Boundary topological orders of (4+1)d fermionic $\mathbb{Z}_{2N}^{\mathrm{F}}$ SPT states

This paper investigates (3+1)d topological orders with anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry by microscopically constructing symmetry-preserving gapped boundary states for related (4+1)d SPT phases, demonstrating that such states admit a topological $\mathbb{Z}_4$ gauge theory description when $N=\nu$, a non-TQFT solution when $\nu=N/2$, and no symmetric gapped state otherwise, thereby confirming existing no-go theorems.

The Gist

Imagine you are trying to build a house (a physical system) that is perfectly stable and quiet (gapped) on the inside, but the blueprint for the house has a "glitch" in its foundation. This glitch is called an anomaly. In the world of quantum physics, an anomaly means that the rules of symmetry (like rotating the house or flipping a switch) work perfectly in the middle of the room, but if you try to apply them to the very edge of the room, things break down.

Usually, when a blueprint has a glitch, the house can't be quiet and stable at the edges. It must either be noisy (gapless) or the walls must crumble (symmetry breaking).

This paper asks a big question: Can we build a special kind of "magic wall" at the edge of this glitchy house that is quiet, stable, and still respects the symmetry rules?

Here is the breakdown of their journey, using everyday analogies:

1. The Glitchy Blueprint (The Anomaly)

The authors are studying a specific type of quantum system made of Weyl fermions (think of them as tiny, one-way traffic cars). These cars have a special rule: they can only drive in a circle if the circle is a multiple of $N$ steps.

• The Problem: If you try to park these cars in a 4D space (a hyper-house) and look at the 3D surface (the wall), the symmetry rule creates a "leak." The cars can't just sit still; they want to keep moving.
• The Constraint: A famous rule (the Cordova-Ohmori theorem) says: "If the number of cars ($\nu$) isn't a perfect multiple of the step size ($N$), you cannot build a quiet, stable wall. The edge must remain noisy."

2. The Magic Trick: Crystalline Correspondence

The authors realized they couldn't fix the wall directly, so they changed the shape of the room.

• The Analogy: Imagine you have a spinning top that is wobbling because of a weird magnetic field. Instead of trying to stop the wobble, you put the top inside a box with mirrors arranged in a circle.
• The Deformation: They took the "wobbly" symmetry and bent it into a shape that looks like a crystal rotation ($C_N$). They turned the abstract "internal" rule into a physical "spinning" rule.
• The Result: The problem transformed. Instead of a messy 3D wall, they now had a 2D floor with a spinning axis in the middle. The "wobble" (anomaly) was now concentrated on a single line (the axis), making it easier to see and fix.

3. Building the Quiet Wall (The Solutions)

Now, they tried to build a quiet, stable wall for different numbers of cars ($\nu$). They found three different outcomes:

Case A: The "Perfect Fit" ($\nu = N$)

• The Situation: You have exactly $N$ cars, which matches the step size perfectly.
• The Solution: They built a $Z_4$ Gauge Theory.
• The Analogy: Imagine the wall is made of a special, invisible fabric. The "cars" on the edge are actually knots in this fabric. The authors showed that if you tie the knots in a specific, repeating pattern (a $Z_4$ pattern), the tension cancels out perfectly. The wall becomes quiet and stable.
• The Catch: The symmetry (the rotation) doesn't just sit there; it acts like a "ghost" that moves through the fabric, shifting the knots. This is a Topological Quantum Field Theory (TQFT)—a state of matter that is defined by its global shape, not its local parts.

Case B: The "Half-Fit" ($\nu = N/2$)

• The Situation: You have half the number of cars needed for a perfect match.
• The Solution: They built a Non-TQFT Gapped State.
• The Analogy: This is like building a wall out of different materials stacked unevenly. On one side, you have a layer of ice; on the other, a layer of rubber. It's stable and quiet, but it's "crunchy" and anisotropic (it feels different depending on which way you touch it).
• The Catch: This wall is too weird to be described by the standard "fabric" math (TQFT). It's a messy, highly specific construction that works, but it's not a "perfect" topological order.

Case C: The "Mismatch" (Other values of $\nu$)

• The Situation: The number of cars doesn't fit the step size at all.
• The Result: No Solution.
• The Analogy: You try to build the wall, but the bricks just won't stack. The wall either stays noisy (gapless) or falls apart (symmetry breaking). This confirms the "No-Go" theorem: some glitches are unfixable.

