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Abstract
For twodimensional topological insulators, the integer and intrinsic (without external magnetic field) quantum Hall effect is described by the gauge anomalous (2+1)dimensional [(2+1)d] ChernSimons (CS) response for the background gauge potential of the electromagnetic U(1) field. The Hall conductance is given by the quantized prefactor of the CS term, which is a momentumspace topological invariant. Here, we show that threedimensional crystalline topological insulators with no other symmetries are described by a topological (3+1)dimensional [(3+1)d] mixed CS term. In addition to the electromagnetic U(1) gauge field, this term contains elasticity tetrad fields Emu(a) (r, t) = partial derivative Xmu(a) (r, t) which are gradients of crystalline U(1) phase fields Xa (r, t) and describe the deformations of the crystal. For a crystal in three spatial dimensions a = 1, 2, 3 and the mixed axialgravitational response contains three parameters protected by crystalline symmetries: the weak momentumspace topological invariants. The response of the Hall conductance to the deformations of the crystal is quantized in terms of these invariants. In the presence of dislocations, the anomalous (3+1)d CS term describes the CallanHarvey anomaly inflow mechanism. The response can be extended to all odd spatial dimensions. The elasticity tetrads, being the gradients of the lattice U(1) fields, have canonical dimension of inverse length. Similarly, if such tetrad fields enter general relativity, the metric becomes dimensionful, but the physical parameters, such as Newton's constant, the cosmological constant, and masses of particles, become dimensionless.
Original language  English 

Article number  023007 
Number of pages  9 
Journal  PHYSICAL REVIEW RESEARCH 
Volume  1 
Issue number  2 
DOIs  
Publication status  Published  6 Sept 2019 
MoE publication type  A1 Journal articlerefereed 
Keywords
 POISSON BRACKETS
 MOMENTUMSPACE
 FIELDTHEORY
 ZERO MODES
 FERMIONS
 GRAVITY
 LATTICE
 STRINGS
 PARITY
 STATES
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 1 Finished

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