Scarred ferromagnetic phase in the long-range transverse-field Ising model

This paper reports the discovery of a "scarred ferromagnetic phase" in the long-range transverse-field Ising model, where specific initial states with small magnetic domains selectively evolve into ferromagnetic equilibrium despite the absence of a thermal ferromagnetic phase, while other initial conditions relax to a paramagnetic state.

The Gist

Imagine a long line of people holding hands, each holding a tiny magnet that can point either Up or Down. In physics, we call this an "Ising model." Usually, if you shake this line up enough (add heat or energy), everyone starts pointing in random directions. The magnets cancel each other out, and the whole line becomes "disordered" or paramagnetic. It's like a crowd of people at a chaotic concert, all looking in different directions.

However, in this new paper, the researchers discovered a strange exception to this rule. They found that even in a chaotic, high-energy crowd, there are specific, hidden "secret clubs" where people spontaneously decide to stand up and face the same way, creating an ordered or ferromagnetic state.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Long-Range Dance Floor

The scientists studied a special version of this magnet line where the people can "talk" to each other even if they are far apart (long-range interactions).

• The Rule: Usually, in a 1D line (a single file), you can't have a permanent order if things are hot or energetic. The Mermin-Wagner theorem says order should be impossible.
• The Twist: Because these magnets can reach across the whole line to talk to each other, they can sometimes stay ordered. But the researchers looked at a specific "hot" zone where, according to all the old textbooks, order should be impossible.

2. The Discovery: The "Scarred" States

In a chaotic system, you expect everything to eventually mix up and become random (thermalize). But the researchers found a set of special energy states they call "Ferromagnetic Scars."

• The Analogy: Imagine a giant, chaotic ballroom dance. Most couples are spinning wildly in random directions (the "paramagnetic sea"). But hidden within this chaos are a few specific dance moves (the "scars") where, if you start with the right steps, the whole room suddenly locks into a synchronized, orderly formation.
• Why "Scarred"? In physics, a "scar" usually refers to a defect or a mark left behind. Here, it means these states are "marked" by order. They are like scars on a chaotic system that refuse to heal into randomness. They violate the usual rule that says "chaos wins."

3. The Secret Ingredient: How to Find the Club

The most exciting part is how you get into these secret clubs. It depends entirely on how you start the dance.

• The "Small Domains" Trick:
  • Scenario A (The Winner): You start with a few tiny groups of people standing together (small magnetic domains). For example, a few "Up" people, then a few "Down," then a few "Up."
  • Result: If you start this way, the system recognizes the pattern and jumps into one of those special "Scarred" states. The whole line stays ordered and magnetized, even though it's supposed to be chaotic.
  • Scenario B (The Loser): You start with one giant block of "Up" people or a completely random mess.
  • Result: The system ignores the secret club. It falls back into the chaotic "paramagnetic sea," and the order disappears.

The Metaphor: Think of it like a game of "Simon Says."

• If Simon says, "Do a tiny, specific wiggle" (small domains), the whole crowd suddenly locks into a perfect, synchronized pose (Ferromagnetic Scar).
• If Simon says, "Spin wildly" or "Do a giant jump" (large domains/random), the crowd just goes back to being a chaotic mess.

4. The "Scarred Ferromagnetic Phase"

The authors propose a new "phase of matter" called the Scarred Ferromagnetic Phase.

• Normal Physics: If you have a hot system, it cools down to a random state.
• This New Phase: If you prepare the system with the right small starting pattern, it gets "stuck" in a permanent ordered state, defying the heat. It's like a cup of coffee that, if you stir it just right, stays hot forever without a heater.

Why Does This Matter?

1. It Breaks the Rules: It shows that even in systems where chaos is expected, order can survive if you know the "secret handshake" (the initial conditions).
2. It Explains Past Mysteries: Scientists had seen strange, long-lasting order in similar experiments but couldn't explain why. This paper says, "Ah, you were just accidentally hitting the 'Scarred' states!"
3. Future Tech: This could help us build better quantum computers. If we can control these "scarred" states, we might be able to store information (order) for a very long time, even in noisy, chaotic environments.

Summary

The paper reveals that in a long line of magnets, chaos isn't the only option. If you start with a few small, specific patterns, the system can get "scarred" with order, refusing to become random. It's a hidden pocket of stability in a world of chaos, waiting to be found by the right starting move.

Technical Summary

1. Problem Statement

The paper addresses a fundamental paradox in quantum many-body physics: the existence of ordered (ferromagnetic) equilibrium states in regimes where standard statistical mechanics predicts a disordered (paramagnetic) phase.

• Context: In one-dimensional (1D) systems with long-range interactions (power-law decay $1/r^\alpha$), a ferromagnetic phase at finite temperature is theoretically forbidden for $\alpha > 2$ due to the Mermin-Wagner theorem and standard thermalization (Eigenstate Thermalization Hypothesis, ETH).
• The Conflict: Previous studies and experiments have observed long-lived ferromagnetic states in these regimes, but the mechanism behind their stability as equilibrium states (rather than just prethermal transients) was not fully understood.
• Goal: The authors aim to identify the microscopic origin of these stable ferromagnetic states in the 1D transverse-field Ising model (TFIM) with long-range interactions, specifically in the regime where $\alpha > 2$ and $T > 0$.

2. Methodology

The authors employ a combination of exact diagonalization, spectral analysis, and dynamical evolution simulations on the 1D TFIM with power-law interactions.

