🔬 condensed matter

Gapped Parent Hamiltonians for the Strongly Deformed Toric Code

This paper rigorously constructs local gapped parent Hamiltonians with exponentially decaying long-range terms for strongly deformed toric code states, demonstrating that these states realize a trivial gapped phase where perimeter-law Wilson loop scaling coexists with an exact 1-form symmetry, thereby evading recent no-go theorems by relaxing standard locality assumptions.

Original authors: Nandagopal Manoj, Zack Weinstein, Jason Alicea

Published 2026-08-03

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Nandagopal Manoj, Zack Weinstein, Jason Alicea

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible Lego set. Most of the time, when you build something with these Legos, the pieces snap together in predictable ways. If you pull one piece, the whole structure might wobble, but it usually stays put. In the world of quantum physics, scientists study "gapped" states—special arrangements of particles that are incredibly stable, like a perfectly locked Lego castle that refuses to fall apart even if you shake the table. These stable states are the foundation of future technologies, like super-powerful quantum computers that don't make mistakes.

But sometimes, nature plays a trick. There are "topological" states, which are like Lego castles built with invisible glue. You can't tell they are special just by looking at a single brick; you have to look at the whole shape. Usually, if you try to twist or deform these shapes, they either stay the same or break completely. However, physicists have been staring at a weird, twisted version of a famous topological shape called the "Toric Code." It's like taking that perfect Lego castle and squishing it with a giant, invisible hand. The result is a state that acts like it's broken in some ways, but perfectly stable in others. It's a puzzle that has confused experts because it seems to break the rules of how stable quantum matter is supposed to behave.

This paper is about solving that puzzle. The authors, Nandagopal Manoj, Zack Weinstein, and Jason Alicea, decided to build a new set of "rules" (mathematical equations called Hamiltonians) to describe this squished, weird state. They found that if you look at the rules closely, this state isn't actually broken or unstable at all. It's just a very ordinary, stable state that happens to look strange because of how we usually measure it.

The Squished Lego Castle

To understand what the authors did, let's go back to our Lego castle. The "Toric Code" is a specific, highly ordered way of arranging these blocks. It's famous because it's a "topological" state, meaning it has a special kind of memory. If you try to mess with it locally (like pulling one block), the whole thing resists. But, scientists found a way to "deform" this state. Imagine taking the Toric Code and running it through a machine that stretches and squishes it. This machine is controlled by a knob called $\beta$ .

When you turn the knob just a little, the castle stays mostly the same. But when you turn it all the way up (a "strongly deformed" state), something weird happens. The castle seems to lose its topological memory (the special glue disappears), and the "magnetic" particles inside it start to clump together. In the world of physics, this usually means the castle is falling apart or becoming "gapless"—a fancy way of saying it's unstable and will collapse if you poke it.

However, there was a catch. Even though the castle looked like it was falling apart, the "loops" of energy running through it (called Wilson loops) were behaving strangely. They were shrinking in a way that usually signals a broken symmetry, which should mean the state is unstable. A recent study (by Sahay et al.) even proved a "no-go theorem," which basically said, "It is impossible for a stable, gapped state to look like this." They argued that if you see these specific signs, the state must be gapless (unstable).

The New Blueprint

The authors of this paper said, "Wait a minute. Let's try to build a blueprint for this state and see if it actually holds up."

They constructed a new set of rules, a "parent Hamiltonian," which is essentially the instruction manual for how the particles in this state interact. Their goal was to prove that this squished state is actually a valid, stable, gapped ground state, just like a normal Lego castle.

Here is the twist: The rules they wrote down aren't simple, short-range instructions. Usually, in physics, we assume particles only talk to their immediate neighbors. But in this new blueprint, the instructions involve "sums of Wilson loop operators." Think of this as an instruction that says, "If you are a block here, you need to check in with a block way over there."

Crucially, the authors found that while these instructions reach far, the strength of the connection drops off incredibly fast. It's like a whisper that gets quieter and quieter the further it travels. If you are two blocks away, the whisper is loud. If you are a hundred blocks away, the whisper is so faint it's basically silence. Mathematically, this strength decays exponentially with the distance (the diameter).

