Non-onsite symmetry breaking: topological phase coexistence and criticality

This paper investigates the spontaneous symmetry breaking of non-onsite symmetries in one and two spatial dimensions, revealing novel phases characterized by the coexistence of distinct symmetry-protected topological orders, long-range entanglement, and topological quantum criticality.

The Gist

The Big Idea: Breaking Rules to Make New Worlds

Imagine you are organizing a massive dance party. Usually, the rules of the dance are simple: everyone spins in place (an "onsite" rule). If everyone breaks this rule at the same time, the party changes from a chaotic mess into an orderly line dance. This is what physicists call Spontaneous Symmetry Breaking (SSB). It's how materials decide to become magnets or superconductors.

But in this paper, the authors ask a weird question: What happens if the rule itself is weird?

Instead of telling everyone to spin in place, imagine the rule is: "If you and your neighbor are holding hands, you must swap places." This rule involves two people at once; it's not just about one person. The authors call this a "non-onsite" symmetry. They wanted to see what kind of "dance" (state of matter) emerges when this weird, two-person rule is broken.

Part 1: The One-Dimensional Party (The 1D Chain)

The Setup:
Imagine a single line of people holding hands. The "weird rule" is that if you look at the whole line, the pattern of who is holding hands with whom flips.

The Discovery:
Usually, when a symmetry breaks, you get one specific outcome (like everyone facing North). But here, the authors found something magical: The ground state (the most comfortable resting position) is a superposition of two completely different worlds.

• World A: Everyone is just standing still in a simple line (a "trivial" state).
• World B: Everyone is holding hands in a complex, knotted pattern (a "cluster state" or SPT order).

The Analogy:
Imagine a coin that, when it lands, doesn't just show Heads or Tails. Instead, it lands in a state where it is both a perfectly smooth coin (World A) and a coin with a complex, knotted string wrapped around it (World B) at the same time.

The paper proves that:

1. They coexist: The system doesn't have to choose one or the other; it lives in a superposition of both.
2. It's stable: Even if you wiggle the system slightly (add a small perturbation), this weird "both/and" state stays stable up to a certain point.
3. The Critical Point: If you wiggle it too hard, the system snaps. It loses its "both/and" nature and becomes a "critical" phase. Think of this like a bridge that is perfectly balanced between two cliffs. If you push it too far, it falls into a river of pure chaos (a gapless phase described by a Conformal Field Theory), where things are constantly fluctuating and never settle down.

The "Charge" Problem:
In normal physics, if you break a rule, you can find a "charged" object that proves it (like a magnet pole). But because this rule is so weird (non-onsite), the "charged object" needed to prove the breakage cannot be a normal, reversible object. It's like trying to use a key that only works one way and then disappears. The authors found a specific "non-invertible" operator that acts as this proof, showing that the system has long-range connections that can't be explained by simple local rules.

Part 2: The Two-Dimensional Party (The 2D Grid)

The Setup:
Now imagine the dancers are on a honeycomb grid (like a beehive). The rule is even stranger: "If you form a closed loop with your neighbors, you must swap places."

The Discovery:
When this rule breaks, the system doesn't just pick one pattern. It creates a "Soup."

The Analogy:
Imagine a pot of soup where the ingredients are loops of 1D "knotted" strings.

• In a normal soup, you have random noodles.
• In this "SPT Soup," the noodles are actually tiny, knotted 1D quantum states (SPTs).
• These knotted loops are floating everywhere, overlapping and condensing.

The Result:
On a torus (a donut shape), there are four different versions of this soup that look exactly the same if you only look at a small spoonful (local view). You can't tell them apart unless you look at the whole donut.

• The Critical Twist: Unlike normal topological order (like the Toric Code) which is rigid and has a gap (a "hard" energy cost to change), this soup is critical. The connections between the loops decay slowly (algebraically), like a signal fading over a long distance rather than vanishing instantly. It's a "liquid" topological state that is constantly fluctuating, sitting right on the edge of a phase transition.

