Lieb-Robinson Bounds with Exponential-in-Volume Tails

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>r</a:mi> </a:math> after a time <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>t</c:mi> </c:math> decays as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>exp</e:mi> <e:mo></e:mo> <e:mo stretchy="false">(</e:mo> <e:mi>v</e:mi> <e:mi>t</e:mi> <e:mo>−</e:mo> <e:mi>r</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> , where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>v</i:mi> </i:math> is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>r</k:mi> <k:mi>d</k:mi> </k:msup> </k:math> sites in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>d</m:mi> </m:math> spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>exp</o:mi> <o:mo></o:mo> <o:mo stretchy="false">(</o:mo> <o:mo>−</o:mo> <o:msup> <o:mi>r</o:mi> <o:mi>d</o:mi> </o:msup> <o:mo stretchy="false">)</o:mo> </o:math> . We confirm this intuition, showing that, for <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>r</s:mi> <s:mo>&gt;</s:mo> <s:mi>v</s:mi> <s:mi>t</s:mi> </s:math> , the volume-filling operator is suppressed by <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>exp</u:mi> <u:mo></u:mo> <u:mo stretchy="false">(</u:mo> <u:mo>−</u:mo> <u:mo stretchy="false">(</u:mo> <u:mi>r</u:mi> <u:mo>−</u:mo> <u:mi>v</u:mi> <u:mi>t</u:mi> <u:msup> <u:mo stretchy="false">)</u:mo> <u:mi>d</u:mi> </u:msup> <u:mo>/</u:mo> <u:mo stretchy="false">(</u:mo> <u:mi>v</u:mi> <u:mi>t</u:mi> <u:msup> <u:mo stretchy="false">)</u:mo> <u:mrow> <u:mi>d</u:mi> <u:mo>−</u:mo> <u:mn>1</u:mn> </u:mrow> </u:msup> <u:mo stretchy="false">)</u:mo> </u:math> . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>ε</cb:mi> </cb:math> for any finite time <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>t</eb:mi> </eb:math> : as <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>ε</gb:mi> </gb:math> becomes sufficiently small, only <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:msup> <ib:mi>ε</ib:mi> <ib:mrow> <ib:mo>−</ib:mo> <ib:mrow> <ib:mi mathvariant="normal">O</ib:mi> </ib:mrow> <ib:mo stretchy="false">(</ib:mo> <ib:msup> <ib:mi>t</ib:mi> <ib:mrow> <ib:mi>d</ib:mi> <ib:mo>−</ib:mo> <ib:mn>1</ib:mn> </ib:mrow> </ib:msup> <ib:mo stretchy="false">)</ib:mo> </ib:mrow> </ib:msup> </ib:math> resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.