Beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion

The motion of dopants in magnetic spin lattices has received tremendous attention for at least four decades due to its connection to high-temperature superconductivity. Despite these efforts, we lack a complete understanding of their behavior, especially out of the equilibrium and at nonzero temperatures. In this paper, we take a significant step towards a much deeper understanding based on state-of-the-art matrix-product-state calculations. In particular, we investigate the nonequilibrium dynamics of a dopant in two-leg <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>t</a:mi><a:mtext>−</a:mtext><a:mi>J</a:mi></a:mrow></a:math> ladders with antiferromagnetic XXZ spin interactions. In the Ising limit, we find that the dopant is for all investigated temperatures due to an emergent disordered potential, with a localization length controlled by the underlying correlation length of the spin lattice, which increases exponentially with decreasing temperature. The dopant, hereby, only delocalizes asymptotically in the zero temperature limit. This greatly generalizes the localization effect discovered recently in Hilbert space fragmented models [, ]. In the presence of spin-exchange processes at rate <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>α</b:mi></b:math>, the dopant diffuses with a diffusion coefficient, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>D</c:mi><c:mi>h</c:mi></c:msub></c:math>, depending nonmonotonically on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>α</d:mi></d:math>. It initially increases linearly as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>D</e:mi><e:mi>h</e:mi></e:msub><e:mo>∝</e:mo><e:mi>α</e:mi></e:mrow></e:math> for <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>α</f:mi><f:mo>≪</f:mo><f:mn>1</f:mn></f:mrow></f:math> before dropping off as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msup><g:mi>α</g:mi><g:mrow><g:mo>−</g:mo><g:mn>1</g:mn></g:mrow></g:msup></g:math> for <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>α</h:mi><h:mo>&gt;</h:mo><h:mn>1</h:mn></h:mrow></h:math>. Moreover, we show that the underlying spin dynamics at infinite temperature behaves qualitatively the same, albeit with important quantitative differences. We substantiate these findings by showing that the dynamics features self-similar scaling behavior, which strongly deviates from the Gaussian behavior of regular diffusion, especially for weak spin exchange. Finally, we show that the diffusion coefficient <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>D</i:mi><i:mi>h</i:mi></i:msub></i:math> follows an Arrhenius relation at high temperatures, whereby it is exponentially suppressed upon cooling.