⚛️ high-energy theory

Generalized Hall Conductivities in Local Commuting Projector Models: Generalized Symmetries and Protected Surface Modes

This paper constructs local commuting projector lattice models in (2+1)D and (3+1)D that realize nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries by utilizing a standard $\mathbb{Z}_N$ toric code with non-onsite symmetries, thereby circumventing traditional no-go theorems while supporting protected gapless boundaries consistent with continuum field theories.

Original authors: Po-Shen Hsin, Ryohei Kobayashi

Published 2026-02-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Po-Shen Hsin, Ryohei Kobayashi

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

The Big Picture: Breaking the "No-Go" Rule

Imagine you are a physicist trying to build a digital simulation of a very special material called a Quantum Hall State. These are materials that conduct electricity in a very strange, "frictionless" way on their edges, but act like insulators in their middle.

For a long time, there was a strict rule (a "no-go theorem") in the world of lattice simulations (which are like pixelated grids used to model physics). The rule said: "You cannot build a Quantum Hall state on a simple, finite grid if the rules of the grid are perfectly local and commute (don't fight each other)."

Think of it like trying to build a perfect, frictionless ice rink using only Lego bricks. The old rule said, "If your bricks snap together perfectly and don't wobble, you can't make the ice slippery." To get the slipperiness (the Hall conductivity), you usually had to use infinite Lego bricks or bricks that could stretch infinitely far.

What this paper does:
The authors, Po-Shen Hsin and Ryohei Kobayashi, say, "Wait a minute." They have built a new kind of Lego set. They managed to create a model on a simple, finite grid that does have this special "slippery" behavior, without breaking the rules of locality or using infinite bricks. They did this by changing how they define the "symmetries" (the rules of conservation) in their model.

The Core Trick: The "Ghost" Symmetry

To understand how they did it, we need to look at how they define Symmetry.

In most physics models, a symmetry is like a local tax collector. If you have a charge (like an electric bill), you can point to a specific house (a specific spot on the grid) and say, "This house owes 5 dollars." The total charge is just the sum of all the houses.

The Authors' Innovation:
In their model, the "charge" doesn't live in the houses. It lives on the fences between the houses.

Analogy: Imagine a neighborhood where you can't count how much money is inside a house. Instead, the only way to know the total wealth of a neighborhood is to count the money flowing over the fences that surround it.
If you look at a single house in the middle of the neighborhood, it looks like it has zero charge. The charge is "hidden" or "fractionalized" on the boundaries.

Because the charge isn't sitting on a specific local spot (it's not "onsite"), the old "no-go" rule doesn't apply. The rule only banned models where the charge was clearly sitting on a specific local brick. Since this charge is a "ghost" that only appears when you look at a whole region's boundary, the model is allowed to exist.

The Result: A "Fractional" Hall Effect

Because of this trick, their model exhibits a Generalized Hall Conductivity.

What is Hall Conductivity? It's a measure of how much current flows sideways when you push a particle through a magnetic field.
The "Fractional" part: In their model, the amount of current flowing is a fraction (like 1/3 or 1/5) of what you'd expect in a normal material.
The "Generalized" part: They showed this works not just for normal electricity (0-form symmetry) but also for more abstract types of "flows" (higher-form symmetries) in higher dimensions (like 3D space).

How They Proved It: The "Gapless Edge"

How do you know a material is a Quantum Hall state? Usually, you look at its edge. A Quantum Hall material has a special, "gapless" edge.

Analogy: Imagine a solid block of ice. The inside is frozen solid (gapped). But the very edge is a thin layer of liquid water that never freezes, no matter how cold it gets (gapless). This liquid edge is protected by the physics of the ice.
The Paper's Achievement: They built a model where the edge is this "liquid water." They used a mathematical tool called the Modified Villain Formalism (think of it as a special type of glue) to ensure the edge stays liquid and doesn't freeze over, even on a digital grid.

They calculated the "Hall Conductivity" in three different ways to be sure:

Edge Currents: Measuring the flow on the liquid edge.
Flux Insertion: Pushing a magnetic "twist" through the system and seeing how the particles react.
Chern Number: A mathematical count of how the quantum states twist around each other.

All three methods gave the same result: The conductivity is non-zero and fractional.

Why This Matters (According to the Paper)

It breaks a barrier: It proves you don't need infinite complexity or "infinite" degrees of freedom to simulate these exotic quantum states. You can do it with finite, local rules.
It redefines Symmetry: It shows that symmetries in quantum systems can be "weird." They don't always have to be simple sums of local charges. They can be defined by how they behave at the boundaries of regions.
It connects to Real Physics: Even though the model is a lattice (grid), the results match the predictions of continuous field theories used to describe real-world fractional quantum Hall states.

Summary in a Nutshell

The authors built a digital Lego model of a quantum material that was previously thought impossible to build on a simple grid. They achieved this by realizing that the "charge" in the system doesn't live on the bricks themselves, but on the boundaries between them. This allowed them to create a model with "fractional" electrical conductivity and protected liquid-like edges, proving that these exotic quantum phenomena can exist in simple, finite, local systems.

