Exact Quantum Many-Body Scars by a generalized Matrix-Product Ansatz

This paper introduces a generalized matrix-product ansatz based on local error cancellation to construct exact eigenstates for non-frustration-free quantum many-body systems, demonstrating its validity through explicit examples in both one and two spatial dimensions.

The Gist

The Big Picture: Finding Order in Chaos

Imagine a massive, chaotic dance floor filled with thousands of dancers (particles). In most quantum systems, if you start the music, the dancers eventually mix up completely, forgetting their starting positions. This is called "thermalization" or "ergodicity"—everything becomes a hot, random soup.

However, physicists have discovered a few rare cases where some dancers refuse to mix. They keep dancing in a specific, repeating pattern, even though the music is loud and chaotic. These special, stubborn patterns are called Quantum Many-Body Scars. They are like "ghosts" of order that survive in a sea of chaos.

The problem is that finding these scars is usually like finding a needle in a haystack. Most methods to find them only work if the system is perfectly balanced (a condition called "frustration-free"). If the system is slightly unbalanced or "frustrated," the old methods break down.

This paper introduces a new, more flexible tool to find these scars, even in messy, unbalanced systems.

The New Tool: The "Local Error Cancellation" Trick

The authors, Sascha Gehrmann and Fabian H.L. Essler, developed a new mathematical recipe. To understand it, let's use an analogy of a relay race.

1. The Old Way (Frustration-Free): Imagine a relay race where every single runner must run perfectly. If one runner stumbles, the whole team loses. In physics, this means every tiny part of the system must be in a perfect, zero-error state. This is very hard to achieve in complex systems.
2. The New Way (Generalized Ansatz): The authors realized you don't need every runner to be perfect. You just need the mistakes to cancel each other out.
   • Imagine Runner A trips and falls forward (creating an "error").
   • But Runner B, who is right behind them, trips and falls backward in a way that perfectly undoes Runner A's mistake.
   • If you look at the whole team, the errors have vanished, and the team finishes the race perfectly, even though individuals stumbled along the way.

The paper calls this a "local error cancellation ansatz." It's based on an old idea used to study how particles move in a line (the Derrida-Evans-Hakim-Pasquier method), but the authors have upgraded it to work for complex quantum spin systems.

How They Tested It

The authors didn't just talk about the theory; they built specific examples to prove it works. They acted like architects building houses in different neighborhoods:

• One-Dimensional Chains (The Hallway): They built a model of a long line of spins (like a row of dominoes).
  • Example 1: They found a whole family of scar states (a "degenerate multiplet") in a system with a specific type of magnetic twist. It's like finding a whole choir of singers who can all hit the same perfect note, even though the room is noisy.
  • Example 2: They found a single, isolated scar in a different setup.
• Two-Dimensional Grids (The Chessboard): They moved to a square grid (like a checkerboard).
  • They showed that this "cancellation trick" works even when the system is two-dimensional and has complex magnetic fields. They found exact solutions for Spin-2 and Spin-1 models that were previously thought to be too messy to solve exactly.

Why This Matters (According to the Paper)

The paper highlights a few key takeaways:

1. It's Exact: Unlike many computer simulations that give you an approximate answer, this method gives you the exact mathematical description of these special states.
2. It's Simple (Relatively): The resulting states can be written down using a compact mathematical format called a "Matrix Product State" (MPS). Think of this as a highly efficient compression algorithm. Instead of needing a library of books to describe the state, you only need a small notebook.
3. It's Accessible: Because these states are so simple (low "entanglement"), the authors suggest they could be observed on current quantum computers and simulators. You don't need a futuristic machine to see them; you can see them in the dynamics of local observables today.

Summary

The paper presents a clever new mathematical "cancellation trick." It allows physicists to find exact, stable quantum patterns (scars) in systems that are messy and unbalanced. By letting local errors cancel each other out globally, they can construct these states in both 1D lines and 2D grids, opening the door to studying these rare quantum phenomena on real, existing quantum hardware.

Technical Summary

Technical Summary: Exact Quantum Many-Body Scars by a Generalized Matrix-Product Ansatz

Problem Statement
The search for mechanisms that break ergodicity in interacting quantum many-body systems has identified "quantum many-body scars" as a distinct class of non-thermal eigenstates. These states, characterized by low entanglement and simple closed-form expressions, allow for persistent oscillations in non-equilibrium dynamics. While various frameworks exist to construct such states (e.g., embedding, group-theoretic structures, spectrum-generating algebras), a dominant method for obtaining exact Matrix Product State (MPS) eigenstates relies on the "frustration-free" condition. In this standard approach, the local Hamiltonian density $h_{j,k}$ annihilates the state locally ($h_{j,k}|\Psi\rangle = 0$). This condition is restrictive and limits the class of Hamiltonians for which exact MPS eigenstates can be constructed. The authors address the question of whether it is possible to construct exact MPS eigenstates for Hamiltonians that are not frustration-free.

