Universal aspects of high-temperature relaxation dynamics in random spin models

Universality is a crucial concept in modern physics, allowing us to capture the essential features of a system’s behavior using a small set of parameters. In this work, we unveil universal spin relaxation dynamics in anisotropic random Heisenberg models with infinite-range interactions at high temperatures. Starting from a polarized state, the total magnetization can relax monotonically or decay with long-lived oscillations, determined by the sign of a universal single function A=-\xi_1^2+\xi_2^2-4\xi_2\xi_3+\xi_3^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mi>ξ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>ξ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> . Here (\xi_1,\xi_3,\xi_3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> characterizes the anisotropy of the Heisenberg interaction. Furthermore, the oscillation shows up only for A&gt;0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , with frequency \Omega \propto \sqrt{A} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>∝</mml:mo> <mml:msqrt> <mml:mi>A</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> . This result is derived from the Kadanoff-Baym equation under the melon diagram approximation, which is consistent with numerical solutions. Furthermore, we verify our theory and approximation using exact diagonalization, albeit for a small system size of N=8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . Our study sheds light on the universal aspect of quantum many-body dynamics beyond low energy limit.