Tensor Networks for Noninvertible Symmetries in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> </mml:math> and Beyond
Pranay Gorantla, Shu-Heng Shao, Nathanan Tantivasadakarn
Abstract
Tensor networks provide a natural language for noninvertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a noninvertible operator implementing the Wegner duality in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>3</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> <a:mi mathvariant="normal">D</a:mi> </a:mrow> </a:math> lattice <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mi mathvariant="double-struck">Z</d:mi> <d:mn>2</d:mn> </d:msub> </d:math> gauge theory. The noninvertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi mathvariant="double-struck">Z</g:mi> <g:mn>2</g:mn> </g:msub> </g:math> gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the noninvertible duality operators (including noninvertible parity and reflection symmetries) of generalized Ising models based on graphs, encompassing the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:mn>1</j:mn> <j:mo>+</j:mo> <j:mn>1</j:mn> <j:mi mathvariant="normal">D</j:mi> </j:mrow> </j:math> Ising model, the three-spin Ising model, the Ashkin-Teller model, and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mn>2</m:mn> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mi mathvariant="normal">D</m:mi> </m:mrow> </m:math> plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.