Level statistics of the disordered Haldane-Shastry model with $1/r^\alpha$ interaction

This study demonstrates that while position disorder or random magnetic fields alone fail to induce many-body localization in the long-range interacting Haldane-Shastry model, their combined presence leads to Poisson level statistics and an MBL phase, with the transition governed by a scaling parameter $\alpha\delta$ when $SU(2)$ symmetry is broken.

The Gist

Imagine a crowded dance floor where everyone is holding hands with everyone else, but the strength of their grip depends on how far apart they are standing. This is the Haldane-Shastry model, a famous physics problem describing a line of tiny magnets (spins) that interact with each other.

In the "clean" version of this dance floor, the magnets are perfectly spaced in a circle. Because they are so perfectly organized, they follow strict rules (symmetries) that make their movements predictable and repetitive, like a perfectly choreographed ballet. In physics terms, this is called an integrable system.

Now, imagine we want to see what happens when we mess things up. We introduce disorder:

1. Position Disorder: We randomly shuffle the dancers' feet so they aren't standing in perfect spots anymore.
2. Magnetic Disorder: We turn on a random, chaotic wind (a magnetic field) that pushes individual dancers in different directions.

The big question the paper asks is: Does this chaos make the dancers forget their choreography and act like a random, thermal crowd (ergodic), or does it freeze them into a rigid, stuck position where they remember exactly where they started (Many-Body Localization or MBL)?

Here is what the researchers found, broken down into simple concepts:

1. The "Goldilocks" of Chaos

In most physics systems, if you add enough disorder, the system freezes (localizes). The energy levels of the system stop mixing, and the statistics of the energy gaps look like Poisson statistics (think of raindrops hitting a roof at random times). If the system is chaotic and thermal, the statistics look like GOE (Gaussian Orthogonal Ensemble), which is like a crowded room where everyone is bumping into each other constantly.

2. The Surprise: One Disorder Isn't Enough

The researchers tried two things separately:

• Just shuffling the feet (Position Disorder): Even when they messed up the spacing, the dancers kept their perfect choreography. The system didn't freeze. It stayed in a "chaotic" state, but not quite the standard chaotic state. It was stuck in a weird middle ground.
• Just the random wind (Magnetic Field): Similarly, blowing random wind on the dancers didn't freeze them either. They kept dancing chaotically.

The Analogy: Imagine trying to stop a spinning top. If you just wiggle the table (position disorder) or just blow on it (magnetic field), it keeps spinning. It refuses to stop.

3. The Magic Combination: The "Double Whammy"

The breakthrough happened when they did both at the same time.

• They shuffled the feet and blew the random wind.
• Result: The dancers finally stopped! They froze in place. The system transitioned from a chaotic, thermal state to a localized state (MBL).

The Analogy: It's like trying to stop a spinning top. Wiggle the table and blow on it, and suddenly, the top wobbles and falls over. The combination of the two disturbances was the key to breaking the system's perfect symmetry and freezing it.

4. The "Long-Range" Twist

The model has a special parameter called $\alpha$ (alpha).

• Low $\alpha$ (Long-Range): Everyone holds hands with everyone, even the people far away. It's like a giant web. In this case, the system is very hard to freeze. Even with both disorders, it stays chaotic unless the wind is incredibly strong.
• High $\alpha$ (Short-Range): People only hold hands with their immediate neighbors. It's like a line of people holding hands. Here, it's much easier to freeze the system.

The researchers found a clever "knob" to turn: $\alpha \times \delta$ (Interaction Range $\times$ Disorder Strength).

• If you increase the disorder strength ($\delta$), you can freeze the system even if the interactions are long-range.
• If you decrease the interaction range ($\alpha$), the system freezes more easily.
• They discovered that these two factors work together in a simple mathematical relationship to predict exactly when the system will freeze.

5. The "Hidden Symmetry" Mystery

There was one weird part: At specific settings (where $\alpha=0$ or $\alpha=2$), the system behaved strangely, even without disorder.

• The Analogy: It's like a deck of cards that, when perfectly shuffled, always deals the exact same hand every time, no matter how many times you shuffle. The researchers realized this was because of "hidden symmetries" (like a secret rule in the deck) that made the energy levels overlap perfectly. Once they removed these overlaps (by looking only at unique energy levels), the system behaved normally.

