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Abstract
I discuss aspects of higherorder topological field theory of crystalline insulators with no other symmetries. I show how the topology and geometry of the crystalline lattice is organized in terms of socalled elasticity tetrads which are ground state degrees of freedom labelling translational lattice topological charges, higherform conservation laws and responses on subdimensional manifolds of the bulk insulator. The quasitopological responses obtained in this way depend on the lattice and its embedding in space, as expected for weak topology. In a topological crystalline insulator, they classify higherorder responses and global symmetries in a transparent fashion in generic dimensions. This hierarchy coincides with the dimensional hierarchy of topological terms, the multipole expansion, and anomaly inflow, related to a mixed number of elasticity tetrads and electromagnetic gauge fields. In the continuum limit of the elasticity tetrads, the semiclassical expansion in momentum space can be used to derive the higherorder or subdimensional topological responses to local U(1) symmetries, such as electromagnetic gauge fields, with explicit formulas for the higherorder quasitopological invariants in terms of the elasticity tetrads and Green's functions. The topological responses in arbitrary dimensions are readily generalized to parameter space to allow for e.g. multipole pumping. The simple results further bridge the recently appreciated connections between topological field theory, higherform symmetries and gauge fields and their relation to fractonic excitations and topological defects with restricted mobility in the elasticity of crystalline insulators.
Original language  English 

Article number  169139 
Number of pages  21 
Journal  Annals of Physics 
Volume  447 
Early online date  2022 
DOIs  
Publication status  Published  Dec 2022 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Effective field theory
 Elasticity
 Generalized global symmetries
 Higherorder topology
 Topological crystalline insulator
 Topological multipole crystalline insulator
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 1 Finished

TOPVAC: From Topological Matter to Relativistic Quantum Vacuum
Volovik, G., Nissinen, J., Eltsov, V., Rysti, J., Rantanen, R., Mäkinen, J., Zhang, K., Laurila, S., Volard, M. & Kamppinen, T.
01/10/2016 → 30/09/2022
Project: EU: ERC grants