[Paper Review] Thermalization and non-monotonic entanglement growth in an exactly solvable model

Shruti Paranjape, Nilakash Sorokhaibam|arXiv (Cornell University)|Sep 9, 2016

Quantum many-body systems22 references3 citations

TL;DR

This paper studies thermalization and entanglement dynamics in a critical quantum quench of free Dirac fermions, showing exact thermalization in Calabrese-Cardy (CC) and generalized CC (gCC) states via exact correlator computations. It reveals a dynamical phase transition in entanglement entropy growth from monotonic to non-monotonic when the effective chemical potential exceeds a critical value, driven by higher-spin charges in gCC states.

ABSTRACT

We study quantum quenches and subsequent non-equilibrium dynamics of free Dirac fermions in 1+1 spacetime dimensions using time dependent mass. The final state is a normalized boundary state which is called generalized Calabrese-Cardy (gCC) state and the system thermalizes to a generalized Gibb's Ensemble(GGE). We can also tune the initial states so that the final states are exact Calabrese-Cardy (CC) state and special gCC states. The system in the CC state thermalizes to a Gibb's ensemble. We derive closed-form analytic expressions for the growth of entanglement entropy of subsystems consisting of arbitrary number of disjoint intervals in CC state. We show that the entanglement entropy of a single interval grows monotonically before saturation. In case of certain gCC states, for particular charges, the entanglement entropy of a single interval grows non-monotonically when the effective chemical potential is increased beyond a critical value. We argue that the non-monotonic growth of entanglement entropy is due to increase in long range correlation and decrease in short range correlation at early times.

Motivation & Objective

• To investigate thermalization in critical quantum quenches of free Dirac fermions using exactly solvable models.
• To examine the role of higher-spin conserved charges in modifying entanglement entropy dynamics beyond monotonic growth.
• To establish the validity of generalized Calabrese-Cardy (gCC) states in fermionic systems and their thermalization to generalized Gibbs ensembles (GGE).
• To compute exact time evolution of correlation functions and entanglement entropy in quenched fermionic systems.
• To identify a dynamical phase transition in entanglement entropy growth triggered by increasing effective chemical potential in gCC states.

Proposed method

• Constructing quenched states from ground and specially prepared squeezed states of the massive Dirac theory, mapping them to CC and gCC states in the massless CFT limit.
• Using Bogoliubov transformations and exact solutions of time-dependent fermionic oscillators to derive time-evolved states and correlation functions.
• Computing energy density and correlation functions explicitly to demonstrate thermalization to thermal and generalized Gibbs ensembles.
• Applying the Baker-Campbell-Hausdorff (BCH) formula to relate squeezed states to exponential operators of Hamiltonian and higher-spin charges.
• Refermionizing the bosonic $ ilde{W}_4$ current to express higher-spin charges in terms of fermionic bilinears and compute their contributions to the gCC state.
• Analyzing entanglement entropy growth in both CC and gCC states, identifying a critical threshold in the effective chemical potential for non-monotonic behavior.

Experimental results

Research questions

• RQ1 Does the Calabrese-Cardy (CC) state formed after a critical quench of free Dirac fermions exhibit exact thermalization, as confirmed by correlation functions?
• RQ2 How do higher-spin conserved charges ($W_4, W_6, ext{etc.}$) in generalized CC (gCC) states affect the time evolution of entanglement entropy?
• RQ3 At what critical value of the effective chemical potential does the entanglement entropy growth transition from monotonic to non-monotonic in gCC states?
• RQ4 Can gCC states with finite numbers of higher-spin charges be prepared via sudden quench from specially engineered squeezed states in the massive fermionic theory?
• RQ5 What is the exact time dependence of entanglement entropy in CC and gCC states, and how does it differ under varying conserved charge parameters?

Key findings

• Exact thermalization is demonstrated in CC states through explicit computation of time-dependent correlation functions, showing agreement with thermal ensemble predictions.
• In gCC states with finite higher-spin charges, entanglement entropy growth transitions from monotonic to non-monotonic when the effective chemical potential exceeds a critical threshold.
• The critical value for the dynamical phase transition in entanglement entropy growth is determined by the balance between the $ ilde{W}_4$ charge and the Hamiltonian in the gCC state.
• Squeezed initial states in the massive Dirac theory map exactly to gCC states in the massless CFT limit, enabling preparation of states with finite higher-spin charges.
• The energy density and higher-spin charges ($W_4, W_6, ext{etc.}$) diverge in the sudden quench limit from the ground state, indicating non-normalizability.
• The refermionized $ ilde{W}_4$ current is derived explicitly, showing that only specific four-fermion terms survive due to anti-commutation relations, enabling exact charge quantization in the gCC state.