Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices

This paper investigates the entanglement asymmetry of free fermions on a honeycomb lattice with sublattice energy imbalance, revealing that the asymmetry exhibits nonanalytic dependence on the imbalance due to Dirac points and persists after a quench to the symmetric point because of a flat energy dispersion in a specific direction.

The Gist

The Big Picture: A Dance of Quantum Particles

Imagine a crowded dance floor made of a honeycomb pattern (like a beehive). On this floor, there are two types of dancers: Team A and Team B. They are "free fermions," which is a fancy way of saying they are quantum particles that don't like to stand on top of each other and follow strict rules about how they move.

In this story, the researchers are studying what happens when the music changes suddenly. They want to know: If the dance floor starts out unfair, does it eventually become fair again?

The Setup: The "Unfair" Dance Floor

1. The Imbalance: At the start, the researchers make the dance floor unfair. They give Team A a little extra energy (like giving them better shoes) and take some away from Team B. This is called an "energy imbalance." Because of this, the dance floor loses its Space-Inversion Symmetry.

   • Analogy: Imagine a seesaw where one side is heavy and the other is light. It's tilted. If you look at it in a mirror, the mirror image doesn't match the real thing because the heavy side is on the left, not the right. The system is "broken."

2. The "Entanglement Asymmetry": The researchers use a special measuring stick called Entanglement Asymmetry. Think of this as a "Fairness Meter."

   • If the meter reads 0, the system is perfectly balanced and symmetric.
   • If the meter reads high, the system is broken and unfair.

The Experiment: The Sudden Switch (The Quench)

The researchers perform a "Quantum Quench." This is like suddenly snapping your fingers and changing the rules of the game instantly.

• Before the snap: The floor is tilted (Unfair).
• After the snap: They remove the extra energy. The floor is now perfectly flat and fair (Symmetric). The music is now the same for everyone.

The Question: Now that the rules are fair, will the dancers naturally settle down into a fair arrangement? Or will the memory of the initial tilt stay with them forever?

The Surprising Discovery: It Depends on the Size of the Room

The researchers found that the answer depends entirely on the shape and size of the specific section of the dance floor they are watching (the "subsystem").

Case 1: The Odd-Sized Room (Symmetry Restored)

If the dance floor section has an odd number of rows across the width:

• What happens: The dancers start moving. They pair up and run in opposite directions. Eventually, they spread out evenly.
• The Result: The "Fairness Meter" drops to zero. The system forgets the initial tilt and becomes symmetric again.
• Analogy: Imagine a group of people running in a circle. If the circle is an odd size, they eventually mix up perfectly, and you can't tell who started where. The "tilt" disappears.

Case 2: The Even-Sized Room (Symmetry Stuck)

If the dance floor section has an even number of rows across the width:

• What happens: Something strange occurs. A specific group of dancers gets stuck in a "flat band."
• The Flat Band: Imagine a section of the dance floor that is perfectly flat and smooth. On this smooth patch, the dancers have zero speed. They are excited by the music, but they literally cannot move forward or backward. They are stuck in place, vibrating but not traveling.
• The Result: Because these "stuck" dancers never leave their spot, they carry the memory of the initial tilt with them forever. The "Fairness Meter" stays high. The system never becomes symmetric, even though the rules of the game are now fair.
• Analogy: Imagine a conveyor belt that suddenly stops. If you are standing on a section of the belt that has stopped moving, you stay exactly where you were, even if the rest of the factory is running smoothly. You never get to the "fair" destination.

Why Does This Matter?

This paper reveals a deep secret about how quantum systems relax (calm down) after a shock.

1. Geometry is King: It's not just about the physics of the particles; it's about the shape of the container. A tiny change in the size of the room (odd vs. even) completely changes the outcome.
2. The "Flat Band" Trap: The researchers discovered that "flat bands" (energy levels where particles have zero speed) act like a prison for symmetry. If particles get stuck in these flat bands, the system can never forget its past, even if the future is perfect.
3. Real-World Application: This isn't just theory. Scientists can build these honeycomb lattices using ultracold atoms in labs. They can actually set up this experiment, change the "tilt," and watch to see if the atoms remember the unfairness.

Summary in One Sentence

When you suddenly fix a broken quantum system, it usually heals itself, unless the system's shape traps some particles in a "speedless zone," in which case the system remains broken forever, carrying the scar of its past.

Technical Summary

1. Problem Statement

The paper investigates the relaxation dynamics of entanglement asymmetry in a non-equilibrium quantum many-body system. Specifically, it addresses the following gaps in current understanding:

• Symmetry Restoration: In standard quantum quenches where the post-quench Hamiltonian preserves a symmetry broken by the initial state, local subsystems typically relax into a Generalized Gibbs Ensemble (GGE) that restores the symmetry. However, cases where symmetry restoration fails in higher dimensions are not well understood.
• Lattice Geometry: Previous studies on entanglement asymmetry have focused primarily on square lattices. The role of lattice geometry (specifically the honeycomb lattice) and its unique band structure (Dirac points, flat bands) on symmetry dynamics remains an open question.
• Symmetry Type: While entanglement asymmetry has been studied for internal symmetries (e.g., $U(1)$, non-Abelian charges), its application to space-inversion symmetry (parity) in condensed matter systems has been unexplored.

The authors study spinless free fermions on a 2D honeycomb lattice with an on-site energy imbalance ($M$) between the two sublattices ($A$ and $B$). They analyze the system's evolution after a global quantum quench from an asymmetric state ($M \neq 0$) to a symmetric state ($M = 0$).

