Construction of asymptotic quantum many-body scar states in the SU($N$) Hubbard model

This paper constructs asymptotic quantum many-body scar states in one-dimensional SU($N$) Hubbard chains ($N\geq 3$) by embedding them into an auxiliary subspace governed by an SU($N$) ferromagnetic Heisenberg parent Hamiltonian, thereby demonstrating that gapless magnons in this model yield explicit scars with vanishing energy variance and subvolume entanglement entropy in the thermodynamic limit.

The Gist

The Big Picture: Finding "Ghost" Particles in a Chaotic Room

Imagine you throw a handful of marbles into a giant, chaotic room filled with obstacles. Usually, the marbles bounce around wildly, mixing with everything until they settle into a random, messy equilibrium. In physics, this is called thermalization. It's like stirring sugar into hot coffee; eventually, the sugar dissolves completely, and you can't tell where the sugar was anymore.

However, scientists have discovered a special class of systems where some particles refuse to mix. They bounce around in a very specific, predictable pattern, almost like a ghost that refuses to interact with the rest of the room. These are called Quantum Many-Body Scars (QMBS). They are the "special kids" in the chaotic classroom who somehow remember their original seat.

This paper introduces a new type of "ghost" called Asymptotic Quantum Many-Body Scars (AQMBS). These aren't perfect ghosts that stay in one spot forever; rather, they are "almost ghosts" that behave like perfect ghosts when the room gets infinitely large.

The Problem: The "Parent" was always the same

To find these special states, physicists use a clever trick called the Parent Hamiltonian method.

• The Analogy: Imagine you want to find a specific type of rare bird (the Scar State) in a forest. Instead of searching the whole forest, you build a special cage (the Parent Hamiltonian) that only lets that specific bird sit comfortably on the floor.
• The Twist: Once you find the cage, you look at the birds that are just above the floor (the excited states). In previous studies, this cage always turned out to be a very simple, standard cage (a "Spin-1/2 Ferromagnetic Heisenberg Model"). It was like finding that every rare bird in every forest lived in the exact same type of wooden crate.

The authors of this paper asked: "What if we build a cage for a more complex bird? Will the crate look different?"

The Solution: A New Cage for a New Bird

The authors studied a specific type of quantum system called the SU(N) Hubbard Model.

• The Setting: Think of this as a one-dimensional train track where "passengers" (electrons) can hop between cars.
• The Complexity: Unlike standard models where passengers only have two options (like "Up" or "Down" spin), these passengers have N different flavors (colors, types, or identities). The authors looked at cases where there are 3 or more flavors ($N \ge 3$).

They applied their "cage-building" recipe to this complex system. Here is what they found:

1. The New Cage: When they built the Parent Hamiltonian for this complex system, it didn't turn into the simple wooden crate from before. Instead, it became a Ferromagnetic SU(N) Heisenberg Model.

   • Metaphor: Instead of a simple wooden crate, they built a high-tech, multi-colored, magnetic cage that respects all the different flavors of the passengers. It's a much more sophisticated structure.

2. The New Ghosts (AQMBS): Inside this new, complex cage, the "excited states" (the birds just above the floor) turned out to be Magnons.

   • Metaphor: Imagine a line of people holding hands. If one person lets go and moves down the line, that moving "hole" or "wave" is a magnon. In this paper, these waves are the AQMBS. They are low-energy ripples that travel through the system without losing their shape.

3. Why They Are Special:

   • They don't mix: Even though the whole system is chaotic, these specific ripples (AQMBS) stay distinct.
   • They are "almost" perfect: If you look at them in a small system, they might wiggle a bit. But as the system gets huge (the thermodynamic limit), they become perfectly stable. Their energy uncertainty vanishes.
   • They are simple: Despite being in a complex quantum system, these states are surprisingly simple to describe mathematically. They don't get "entangled" (tangled up with each other) in a messy way. They follow a "sub-volume law," meaning their complexity grows slowly, not explosively.

