Breaking of clustering and macroscopic coherence under the lens of asymmetry measures

This paper investigates how local perturbations in an interacting one-dimensional system with conserved domain walls amplify quantum interferences to produce macroscopic magnetization profiles and quantum coherence, characterizing these phenomena using Entanglement Asymmetry and Quantum Fisher Information while establishing a generalized inequality between them for mixed states.

The Gist

The Big Picture: Can a "Schrödinger's Cat" Survive in a Crowd?

Imagine you have a row of light switches (spins) all flipped to "ON." This is a perfectly ordered state. Now, imagine you flip just one switch in the middle. In a simple, non-interacting world, that single flip would just create a small ripple that travels out and fades away. The system would eventually calm down, and the switches far away wouldn't know anything happened. This is called "clustering"—things far apart act independently.

However, this paper asks a fascinating question: What if the switches are talking to each other? What if they are "interacting"?

The author, Florent Ferro, investigates what happens when you disturb a highly ordered system where the particles are strongly connected. He discovers that instead of a small ripple fading away, the disturbance creates a giant, macroscopic quantum superposition.

Think of it like this: In a normal crowd, if one person claps, the noise dies out quickly. But in this specific quantum crowd, if one person claps, the entire crowd suddenly enters a state where they are simultaneously clapping and not clapping, and this "both/and" state stretches across the whole room. This is the quantum version of Schrödinger's Cat, but instead of a cat being alive and dead, the whole system is in two different magnetic states at once.

The Cast of Characters

To understand the experiment, let's meet the players:

1. The System (The Dual XXZ Chain): Imagine a long line of dancers holding hands. They can face North or South. They have a rule: they want to face the same direction as their neighbors (ferromagnetic order).
2. The Ground State (The Calm): Initially, everyone is facing North. This is the "ground state." It's stable and ordered.
3. The Local Quench (The Spark): The researcher flips a small group of dancers in the middle to face South. This is the "perturbation."
4. The Domain Walls (The Messengers): Because the dancers are holding hands, you can't just have a South dancer next to a North dancer without a "boundary" between them. These boundaries are called Domain Walls.
   • Analogy: Imagine a line of people wearing red hats (North) and blue hats (South). The spot where a red hat meets a blue hat is a "domain wall." When you flip the middle, you create two of these walls. They are like two messengers sent out from the center.

The Journey: What Happens Next?

In a simple world, these two messengers (domain walls) would just run away from each other at a constant speed. But in this interacting world, they are special. They can scatter off each other, bounce, or even stick together to form a "bound state" (like two dancers holding hands and spinning together).

The Discovery:
As these messengers travel outward, they don't just carry information; they carry quantum interference. Because they are quantum particles, they act like waves. When they spread out, they create a giant interference pattern.

The paper shows that this pattern is so strong that it creates a "macroscopic superposition."

• The Result: A large chunk of the system (the "light cone" or the area the messengers have reached) is now in a state where it is simultaneously "mostly North" and "mostly South."
• Why it's cool: Usually, big things (macroscopic objects) can't be in two states at once because they interact with the environment and "collapse." But here, the interaction between the particles protects this weird quantum state, making it robust. It's like a giant wave that refuses to break.

The Tools: How Did They Measure It?

The author didn't just guess; he used two special "rulers" to measure how "quantum" this superposition is.

1. Entanglement Asymmetry (EA): The "Confusion Meter"

Imagine you have a bag of marbles. Some are red, some are blue.

• If the bag is a simple mix (50% red, 50% blue), you know the odds, but there's no quantum weirdness.
• If the bag is a quantum superposition, the bag is simultaneously 100% red and 100% blue until you look.

Entanglement Asymmetry measures how "confused" the system is about which state it is in.

• The Finding: As time goes on, the "confusion" grows logarithmically. The system isn't just in one state or another; it is coherently spanning a huge number of different magnetization states. The number of possibilities grows with time, like a tree branching out infinitely.

2. Quantum Fisher Information (QFI): The "Sensitivity Meter"

Imagine you are trying to measure the weight of a feather. If you use a heavy scale, you can't tell the difference. If you use a super-sensitive scale, you can.