Summary

This paper is a detective story about how disorder affects quantum systems.

• The Mystery: Can we freeze a long-range interacting quantum system?
• The Clue: Doing it with just one type of disorder fails.
• The Solution: You need a "double dose" of disorder (shuffling positions + random magnetic fields) to break the system's perfect symmetry and freeze it into a localized state.
• The Takeaway: Nature is tricky. Sometimes, you need two different kinds of chaos working together to stop a system from moving, rather than just one big push.

This is important because understanding how systems freeze (localize) helps us build better quantum computers that don't lose their information to heat and chaos.

Technical Summary

1. Problem Statement

The paper investigates the interplay between long-range interactions and disorder in quantum many-body systems, specifically focusing on the emergence of Many-Body Localization (MBL). While MBL is well-understood in short-range interacting systems, its behavior in long-range interacting systems remains an open frontier.

The authors focus on a variant of the spin-1/2 Haldane-Shastry (HS) model, an integrable model with all-to-all interactions decaying as $1/r^\alpha$. The HS model possesses a high degree of symmetry (Yangian $Y(sl_2)$ and $SU(2)$ spin symmetry), leading to massive spectral degeneracies that obscure standard level statistics. The central questions addressed are:

• How does the interaction range parameter $\alpha$ affect level statistics in the presence of disorder?
• Can position disorder or random magnetic fields alone induce MBL (characterized by Poisson statistics) in this long-range system?
• What is the role of symmetry breaking (specifically $SU(2)$) in the transition from ergodic (GOE) to localized (Poisson) phases?

2. Methodology

The authors employ Exact Diagonalization (ED) to study the spectral statistics of the system.

• Model Hamiltonian:

  • Clean System ($H_0$): A spin-1/2 chain on a unit circle with $N$ sites. The interaction strength between spins $p$ and $q$ scales as $1/|\eta_p - \eta_q|^\alpha$, where $\eta$ represents the chord distance. The parameter $\alpha$ controls the range: $\alpha=0$ is infinite-range (all-to-all), $\alpha=2$ is the standard HS model, and $\alpha \to \infty$ approaches nearest-neighbor interactions.
  • Position Disorder ($H_\delta$): Spins are randomly displaced from their lattice sites within a range $\delta$, breaking translational invariance but preserving $SU(2)$ symmetry.
  • Magnetic Field Disorder ($H_h$): A random longitudinal magnetic field is added, breaking $SU(2)$ symmetry down to $U(1)$.
  • Combined Disorder ($H$): The system includes both position disorder and random magnetic fields.

• Numerical Techniques:

  • Symmetry Resolution: Calculations are performed in symmetry-resolved sectors defined by total magnetization ($m$) and pseudo-momentum ($k$) to avoid trivial degeneracies from global symmetries.
  • Level Statistics Metric: The primary diagnostic is the average gap ratio $\langle \tilde{r} \rangle$, defined as the average of $\min(d_n, d_{n-1}) / \max(d_n, d_{n-1})$ over the spectrum.
    • $\langle \tilde{r} \rangle \approx 0.536$ indicates GOE (ergodic/chaotic).
    • $\langle \tilde{r} \rangle \approx 0.386$ indicates Poisson (localized/integrable).
  • Finite-Size Scaling: Simulations are performed for system sizes $N \in [10, 22]$ (disordered) and up to $N=50$ (clean HS via analytical methods), with thermodynamic extrapolation ($1/N \to 0$).
  • Degeneracy Handling: The authors distinguish between "raw energies" (including degeneracies) and "unique energies" (filtering out degeneracies) to isolate the intrinsic spectral statistics from symmetry-induced degeneracies.

