Exact bosonization in all dimensions and the duality between supercohomology fermionic SPT and higher-group bosonic SPT phases



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Recording Details

Speaker(s): 
Scientific Areas: 
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PIRSA Number: 
19120017

Abstract

The first part of this talk will introduce generalized Jordan–Wigner
transformation on arbitrary triangulation of any simply connected
manifold in 2d, 3d and general dimensions. This gives a duality
between all fermionic systems and a new class of Z2 lattice gauge
theories. This map preserves the locality and has an explicit
dependence on the second Stiefel–Whitney class and a choice of spin
structure on the manifold. In the Euclidean picture, this mapping is
exactly equivalent to introducing topological terms (Chern-Simon term
in 2d or the Steenrod square term in general) to the Euclidean action.
We can increase the code distance of this mapping, such that this
mapping can correct all 1-qubit and 2-qubits errors and is useful for
the simulation of fermions on the quantum computer. The second part of
my talk is about SPT phases. By the boson-fermion duality, we are able
to show the equivalent between any supercohomology fermionic SPT and
some higher-group bosonic SPT phases. Particularly in (3+1)D, we have
constructed a unitary quantum circuit for any supercohomology
fermionic SPT state with gapped boundary construction. This fermionic
SPT state is derived by gauging higher-form Z2 symmetry in the
higher-group bosonic SPT and apply the boson-fermion duality.