• QFI measures how sensitive the system is to changes.
• The Finding: In a normal system, sensitivity grows slowly. But in this system, the sensitivity grows quadratically (very fast). This proves that the system is in a "cat state"—a delicate superposition where a tiny change affects the whole system. It confirms that the "messengers" have created a state where the whole region is entangled.

The "Secret Sauce": Interactions

The most important part of this paper is that it proves this phenomenon survives even when the particles interact.

• In simple models (where particles don't talk), this effect was known.
• In complex models (where particles bounce off each other and form bound states), many physicists thought the "noise" of interactions would destroy the delicate quantum superposition.
• The Verdict: No! The interactions actually help. They create a "scattering phase" (a specific quantum signature) that gets imprinted on the magnetization profile. The system remains robust. Even if some particles stick together (bound states), the rest of the system still forms a massive, coherent quantum superposition.

The Takeaway

This paper tells us that macroscopic quantum superpositions (the "Schrödinger's Cats" of the quantum world) are not just fragile lab curiosities. They can emerge naturally from simple local disturbances in interacting systems.

The Analogy Summary:
Imagine a stadium wave.

• Normal Physics: You start a wave in one section. It travels, fades, and the rest of the stadium goes back to sitting still.
• This Paper: You start a wave, but because the fans are holding hands and reacting to each other, the wave doesn't fade. Instead, the entire stadium enters a state where they are simultaneously standing and sitting. This "standing/sitting" state is robust, spreads across the whole stadium, and is detectable by how sensitive the crowd is to a single person's movement.

The author has provided the mathematical proof and the tools (EA and QFI) to measure this "standing/sitting" state, showing that nature is more quantum and more interconnected than we previously thought.

Technical Summary

1. Problem Statement

The paper addresses the stability and emergence of macroscopic quantum superpositions (often called "cat states") in one-dimensional quantum systems.

• Context: In equilibrium, local Hamiltonians typically satisfy "clustering properties," meaning correlations decay at large distances, preventing macroscopic superpositions. However, out-of-equilibrium scenarios, such as local quenches on spontaneously symmetry-broken (SSB) ground states, can generate robust macroscopic superpositions.
• Gap: Previous work (specifically Ref [24]) demonstrated that local quenches on the SSB ground state of the free Ising model generate robust macroscopic superpositions (similar to $W$ states rather than fragile $GHZ$ states). However, it remained unclear whether this phenomenology survives in the presence of interactions, which introduce non-trivial scattering and bound state formation.
• Goal: To investigate how interactions affect the breaking of clustering and the emergence of macroscopic quantum coherence in a system with a conserved domain wall number, specifically analyzing the role of scattering phases and bound states.

2. Methodology

The author employs a combination of analytical techniques based on integrability and numerical simulations.

• Model System: The study utilizes the dual XXZ spin chain (derived from the standard XXZ chain via a Kramers-Wannier transformation).
  • Hamiltonian: $H = \frac{1}{4}\sum (\sigma^x_j - \Delta)(1 - \sigma^z_{j-1}\sigma^z_{j+1}) - h\sum \sigma^z_j \sigma^z_{j+1}$.
  • Symmetries: The model possesses a $U(1)$ symmetry (conserving the number of domain walls, $Q$) and a $Z_2$ symmetry (spin flip).
  • Ground State: In the ordered phase (high $h$), the ground state breaks $Z_2$ symmetry, existing as a superposition of ferromagnetic states $|\uparrow\dots\uparrow\rangle$ and $|\downarrow\dots\downarrow\rangle$.
• Initial Perturbation: A local quench is performed by creating a small ferromagnetic domain (a "kink" or pair of domain walls) in the symmetry-broken ground state $|\Uparrow\rangle$. This excitation creates two domain walls that propagate and interact.
• Analytical Tool: The dynamics are solved exactly using the Coordinate Bethe Ansatz (CBA). The wavefunction is decomposed into:
  1. Scattering States: Pairs of domain walls with real momenta that scatter off each other.
  2. Bound States: Pairs of domain walls with complex-conjugate momenta that form a bound state (a localized ferromagnetic domain).
• Scaling Limit: The analysis focuses on the thermodynamic limit ($N \to \infty$) and the long-time limit ($t \to \infty$) while keeping the ratio of subsystem size to time ($\tau = 2t/|A|$) fixed. This allows for the derivation of scaling limits for correlation functions and asymmetry measures.
• Asymmetry Measures: Two key quantities are computed to characterize quantum coherence:
  1. Entanglement Asymmetry (EA): Measures the relative entropy between a state and its "twirled" version (decohered in the charge basis). It quantifies coherence between different magnetization sectors.
  2. Quantum Fisher Information (QFI): Specifically the rescaled QFI (rQFI), which acts as a witness for multipartite entanglement and is sensitive to the magnitude of fluctuations between charge sectors.