4. Why Should We Care? (The Real-World Connection)

The paper ends with a fascinating connection to our actual universe: The Standard Model of Particle Physics.

• The universe has a specific "glitch" involving protons and neutrons (baryons) and electrons (leptons).
• The math shows that our universe has a "mismatch" of -3 (mod 16).
• The Big Idea: If the universe is a 4D "hyper-house" with a glitch, maybe the edge of our universe (or a hidden extra dimension) is filled with one of these magic walls (like the $Z_4$ gauge theory) to cancel out the glitch!
• This could explain why protons don't decay instantly and might even point to a hidden form of Dark Matter living in these topological "knots" on the edge of reality.

Summary

The authors took a complex, unsolvable quantum puzzle (a glitchy symmetry in 4D space), bent it into a crystal shape to make it visible, and then built three types of "magic walls" to fix it:

1. A perfect, invisible fabric ($Z_4$ gauge theory) for the best-case scenario.
2. A weird, stacked wall for the half-case scenario.
3. A proof that some walls can't be built for the mismatched scenarios.

They showed us that the universe might be using these exact "magic walls" to stay stable, potentially hiding a new form of matter (Dark Matter) in the process.

Technical Summary

Here is a detailed technical summary of the paper "Boundary topological orders of (4+1)d fermionic $\mathbb{Z}^F_{2N}$ SPT states" by Meng Cheng, Juven Wang, and Xinping Yang.

1. Problem Statement

The paper investigates the low-energy dynamics of (3+1)d fermionic systems possessing an anomalous discrete chiral symmetry $\mathbb{Z}^F_{2N}$, where the $\mathbb{Z}^F_2$ subgroup corresponds to fermion parity. This anomaly arises as the discrete subgroup of the chiral $U(1)$ symmetry of $\nu$ copies of Weyl fermions of the same chirality.

The central question is: What is the minimal topological order required to saturate the $\mathbb{Z}^F_{2N}$ anomaly at the boundary of a (4+1)d Fermionic Symmetry-Protected Topological (FSPT) state?

According to the no-go theorem by Cordova and Ohmori:

• If $N \nmid \nu$ (N does not divide $\nu$), no symmetry-preserving gapped boundary described by a Topological Quantum Field Theory (TQFT) exists.
• If $N \mid \nu$, the existence of such a boundary is not forbidden, but the specific nature of the topological order remains to be constructed and classified.

The authors aim to explicitly construct these symmetric gapped boundary states and determine their topological nature (TQFT vs. non-TQFT) for various values of $\nu$.

2. Methodology

The authors employ a multi-step strategy combining crystalline correspondence, block-state constructions, and symmetry extension techniques:

1. Crystalline Correspondence Principle:
   Instead of working directly with the internal $\mathbb{Z}^F_{2N}$ symmetry, the authors deform the (3+1)d Weyl fermion system by introducing a spatially inhomogeneous superconducting mass term containing a vortex line.

   • This breaks the continuous chiral $U(1)$ and rotational $SO(3)$ symmetries but preserves a diagonal subgroup: $C_N \times \mathbb{Z}^F_2$ (where $C_N$ is $N$-fold rotation).
   • The deformation maps the (3+1)d Weyl fermions to (1+1)d chiral Majorana-Weyl fermions localized on the rotation axis (vortex line), while the bulk is gapped.
   • This establishes an isomorphism between the classification of $\mathbb{Z}^F_{2N}$ SPTs and $C_N \times \mathbb{Z}^F_2$ FSPTs.

2. Block-State Construction:
   The (4+1)d bulk FSPT is modeled as a "block state" where $N$ half-planes (meeting at the rotation axis) are decorated with (2+1)d topological orders ($\mathcal{B}$), and the rotation center (a 2D plane in 4D bulk) is decorated with a $p_x + i p_y$ superconductor.

   • The goal is to gap out the (1+1)d chiral edge modes at the rotation axis while preserving the $C_N$ symmetry.

3. Symmetry Extension and Gauging:
   To gap the modes, the authors utilize the symmetry extension method. They extend the global symmetry $C_N$ to a larger group $C_{N'}$ (specifically $C_{4N}$ or $C_{2N}$) such that the anomaly becomes trivial in the extended group.

   • They decorate the system with invertible phases (SPTs) protected by the extended symmetry.
   • Finally, they gauge the extra symmetry group (e.g., $\mathbb{Z}_4$ or $\mathbb{Z}_2$) to restore the physical $C_N$ symmetry. This process converts the invertible phases into invertible topological defects within a (3+1)d TQFT.

