Breakdown of chiral anomaly and emergent phases in Weyl semimetals under orbital magnetic fields

This paper investigates how an orbital magnetic field induces gap-opening and layered Chern insulating states in lattice-based Weyl semimetals, revealing rich phenomenology driven by cone anisotropy and distinct surface Fermi-arc evolution that differs from continuum predictions.

The Gist

Imagine a world made of a special kind of crystal called a Weyl Semimetal. Inside this crystal, electrons don't behave like normal cars on a highway; they act like massless, super-fast particles called "Weyl fermions."

In this crystal, these particles live in pairs. Think of them as two distinct neighborhoods, Neighborhood A and Neighborhood B, located far apart in a "map" of momentum (a way to describe how fast and in what direction the electrons are moving). These two neighborhoods have opposite "handedness" (chirality), like a left hand and a right hand.

Normally, these two neighborhoods are separated by a wide, empty valley. Electrons can't jump from one to the other, so the material stays "gapless"—meaning electricity can flow freely without resistance. This is the Weyl Semimetal state.

The Magnetic "Bridge"

Now, imagine you apply a strong magnetic field from the side, perpendicular to the line connecting these two neighborhoods.

In the old, simplified view of physics (the "continuum" picture), this magnetic field acts like a magical bridge. It allows electrons to tunnel (jump) from Neighborhood A to Neighborhood B. If they jump, the two neighborhoods merge, the valley disappears, and a "gap" opens up. The material stops being a super-conductor and becomes an insulator (a material that blocks electricity).

The old theory said this bridge forms smoothly and predictably: the stronger the magnetic field, the wider the bridge, and the bigger the gap.

The New Discovery: The "Pixelated" Map

The authors of this paper, Faruk Abdulla, Anna Keselman, and Daniel Podolsky, decided to look closer. They realized that real crystals aren't smooth, continuous maps; they are pixelated grids (lattices), like a chessboard or a video game world.

When you zoom in on this pixelated grid, the story changes dramatically. The magnetic field doesn't just build one smooth bridge; it creates a complex, shifting landscape with two very different behaviors depending on the shape of the neighborhoods.

Scenario 1: The "Egg-Shaped" Neighborhoods (Simple Case)

Imagine the neighborhoods are shaped like smooth, round eggs.

• What happens: When you turn on the magnetic field, the electrons can jump across.
• The Result: The material simply switches from a super-conductor to a normal insulator. It's a clean, one-step switch. The gap opens up and stays open.

Scenario 2: The "Crescent-Moon" Neighborhoods (The Complex Case)

Now, imagine the neighborhoods are shaped like crescent moons, curving around each other.

• What happens: The magnetic field creates two different paths for electrons to jump: one path going "inside" the curve and one going "outside."
• The Interference: These two paths act like waves in a pond. Sometimes they crash into each other and cancel out (destructive interference). When they cancel out, the "bridge" disappears!
• The Result: As you increase the magnetic field, the material doesn't just stay insulating. It oscillates. It opens a gap, then closes it, then opens it again.
  • It flips back and forth between being a Normal Insulator (blocks electricity) and a weird, special state called LCI' (a layered insulator that is protected by the crystal's symmetry).
  • It's like a light switch that flickers on and off rapidly as you turn the knob, rather than just getting brighter.

The "Layered" Insulators and Surface Ghosts

The paper also discovers a new type of insulator called a Layered Chern Insulator (LCI).

• The Analogy: Imagine a stack of pancakes. In a normal insulator, the pancakes are just stacked. In an LCI, each pancake is a tiny, magical island where electricity can only flow in a circle around the edge, never getting stuck in the middle.
• The Surface Ghosts (Fermi Arcs): In the original Weyl Semimetal, there are "ghost roads" (Fermi arcs) on the surface of the crystal that connect the two neighborhoods.
  • In the Simple Case, when the gap opens, these ghost roads vanish instantly.
  • In the Complex Case, the ghost roads stretch out, wrap around the edge of the crystal, and turn into a complete loop. They don't disappear; they just change shape to fit the new "layered" world.

