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Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 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: 2021-09-30 10:56
Submitted by: Scopa, Stefano
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature 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 non-trivial 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 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.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission scipost_202109_00036v2 on 10 January 2022

Reports on this Submission

Anonymous Report 2 on 2021-12-4 (Invited Report)


- exact calculations
- analytical solution of a non-trivial GHD evolution
- entanglement entropy is extracted and well compare with numerics


- more explications needed in the interacting case and lack of discussion on higher order effects


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.

  • validity: high
  • significance: high
  • originality: high
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  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2021-11-2 (Invited Report)


The paper is clear and thorough and applies recently developed techniques in the non-equilibrium dynamics of solvable models to describe evolution from a domain-wall pair in the spin-1/2 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. 59-60.

Sec 3.2 and 4.5: The scaling of leading-order corrections to GHD in integrable systems is reasonably well-understood 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 re-quantization of the Fermi contour of Sec. 3.1 in terms of a Luttinger-liquid”. 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 coarse-graining). So it seems that “diluteness” of the two-wall 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: Fisher-Hartwing - > Fisher-Hartwig

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