3. Key Contributions and Results

A. Magnetization Profiles and Scattering Phases

• Macroscopic Deformation: The local quench induces a macroscopic deformation in the transverse magnetization profile across the entire light-cone ($\sim 2t$).
• Scattering Phase Imprint: The magnetization profile $M(\xi)$ directly reflects the scattering phase $S(k_1, k_2)$ of the model. The interference pattern between semi-classical trajectories (with and without particle exchange) is amplified by interactions.
• Clustering Breaking: The two-point correlation function does not factorize into a product of one-point functions, even at large distances, confirming the breaking of clustering properties.
• Bound States: Interactions create bound states (for $\Delta > 1$). These states move slower than scattering states and do not contribute to the breaking of clustering in the same way; they effectively restore the ground state magnetization locally as $\Delta \to \infty$.

B. Entanglement Asymmetry (EA)

• Logarithmic Growth: The EA, $\Delta S(\rho_A, Z_A)$, grows logarithmically with time ($\sim \log t$) for subsystems within the light-cone.
• Physical Interpretation: This growth signifies that the subsystem enters a coherent superposition of a number of magnetization sectors that diverges linearly with time. The state is not a simple superposition of two states (like a cat state) but a complex superposition of many sectors (resembling a $W$-state structure).
• Scaling Limit: The leading behavior is universal and determined by the probability of finding scattering states within the subsystem, independent of the specific details of the bound states (which only contribute $O(1)$ corrections).

C. Quantum Fisher Information (QFI)

• Macroscopic Coherence: The rescaled QFI, $\chi(\rho_A, Z_A)$, remains of order $O(1)$ in the scaling limit for intermediate times.
• Significance: A non-zero rQFI in the scaling limit is a rigorous witness that the state is a coherent superposition of magnetization sectors with charges differing by an amount proportional to the system size. This confirms the robustness of the macroscopic superposition against interactions.
• Dynamics: The rQFI grows as $\tau^2$ (where $\tau = 2t/|A|$) until the light-cone fills the subsystem, then decays as $1/\tau$ as particles escape.

D. Theoretical Link between EA and QFI

• Generalized Inequality: The paper derives a new inequality relating EA and QFI for mixed states, generalizing the known variance/entropy inequality for pure states.
  $$\Delta S(\rho, O) \leq \frac{1}{2} \log \left[ 2\pi e \left( \frac{1}{4} F_Q(\rho, O) + \frac{1}{12} \right) \right]$$
• Implication: This inequality provides a lower bound on the QFI based on the EA. It suggests that if the EA scales logarithmically with system size (indicating many sectors), the QFI must scale quadratically (indicating macroscopic entanglement), provided the state is sufficiently "pure" or the bound states are negligible.

4. Significance

• Robustness of Macroscopic Superpositions: The work demonstrates that macroscopic quantum superpositions generated by local quenches are robust against interactions. Even with non-trivial scattering and bound state formation, the system maintains macroscopic coherence.
• Beyond Cat States: It confirms that these states are structurally different from fragile $GHZ$ (cat) states; they are more akin to $W$ states, surviving the loss of parts of the system.
• Diagnostic Tools: The paper establishes EA and QFI as powerful, experimentally relevant tools for detecting symmetry breaking and macroscopic coherence in non-equilibrium many-body systems.
• Integrability and Universality: By solving the interacting case exactly via Bethe Ansatz, the paper bridges the gap between free models and interacting systems, suggesting that the phenomenology of "kink" excitations is a universal feature of SSB systems with domain wall excitations, potentially applicable to non-integrable models as well.

In summary, Ferro shows that interactions do not destroy the macroscopic quantum coherence induced by local quenches in symmetry-broken systems; rather, they imprint the scattering physics onto the macroscopic observables, while the system retains a robust, multipartite entangled structure characterized by logarithmic growth in asymmetry and finite rescaled QFI.