The "N=3" Surprise

The paper also notes a funny quirk. When the number of flavors is exactly 3 ($N=3$), the math behaves in a weirdly symmetric way. It turns out that the "cage" for 3 flavors looks mathematically identical to the cage for 4 flavors. It's like discovering that a 3-legged stool is structurally identical to a 4-legged stool if you look at it from a specific angle. This allows them to use the results for 4 flavors to solve the 3-flavor problem instantly.

Why Does This Matter?

1. Breaking the Pattern: It proves that the "Parent Hamiltonian" trick isn't limited to simple, 2-option systems. It works for complex, multi-flavor systems too.
2. New Tools: It gives physicists a new way to find stable, non-chaotic states in complex materials. This is crucial for quantum computing, where we need states that don't get messed up by noise (chaos).
3. Experimental Hope: While these are theoretical constructs, the authors suggest that with advanced technology (like optical lattices where atoms are trapped in light), we might be able to create these "SU(N) scars" in a lab. If we can, we could observe these "ghost waves" (magnons) that refuse to thermalize.

Summary in One Sentence

The authors discovered that in complex quantum systems with many particle types, you can find special, stable "ghost waves" (AQMBS) by building a sophisticated magnetic cage (the SU(N) Heisenberg model), proving that these non-chaotic states exist even in the most complicated quantum environments.

Technical Summary

1. Problem Statement

The paper addresses the phenomenon of Quantum Many-Body Scars (QMBS) and their asymptotic counterpart, Asymptotic Quantum Many-Body Scars (AQMBS), in non-integrable quantum systems.

• Context: While the Eigenstate Thermalization Hypothesis (ETH) predicts thermalization in generic non-integrable systems, QMBS are special high-energy eigenstates that violate ETH, exhibiting low entanglement and persistent revivals.
• Gap: Previous systematic constructions of AQMBS (states that are not exact eigenstates but have vanishing energy variance in the thermodynamic limit) relied on a "parent Hamiltonian" scheme where the resulting parent Hamiltonian was invariably the spin-1/2 ferromagnetic Heisenberg model.
• Objective: The authors aim to determine if this framework can be extended to systems with higher internal symmetries, specifically the SU(N) Hubbard model ($N \ge 3$), and whether a different class of parent Hamiltonians emerges.

2. Methodology

The authors employ a systematic algebraic and symmetry-based framework to construct AQMBS:

• Multiladder Restricted Spectrum-Generating Algebra (MLRSGA):
  • They generalize the standard Restricted Spectrum-Generating Algebra (RSGA), which typically uses a single ladder operator, to a multiladder case.
  • For the SU(N) Hubbard model, they identify $N-1$ independent ladder operators (derived from $\eta$-pairing operators) that generate a tower of exact QMBS states starting from a vacuum reference state.
• Symmetry-Based Formalism:
  • The Hamiltonian is decomposed into an annihilating part ($\hat{H}_A$), a spectrum-generating part ($\hat{H}_{SG}$), and a symmetry-preserving part ($\hat{H}_{sym}$).
  • They define an auxiliary Hilbert subspace $\mathcal{H}_P$ (spanned by holons and specific doublons involving a reference flavor) where the scar states reside.
• Parent Hamiltonian Construction:
  • Following the scheme by Kunimi et al., they construct a parent Hamiltonian $\hat{H}_p = \hat{P} \hat{H}_0^2 \hat{P}$, where $\hat{P}$ projects onto $\mathcal{H}_P$ and $\hat{H}_0$ is the local annihilator.
  • The ground states of $\hat{H}_p$ coincide with the exact QMBS of the original model.
  • The low-energy gapless excitations of $\hat{H}_p$ are identified as the candidate AQMBS states for the original system.
• Analytical Derivation:
  • They explicitly calculate the action of the squared hopping operator on the constrained subspace to derive the explicit form of $\hat{H}_p$.
  • They analyze the dispersion relations for periodic and open boundary conditions.
  • They utilize Matrix Product State (MPS) and Matrix Product Operator (MPO) techniques to rigorously bound the entanglement entropy.

