Fermionic quantum criticality through the lens of topological holography

We utilize the topological holographic framework to characterize and gain insights into the nature of quantum critical points and gapless phases in fermionic quantum systems. Topological holography is a general framework that describes the generalized global symmetry and the symmetry charges of a local quantum system in terms of a slab of a topological order, termed as the symmetry topological field theory (SymTFT), in one higher dimension. In this work, we consider a generalization of the topological holographic picture for (1+1)<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math> fermionic quantum phases of matter. We discuss how spin structures are encoded in the SymTFT, and we establish the connection between the formal fermionization formula in quantum field theory and the choice of fermionic gapped boundary conditions of the SymTFT. We demonstrate the identification and the characterization of the fermionic gapped phases and phase transitions through detailed analysis of various examples, including the fermionic systems with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msubsup><b:mi mathvariant="double-struck">Z</b:mi><b:mn>2</b:mn><b:mi>F</b:mi></b:msubsup><b:mo>,</b:mo><b:mo> </b:mo><b:mrow><b:msub><b:mi mathvariant="double-struck">Z</b:mi><b:mn>2</b:mn></b:msub><b:mo>×</b:mo><b:msubsup><b:mi mathvariant="double-struck">Z</b:mi><b:mn>2</b:mn><b:mi>F</b:mi></b:msubsup></b:mrow><b:mo>,</b:mo><b:mo> </b:mo><b:msubsup><b:mi mathvariant="double-struck">Z</b:mi><b:mrow><b:mn>4</b:mn></b:mrow><b:mi>F</b:mi></b:msubsup></b:math>, and the fermionic version of the noninvertible <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mtext>Rep</g:mtext><g:mo>(</g:mo><g:msub><g:mi>S</g:mi><g:mn>3</g:mn></g:msub><g:mo>)</g:mo></g:mrow></g:math> symmetry. Our work uncovers many exotic fermionic gapped phases, quantum critical points, and gapless phases. These include gapped phases with fermionic noninvertible <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mtext>Rep</h:mtext><h:mo>(</h:mo><h:msub><h:mi>S</h:mi><h:mn>3</h:mn></h:msub><h:mo>)</h:mo></h:mrow></h:math> symmetry, two kinds of fermionic symmetry-enriched quantum critical points, a fermionic gapless symmetry-protected topological phase, and a fermionic gapless spontaneous symmetry-breaking phase that breaks the fermionic noninvertible symmetry.