3. Key Contributions and Results

A. Clean Limit ($H_0$)

• Transition Behavior: As $\alpha$ increases from 0, the system transitions from GOE statistics (chaotic, $\alpha < 1$) to Poisson-like statistics ($\alpha > 1$).
• Anomalies at $\alpha=0, 2$:
  • At $\alpha=0$ (infinite-range), the system behaves like a single large spin, resulting in anomalous statistics ($\langle \tilde{r} \rangle \to 1$) due to massive degeneracies.
  • At $\alpha=2$ (standard HS), the Yangian symmetry causes large degeneracies, forcing $\langle \tilde{r} \rangle \approx 0$.
  • Crucial Finding: Even after removing degeneracies to look at "unique energies," $\alpha=0$ and $\alpha=2$ exhibit anomalous statistics distinct from standard integrable models, suggesting these points are fundamentally different from the generic $\alpha$ behavior.
• Short-Range Regime ($\alpha > 1$): The system exhibits Poisson statistics, suggesting an underlying integrability or unresolved symmetry, even though the ground state wavefunction retains a high overlap (>99%) with the Jastrow form of the $\alpha=2$ HS model.

B. Position Disorder Only ($H_\delta$)

• No MBL Transition: Introducing position disorder alone does not lead to pristine Poisson statistics.
• Saturation: For $\alpha \ge 1$, $\langle \tilde{r} \rangle$ saturates at a value slightly higher than the Poisson limit ($\approx 0.399$).
• Symmetry Constraint: The persistence of $SU(2)$ symmetry prevents the system from localizing in the conventional MBL sense (which requires area-law entanglement scaling). The results support the theoretical incompatibility between $SU(2)$ symmetry and standard MBL.

C. Random Magnetic Field Only ($H_h$)

• Ergodic Phase: For weak fields ($h/J_{NN} < 1$), the system remains in the ergodic (GOE) phase for all $\alpha$. The field breaks $SU(2)$ but is not strong enough to induce localization.
• Localization at Strong Fields: When $h/J_{NN} \gtrsim 1$, the system transitions to Poisson statistics. This is attributed to single-particle Anderson localization dominating the spin-spin interactions.
• Symmetry Note: Despite breaking time-reversal symmetry via the magnetic field, the system retains an "unconventional" time-reversal symmetry ($e^{i\pi S_x}T_0$), keeping the statistics in the GOE class rather than GUE for weak fields.

D. Combined Disorder ($H_\delta + H_h$)

• Emergence of MBL: The most significant finding is that Poisson statistics (MBL) only emerge when both position disorder and random magnetic fields are present.
• Scaling Collapse: The transition point from GOE to Poisson depends inversely on the position disorder strength $\delta$. The authors identify a single effective control parameter $\alpha\delta$.
  • Data for different $\delta$ and $\alpha$ collapse onto a universal curve when plotted against $\alpha\delta$.
  • This suggests that once $SU(2)$ is broken by the magnetic field, the position disorder strength $\delta$ determines the critical interaction range $\alpha_c \sim 1/\delta$ for the onset of localization.
• Long-Range Stability: Even with maximum position disorder, the system remains ergodic (GOE) in the long-range regime ($\alpha < 1$). Localization only occurs in the short-range regime ($\alpha > 1$) when combined with symmetry-breaking fields.

4. Significance and Implications

• Symmetry and Localization: The work highlights that $SU(2)$ symmetry is a robust barrier to MBL in long-range interacting systems. Breaking this symmetry is a prerequisite for observing localization in this specific model.
• Universality of Short-Range Regime: The observation that the ground state overlap with the Jastrow wavefunction remains high for $\alpha > 1$ (even with disorder) suggests a common universality class for the short-range regime of the HS model.
• New Control Parameter: The identification of the scaling parameter $\alpha\delta$ provides a novel way to characterize the crossover between delocalized and localized phases in long-range disordered systems.
• Limitations: The authors note that their analysis relies solely on spectral statistics. While robust for detecting ergodicity breaking, a full characterization of MBL would require complementary diagnostics like entanglement entropy scaling and dynamical transport properties.

Conclusion

The paper demonstrates that in the long-range interacting Haldane-Shastry model, neither position disorder nor random magnetic fields alone are sufficient to induce Many-Body Localization. MBL emerges only when both types of disorder are present, which breaks the protective $SU(2)$ symmetry and allows the system to localize. The transition is governed by a scaling relation involving the interaction range and disorder strength ($\alpha\delta$), offering new insights into the stability of ergodicity in long-range quantum systems.