A complete classification of 2d symmetry protected states with symmetric entanglers

This paper proves that the classification of two-dimensional symmetry-protected topological states by the cohomology group $H^3(G,U(1))$ is complete for the specific subset of states that can be prepared from a product state via a symmetric entangler.

The Gist

Imagine you have a giant, infinite grid of tiny magnets (or "spins"). In physics, we often look at how these magnets arrange themselves in their lowest energy state. Sometimes, they arrange themselves in a way that looks completely boring and simple, like everyone just pointing in the same direction. We call this a "product state."

But sometimes, the magnets arrange themselves in a way that looks simple from a distance, but if you look closely, they are secretly holding hands in a complex, entangled pattern. This is called a Symmetry Protected Topological (SPT) state.

The "catch" is that this secret pattern only exists because of a rule: Symmetry. If you break the rule (like forcing the magnets to point in a specific direction that breaks the symmetry), the secret pattern disappears, and the system becomes boring again.

The Big Question

For a long time, physicists have been trying to answer: "How many different types of these secret patterns exist?"

Mathematicians have a tool called Group Cohomology (specifically $H^3(G, U(1))$ for 2D systems) that they think perfectly counts these patterns. It's like having a catalog that says, "There are exactly 5 types of secret patterns for this specific rule."

However, there was a nagging doubt: Is this catalog complete? Could there be a weird, hidden pattern that the catalog missed? This was an open problem for 2D systems (flat surfaces like a sheet of paper).

The Paper's Solution: The "Symmetric Entangler"

The authors of this paper decided to tackle the problem by focusing on a specific, manageable way to create these patterns.

Think of a Product State as a blank canvas. To turn it into an SPT state, you need to apply a "painter" or a "sculptor." In physics, this is a Quantum Circuit (a sequence of operations).

The authors restricted their study to Symmetric Entanglers.

• The Analogy: Imagine you have a blank canvas (the product state) and a set of rules (the symmetry). A "Symmetric Entangler" is a machine that takes the blank canvas and twists it into a complex pattern, but the machine itself obeys the rules perfectly at every step. It doesn't cheat.

The authors proved a massive result: If you only look at patterns created by these "honest" machines, the catalog is 100% complete. There are no hidden patterns. The number of patterns is exactly what the math predicted ($H^3(G, U(1))$).

How Did They Prove It? (The "Blending" Trick)

To prove the catalog was complete, they had to show that if you have a pattern that looks like it belongs to the catalog (has a "trivial" index), it is actually just a boring product state in disguise.

They used a clever technique called Symmetric Blending.

The Metaphor:
Imagine you have a long hallway.

• On the left side of the hallway, the walls are twisted in a complex, secret pattern (the SPT state).
• On the right side, the walls are perfectly straight and boring (the identity/empty state).
• In the middle, you want to smoothly blend the twisted walls into the straight walls without breaking the symmetry rules.

The authors showed that if the "twist" on the left side is mathematically "trivial" (meaning it belongs to the zero category of the catalog), you can build a machine that smoothly transitions from the twist to the straight wall.

The "Swindle" (The Eilenberg-Mazur Swindle):
To prove this, they used a mathematical trick called the "Eilenberg-Mazur Swindle."

• Imagine: You have a long line of people holding hands. Some are holding hands in a "twisted" way, some in a "straight" way.
• The Trick: If you have an infinite line, you can pair them up in a specific way (like a zipper) to cancel out all the twists. You pair a "twisted" person with an "anti-twisted" person, and they cancel each other out, leaving you with a straight line.
• Because the system is infinite, you can keep doing this pairing forever, effectively "zipping" away all the complexity until nothing is left but the boring, straight state.

Why Does This Matter?

1. It Solves a Mystery: It confirms that for 2D systems created by these specific "honest" machines, the mathematical catalog is perfect. We don't need to look for new, hidden types of matter.
2. It Connects Dimensions: The proof relies on understanding what happens at the edge of the system. A 2D system's edge acts like a 1D system. The authors used their knowledge of 1D physics (which was already solved) to solve the 2D puzzle.
3. It's a Stepping Stone: While they proved it for "Symmetric Entanglers," they suspect this covers all SPT states. If true, this means our understanding of 2D quantum matter is solid.