2. Methodology

The authors employ a combination of analytical derivations and numerical simulations based on the following framework:

• Model: Free fermions on a honeycomb lattice mapped to a "brickwork" lattice topology. The Hamiltonian includes nearest-neighbor hopping ($J$) and a sublattice energy difference ($M$).
• Quench Protocol: The system is initialized in the ground state $|\Psi_0\rangle$ of the Hamiltonian with $M \neq 0$. At $t=0$, the parameter $M$ is suddenly set to 0, and the system evolves unitarily under the inversion-symmetric Hamiltonian.
• Entanglement Asymmetry ($\Delta S_A$): Defined as the Kullback-Leibler divergence between the reduced density matrix of a subsystem ($\rho_A$) and its symmetrized version ($\bar{\rho}_A = (\rho_A + P\rho_A P^{-1})/2$), where $P$ is the space-inversion operator.
• Dimensional Reduction: To handle the 2D system, the authors utilize a technique that exploits the translational invariance of the subsystem in the transverse direction. This decomposes the 2D problem into a set of independent 1D problems labeled by transverse momentum $k_y$.
• Quasiparticle Picture: For the time evolution, they apply the quasiparticle picture of integrable systems. They calculate the "charged moments" of the density matrix, which are expressed via determinants of block-Toeplitz matrices derived from two-point correlation functions.
• Analytical Tools:
  • Widom-Szegő Theorem: Used to evaluate the asymptotic behavior of the determinants of block-Toeplitz matrices for large subsystem sizes ($\ell \to \infty$).
  • Ballistic Limit: Analysis performed in the limit where time $t$ and subsystem size $\ell$ go to infinity with the ratio $t/\ell$ fixed.

3. Key Contributions and Results

A. Nonanalytic Dependence on Energy Imbalance (Ground State)

In the ground state ($t=0$), the entanglement asymmetry exhibits a nonanalytic dependence on the energy imbalance $M$ near $M=0$.

• Dirac Points: When the transverse system size $L_y$ is a multiple of 3, the quantized momenta include the Dirac points ($k = \pm(2\pi/3, 4\pi/3)$) where the energy gap vanishes. In this case, the asymmetry scales as $e^{-\ell|M|/J}$, showing a cusp-like nonanalyticity at $M=0$.
• Gapped Case: When $L_y$ is not a multiple of 3, the Dirac points are missed, the system is effectively gapped, and the asymmetry depends analytically on $M$.

B. Geometry-Dependent Symmetry Restoration (Quench Dynamics)

The most significant finding concerns the long-time behavior of the entanglement asymmetry after the quench ($M \to 0$). The outcome depends critically on the parity of the transverse subsystem size ($L_y$):

1. Odd $L_y$ (Symmetry Restoration):

   • The entanglement asymmetry decays to zero as $t \to \infty$.
   • The subsystem relaxes into the GGE of the symmetric post-quench Hamiltonian.
   • The decay follows a power law: $\Delta S_A(t) \sim (Jt)^{-1}$ (or $(Jt)^{-1}\ln(Jt)$ for the von Neumann limit).

2. Even $L_y$ (Absence of Symmetry Restoration):

   • The entanglement asymmetry relaxes to a finite, non-zero value even as $t \to \infty$.
   • The inversion symmetry broken in the initial state persists despite the symmetric dynamics.
   • The saturation value is given by $\lim_{t\to\infty} \Delta S_A(t) = H^{(n)}([1+(M/J)^2]^{-\ell/2})$, which remains finite for large $\ell$.

C. Physical Mechanism: Flat Bands and Zero-Velocity Modes

The authors attribute the failure of symmetry restoration in the even-$L_y$ case to the specific band structure of the honeycomb lattice:

• Flat Dispersion: For even $L_y$, the transverse momentum $k_y = \pi$ is allowed. At this momentum, the energy dispersion $\varepsilon_k$ becomes independent of the longitudinal momentum $k_x$ (a flat band in the $k_x$ direction).
• Zero Group Velocity: Consequently, quasiparticle modes at $k_y = \pi$ have zero longitudinal group velocity ($v_g = \partial_{k_x}\varepsilon_k = 0$).
• Macroscopic Occupation: The quench excites a macroscopic number of these zero-velocity quasiparticle pairs. Because their group velocity is zero, they never leave the local subsystem.
• Persistence of Asymmetry: These "stuck" quasiparticles carry the memory of the initial symmetry breaking, preventing the reduced density matrix from relaxing to a symmetric GGE.

4. Significance and Implications

• Role of Geometry: The study demonstrates that lattice geometry and boundary conditions (which determine allowed momenta) are as crucial as the bulk Hamiltonian in determining relaxation dynamics. A simple change in system size parity ($L_y$ odd vs. even) fundamentally alters the long-time fate of the system.
• New Mechanism for Symmetry Breaking: This identifies a new mechanism for the absence of symmetry restoration distinct from previous findings involving non-Abelian charges or Bose-Einstein condensation. It relies purely on the existence of flat bands and zero-velocity modes in the post-quench spectrum.
• Experimental Feasibility: The authors propose that this phenomenon is experimentally observable in ultracold atoms in optical lattices. The honeycomb lattice and sublattice energy imbalance can be precisely tuned, and the $n=2$ Rényi entanglement asymmetry can be measured using beam-splitter interference and site-resolved imaging.
• Theoretical Framework: The work extends the dimensional reduction technique and the quasiparticle picture of entanglement asymmetry to 2D systems with complex band structures, providing a rigorous analytical tool for studying non-equilibrium symmetry dynamics in higher dimensions.

In summary, the paper reveals that in honeycomb lattice systems, the restoration of space-inversion symmetry after a quantum quench is not guaranteed; it can be permanently suppressed by the presence of flat bands induced by specific system geometries, leading to a persistent memory of the initial symmetry-breaking state.