Non-invertible symmetries out of equilibrium: Eigenstate order and Floquet physics

This paper demonstrates how Rep($D_8$) non-invertible symmetries manifest in non-equilibrium dynamics by inducing unique spectral degeneracies, distinct eigenstate orders in disordered Hamiltonians, and novel edge modes that exhibit temperature-dependent oscillations or period-doubling in Floquet systems, all while remaining symmetric under invertible symmetries yet charged under the non-invertible symmetry.

The Gist

Imagine a vast, complex dance floor where particles (qubits) are constantly moving. In the world of quantum physics, scientists usually study the "ground state" of this dance floor—the calm, quiet moment when everyone is standing still in their most comfortable positions. This paper, however, asks a different question: What happens when the music is loud, the dancers are moving fast, and the system is far from calm?

The authors, Yabo Li and Aditi Mitra, explore a strange new type of "rule" that governs this chaotic dance, called a non-invertible symmetry.

The Magic Mirror vs. The Broken Mirror

To understand this, let's use a mirror analogy.

• Normal (Invertible) Symmetry: Imagine a perfect mirror. If you look in it, you see a reflection. If you look at the reflection in a second mirror, you get back to yourself. You can undo the action. This is like a standard symmetry in physics.
• Non-Invertible Symmetry: Now, imagine a "magic mirror" that doesn't just reflect you; it splits you into two versions or projects you into a specific group. If you try to look in a second mirror to undo it, you don't get back to your original self. You might get a projection of yourself, or nothing at all. You cannot simply "undo" the action. This is what the authors call non-invertible.

The paper focuses on a specific type of these magic mirrors called Rep(D8).

The Dance of Disorder

The researchers studied what happens when they introduce "disorder" into the system—like shaking the dance floor randomly.

• The Finding: Even in this chaotic, noisy environment, the "magic mirror" rules create special patterns.
• The Analogy: Imagine a crowd of people dancing. Usually, if you shake the floor, everyone gets confused and the patterns disappear. But with these special rules, the dancers form pairs that stay perfectly synchronized, even when the floor is shaking. These pairs are "degenerate," meaning they have the exact same energy, and the disorder can't easily break them apart. It takes a massive amount of effort (scaling with the size of the whole room) to finally break this perfect sync.

The "String" of Order

How do they know these patterns exist? They use a tool called a string order parameter.

• The Analogy: Imagine a long string of beads. In a normal, chaotic system, if you pull on one end, the whole string wiggles randomly. But in these special quantum states, the string holds a secret message. Even if you look at beads far apart from each other, they still "know" what the others are doing. The paper shows that in these non-invertible states, this "string" of connection remains strong and visible, acting like a fingerprint that proves the special symmetry is still there, even in the excited, noisy states.

The Edge Dancers: Zero and Double-Time

The most exciting part of the paper happens at the edges of the system (the boundaries of the dance floor).

• The Setup: The researchers created a scenario where one side of the floor follows one set of dance rules, and the other side follows a different set. Where they meet is an "interface."
• The Result: At this interface, a special dancer (an "edge mode") appears.
  1. The Zero Mode: In a standard, calm system, this dancer stands perfectly still (zero energy).
  2. The Period-Doubled Mode: In a "Floquet" system (where the rules of the dance floor change rhythmically, like a strobe light), this dancer doesn't just stand still. They start dancing in a rhythm that is twice as slow as the music. If the music beats every second, the dancer moves every two seconds.

The Twist: Who is the Dancer?

Here is the unique twist the paper discovered.

• In previous studies of similar "slow-dancing" edge modes, the dancer was charged with a "normal" symmetry charge (like wearing a specific color shirt that matches the music).
• In this paper: The dancer is neutral to the normal rules (they don't wear the color shirt), but they are charged by the "magic mirror" (non-invertible) rule.
• The Metaphor: Imagine a security guard at a club. Usually, the guard checks for a specific ID card (normal symmetry). But in this new club, the guard ignores the ID card and instead checks for a secret handshake (non-invertible symmetry). The edge mode is the only one who knows the secret handshake, making it protected and unique.

Summary

In simple terms, this paper shows that even when a quantum system is chaotic, noisy, and far from equilibrium, these strange "non-invertible" rules act like a hidden safety net. They:

1. Protect specific energy levels from being broken by disorder.
2. Create long-range connections (strings) that survive the chaos.
3. Create special "edge dancers" at the boundaries that move in unique, slow rhythms, protected by the magic mirror rules rather than standard ones.

The authors conclude that these symmetries are not just theoretical curiosities for calm, quiet systems; they are robust and active even in the wildest, most energetic parts of the quantum world.