The Verdict: It's Stable!

By using this new blueprint, the authors proved that for strong deformations (when the knob $\beta$ is large enough), this state is stable. It has a "spectral gap," meaning it is a solid, gapped quantum state.

This finding has a few massive implications:

The "No-Go" Theorem Wasn't Wrong, Just Limited: The previous "no-go" theorem said this state couldn't exist. The authors showed that the theorem relied on a specific assumption: that the rules of the game must be "local" in a very strict way (independent of the shape of the system). The authors' blueprint breaks this strict rule by allowing those far-away whispers, but only in a way that fades away so quickly it doesn't matter in the real world. They showed that if you relax this strict rule just a tiny bit, the "impossible" state becomes perfectly possible.
Perimeter Laws Don't Always Mean Broken Symmetry: In physics, a "perimeter law" (where energy scales with the edge of a shape) is usually a sign that a symmetry is broken. The authors showed that in this specific case, you can have a perimeter law without the symmetry being broken. It's like seeing a shadow that looks like a broken object, but when you turn on the light, the object is actually whole.
The "Dual" Surprise: They also looked at this from the opposite side (a "dual" perspective). They showed that you can have a state that looks like a magnet with long-range order (everyone agreeing on a direction) but also has "disorder" that scales with the perimeter. This combination was thought to be impossible in a stable state, but their blueprint proves it can happen.

Why This Matters

The authors didn't just find a loophole; they highlighted that our definition of "locality" (how close things need to be to interact) is a subtle but powerful choice. By choosing a slightly more flexible definition—one that allows for those rapidly fading long-distance whispers—we can describe states that were previously thought to be impossible or unstable.

They also applied this logic to a "decohered" Toric Code, which is a state that has been messed up by noise (like a Lego castle being shaken by a toddler). They proved that even after this noise, the state can be broken down into simple, stable pieces, meaning it's not as "entangled" or messy as we thought.

In short, the paper takes a weird, confusing quantum state that everyone thought was a glitch or an impossibility, builds a new set of rules for it, and shows that it's actually a perfectly normal, stable, and gapped state. It's a reminder that sometimes, the universe isn't breaking the rules; we just need to write the rules a little differently to see the full picture.

Problem Statement
The paper addresses a fundamental tension in the classification of gapped quantum phases of matter. Ground states of local gapped Hamiltonians are rigorously known to exhibit exponentially decaying correlations. However, the converse—determining sufficient conditions for a quantum state to be the ground state of a local gapped Hamiltonian—remains an open problem in dimensions greater than one. Specifically, the authors investigate "strongly deformed toric code" states, which arise from applying a non-unitary operator to the toric code ground state. These states present a paradoxical combination of properties: (i) condensed $m$ anyons (indicating a trivial phase), (ii) an exact 1-form symmetry, and (iii) perimeter-law scaling of Wilson loops. Typically, perimeter-law scaling is interpreted as a signature of deconfined excitations and spontaneously broken 1-form symmetry, which is incompatible with a condensed, trivial phase. A recent no-go theorem (Ref. 18) suggested that such states are intrinsically gapless and cannot be the ground states of local gapped Hamiltonians, provided specific technical assumptions regarding locality and symmetry are met.

Methodology
The authors construct explicit parent Hamiltonians for the strongly deformed toric code states $|\psi(\beta)\rangle$ , defined as $|\psi(\beta)\rangle \propto e^{\frac{\beta}{2}\sum_e Z_e} |TC\rangle$ . Their approach involves two distinct perspectives:

Small $\beta$ Regime: They acknowledge the Castelnovo-Chamon Hamiltonian ( $H_{CC}$ ), which is a finite-range parent Hamiltonian for these states. However, $H_{CC}$ is gapless for $\beta > \beta_c$ , failing to provide a gapped description in the strongly deformed phase.
Large $\beta$ Regime: The authors treat $|\psi(\beta)\rangle$ $∣ ψ (β)⟩$ as a deformation of the trivial product state $|0\rangle$ $∣0 ⟩$ . They express the state as $|\psi(\beta)\rangle = e^W |0\rangle$ $∣ ψ (β)⟩ = e^{W} ∣0 ⟩$ , where $W = \log Z$ $W = lo g Z$ and $Z$ $Z$ is an operator-valued partition function related to a bond-disordered Ising model.
- They utilize the Mayer cluster expansion to decompose the operator $W$ into a sum of connected clusters of loop operators.
- They define a new parent Hamiltonian $H(\beta) = \sum_e e^{-W_e}(1-Z_e)e^{-W_e}$ , where $W_e$ contains the components of $W$ that anticommute with the local $Z_e$ .
- To prove locality, they map the operator $Z$ to a hard-core polymer model and rigorously bound the coefficients of the cluster expansion. They demonstrate that the coefficients decay exponentially with the size of the polymer clusters.
- They establish that $H(\beta)$ can be written as $H_0 + V(\beta)$ , where $V(\beta)$ is a perturbation composed of terms whose operator norms decay exponentially with the diameter of their support.
- Using gap stability theorems (Refs. 29, 30), they prove that for sufficiently large $\beta$ , $H(\beta)$ retains a finite spectral gap.

Key Contributions and Results

Construction of Gapped Parent Hamiltonians: The authors rigorously construct local parent Hamiltonians $H(\beta)$ for which the strongly deformed toric code states are the unique, gapped ground states in the large $\beta$ limit.
Resolution of the No-Go Theorem: The construction circumvents the no-go theorem of Ref. 18 by relaxing its assumptions. The theorem requires an aspect-ratio-independent locality bound. The authors show that while their Hamiltonian satisfies standard locality bounds (exponential decay in diameter), it violates the stricter aspect-ratio-independent bound for the "no-go" state $|\psi_{00}(\beta)\rangle$ (which includes only contractible loops). Specifically, the Hamiltonian contains terms creating far-separated non-contractible loops; while their total norm is exponentially suppressed in system size, they violate the uniform bound required by the theorem.
Reinterpretation of Wilson Loop Scaling: The paper demonstrates that within the standard definition of locality (allowing exponential-in-diameter decay), perimeter-law scaling of Wilson loops does not imply spontaneous symmetry breaking (SSB) of a 1-form symmetry. The constructed states are in a trivial gapped phase despite exhibiting perimeter-law scaling.
Dual Perspective: Through Wegner duality, the results imply that a 2D gapped ground state can simultaneously exhibit long-range ferromagnetic order and perimeter-law disorder parameter correlations. This contradicts intuition derived from 1D constraints and standard 2D Ising models, highlighting the necessity of stronger locality notions (e.g., exponential-in-volume) to enforce such constraints.
Separability of Decohered Toric Code: The authors extend their construction to the strongly Pauli-Z decohered toric code. They prove that for large decoherence strengths, the mixed state is separable (a convex sum of short-range entangled pure states), completing the argument that the decohered toric code does not retain topological order.

Significance
The paper's primary significance lies in challenging the robustness of standard diagnostics for 1-form symmetry breaking and topological order. It establishes that the coexistence of perimeter-law Wilson loops and a trivial gapped phase is possible if one adopts the standard, diameter-based definition of locality. The authors argue that this implies stronger notions of locality are necessary to prohibit such "counterintuitive" properties. By quantifying the relaxation of assumptions required to evade the no-go theorem, the work clarifies the boundary between gapless and gapped phases in deformed topological systems. Furthermore, it provides a rigorous proof that the strongly decohered toric code is short-range entangled, resolving questions regarding the separability of such mixed states. The results suggest that Hamiltonian-agnostic criteria, such as entanglement bootstrap axioms, may be required to robustly define gapped phases in the presence of such deformations.

The Squished Lego Castle

The New Blueprint

The Verdict: It's Stable!

Why This Matters

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