Technical Summary: Generalized Hall Conductivities in Local Commuting Projector Models

Problem Statement
Hall conductivities are fundamental characterizations of topological phases of matter, such as fractional quantum Hall (FQH) states. However, a significant theoretical obstacle exists: "no-go theorems" establish that local commuting projector lattice models with finite-dimensional local Hilbert spaces and onsite $U(1)$ symmetries must exhibit vanishing Hall conductivities. Previous attempts to evade these theorems relied on infinite-dimensional local Hilbert spaces (e.g., rotor degrees of freedom) or local Hamiltonians with exponentially decaying tails. The central problem addressed in this work is whether it is possible to construct local commuting projector models with finite-dimensional local Hilbert spaces that exhibit non-zero generalized Hall conductivities for ordinary and higher-form continuous symmetries, without resorting to infinite dimensions or non-local interactions.

Methodology
The authors construct explicit lattice models based on the $\mathbb{Z}_N$ toric code in $(2+1)$ D and $(3+1)$ D dimensions. The methodology involves three primary components:

Symmetry Definition via Non-Onsite Truncation: The authors define continuous $U(1)$ symmetries that do not act faithfully on the Hilbert space. Specifically, the symmetry operators $U(\Sigma)$ on closed submanifolds are trivial (identity), but the operators defined on open submanifolds with boundaries create non-trivial symmetry defects. These symmetries cannot be expressed as sums of onsite charge operators. Instead, they are defined as depth-1 or depth-2 local unitary circuits involving fractional powers of Pauli operators (e.g., $Z^{[\theta]/2\pi}$ ). The non-trivial information of the symmetry is encoded in the "junction operators" where symmetry transformations meet, which correspond to symmetry defects (visons).
Modified Villain Formalism for Boundaries: To detect the Hall conductivity, the authors construct protected gapless boundary states. Standard lattice realizations of gapless $U(1)$ gauge theories or compact bosons are typically gapped due to monopole perturbations (Polyakov mechanism). The authors employ the modified Villain formalism to suppress these monopole perturbations, thereby constructing gapless boundary theories (compact bosons in $(1+1)$ D and $U(1)$ gauge theories in $(2+1)$ D) that preserve the required mixed 't Hooft anomalies.
Multiple Characterization Methods: The generalized Hall conductivities are computed using three distinct methods to ensure robustness:
- Surface Currents: Calculating the commutator of charge density operators on the gapless boundary to identify mixed anomalies.
- Many-Body Chern Number: Evaluating the Berry phase accumulated by the ground state under twisted boundary conditions (flux insertion) for the symmetry defects.
- Flux Insertion: Analyzing the fractional charge induced at the junction of symmetry defects on a sphere, relating the magnetic flux of one symmetry to the electric charge of another.

Key Contributions and Results

Construction of Non-Zero Hall Models: The authors successfully construct local commuting projector models in $(2+1)$ D and $(3+1)$ D with finite-dimensional local Hilbert spaces that possess non-zero generalized Hall conductivities.
- In $(2+1)$ D, the model exhibits a fractional Hall conductivity $\sigma_{xy} = -1/N$ for ordinary $U(1)$ symmetries.
- In $(3+1)$ D, the model exhibits a generalized fractional Hall conductivity $\xi = -1/N$ relating the ordinary $U(1)$ symmetry to a $U(1)$ one-form symmetry.
Evasion of No-Go Theorems: The work demonstrates that the no-go theorems for vanishing Hall conductivity are evaded because the $U(1)$ symmetries in these models are not onsite. The charge operators are not sums of local terms with well-defined integer eigenvalues; rather, the charge is localized on the boundary of the region. Consequently, the standard assumptions of the no-go theorems (existence of local charge operators) do not hold. The symmetry operators are trivial on closed manifolds but non-trivial on manifolds with boundaries, leading to non-trivial symmetry fractionalization.
Symmetry Fractionalization and Junctions: The paper clarifies that the non-trivial Hall response arises from the specific choice of symmetry truncation. While the symmetry operator on a closed surface is trivial, the junction of symmetry transformations (e.g., $U_{\theta_1} U_{\theta_2} U_{\theta_1+\theta_2}^{-1}$ ) creates a non-trivial line operator (a vison) that detects fractional charge. This "symmetry junction operator" is the lattice counterpart of the continuum topological response.
Protected Gapless Boundaries: The authors show that the bulk models necessarily possess gapless boundaries protected by the mixed anomalies of the $U(1)$ symmetries. The boundary theories are realized using the modified Villain formalism, which successfully maintains the gapless nature and the required symmetry structure without fine-tuning.

Significance
The paper establishes that infinite-dimensional local Hilbert spaces or non-local Hamiltonian tails are not essential prerequisites for realizing non-vanishing Hall conductivities in commuting projector models. By redefining how continuous symmetries are realized on the lattice—specifically through non-onsite, depth-1/2 circuits that act trivially on closed manifolds but non-trivially on boundaries—the authors demonstrate that local commuting projector models can host generalized fractional quantum Hall states. This expands the known landscape of exactly solvable lattice models capable of describing topological phases with continuous symmetries and provides a concrete mechanism for symmetry fractionalization that leads to non-trivial transport coefficients. The results suggest that the "locality" of the symmetry operator, rather than just the locality of the Hamiltonian, is the critical factor determining the possibility of non-zero Hall conductance in these systems.