3. Key Contributions

• Discovery of a New Parent Hamiltonian Family: The paper demonstrates that for SU(N) Hubbard models, the parent Hamiltonian is not the spin-1/2 ferromagnetic Heisenberg model, but rather the ferromagnetic SU(N) Heisenberg model. This breaks the universality of the spin-1/2 case in previous AQMBS constructions.
• Generalization to Multiladder Systems: The authors successfully extend the RSGA framework to handle multiple independent ladder operators, a necessary step for SU(N) systems with $N \ge 3$.
• Explicit Construction of AQMBS: They provide an analytic construction of AQMBS states as gapless magnon excitations of the SU(N) ferromagnetic Heisenberg parent Hamiltonian.

4. Key Results

A. The Parent Hamiltonian

By projecting the squared local hopping operator onto the subspace $\mathcal{H}_P$ (containing holons and doublons formed with a reference flavor $\alpha_1$), the authors derive:
$$\hat{H}_p = 2J^2 \sum_j \left( \hat{I}_{j,j+1} - \sum_{\mu,\nu=0}^{N-1} \hat{F}^{\mu\nu}_j \hat{F}^{\nu\mu}_{j+1} \right)$$
This is identified as the ferromagnetic SU(N) Heisenberg Hamiltonian (up to a constant shift), where $\hat{F}^{\mu\nu}$ are local SU(N) spin operators.

B. Gapless Excitations (AQMBS)

The low-energy excitations of this parent Hamiltonian are magnons.

• Periodic Boundary Conditions (PBC): One-magnon states created by $\hat{F}^{\mu0}_{PBC}(k)$ have the dispersion:
  $$E(k) = 4J^2 (1 - \cos k) \approx 2J^2 k^2 \quad (\text{near } k=0)$$
• Open Boundary Conditions (OBC): Standing-wave magnons have dispersion:
  $$E(p) = 4J^2 \left[ 1 - \cos\left(\frac{\pi p}{L}\right) \right]$$
• AQMBS Properties:
  1. Orthogonality: These magnon states are orthogonal to the exact QMBS tower.
  2. Vanishing Variance: In the thermodynamic limit ($L \to \infty$), the energy variance of these states with respect to the original Hubbard Hamiltonian vanishes.
  3. Slow Relaxation: They exhibit parametrically slow relaxation characterized by a diverging Mandelstam–Tamm bound.

C. Entanglement Scaling

Using MPS/MPO representations, the authors prove that these excited states obey a sub-volume law for entanglement entropy ($S_{vN}$):
$$S_{vN} \le (N-1) \ln L + O(1)$$
This confirms that the AQMBS states are low-entanglement states, a defining characteristic of scars, despite being high-energy excitations.

D. Special Case: N=3

The authors note that for $N=3$, the local doublon states transform under the conjugate fundamental representation, allowing for a more symmetric construction. They show that the parent Hamiltonian for $N=3$ is equivalent to the SU(4) ferromagnetic Heisenberg model after a unitary transformation.

5. Significance

• Theoretical Advancement: This work significantly broadens the scope of the parent-Hamiltonian approach for AQMBS. It proves that higher-symmetry settings (SU(N)) naturally emerge in this construction, moving beyond the ubiquitous spin-1/2 case.
• Analytic Control: It provides an analytically tractable model for studying AQMBS in strongly correlated systems with high symmetry, offering explicit formulas for dispersion and entanglement.
• Experimental Relevance: The paper discusses the potential for observing these states in optical lattice experiments with ultracold atoms (specifically alkaline-earth-like atoms with SU(N) symmetry). While preparing the initial $\eta$-pairing scar state remains a challenge, the identification of these states as magnon excitations offers a clear theoretical target for future experimental verification.
• Future Directions: The framework opens avenues for exploring AQMBS in other multiladder systems, such as pyramid scar states, and investigating the dynamics of SU(N) systems in the scar regime.