Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model
Stefano Scopa, Pasquale Calabrese, Jérôme Dubail
Abstract
We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature in the regime |\Delta|< 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , initially prepared in a product state with two domain walls i.e, |\downarrow\dots\downarrow\uparrow\dots\uparrow\downarrow\dots\downarrow \rangle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo>↓</mml:mo> <mml:mi>…</mml:mi> <mml:mo>↓</mml:mo> <mml:mo>↑</mml:mo> <mml:mi>…</mml:mi> <mml:mo>↑</mml:mo> <mml:mo>↓</mml:mo> <mml:mi>…</mml:mi> <mml:mo>↓</mml:mo> <mml:mo stretchy="false" form="postfix">⟩</mml:mo> </mml:mrow> </mml:math> . At early times, the two domain walls evolve independently and only after a calculable time a non-trivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For \Delta=0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we derive exact asymptotic results for the magnetization and the spin current by using a semi-classical Wigner function approach, and we exactly determine the spreading of entanglement entropy exploiting the recently developed tools of quantum fluctuating hydrodynamics. In the interacting case, we analytically solve the Generalized Hydrodynamics equation providing exact expressions for the conserved quantities. We display some numerical results for the entanglement entropy also in the interacting case and we propose a conjecture for its asymptotic value.