boundary Luttinger theorem
• Inequivalent Berry phases for the bulk polarization
• Topological phase transition and ${\mathbb{Z}}_2$ index for $S=1$ quantum spin chains
• Insulator, conductor and commensurability: a topological approach
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Scaling of polarization amplitude in quantum many-body systems in one dimension
• Quantum phase transitions from analysis of the polarization amplitude
• Generic Local Hamiltonians are Gapless
• Kohn's localisation in disordered fermionic systems with and without interactions
• Spin-Chirality Separation and S_3-Symmetry Breakings in the Magnetization Plateau of the Quantum Spin Tube
• Reconstruction of the polarization distribution of the Rice-Mele model
• Two No-Go Theorems on Superconductivity
• Bosonic integer quantum Hall effect as topological pumping
• Polarization amplitude near quantum critical points
• Lieb-Schultz-Mattis theorem in higher dimensions from approximate magnetic translation symmetry
• Many-body multipole index and bulk-boundary correspondence
• Correlated Zak insulator in organic antiferromagnets
• Quantized polarization in a generalized Rice-Mele model at arbitrary filling
• Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
• Identification of gapless phases by a twisting operator
• Diagnosing energy gap in quantum spin liquids via polarization amplitude