Unveiling the scaling laws behind spin-phonon enhanced decorrelation in spin chains

To study the spin-Peierls transition at finite temperatures, we use path integral molecular dynamics (PIMD) in combination with stochastic series expansion or exact diagonalization to sample the quantum Boltzmann distribution of the spin-phonon-coupled one-dimensional isotropic Heisenberg chains. The use of PIMD in this context has apparently been overlooked up to now and offers several advantages over other approaches, including no restriction on the functional forms of the phonon potential or the exchange coupling. This approach allows us to study the spin-spin correlation in such systems in detail. Looking at <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>S</a:mi> <a:mo>=</a:mo> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:math> , we find that it exhibits algebraic decay in the undimerized phase and exponential decay in the dimerized phase, but with many nuanced trends present. The decay exponentials in all cases roughly obey the scaling of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msup> <b:mrow> <b:mo>(</b:mo> <b:msup> <b:mi>g</b:mi> <b:mn>2</b:mn> </b:msup> <b:mo>/</b:mo> <b:mrow> <b:mo>(</b:mo> <b:mi>m</b:mi> <b:msubsup> <b:mi>ω</b:mi> <b:mrow> <b:mi>π</b:mi> </b:mrow> <b:mn>2</b:mn> </b:msubsup> <b:mo>)</b:mo> </b:mrow> <b:mo>)</b:mo> </b:mrow> <b:mi>γ</b:mi> </b:msup> </b:math> , where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>g</c:mi> </c:math> is the spin-phonon coupling constant, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>m</d:mi> </d:math> is mass, and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>ω</e:mi> <e:mi>π</e:mi> </e:msub> </e:math> is the frequency of the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>k</f:mi> <f:mo>=</f:mo> <f:mi>π</f:mi> </f:mrow> </f:math> phonon.