Scaling of many-body localization transitions: Quantum dynamics in Fock space and real space

Many-body-localization (MBL) transitions are studied in a family of single-spin-flip spin-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mfrac><a:mn>1</a:mn><a:mn>2</a:mn></a:mfrac></a:math> models, including the one-dimensional (1D) chain with nearest-neighbor interactions, the quantum dot (QD) model with all-to-all pair interactions, and the quantum random energy model (QREM). We investigate the generalized imbalance that characterizes propagation in Fock space out of an initial basis state and, at the same time, can be efficiently probed by real-space measurements. For all models considered, the average imbalance and its quantum and mesoscopic fluctuations provide excellent indicators for the position of the MBL transition <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>W</b:mi><b:mi>c</b:mi></b:msub><b:mrow><b:mo>(</b:mo><b:mi>n</b:mi><b:mo>)</b:mo></b:mrow></b:mrow></b:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>n</c:mi></c:math> is the number of spins. Combining these findings with earlier results on level statistics, we determine phase diagrams of the MBL transitions in the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>n</d:mi><d:mtext>−</d:mtext><d:mi>W</d:mi></d:mrow></d:math> plane. Our results provide evidence for a direct transition between the ergodic and MBL phases for each of the models, without any intermediate phase. For QREM and QD model, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>W</e:mi><e:mi>c</e:mi></e:msub><e:mrow><e:mo>(</e:mo><e:mi>n</e:mi><e:mo>)</e:mo></e:mrow></e:mrow></e:math> grows as a power law of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>n</f:mi></f:math> (with logarithmic corrections), in agreement with analytical predictions <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msubsup><g:mi>W</g:mi><g:mi>c</g:mi><g:mi>QREM</g:mi></g:msubsup><g:mrow><g:mo>(</g:mo><g:mi>n</g:mi><g:mo>)</g:mo></g:mrow><g:mo>∼</g:mo><g:msup><g:mi>n</g:mi><g:mrow><g:mn>1</g:mn><g:mo>/</g:mo><g:mn>2</g:mn></g:mrow></g:msup><g:mo form="prefix">ln</g:mo><g:mi>n</g:mi></g:mrow></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msubsup><i:mi>W</i:mi><i:mi>c</i:mi><i:mi>QD</i:mi></i:msubsup><i:mrow><i:mo>(</i:mo><i:mi>n</i:mi><i:mo>)</i:mo></i:mrow><i:mo>≳</i:mo><i:msup><i:mi>n</i:mi><i:mrow><i:mn>3</i:mn><i:mo>/</i:mo><i:mn>4</i:mn></i:mrow></i:msup><i:msup><i:mo form="prefix">ln</i:mo><i:mrow><i:mn>1</i:mn><i:mo>/</i:mo><i:mn>2</i:mn></i:mrow></i:msup><i:mi>n</i:mi></i:mrow></i:math>. This growth is in stark contrast to the 1D model, where <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msub><k:mi>W</k:mi><k:mi>c</k:mi></k:msub><k:mrow><k:mo>(</k:mo><k:mi>n</k:mi><k:mo>)</k:mo></k:mrow></k:mrow></k:math> is essentially independent of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>n</l:mi></l:math>, consistent with the analytic expectation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msubsup><m:mi>W</m:mi><m:mi>c</m:mi><m:mrow><m:mn>1</m:mn><m:mi mathvariant="normal">D</m:mi></m:mrow></m:msubsup><m:mrow><m:mo>(</m:mo><m:mi>n</m:mi><m:mo>→</m:mo><m:mi>∞</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>const</m:mi></m:mrow></m:math>. We also determine the scaling of the transition width <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mi mathvariant="normal">Δ</o:mi><o:mi>W</o:mi><o:mrow><o:mo>(</o:mo><o:mi>n</o:mi><o:mo>)</o:mo></o:mrow><o:mo>/</o:mo><o:msub><o:mi>W</o:mi><o:mi>c</o:mi></o:msub><o:mrow><o:mo>(</o:mo><o:mi>n</o:mi><o:mo>)</o:mo></o:mrow></o:mrow></o:math> and estimate the system size <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mi>n</q:mi></q:math> needed to study the asymptotic scaling behavior. Although the corresponding values of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mi>n</r:mi></r:math> are larger than those accessible to exact simulations on a classical computer, they are within the reach of quantum simulators. Our results indicate feasibility of experimental studies of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mrow><s:mi>n</s:mi><s:mtext>−</s:mtext><s:mi>W</s:mi></s:mrow></s:math> phase diagrams and scaling properties of MBL transitions in models of 1D and QD type and in their extensions to other spatial geometry or distance-dependent interactions.