[Paper Review] Entanglement Hamiltonian during a domain wall melting in the free Fermi chain

Federico Rottoli, Stefano Scopa+1 more|arXiv (Cornell University)|Feb 9, 2022

Quantum many-body systems107 references29 citations

TL;DR

This paper derives the entanglement Hamiltonian for a one-dimensional free Fermi gas undergoing domain wall melting using quantum fluctuating hydrodynamics. By combining effective field theory for low-energy dynamics with the annulus method, it obtains an exact asymptotic expression for the entanglement Hamiltonian in the Euler scaling limit, validated by high-precision lattice numerics showing excellent agreement across moderate system sizes and times.

ABSTRACT

We study the unitary time evolution of the entanglement Hamiltonian of a free Fermi lattice gas in one dimension initially prepared in a domain wall configuration. To this aim, we exploit the recent development of quantum fluctuating hydrodynamics. Our findings for the entanglement Hamiltonian are based on the effective field theory description of the domain wall melting and are expected to exactly describe the Euler scaling limit of the lattice gas. However, such field theoretical results can be recovered from high-precision numerical lattice calculations only when summing appropriately over all the hoppings up to distant sites.

Motivation & Objective

• To extend quantum fluctuating hydrodynamics to compute the entanglement Hamiltonian in inhomogeneous, non-equilibrium settings.
• To address the lack of analytical predictions for the entanglement Hamiltonian in non-equilibrium, inhomogeneous quantum systems.
• To test the validity of effective field theory in describing the entanglement structure during the dynamical melting of a domain wall.
• To bridge field-theoretic hydrodynamic predictions with exact lattice calculations for a prototypical non-equilibrium quantum system.

Proposed method

• Formulates the domain wall quench as a semi-classical hydrodynamic problem using fermionic occupation functions in phase space.
• Applies an effective field theory for a massless Dirac fermion in curved spacetime to describe low-energy quantum fluctuations during melting.
• Employs the annulus method to compute the Rényi entropy and derive the entanglement Hamiltonian in the continuum limit.
• Derives an explicit expression for the local inverse temperature β(x,t) as a sum over all hopping terms up to distance rmax, weighted by local Fermi momentum sin(πρ(x,t)ra).
• Performs high-precision numerical calculations on the lattice model to test the field-theoretic prediction.
• Compares the continuum prediction with lattice data in the Euler scaling limit, using a working precision of 500 digits and rmax=8.

Experimental results

Research questions

• RQ1 Can the entanglement Hamiltonian be exactly computed in a non-equilibrium, inhomogeneous quantum system using effective field theory?
• RQ2 How does the entanglement Hamiltonian evolve during the domain wall melting process in a free Fermi chain?
• RQ3 What is the role of long-range hoppings and local Fermi momentum in shaping the entanglement Hamiltonian in the hydrodynamic limit?
• RQ4 To what extent do field-theoretic predictions for the entanglement Hamiltonian match exact lattice results in the Euler scaling regime?
• RQ5 Can the hydrodynamic framework accurately describe the entanglement structure beyond entanglement entropy, including the full modular Hamiltonian?

Key findings

• The entanglement Hamiltonian in the domain wall melting process is given by ˆKA ∼ ∫ dx/(2π) [2a/rmax ∑_{r=1}^{rmax} r sin(πρ(x,t)ra) |hx,x+r(t)| ] (TL + TR), with a precise expression for the local inverse temperature β(x,t).
• The field-theoretic prediction for β(x,t) matches high-precision lattice numerics with excellent agreement, even at modest system sizes (L=200) and times up to t=65.
• The inclusion of all hopping terms up to rmax=8 is essential for quantitative agreement, as truncating the sum leads to significant deviations.
• The local Fermi momentum factor sin(πρ(x,t)ra) introduces a non-monotonic behavior in β(x,t), correctly capturing the density vanishing at the light cone x=t.
• The effective inverse temperature β(x,t) exhibits perfect data collapse in the scaling variable x/t, confirming the universality of the hydrodynamic prediction.
• The result confirms that quantum fluctuating hydrodynamics provides an asymptotically exact description of the entanglement Hamiltonian in non-equilibrium, inhomogeneous settings.