• Model Hamiltonian:
  $$\hat{H}_{\text{TFIM}} = -\frac{1}{K(\alpha)} \sum_{i<j} \frac{J}{|i-j|^\alpha} \hat{\sigma}^x_i \hat{\sigma}^x_j + h \sum_i \hat{\sigma}^z_i$$
  Where $J$ is the coupling, $h$ is the transverse field, $\alpha$ is the interaction exponent, and $K(\alpha)$ is the Kac factor ensuring extensivity.

• Symmetry Analysis & Generalized ETH:

  • The system possesses a $\mathbb{Z}_2$ parity symmetry ($\hat{\Pi} = \prod \hat{\sigma}^z_i$).
  • Standard ETH suggests that in chaotic systems, eigenstates are thermal, and the order parameter (magnetization $\hat{m}$) should vanish ($\langle \hat{m} \rangle = 0$) because $\hat{m}$ flips parity.
  • The authors utilize a generalized ETH framework (based on Ref. [63]) which considers 2D subspaces spanned by pairs of eigenstates with opposite parity ($|E_{n,+}\rangle$ and $|E_{n,-}\rangle$) that are nearly degenerate (small energy gap).
  • They calculate the eigenvalues ($\lambda_n$) of the magnetization operator restricted to these subspaces. If $|\lambda_n| \neq 0$ in the thermodynamic limit, symmetry breaking is possible.

• Spectral Scanning:

  • They compute the eigenvalues of the normalized total magnetization across the entire energy spectrum for various system sizes ($N$) and interaction exponents ($\alpha$).
  • They identify "scarred states" as eigenstates (or subspaces) where $|\lambda_n|$ is significantly non-zero, despite the surrounding spectrum being paramagnetic.

• Dynamical Simulations:

  • They simulate time evolution starting from specific initial states with defined magnetic domain structures.
  • They analyze the Local Density of States (LDOS) to see which eigenstates are populated.
  • They track the long-time average of the magnetization $\langle \hat{m}(t) \rangle$ to determine the final equilibrium phase.

3. Key Contributions

1. Discovery of Ferromagnetic Scars: The authors identify a large set of "ferromagnetic scarred states" in the spectrum of the long-range TFIM. These are high-energy eigenstates that violate the generalized ETH by maintaining non-zero magnetization.
2. Existence in the "Forbidden" Regime: They demonstrate that these scars exist and are exponentially numerous even for $\alpha > 2$ (specifically $\alpha \approx 2.2$), a regime where no thermal ferromagnetic phase exists.
3. Selective Population Mechanism: They establish that simple, experimentally accessible initial conditions—specifically those with a small number of small magnetic domains—selectively populate these scarred states.
4. Dynamical Phase Transition: They propose a new type of dynamical phase transition controlled by the size and number of magnetic domains in the initial state, rather than just temperature or external field strength.

4. Key Results

• Spectral Structure (Fig. 1):

  • For $\alpha = 0.6$ (strong long-range), a low-energy ferromagnetic phase exists.
  • For $\alpha = 2.2$ (where thermal ferromagnetism is forbidden), the spectrum is not uniformly paramagnetic. Instead, it exhibits bands of scarred states with non-zero magnetization eigenvalues ($\lambda_n \neq 0$) interspersed with paramagnetic regions.
  • The number of these scarred states scales exponentially with system size $N$ for $\alpha \lesssim 3$, indicating they are not rare anomalies but a dominant feature of the Hilbert space.

• Dynamical Evolution (Fig. 2):

  • State $|\psi_1\rangle$: Composed of three small magnetic domains. Its LDOS overlaps heavily with the ferromagnetic scarred bands. Result: The system evolves to a ferromagnetic equilibrium ($\langle \hat{m} \rangle \neq 0$).
  • State $|\psi_2\rangle$: Composed of two large magnetic domains. Its LDOS overlaps primarily with paramagnetic states. Result: The system relaxes to a paramagnetic equilibrium ($\langle \hat{m} \rangle \approx 0$).
  • This proves that the final state depends on the initial domain structure, not just energy.

• Phase Diagram (Fig. 3):

  • The authors map the equilibrium magnetization as a function of the ratio of up-spins ($N_\uparrow/N$) and the number of domains.
  • A critical threshold exists: Initial states with a low density of spins and few domains remain ferromagnetic. States with larger domains or higher spin density relax to the paramagnetic state.
  • This confirms the existence of a "Scarred Ferromagnetic Phase" distinct from the thermal phase.

• Robustness: The effect persists for various magnetic fields ($h$) and system sizes, though it diminishes as $\alpha \to \infty$ (short-range limit) or $h/J$ becomes too large.

5. Significance and Implications

• Resolution of Long-standing Anomalies: The paper provides a theoretical explanation for previous experimental observations (e.g., in trapped ions and Rydberg atoms) where ferromagnetic order persisted in 1D long-range systems at finite temperatures, a phenomenon previously attributed to prethermalization or confinement. The authors argue these are true equilibrium states stabilized by scars.
• Beyond ETH: It demonstrates that the Eigenstate Thermalization Hypothesis can be violated in chaotic long-range systems not just by rare "scar" states, but by a macroscopic set of them that can be dynamically accessed.
• New Dynamical Control: It introduces a mechanism to control the final state of a quantum system by engineering the initial domain structure (number and size of domains), offering a potential protocol for quantum simulation and memory storage.
• Connection to Confinement: The results link the existence of these scars to the phenomenon of domain-wall confinement, suggesting that symmetry-breaking equilibrium states in these models are intrinsically tied to the existence of constants of motion that restrict domain wall propagation.

In conclusion, the paper establishes that long-range interactions induce a "scarred ferromagnetic phase" where specific initial conditions lead to stable, ordered equilibrium states in regimes traditionally expected to be disordered, fundamentally altering our understanding of thermalization in quantum many-body systems.