Bipartite Fluctuations of Critical Fermi Surfaces

Fluctuations of conserved quantities within a subsystem are nonlocal observables that provide unique insights into quantum many-body systems. In this paper, we study bipartite charge (and spin) fluctuations across interaction-driven “metal-insulator transitions” out of Landau Fermi liquids. The “charge insulators” include a class of non-Fermi-liquid states of fractionalized degrees of freedom, such as compressible composite Fermi liquids (for spinless electrons) and incompressible spin-liquid Mott insulators (for spin-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>1</a:mn><a:mo>/</a:mo><a:mn>2</a:mn></a:math> electrons). We find that charge fluctuations <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="script">F</c:mi></c:math> exhibit distinct leading-order scalings across the transition: <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="script">F</f:mi><f:mo>∼</f:mo><f:mi>L</f:mi><f:mi>log</f:mi><f:mo stretchy="false">(</f:mo><f:mi>L</f:mi><f:mo stretchy="false">)</f:mo></f:math> in Landau Fermi liquids and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="script">F</k:mi><k:mo>∼</k:mo><k:mi>L</k:mi></k:math> in charge insulators, where <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>L</n:mi></n:math> is the linear size of the subsystem. In composite Fermi liquids, under certain conditions, we also identify a universal constant term <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mo>−</p:mo><p:mi mathvariant="normal">f</p:mi><p:mo stretchy="false">(</p:mo><p:mi>θ</p:mi><p:mo stretchy="false">)</p:mo><p:mo stretchy="false">|</p:mo><p:msub><p:mi>σ</p:mi><p:mrow><p:mi>x</p:mi><p:mi>y</p:mi></p:mrow></p:msub><p:mo stretchy="false">|</p:mo><p:mo>/</p:mo><p:mo stretchy="false">(</p:mo><p:mn>2</p:mn><p:mi>π</p:mi><p:mo stretchy="false">)</p:mo></p:math> when the subsystem geometry contains a sharp corner, where <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi mathvariant="normal">f</y:mi><y:mo stretchy="false">(</y:mo><y:mi>θ</y:mi><y:mo stretchy="false">)</y:mo></y:math> denotes a function of the corner angle and <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msub><db:mi>σ</db:mi><db:mrow><db:mi>x</db:mi><db:mi>y</db:mi></db:mrow></db:msub></db:math> is the Hall conductivity. At the critical point, provided the transition is continuous, the leading scaling <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mi mathvariant="script">F</fb:mi><fb:mo>∼</fb:mo><fb:mi>L</fb:mi></fb:math> is accompanied by a subleading universal corner contribution <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mo>−</ib:mo><ib:mi>log</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:mi>L</ib:mi><ib:mo stretchy="false">)</ib:mo><ib:mi mathvariant="normal">f</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:mi>θ</ib:mi><ib:mo stretchy="false">)</ib:mo><ib:msub><ib:mi>C</ib:mi><ib:mi>ρ</ib:mi></ib:msub><ib:mo>/</ib:mo><ib:mn>2</ib:mn></ib:math> with the same angle dependence <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi mathvariant="normal">f</pb:mi><pb:mo stretchy="false">(</pb:mo><pb:mi>θ</pb:mi><pb:mo stretchy="false">)</pb:mo></pb:math>, and the universal coefficient <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msub><ub:mi>C</ub:mi><ub:mi>ρ</ub:mi></ub:msub></ub:math> is directly related to the predicted universal jumps in longitudinal and Hall resistivities. These results establish fluctuation-transport relations, paving the way for numerical and experimental studies of unconventional quantum criticalities in metals.