Electric polarization as a nonquantized topological response and boundary Luttinger theorem

Abstract

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in $d\ge 1$ dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in $2d$ the monopole is a source of $2\pi$-flux, and the polarization is determined by the crystal momentum of the $2\pi$-flux. In $3d$ the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a concrete scheme to calculate polarization in $2d$, which in principle can be applied even to strongly interacting systems. For open boundary condition, the bulk polarization leads to an altered `boundary' Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.

arXiv comment7+8 pages, 3 figures, updated discussion on magnetic translations

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• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
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• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
• Non-zero momentum requires long-range entanglement
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Higher-form Gauge Symmetries in Multipole Topological Phases
• Mixed-state quantum anomaly and multipartite entanglement
• Emergent anomalies and generalized Luttinger theorems in metals and semimetals
• Unquantized anomalies in topological semimetals
• Topological invariants based on generalized position operators and application to the interacting Rice-Mele model
• Topological polarization, dual invariants, and surface flat band in crystalline insulators
• Poynting vector controversy in axion modified electrodynamics
• Theory of polarization textures in crystal supercells
• Dimensionless physics
• Fermi Surface Symmetric Mass Generation
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• Definition and Classification of Fermi Surface Anomalies
• Topological order in interacting semimetals
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• LSM and CPT
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• Dual Magnetic and Electric Dipole Symmetry: Pseudo Angular Momentum in Parity Space and the Electric Landé $g$-Factor