Eigenstate thermalization hypothesis for off-diagonal matrix elements in integrable spin chains

We investigate off-diagonal matrix elements of local operators in integrable spin chains, focusing on the isotropic spin- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:math> Heisenberg chain ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>X</b:mi> <b:mi>X</b:mi> <b:mi>X</b:mi> </b:mrow> </b:math> chain). We employ state-of-the-art algebraic Bethe ansatz results, which allow us to efficiently compute matrix elements of operators with support up to two sites between generic energy eigenstates. We consider both matrix elements between eigenstates that are in the same thermodynamic macrostate, as well as eigenstates that belong to different macrostates. In the former case, focusing on thermal states we numerically show that matrix elements are compatible with the exponential decay as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mo form="prefix">exp</c:mo> <c:mo>(</c:mo> <c:mo>−</c:mo> <c:mi>L</c:mi> <c:mo>|</c:mo> <c:msubsup> <c:mrow> <c:mi>M</c:mi> </c:mrow> <c:mrow> <c:mi>i</c:mi> <c:mi>j</c:mi> </c:mrow> <c:mstyle scriptlevel="2" displaystyle="false"> <c:mi mathvariant="script">O</c:mi> </c:mstyle> </c:msubsup> <c:mo>|</c:mo> <c:mo>)</c:mo> </c:mrow> </c:math> . The probability distribution functions of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msubsup> <h:mi>M</h:mi> <h:mrow> <h:mi>i</h:mi> <h:mi>j</h:mi> </h:mrow> <h:mstyle scriptlevel="2" displaystyle="false"> <h:mi mathvariant="script">O</h:mi> </h:mstyle> </h:msubsup> </h:math> depend on the observable and on the macrostate, and are well described by Gumbel distributions. On the other hand, matrix elements between eigenstates in different macrostates decay faster as <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mo form="prefix">exp</l:mo> <l:mo>(</l:mo> <l:mo>−</l:mo> <l:mo>|</l:mo> <l:msubsup> <l:mrow> <l:msup> <l:mi>M</l:mi> <l:mo>′</l:mo> </l:msup> </l:mrow> <l:mrow> <l:mi>i</l:mi> <l:mi>j</l:mi> </l:mrow> <l:mstyle scriptlevel="2" displaystyle="false"> <l:mi mathvariant="script">O</l:mi> </l:mstyle> </l:msubsup> <l:mo>|</l:mo> <l:msup> <l:mi>L</l:mi> <l:mn>2</l:mn> </l:msup> <l:mo>)</l:mo> </l:mrow> </l:math> , with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msubsup> <q:mrow> <q:msup> <q:mi>M</q:mi> <q:mo>′</q:mo> </q:msup> </q:mrow> <q:mrow> <q:mi>i</q:mi> <q:mi>j</q:mi> </q:mrow> <q:mstyle scriptlevel="2" displaystyle="false"> <q:mi mathvariant="script">O</q:mi> </q:mstyle> </q:msubsup> </q:math> , again, compatible with a Gumbel distribution.