Summary in One Sentence

The authors proved that for 2D quantum systems built by "rule-abiding" machines, the number of possible secret topological patterns is exactly what the math predicted, using a clever "infinite zipper" trick to show that any "fake" pattern can be untangled into nothingness.

Technical Summary

Here is a detailed technical summary of the paper "A complete classification of 2d symmetry protected states with symmetric entanglers" by Alex Bols, Wojciech De Roeck, Michiel De Wilde, and Bruno de O. Carvalho.

1. Problem Statement and Context

The Core Problem:
The paper addresses the classification of Symmetry Protected Topological (SPT) phases in 2-dimensional (2D) quantum spin systems with a finite on-site symmetry group $G$.

• SPT States: These are gapped ground states that are short-range entangled (trivial in the absence of symmetry) but cannot be adiabatically connected to a product state without breaking the symmetry $G$.
• The Classification Conjecture: It is widely conjectured that 2D SPT phases are classified by the third group cohomology group, $H^3(G, U(1))$.
• The Open Question: While this classification is known to be complete for dimensions $d=0$ and $d=1$, and known to be incomplete for $d \geq 3$ (due to the existence of non-group-cohomology SPTs), the completeness of the $H^3(G, U(1))$ classification for $d=2$ remained an open problem.

The Specific Restriction:
The authors restrict their analysis to a specific subclass of SPT states: SPT states with symmetric entanglers.

• A state $|\psi\rangle$ is prepared from a product state $|\phi\rangle$ via a finite-depth quantum circuit (FDQC) $C$: $|\psi\rangle = C|\phi\rangle$.
• A symmetric entangler requires that both the product state $|\phi\rangle$ and the unitary circuit $C$ are individually symmetric under the group action $G$ (i.e., $U_g|\phi\rangle \propto |\phi\rangle$ and $[C, U_g] = 0$).
• Note: The circuit $C$ does not necessarily consist of symmetric gates; if it did, the state would be trivial. The symmetry of the entire operator $C$ is the key constraint.

The authors do not prove that all 2D SPT states admit symmetric entanglers, but they prove that within this subclass, the classification by $H^3(G, U(1))$ is complete.

2. Methodology

The proof strategy relies on a reductionist approach, moving from the classification of states to the classification of the operators (entanglers) that create them.

A. Mathematical Framework

• Algebraic Setting: The authors work within the framework of $C^*$-algebras on infinite lattices ($\mathbb{Z}^d$) to rigorously handle the thermodynamic limit, avoiding the ill-defined nature of state vectors in infinite volume.
• Equivalence Relations: They define stable equivalence for SPT states and symmetric entanglers. Two objects are stably equivalent if they become equivalent after stacking with auxiliary product states (or identity operators) and applying symmetric FDQCs.
• Monoid Structure: The set of equivalence classes forms a monoid under the "stacking" (tensor product) operation. The goal is to show this monoid is isomorphic to the group $H^3(G, U(1))$.

B. The Index Map

The core of the methodology is the construction of an index map:
$$\text{ind}: \text{SE}_{G,d} / \sim \longrightarrow H^{d+1}(G, U(1))$$
where $\text{SE}_{G,d}$ is the set of symmetric entanglers in dimension $d$.

• Surjectivity: The authors construct explicit examples of symmetric entanglers for every cohomology class, proving the map is surjective.
• Injectivity: This is the novel and most difficult part. They must prove that if a symmetric entangler has a trivial index (maps to the identity in cohomology), it is stably equivalent to the identity (trivial).