Part 3: How to Make It (The Recipe)

The authors also figured out how to cook this up in a lab using quantum computers.

The Protocol:

1. Start: Put all qubits (quantum bits) in a simple state.
2. Measure: Perform measurements on some parts of the system.
3. Feedback: Based on the measurement results, apply a quick "correction" (a unitary gate).
4. Result: With a 50% chance, you end up with the perfect "Superposition of Trivial and Knotted" state.

The Catch:
While they can make the 1D version reliably, making the 2D "SPT Soup" is much harder. It's like trying to untangle a knot in a ball of yarn by only looking at a tiny section of it. The "defects" (mistakes in the pattern) in this 2D soup are stubborn; they can't be easily fixed with a simple, quick move, making the 2D version harder to prepare perfectly.

Summary of the "New Physics"

• Non-Onsite Symmetries: These are rules that involve groups of neighbors, not just individuals.
• Coexistence: Breaking these rules creates a state that is simultaneously "simple" and "complex" (trivial and topological).
• Criticality: This state is fragile. Push it too hard, and it turns into a critical, fluctuating phase (CFT) rather than a solid, stable phase.
• SPT Soup: In 2D, breaking these rules creates a "condensate" of 1D knots, resulting in a state with long-range algebraic correlations (power-law decay) rather than short-range order.

In short, the paper discovers a new class of quantum matter where the "rules of the game" are so interconnected that breaking them creates a hybrid world of order and chaos, existing in a delicate, critical balance.

Technical Summary

Technical Summary: Non-onsite Symmetry Breaking: Topological Phase Coexistence and Criticality

Problem Statement
Symmetry serves as a fundamental principle for characterizing quantum phases of matter. While conventional global symmetries are typically realized "onsite" (as products of single-site unitary operators), recent developments have highlighted the importance of "non-onsite" symmetries, where the symmetry generator involves overlapping multi-qubit operations (e.g., controlled-Z gates). The central question addressed in this work is: What states of matter emerge when such non-onsite symmetries undergo spontaneous symmetry breaking (SSB)? Specifically, the authors investigate the consequences of SSB for global $Z_2$ non-onsite symmetries in one dimension (1D) and 1-form non-onsite symmetries in two dimensions (2D).

Methodology
The authors employ a combination of analytical constructions, rigorous proofs, and numerical simulations (Density Matrix Renormalization Group - DMRG) to explore these phases.

1. 1D Model Construction: They define a global non-onsite $Z_2$ symmetry $U_{CZ} = \prod_i CZ_{i,i+1}$ on a periodic 1D chain. To study its SSB, they construct a frustration-free, local Hamiltonian inspired by the O'Brien-Fendley model. This Hamiltonian is designed to have two degenerate ground states: a trivial product state $|+\rangle^{\otimes N}$ and a non-trivial symmetry-protected topological (SPT) cluster state $|cluster\rangle = U_{CZ}|+\rangle^{\otimes N}$.
2. Analytical Proofs: The authors analytically prove the two-fold ground-state degeneracy and the existence of a finite energy gap in the thermodynamic limit using techniques adapted from Ref. [8] and Knabe's argument. They also demonstrate that the resulting long-range entangled ground state (a superposition of the two degenerate states) cannot be connected to a product state via finite-depth unitary circuits.
3. Order Parameters: A key methodological challenge is identifying an order parameter for non-onsite SSB. The authors prove that any charged operator for $U_{CZ}$ must be non-invertible (due to the non-zero trace of the symmetry generator). They explicitly construct a non-invertible charged operator $O_i = X_i(1 - Z_{i-1}Z_{i+1})$ and calculate its long-range correlations.
4. Phase Diagram and Criticality: They introduce a symmetric perturbation to the Hamiltonian to map out the phase diagram. They analyze the transition from the gapped SSB phase to a gapless phase using DMRG, fitting entanglement entropy scaling to extract the central charge and analyzing correlation functions to determine critical exponents.
5. 2D Extension: The authors extend the analysis to 2D honeycomb lattices with 1-form non-onsite symmetries (products of $CZ$ gates along closed loops). They construct a "SPT soup" state—a condensate of 1D SPT orders along loops—and map the correlation functions of this state to the critical $O(2)$ loop model.
6. State Preparation: They propose a constant-depth measurement-feedback protocol to prepare the 1D superposition state with constant success probability in the thermodynamic limit.

