Operator growth in disordered spin chains: Indications for the absence of many-body localization

We consider the spreading of a local operator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>A</a:mi></a:math> in one-dimensional many-body systems with Hamiltonian <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>H</b:mi></b:math> by calculating the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>k</c:mi></c:math>-fold commutator <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>[</d:mo><d:mi>H</d:mi><d:mo>,</d:mo><d:mo>[</d:mo><d:mi>H</d:mi><d:mo>,</d:mo><d:mo>[</d:mo><d:mo>.</d:mo><d:mo>.</d:mo><d:mo>.</d:mo><d:mo>,</d:mo><d:mo>[</d:mo><d:mi>H</d:mi><d:mo>,</d:mo><d:mi>A</d:mi><d:mo>]</d:mo><d:mo>]</d:mo><d:mo>]</d:mo><d:mo>]</d:mo></d:mrow></d:math>. We derive bounds for the operator norm of this commutator in free and interacting systems with and without disorder thus directly connecting the with questions of localization. We analytically show, in particular, that an almost-factorial growth of the operator norm—as recently proven for the random Ising model and strongly suggested here to also hold for the Heisenberg model with random fields—is inconsistent with an exponential localization of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>A</e:mi></e:math>. Assuming there exists a quasilocal unitary <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>U</f:mi></f:math> which maps <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>H</g:mi></g:math> onto an effective Hamiltonian <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mover accent="true"><h:mi>H</h:mi><h:mo>̃</h:mo></h:mover><h:mo>=</h:mo><h:mi>U</h:mi><h:mi>H</h:mi><h:msup><h:mi>U</h:mi><h:mo>†</h:mo></h:msup><h:mo>=</h:mo><h:msub><h:mo>∑</h:mo><h:mi>n</h:mi></h:msub><h:msub><h:mi>E</h:mi><h:mi>n</h:mi></h:msub><h:msubsup><h:mi>τ</h:mi><h:mi>n</h:mi><h:mi>z</h:mi></h:msubsup><h:mo>+</h:mo><h:msub><h:mo>∑</h:mo><h:mrow><h:mi>i</h:mi><h:mo>,</h:mo><h:mi>j</h:mi></h:mrow></h:msub><h:msub><h:mi>J</h:mi><h:mrow><h:mi>i</h:mi><h:mi>j</h:mi></h:mrow></h:msub><h:msubsup><h:mi>τ</h:mi><h:mi>i</h:mi><h:mi>z</h:mi></h:msubsup><h:msubsup><h:mi>τ</h:mi><h:mi>j</h:mi><h:mi>z</h:mi></h:msubsup><h:mo>+</h:mo><h:mo>⋯</h:mo></h:mrow></h:math>, we show that <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mover accent="true"><j:mi>A</j:mi><j:mo>̃</j:mo></j:mover><j:mo>=</j:mo><j:mi>U</j:mi><j:mi>A</j:mi><j:msup><j:mi>U</j:mi><j:mo>†</j:mo></j:msup></j:mrow></j:math> is a quasilocal operator which, in contrast to the Anderson case, indeed in the general many-body case, leading to an almost-factorial growth of the commutator norm. Therefore, either the unitary <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>U</l:mi></l:math> in many-body systems with maximal norm growth does not exist, and such systems are always ergodic, or unusual nonergodic phases described by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mover accent="true"><m:mi>H</m:mi><m:mo>̃</m:mo></m:mover></m:math> exist which violate the operator growth hypothesis and in which local operators spread over the entire lattice, implying that transport will eventually set in. To investigate this issue further, we concentrate on the chain with random magnetic fields. We analytically and symbolically verify our general results for the noninteracting Anderson and Aubry-André models. For the case, the symbolic calculations are consistent with a maximal norm growth. Furthermore, we find no indication of a weakened exponential localization of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>A</o:mi></o:math>, expected for strong disorder and low commutator orders if the unitary <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mi>U</p:mi></p:math> does exist. Finally, we study the differences between the interacting and noninteracting cases when trying to perturbatively construct <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mi>U</q:mi></q:math> by consecutive Schrieffer-Wolff transformations. While it is straightforward to show that this construction converges in the Anderson case, we find no indications for a convergence in the interacting case, suggesting that <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mi>U</r:mi></r:math> does not exist and that many-body localization is absent.