Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">⟨</a:mo><a:mi mathvariant="bold-italic">μ</a:mi><a:mo stretchy="false">|</a:mo><a:mi mathvariant="script">O</a:mi><a:mo stretchy="false">|</a:mo><a:mi mathvariant="bold-italic">λ</a:mi><a:mo stretchy="false">⟩</a:mo></a:math> in the same macrostate scale as <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>exp</j:mi><j:mo mathvariant="bold" stretchy="false">(</j:mo><j:mo>−</j:mo><j:msup><j:mrow><j:mi>c</j:mi></j:mrow><j:mrow><j:mi mathvariant="script">O</j:mi></j:mrow></j:msup><j:mi>L</j:mi><j:mi>ln</j:mi><j:mo stretchy="false">(</j:mo><j:mi>L</j:mi><j:mo stretchy="false">)</j:mo><j:mo>−</j:mo><j:mi>L</j:mi><j:msubsup><j:mrow><j:mi>M</j:mi></j:mrow><j:mrow><j:mi mathvariant="bold-italic">μ</j:mi><j:mo>,</j:mo><j:mi mathvariant="bold-italic">λ</j:mi></j:mrow><j:mrow><j:mi mathvariant="script">O</j:mi></j:mrow></j:msubsup><j:mo mathvariant="bold" stretchy="false">)</j:mo></j:mrow></j:math>, where the probability distribution function for <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msubsup><v:mi>M</v:mi><v:mrow><v:mi mathvariant="bold-italic">μ</v:mi><v:mo>,</v:mo><v:mi mathvariant="bold-italic">λ</v:mi></v:mrow><v:mi mathvariant="script">O</v:mi></v:msubsup></v:math> is well described by Fréchet distributions and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>c</ab:mi><ab:mi mathvariant="script">O</ab:mi></ab:msup></ab:math> depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>exp</db:mi><db:mo stretchy="false">(</db:mo><db:mo>−</db:mo><db:msup><db:mi>d</db:mi><db:mi mathvariant="script">O</db:mi></db:msup><db:msup><db:mi>L</db:mi><db:mn>2</db:mn></db:msup><db:mo stretchy="false">)</db:mo></db:math>, where <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msup><ib:mi>d</ib:mi><ib:mi mathvariant="script">O</ib:mi></ib:msup></ib:math> depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections.