When Wannier centers jump: Critical points between atomic insulating phases

This paper demonstrates that critical points between distinct, featureless bosonic atomic insulators in $(2+1)$ dimensions can host emergent conformally invariant quantum electrodynamics (QED$_3$) states, stabilized by lattice symmetries that suppress monopole proliferation, thereby revealing rich physics even in transitions between trivial phases.

The Gist

Imagine you are looking at a vast, perfectly organized city made of atoms. In this city, the "residents" (electrons or bosons) are stuck in their homes. They can't move around freely to conduct electricity; they are insulators. Usually, we think of these insulators as boring, simple, and identical to one another—like a city where every house is built exactly the same way, with the furniture sitting right in the center of the room.

But this paper discovers a secret: some of these "boring" cities are actually obstructed.

The "Obstructed" House

In a normal city, the furniture (the electron's wavefunction) sits perfectly on the floorboards (the atomic sites). But in an Obstructed Atomic Insulator, the furniture is shifted. It's sitting in the middle of the hallway or the corner of the room, not on the floorboards.

You might think, "So what? It's still just a house." But here's the catch: You cannot slide the furniture from the hallway back to the center of the room without breaking the rules of the house (symmetry) or smashing the walls (closing the energy gap). These two states—furniture on the floor vs. furniture in the hall—are fundamentally different, even though neither has any "magic" topological powers like a superconductor.

The Great Migration: When the Furniture Jumps

The paper asks: What happens when you try to transform a city with furniture in the hall into a city with furniture on the floor?

Usually, when two different phases of matter meet, they go through a messy transition. But here, the authors found something surprising. When you tune the city just right, the transition point doesn't just get messy; it becomes a party.

At this critical moment, the residents stop acting like individual people. They start behaving like a chaotic, interconnected crowd that creates a new kind of physics entirely.

The "Quantum Electrodynamics" Party

The authors found that for certain city layouts (specifically, a "breathing kagome" lattice, which looks like a honeycomb that breathes in and out), this transition point hosts a state of matter described by QED3 (Quantum Electrodynamics in 2+1 dimensions).

The Analogy:
Imagine the residents of the city suddenly decide to stop being people and start acting like ghosts that can only communicate by sending invisible messages (photons) to each other.

• The Partons: The authors use a trick called "fractionalization." They pretend each resident is actually two ghosts holding hands.
• The Gauge Field: These ghosts are connected by an invisible rubber band (the gauge field). If one ghost moves, the other feels it instantly, no matter how far apart they are.
• The Result: At the transition, the city becomes a "liquid" of these ghosts and rubber bands. It's a state of matter that doesn't exist in the normal world, described by complex math usually reserved for particle physics.

The "Monopole" Gatekeepers

Here is the most critical part of the story. The paper explains that whether this "ghost party" (the QED3 state) can actually happen depends on the shape of the city.

Think of the invisible rubber bands as having a tendency to snap and tie themselves into knots. These knots are called monopoles.

• The Bipartite City (Square/Honeycomb): In cities with a checkerboard pattern (like the square lattice), the rules of the city allow these knots to form easily. As soon as you try to start the party, the knots tie themselves up, the rubber bands snap, and the ghosts get trapped. The "ghost party" collapses, and the transition becomes a boring, abrupt jump (a first-order transition). It's like trying to hold a dance party in a room where the floor is covered in sticky tape; you can't move, so the party never starts.
• The Tripartite City (Breathing Kagome): In the breathing kagome city, the layout is different. The rules of the city forbid these knots from forming. The rubber bands stay loose, the ghosts stay free, and the "ghost party" (the QED3 critical point) survives! This is a rare, stable, and exotic state of matter.

Why This Matters

For a long time, physicists thought that "boring" insulators were just boring. If you moved from one to another, it was a simple, predictable change.

This paper shows that:

1. Boring can be exciting: Even between two "trivial" phases, you can find a hidden, exotic world of quantum entanglement and emergent forces.
2. Architecture matters: The specific shape of the atomic lattice (the city plan) acts as a gatekeeper. It decides whether this exotic physics can exist or if it gets crushed by "knots" (monopoles).
3. New Physics: It proves that we can engineer materials where the transition point itself is a new state of matter, governed by the same laws that describe the fundamental forces of the universe.

In a nutshell: The authors found that if you build your atomic city with the right blueprint (the breathing kagome lattice), you can force the atoms to dance in a way that creates a temporary, magical state of "ghostly" electricity right in the middle of a boring transition. If you build it with the wrong blueprint (a square grid), the magic gets blocked by invisible knots, and the transition stays boring.

Technical Summary

1. Problem Statement

The paper investigates quantum phase transitions between obstructed atomic insulators (OAIs) in $(2+1)$ dimensions.

• Context: OAIs are "trivial" in the sense that they lack topological order, protected edge modes, or long-range entanglement. Their ground states can be adiabatically deformed into product states if symmetry is broken. However, they are distinct from "trivial" atomic insulators because their Wannier centers are displaced from physical atomic sites (e.g., located at bond centers or plaquette centers rather than lattice sites).
• The Question: What is the nature of the critical point when two such distinct, symmetry-preserving trivial insulating phases transition into one another?
• The Challenge: Standard Landau-Ginzburg theory fails here because there is no local order parameter distinguishing the phases (both preserve all symmetries). The authors seek to determine if these transitions can host exotic, non-Landau critical points, specifically those described by emergent gauge theories like Quantum Electrodynamics in $(2+1)$ dimensions ($QED_3$).

2. Methodology

The authors employ a combination of parton constructions, conformal field theory (CFT) analysis, and anomaly matching arguments.

• Parton Construction (Fractionalization): To analyze bosonic systems, the authors fractionalize hard-core bosons ($b$) into two fermionic partons ($d_1, d_2$) via the relation $b = d_1^\dagger d_2$. This introduces an emergent $U(1)$ (or $SU(2)$) gauge field.
• Mean-Field Theory: They construct mean-field Hamiltonians where the partons realize Dirac band touchings at the critical point. The low-energy effective theory is identified as $N_f$ Dirac fermions coupled to an emergent gauge field.
• Symmetry Analysis & Monopole Operators: A crucial step is analyzing the monopole operators (topological defects in the gauge field) in the emergent $QED_3$ theory. The stability of the critical point depends on whether the microscopic lattice symmetries (UV) allow for relevant monopole operators in the infrared (IR) theory. If a symmetry-allowed monopole is relevant, it drives the system to confinement (first-order transition). If all relevant monopoles are forbidden by symmetry, the $QED_3$ fixed point is stable.
• Anomaly Matching: The authors use 't Hooft anomaly matching (specifically Lieb-Schultz-Mattis-Oshikawa-Hastings or LSMOH constraints) to verify that the proposed IR critical theories are consistent with the UV lattice symmetries. This ensures the critical points are "emergeable" from local lattice Hamiltonians.
• Lattice Models: They analyze specific lattice geometries:
  • Square Lattice (BBH Model): A bipartite lattice with $C_4$ symmetry.
  • Breathing Kagome Lattice: A tripartite lattice with $C_3$ symmetry.
  • Breathing Honeycomb Lattice: A bipartite lattice with $C_6$ symmetry.

3. Key Contributions and Results

A. One-Dimensional Warm-up (SSH Chain)

The authors first review the Su-Schrieffer-Heeger (SSH) model.

• Fermionic Case: The transition is described by a massless Dirac fermion.
• Bosonic Case: With interactions, the transition is described by a Luttinger liquid (a CFT with no quasiparticles). This establishes that even between trivial phases, the critical point can be a non-Fermi liquid.

B. Two-Dimensional Transitions: The Role of Lattice Geometry

The central finding is that the stability of the $N_f=4$ $QED_3$ critical point depends critically on whether the underlying lattice is bipartite or tripartite.

1. Bipartite Lattices (Square and Breathing Honeycomb):

• Model: The Benalcazar-Bernevig-Hughes (BBH) model on a square lattice and a similar model on a breathing honeycomb lattice.
• Result: On bipartite lattices, the embedding of UV symmetries into the IR $QED_3$ theory always permits at least one relevant, symmetry-allowed monopole operator.
• Mechanism: The authors argue that the $N_f=4$ $QED_3$ on a bipartite lattice can be adiabatically deformed to $N_f=2$ $QCD_3$. This deformation guarantees the existence of a "trivial" monopole (a singlet under all lattice symmetries).
• Consequence: The presence of this relevant monopole destabilizes the conformal fixed point. The transition is generically first-order (or highly multicritical), rather than a continuous transition described by a stable $QED_3$ CFT.

2. Tripartite Lattices (Breathing Kagome):

• Model: A $C_3$-symmetric model on a breathing kagome lattice.
• Result: The tripartite nature of the lattice prevents the formation of a trivial monopole singlet.
• Mechanism: The specific embedding of $C_3$ rotation and time-reversal symmetries forbids all relevant monopole operators. The only relevant operator is the mass term tuning the transition.
• Consequence: The $N_f=4$ $QED_3$ critical point is stable and conformal. This represents a genuine, continuous quantum phase transition between two trivial insulating phases, described by an interacting CFT with emergent electrodynamics.

C. Anomaly Matching Verification

The authors perform a rigorous check using LSMOH anomaly matching.

• They demonstrate that the IR anomalies of the $N_f=4$ $QED_3$ theory (including the unstable cases on bipartite lattices) correctly pull back to the trivial UV anomalies of the lattice boson models.
• This confirms that the proposed critical theories, even the unstable ones, are physically realizable from local lattice Hamiltonians.

4. Significance and Implications

• Beyond Landau Paradigm: The paper demonstrates that "trivial-to-trivial" transitions can host exotic critical points described by deconfined gauge theories, challenging the notion that such transitions are always first-order or described by simple order parameters.
• Lattice Symmetry as a Determinant: A major conceptual advance is the realization that the microscopic lattice structure (bipartite vs. tripartite) dictates the fate of the critical point. It is not just the band topology that matters, but how lattice symmetries embed into the emergent continuum symmetries.
• Realization of $QED_3$: The breathing kagome lattice is identified as a concrete, experimentally accessible platform (e.g., in cold atom systems or optical lattices) where a stable $N_f=4$ $QED_3$ CFT can be realized.
• Monopole Physics: The work provides a detailed classification of monopole quantum numbers under lattice symmetries, resolving previous ambiguities about the stability of $QED_3$ in lattice models.
• Generalizability: The framework suggests that similar phenomena could occur in higher dimensions (e.g., pyrochlore lattices) or with different internal symmetries, opening a new avenue for exploring "unnecessary" quantum critical points.

In summary, Zhang and Senthil establish that while transitions between obstructed atomic insulators on bipartite lattices are likely first-order due to monopole proliferation, transitions on tripartite lattices can realize stable, conformal $QED_3$ critical points, offering a rare example of a continuous transition between two featureless, symmetry-preserving insulating phases.