SciPost Submission Page
Exact hydrodynamic solution of a double domain wall melting in the spin1/2 XXZ model
by Stefano Scopa , Pasquale Calabrese and Jérôme Dubail
This is not the current version.
Submission summary
As Contributors:  Jerome Dubail · Stefano Scopa 
Preprint link:  scipost_202109_00036v1 
Date submitted:  20210930 10:56 
Submitted by:  Scopa, Stefano 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the nonequilibrium dynamics of a onedimensional spin1/2 XXZ model at zerotemperature in the regime $\Delta< 1$, initially prepared in a product state with two domain walls i.e, $\ket{\downarrow\dots\downarrow\uparrow\dots\uparrow\downarrow\dots\downarrow}$. At early times, the two domain walls evolve independently and only after a calculable time a nontrivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For $\Delta=0$, we derive exact asymptotic results for the magnetization and the spin current by using a semiclassical 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.
Current status:
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 2 on 2021124 (Invited Report)
Strengths
 exact calculations
 analytical solution of a nontrivial GHD evolution
 entanglement entropy is extracted and well compare with numerics
Weaknesses
 more explications needed in the interacting case and lack of discussion on higher order effects
Report
Very interesting paper, generalizing known results on the time evolution of the domain wall state. The only comment I have is to comment on higher order hydrodynamic effects, in particular on diffusion. It is known that there is no diffusion in the pure domain wall case, as the state always remain a split Fermi sea (n=0 or 1 for all x and lambda) but here when the two domain wall start mixing one should expect diffusive effects, due to the fact that now states are not pure Fermi seas (n \neq 0 or 1). If not, could the authors explains why and maybe provide a plot of the evolution of the occupation numbers in the interacting case.
Anonymous Report 1 on 2021112 (Invited Report)
Report
The paper is clear and thorough and applies recently developed techniques in the nonequilibrium dynamics of solvable models to describe evolution from a domainwall pair in the spin1/2 XXZ chain. The authors show that generalized hydrodynamics captures the dynamics of local observables such as spin and spin current in this quench (as expected), and also successfully apply the formalism of Ref. 43 to entanglement dynamics in this system (which is more striking).
I have a few comments:
p2: “a full analytical understanding of the DW dynamics”: this is a bit of an overstatement. The exact hydrodynamic solutions pertain to a particular scaling limit, and the exact microscopic dynamics for a domain wall with Delta != 0 remains unsolved. This lack of analytical understanding is particularly pronounced at Delta = 1. The front dynamics subleading to the Euler hydrodynamics (studied numerically e.g. in JM Stephan, SciPostPhys.6.5.057) is also not understood analytically for Delta != 0.
p2: “Quite interestingly, the physics of the melting process is not modified by the presence of interactions.” Again, this is only true at the ballistic scale and for Delta < 1. As discussed in SciPostPhys.6.5.057, interactions lead to qualitatively different front scaling for Delta > 0 compared to Delta=0. Interactions also (obviously) dictate the dynamics for Delta>= 1. For a finite density of domain walls at Delta > 0, interactions should furthermore eventually thermalize the system to a GGE (more on this below).
p6, eq. 12: the justification for this equation provided in the text is a bit ad hoc, given that the “Wigner function approach” is referred to specifically in the abstract – it might be worth noting explicitly in the text that this equation was derived microscopically for the XX chain in Refs. 5960.
Sec 3.2 and 4.5: The scaling of leadingorder corrections to GHD in integrable systems is reasonably wellunderstood by now. What is less clear to me is the regime of validity of the procedure used by the authors, when they state that the “asymptotic behavior of the entanglement is effectively described by a quantum hydrodynamic theory, obtained after the requantization of the Fermi contour of Sec. 3.1 in terms of a Luttingerliquid”. While the agreement of the authors' predictions with numerics is impressive, the formalism of Ref. 43 makes several more approximations than ballistic GHD. Could the authors discuss in the text what these approximations underlying QGHD are and why they are so accurate for the quench under consideration? e.g. how do the leading corrections to QGHD scale in space and time?
In particular, while a lack of entropy growth is plausible for Delta = 0, for Delta > 0 any finite density of domain walls eventually generates entropy via quasiparticle diffusion, which invalidates the assumptions of Ref. 43, even though the initial state has zero entropy (in a particular coarsegraining). So it seems that “diluteness” of the twowall initial condition is important for the validity of the authors’ proposal.
Finally, I noticed the following typos:
p4: address the reader > refer the reader?
p8: contour ad > contour and?
p12: FisherHartwing  > FisherHartwig