Key Contributions and Results

• 1D Topological Phase Coexistence: The authors demonstrate that the spontaneous breaking of a global non-onsite $Z_2$ symmetry leads to a gapped phase where trivial and non-trivial SPT orders coexist in the ground subspace. They prove analytically that this phase possesses a finite energy gap and two-fold degeneracy.
• Non-Invertible Order Parameters: The work establishes that SSB of non-onsite symmetries is diagnosed by non-invertible charged operators. They show that for the state $|\psi\rangle \propto |+\rangle^{\otimes N} + |cluster\rangle$, the correlation function $\langle O_i O_j \rangle$ of the non-invertible operator $O_i$ saturates to a non-zero constant at long distances, signaling long-range order.
• Criticality and Conformal Field Theory (CFT): Under symmetric deformation, the system undergoes a transition to a gapless phase. Numerical results indicate this phase is described by a $c=1$ Conformal Field Theory. The authors identify the critical point separating the SSB and gapless phases and conjecture it corresponds to the $SU(2)_1$ CFT. They also show that the critical exponents vary continuously within the gapless phase, consistent with a marginal perturbation analysis via bosonization.
• 2D SPT Soup and Algebraic Correlations: In 2D, the SSB of 1-form non-onsite symmetries yields a "SPT soup" state. On a torus, this state exhibits four locally indistinguishable ground states. Unlike topological orders with local gapped parents (like the toric code), the SPT soup has algebraic (power-law) correlations between local operators. By mapping the system to the critical $O(2)$ loop model, the authors derive that the two-point correlation function decays as $1/|i-j|$.
• Measurement-Based Preparation: The authors present a protocol using mid-circuit measurements and feedback to prepare the 1D superposition state $|+\rangle^{\otimes N} \pm |cluster\rangle$ with a constant success probability ($1/2$) in the thermodynamic limit, independent of system size. They note that while the global sign is probabilistic, local observables are deterministically prepared due to local indistinguishability.

Significance
The paper claims to reveal novel quantum states of matter arising from the spontaneous breaking of non-onsite symmetries, which differ fundamentally from conventional SSB.

• Novel Phases: It identifies a gapped phase characterized by the coexistence of distinct SPT orders, a phenomenon not typically associated with standard symmetry breaking.
• New Diagnostics: It introduces non-invertible charged operators as essential tools for diagnosing long-range order in non-onsite symmetry breaking, distinguishing it from standard unitary order parameters.
• Topological Quantum Criticality: The 2D "SPT soup" provides an example of "topological quantum criticality," where a state exhibits long-range entanglement and algebraic correlations without a local gapped parent Hamiltonian, bridging the gap between SPT phases and critical phenomena.
• Experimental Relevance: The proposed constant-depth measurement-feedback protocol offers a feasible pathway to engineer and study these exotic states on near-term quantum devices (e.g., superconducting qubits or trapped ions) that support mid-circuit measurements.

The authors maintain a modest stance regarding future implications, noting that the connection between the 1-form non-onsite symmetry and the criticality of the SPT soup requires further investigation, and that the deterministic preparation of the 2D SPT soup in constant depth remains an open question. They also suggest that the framework of Symmetry Topological Field Theory (SymTFT) offers a promising perspective for understanding these phases, though a complete classification is left for future work.