Methodology: Generalized DHEP Ansatz
The authors propose a generalization of the Derrida-Evans-Hakim-Pasquier (DHEP) ansatz, originally developed for the stationary state of the asymmetric simple exclusion process (ASEP). The core of their method is a "local error cancellation" mechanism.

1. General Framework: For a quantum spin system on a graph $G$ with Hamiltonian $H = \sum h_{j,k}$, the authors seek an MPS $|\Psi\rangle$ defined by tensors $A$.
2. Relaxed Condition: Instead of requiring $h_{j,k}|\Psi\rangle = 0$, they impose a condition where the action of the local Hamiltonian density on the MPS tensors generates "error terms" that are not zero but can be expressed as a telescoping sum. Specifically, for an edge connecting nodes $k$ and $\ell$:
   $$h_{k,\ell} [A_k] [A_\ell] = [E^{(k,\ell)}_k] [A_\ell] + [A_k] [E^{(k,\ell)}_\ell]$$
   Here, $E$ represents a tensor of the same structure as $A$.
3. Global Cancellation: If the sum of these error tensors over all edges connected to a node vanishes ($\sum_{e \in E_v} E^e_v = 0$), the local errors cancel out globally due to translational invariance (or appropriate boundary conditions). Consequently, $H|\Psi\rangle = 0$ (or a constant energy), yielding an exact eigenstate despite the Hamiltonian not being frustration-free.
4. Algebraic Reduction: The problem reduces to finding representations of a quadratic algebra defined by the local cancellation condition.

Key Results and Examples
The authors demonstrate the efficacy of this method by constructing exact MPS scars in both one and two spatial dimensions for non-frustration-free Hamiltonians.

• 1D Model I (Degenerate Multiplet of Scars):

  • System: A spin-$S$ chain with a Hamiltonian combining a generalized Rydberg term (penalizing adjacent excitations) and a Dzyaloshinskii–Moriya (DM) interaction.
  • Result: For $S=1/2$, the authors construct an exact MPS with zero energy. The state exhibits a finite correlation length.
  • Degeneracy: Numerical analysis reveals an $N/2 + 2$-fold degenerate zero-energy multiplet. The authors show that $N/2 + 1$ of these states (all with zero momentum) can be generated by expanding the MPS parameter $b$ in a power series of $1/a$. This expansion yields a tower of states $|v_n\rangle$ where $|v_n\rangle$ contains at most $n$ overturned spins, analogous to structures found in the spin-1/2 XYZ chain.
  • Higher Spins: The construction extends to $S \geq 1$, suggesting an extensive number of translation-invariant zero-energy eigenstates within the Rydberg subspace.
  • Boundary Conditions: The method is also applied to open chains with specific boundary magnetic fields, yielding a non-degenerate exact eigenstate with a tunable energy eigenvalue.

• 1D Model II (Isolated Scar):

  • System: A spin-1 chain with a specific Hamiltonian involving quadratic spin terms and a magnetic field.
  • Result: An exact zero-energy MPS is constructed. Unlike Model I, the zero-energy eigenspace is non-degenerate. The state has a finite, non-vanishing correlation length $\xi = \log(3)$.

• 2D Square Lattice Models:

  • Spin-2 Model: The authors construct a translationally invariant zero-energy MPS for a spin-2 model on a square lattice with a magnetic field. In the absence of the field, the model is frustration-free; the non-zero field breaks this property, turning the state into a scar.
  • Spin-1 XYZ Model with DM Interaction: An exact zero-energy MPS is found for a spin-1 XYZ model with DM interaction and a magnetic field. When the DM interaction is non-zero, the state is no longer a ground state but remains an exact eigenstate. Notably, this MPS can be decomposed into two product states, resulting in a vanishing correlation length in the infinite volume limit.

Significance and Claims
The paper claims to provide a systematic algorithm for obtaining exact MPS eigenstates that relaxes the strict frustration-free condition. By allowing for local error terms that telescope to zero globally, the method generalizes both the DHEP ansatz and the earlier work of Klümper and Zittartz.

The authors emphasize that the constructed scar states possess low bond dimensions, making them theoretically accessible for observation in the dynamics of local observables on present-day quantum computers. The work is modest in its scope, presenting specific illustrative examples in 1D and 2D rather than a universal classification of all scarred systems. The authors note that while they have verified the construction for $S \leq 2$, a proof for $S > 2$ remains an open question. They also acknowledge a recent preprint investigating a more general version of this ansatz in the context of injective MPS.