3. Key Contributions and Results

The paper provides a microscopic construction of the boundary states for different anomaly indices $\nu$:

A. Case: $N \nmid \nu$

• Result: No symmetric gapped TQFT exists.
• Significance: This confirms and refines the Cordova-Ohmori no-go theorem. The authors conjecture that for these cases, the boundary must either be gapless or spontaneously break the $C_N$ symmetry.

B. Case: $\nu = N/2$ (e.g., $N=2, \nu=1$)

• Result: A symmetric gapped state exists, but it is non-TQFT.
• Construction: The authors construct a highly anisotropic state by stacking (2+1)d topological orders (specifically $SO(3)_3$ and semion-fermion theories) on the half-planes.
• Key Feature: The resulting state cannot be described by a standard (3+1)d TQFT because the underlying topological order involves anyons with non-integer quantum dimensions (e.g., $2+\sqrt{2}$) that cannot be trivialized by stacking. This represents a new class of "non-TQFT" gapped boundaries.

C. Case: $\nu = N$ (The Main Construction)

• Result: The minimal topological order saturating the anomaly is a $\mathbb{Z}_4$ gauge theory.
• Mechanism:
  1. The chiral central charge of the edge modes is canceled by adding $N$ layers of $p_x - i p_y$ superconductors.
  2. The symmetry is extended from $C_N$ to $C_{4N}$ (by adding a $\mathbb{Z}_4$ gauge group).
  3. The system is decorated with (2+1)d Majorana FSPTs protected by $\mathbb{Z}_4 \times \mathbb{Z}^F_2$.
  4. Gauging the $\mathbb{Z}_4$ symmetry yields a (3+1)d $\mathbb{Z}_4$ gauge theory.
• Symmetry Action: The $C_N$ rotation symmetry acts on the $\mathbb{Z}_4$ gauge theory via a static network of invertible topological defects.
  • Gauge charges acquire fractionalized angular momentum (projective phase $e^{i\pi/2N}$).
  • Flux loops linked to symmetry defects carry Majorana zero modes.
• Refinement for $N \equiv 0 \pmod 8$: The authors analyze whether a $\mathbb{Z}_2$ gauge theory suffices when $N$ is a multiple of 8. They calculate topological invariants (specifically the topological spin of the twist field) and find that $\Theta_r \neq 1$ for the $\mathbb{Z}_2$ extension. Thus, even for $N \equiv 0 \pmod 8$, the $\mathbb{Z}_4$ gauge theory remains the minimal TQFT required to saturate the anomaly.

4. Significance and Implications

1. Resolution of the Anomaly Saturation Problem: The paper provides the first explicit microscopic construction of symmetric gapped boundaries for (4+1)d fermionic SPTs with $\mathbb{Z}^F_{2N}$ symmetry, distinguishing between TQFT and non-TQFT outcomes.
2. New Class of Gapped States: The identification of a gapped boundary for $\nu = N/2$ that is not a TQFT challenges the assumption that all gapped topological phases in (3+1)d can be described by finite group gauge theories.
3. Standard Model Connection (Appendix B): The authors apply their results to the Standard Model (SM). The SM with $Z_4,X$ symmetry (a specific combination of Baryon minus Lepton number and Hypercharge) exhibits a $\mathbb{Z}^F_{16}$ anomaly with index $\nu = 45 \equiv -3 \pmod{16}$.
   • Since $4 \nmid (-3)$, the no-go theorem applies, suggesting the SM boundary cannot be a simple TQFT.
   • However, the authors speculate that the exotic topological order constructed in this paper (or similar non-invertible orders) could serve as a "BSM sector" to cancel the anomaly without requiring the existence of right-handed neutrinos, potentially offering a candidate for Cold Dark Matter.
4. Methodological Advancement: The work demonstrates the power of combining crystalline correspondence with symmetry extension to solve complex anomaly matching problems in higher dimensions.

In summary, the paper establishes that the boundary of (4+1)d fermionic SPTs with $\mathbb{Z}^F_{2N}$ symmetry can be gapped, but the nature of the resulting phase depends critically on the anomaly index $\nu$: it is a $\mathbb{Z}_4$ gauge theory for $\nu=N$, a non-TQFT anisotropic state for $\nu=N/2$, and impossible (gapless/broken) otherwise.