Technical Summary

Technical Summary: Non-invertible symmetries out of equilibrium: Eigenstate order and Floquet physics

Problem Statement
While non-invertible symmetries (described by fusion categories) have been extensively studied in the context of ground states and Symmetry Protected Topological (SPT) phases, their manifestations in non-equilibrium dynamics remain largely unexplored. Specifically, it is unclear how these symmetries, which do not form a group (their action and conjugate do not yield unity), influence the entire energy spectrum, eigenstate order, and the stability of edge modes in both energy-conserving (Hamiltonian) and discrete-time (Floquet) dynamics. The authors focus on the $\text{Rep}(D_8)$ non-invertible symmetry, an anomaly-free example, to investigate these phenomena beyond the ground state sector.

Methodology
The authors employ a combination of analytical derivations and numerical simulations on a one-dimensional lattice of $L=4N$ qubits.

1. Model Construction: They define the $\text{Rep}(D_8)$ symmetry on the lattice using three generators: two invertible $\mathbb{Z}_2$ symmetries ($\eta_e, \eta_o$) and a non-invertible Kennedy-Tasaki (KT) duality operator ($K_T$). The KT operator is realized via a sequential circuit and a Matrix Product Operator (MPO) construction.
2. Hamiltonian Dynamics: They construct fixed-point Hamiltonians for trivial, odd, and even SPT phases of $\text{Rep}(D_8)$. To study eigenstate order, they introduce quenched disorder to these Hamiltonians. They analyze spectral statistics (specifically the $\langle r \rangle$-ratio) and the behavior of string order parameters defined via truncated MPOs of the KT operator.
3. Floquet Dynamics: They construct Floquet unitaries and Hamiltonians with spatial boundaries separating distinct SPT phases. They derive exact solutions for edge modes by mapping the system to free fermions within specific symmetry sectors using a Jordan-Wigner transformation.
4. Dynamics Analysis: They compute autocorrelation functions of edge operators at various effective temperatures to probe the stability and oscillation frequencies of edge modes.

Key Contributions and Results

• Spectral Degeneracies and Eigenstate Order:

  • The authors demonstrate that the non-invertible $K_T$ symmetry protects spectral degeneracies in disordered Hamiltonians. Unlike standard symmetries where disorder lifts degeneracies at low orders, degeneracies protected by $K_T$ can only be lifted by perturbations of order scaling with the system size. This is because $K_T$ maps eigenstates to degenerate partners, and lifting the degeneracy requires off-diagonal terms in the effective Hamiltonian that involve extensive products of local operators.
  • Eigenstate Order: In disordered systems, the authors find that the spectrum of an odd SPT Hamiltonian contains eigenstates with both trivial and non-trivial SPT order (detected via non-zero expectation values of string order parameters). This contrasts with ordinary SPTs where the spectrum typically exhibits a single type of order. However, in the perturbative regime of the trivial phase, the spectrum reverts to containing only trivial order.
  • Phase Transitions: When coupling odd and even SPT Hamiltonians, the transition between eigenstate orders is accompanied by a change in the $\langle r \rangle$-ratio and the vanishing of the corresponding string order parameter.

• Edge Mode Dynamics:

  • Hamiltonian Dynamics: At the interface between two distinct SPT phases, the system hosts edge modes. In the presence of disorder (fluctuations in coupling strengths), these modes are not exact zero modes but exhibit slow oscillations. The frequency of oscillation is determined by the effective chain length of the cluster model sectors, weighted by the effective temperature. In the zero-temperature limit, the mode becomes an exact zero mode.
  • Floquet Dynamics: The authors construct a Floquet model where the edge mode exhibits period-doubled dynamics (a $\pi$-mode). This occurs when the Floquet unitary is modified by the non-invertible $K_T$ operator.
  • Symmetry Characterization: A crucial distinction is identified between these modes and conventional Floquet SPT edge modes. In standard $\mathbb{Z}_2$ models, zero and $\pi$ modes are charged under the invertible symmetry. In contrast, the edge modes in this $\text{Rep}(D_8)$ model are symmetric under the invertible $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry but are charged under the non-invertible $K_T$ symmetry.

Significance
The paper establishes that non-invertible symmetries have profound implications for non-equilibrium physics, extending beyond ground state properties. The primary significance lies in demonstrating that:

1. Non-invertible symmetries can protect spectral degeneracies against disorder in a manner distinct from invertible symmetries, requiring system-size scaling to lift.
2. Eigenstate order in non-invertible SPTs can be more complex, allowing for the coexistence of different string order parameters within the same spectrum.
3. Edge modes in non-invertible SPTs possess a unique symmetry protection mechanism: they are charged under the non-invertible symmetry while remaining neutral under the associated invertible symmetries. This distinguishes them from the edge modes of conventional Floquet SPTs.

The authors conclude that while these non-equilibrium phases require localization to be stable against heating, the stable dynamics presented (slowly oscillating edge modes and spectral degeneracies) should manifest for long times, particularly in the thermodynamic limit. The work opens the door to understanding non-invertible symmetries in the context of time-crystalline behavior and Thouless pumps, though a full classification of non-invertible Floquet SPTs remains a subject for future research.