C. Key Technical Tools

1. Boundary Algebras: For a symmetric entangler $\alpha$ acting on a half-space $L_x$, they define a boundary algebra $P_x^{[r]} = \alpha(A_{L_x}) \cap A'_{L_x-r}$. This algebra captures the "edge" physics.
2. Symmetry Action on the Boundary: The global symmetry $G$ induces a Locality Preserving Symmetry (LPS) on the boundary algebra.
   • In 1D, the boundary is a point (0D), and the symmetry action is a projective representation classified by $H^2(G, U(1))$.
   • In 2D, the boundary is a 1D chain. The induced symmetry on this chain is an LPS.
3. Reduction to 1D Anomalies: The authors utilize a recent result (from their previous work [33]) that classifies 1D LPSs by $H^3(G, U(1))$. By showing that the boundary of a 2D symmetric entangler carries an LPS with a specific anomaly, they define the 2D index.
4. Symmetric Blending (The "Swindle"): To prove injectivity, they construct a symmetric blend.
   • They take a symmetric entangler $\alpha$ with a trivial index and "blend" it with the identity operator across the lattice.
   • This creates a new entangler that looks like $\alpha$ on the left and the identity on the right.
   • By decomposing the lattice into alternating stripes and using the Eilenberg-Mazur swindle (an infinite stacking argument), they show that the "charges" (cohomology classes) on the stripes can be canceled out pairwise, proving the original entangler is stably equivalent to the identity.

3. Key Contributions

1. Complete Classification of 2D SPTs with Symmetric Entanglers:
   The primary result is Theorem 2.5, which establishes a group isomorphism:
   $$\text{SPT}^{\text{SE}}_{G,2} \cong H^3(G, U(1))$$
   This confirms the group cohomology classification is complete for the specific class of states generated by symmetric entanglers.

2. Classification of Symmetric Entanglers (Proposition 2.1):
   The paper proves that the monoid of stable equivalence classes of symmetric entanglers in dimensions $d=0, 1, 2$ is isomorphic to $H^{d+1}(G, U(1))$.

   • $d=0$: $H^1(G, U(1))$ (charges).
   • $d=1$: $H^2(G, U(1))$ (projective representations).
   • $d=2$: $H^3(G, U(1))$ (anomalous boundary actions).

3. Development of the "Symmetric Blending" Technique:
   The authors generalize the technique of "blending" an operator with the identity to the symmetric setting. This involves constructing an auxiliary system where the symmetry action on the boundary can be disentangled (converted to an on-site action) using FDQCs, allowing for the cancellation of topological indices.

4. Rigorous Algebraic Formulation:
   The paper provides a rigorous $C^*$-algebraic formulation of SPT states and entanglers, clarifying the relationship between finite-depth circuits, locality preserving symmetries, and group cohomology in the thermodynamic limit.

4. Results

• Theorem 2.5: The set of 2D SPT phases with symmetric entanglers, modulo stable equivalence, forms a group isomorphic to $H^3(G, U(1))$.
• Proposition 2.1: The classification holds for $d=0, 1, 2$. The index map is a well-defined group isomorphism.
• Injectivity Proof: The authors successfully demonstrated that any symmetric entangler with a trivial cohomology index is stably equivalent to the identity. This was achieved by:
  1. Showing the boundary of the entangler carries an LPS with a trivial anomaly.
  2. Using the classification of 1D LPSs to disentangle this boundary action.
  3. Constructing a symmetric blend that effectively "pushes" the non-trivial part of the entangler to infinity, where it cancels out via the Eilenberg-Mazur swindle.
• Surjectivity Proof: Explicit constructions were provided for every element in $H^3(G, U(1))$ using layered 1D systems with alternating symmetry actions.

5. Significance

• Resolution of a Major Open Problem: While the completeness of the $H^3$ classification for general 2D SPTs remains open (as it is unknown if all SPTs admit symmetric entanglers), this paper resolves the problem for the physically relevant and mathematically tractable subclass of states generated by symmetric entanglers.
• Bridging Dimensions: The work elegantly connects the classification of 0D, 1D, and 2D systems. It demonstrates that the 2D classification is fundamentally rooted in the classification of anomalous symmetry actions on 1D boundaries.
• Methodological Advancement: The "symmetric blending" technique and the rigorous handling of stable equivalence in the algebraic setting provide a robust toolkit for future investigations into higher-dimensional SPT phases and other topological orders.
• Clarification of "Symmetric Entanglers": The paper clarifies the distinction between SPTs generated by symmetric circuits (trivial) and those generated by symmetric entanglers (potentially non-trivial), highlighting the subtle role of symmetry in the preparation of topological states.

In summary, the paper provides a definitive proof that for 2D quantum spin systems, if an SPT state can be prepared from a product state using a symmetric entangler, its topological phase is uniquely and completely characterized by the third group cohomology of the symmetry group.