Generalized Hall conductivities in local commuting projector models: Generalized symmetries and protected surface modes

Hall conductivities are important characterizations of phases of matter. It is known that nonzero Hall conductivities are difficult to realize in local commuting projector lattice models due to no-go theorems in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mo>(</a:mo> <a:mn>2</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> <a:mo>)</a:mo> <a:mi mathvariant="normal">D</a:mi> </a:mrow> </a:math> . In this work we construct local commuting projector models in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mo>(</c:mo> <c:mn>2</c:mn> <c:mo>+</c:mo> <c:mn>1</c:mn> <c:mo>)</c:mo> <c:mi mathvariant="normal">D</c:mi> </c:mrow> </c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>(</e:mo> <e:mn>3</e:mn> <e:mo>+</e:mo> <e:mn>1</e:mn> <e:mo>)</e:mo> <e:mi mathvariant="normal">D</e:mi> </e:mrow> </e:math> with nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries on tensor product Hilbert space of finite local dimension. The model is given by a standard <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi mathvariant="double-struck">Z</g:mi> <g:mi>N</g:mi> </g:msub> </g:math> toric code, but the symmetries do not admit expression in terms of on-site charge operators. The symmetry do not have local charges or currents on the lattice in the absence of boundaries, but there is still a notion of Hall conductivities that coincide with the continuum field theories. We construct protected gapless boundaries of the lattice models using modified Villain formalism. The generalized Hall conductivities are computed by surface currents as well as bulk flux insertion and many-body Chern number.