Why Does This Matter?

1. It's Not Just Theory: Previous theories assumed the crystal was smooth. This paper shows that the "pixels" of the real crystal matter a lot, especially in strong magnetic fields.
2. New States of Matter: We found a new state of matter (LCI') that only exists because of this pixelated nature and the specific shape of the electron neighborhoods.
3. Experimental Clues: The authors suggest that scientists can tell these different states apart by measuring Hall Conductivity (how electricity moves sideways in a magnetic field).
   • If the conductivity is zero, it's a normal insulator.
   • If it's a specific "quantized" number, it's the Layered Chern Insulator.
   • If it's zero again but with different surface properties, it's the special LCI' state.

The Bottom Line

This paper is like realizing that a smooth road map is actually a grid of city blocks. When you try to drive across the city with a strong wind (magnetic field), the way you get from point A to point B depends entirely on the shape of the blocks and the grid layout. Sometimes the wind helps you cross; sometimes it pushes you back; and sometimes it creates a completely new route that didn't exist before.

The authors have mapped out these routes, showing us that the quantum world is full of surprises, oscillations, and hidden layers that only reveal themselves when we look at the "pixels" of reality.

Technical Summary

1. Problem Statement

Weyl Semimetals (WSMs) are topological phases characterized by pairs of Weyl nodes (WNs) with opposite chiralities separated in momentum space. A key feature of WSMs is the chiral anomaly, which manifests in phenomena like negative magnetoresistance. However, when a strong external orbital magnetic field ($B$) is applied perpendicular to the separation vector of the WNs, it induces quantum tunneling between the nodes, potentially opening a gap in the electronic spectrum and destroying the WSM state.

Previous theoretical investigations relied on low-energy continuum models. These models predicted that:

• The induced gap is non-perturbative (exponentially small) in the field strength.
• The gap behavior depends on the anisotropy of the Weyl cone dispersion.
• In some anisotropic cases, the gap exhibits oscillatory behavior as a function of the magnetic field or node separation.

Limitations of Continuum Models:

1. Lattice Effects Ignored: Continuum models assume $Q \ll \text{BZ size}$ and magnetic length $l_B \gg a$ (lattice constant). They fail to capture phenomena arising from the periodicity of the Brillouin Zone (BZ) and the discrete nature of the lattice, particularly in the "Hofstadter regime" where $l_B \sim a$.
2. Surface States: Continuum models cannot adequately describe the fate of topologically protected surface Fermi-arc states when the bulk gap opens.
3. Topological Classification: Determining whether the gapped state is a trivial insulator or a non-trivial topological phase (e.g., Chern insulator) requires a full lattice treatment.

Objective: The authors aim to investigate the gap-opening mechanism and the resulting emergent phases in WSMs using a lattice model, specifically addressing how lattice periodicity and Landau level (LL) dispersion alter the phase diagram and surface state evolution.

2. Methodology

The authors employ a minimal two-band lattice model for a WSM that breaks time-reversal symmetry but preserves inversion symmetry.

• Hamiltonian: A cubic lattice model with WNs located at $k_x = \pm k_0$ on the $k_x$ axis. The Hamiltonian is:
  $$H(k) = 2\tilde{v}_x m(k)\sigma_x + 2v_y \sin(k_y a)\sigma_y + 2v_z \sin(k_z a)\sigma_z$$
  where $m(k)$ defines the dispersion and $\tilde{v}_x$ is a renormalized velocity.
• Magnetic Field: A uniform orbital magnetic field $B \parallel \hat{z}$ is applied. The system is treated in the Landau gauge $\mathbf{A} = (-y, 0, 0)B$.
• Key Parameter ($\gamma$): The authors define an anisotropy parameter $\gamma = 1 - \frac{v_y^2}{v_x^2 t^2}$ (where $t$ is a model parameter). This parameter dictates the shape of the Fermi pockets around the WNs:
  • $\gamma < 0$: Elliptical pockets.
  • $\gamma > 0$: Crescent-shaped pockets.
• Approach:
  1. Continuum Limit Review: They first revisit the continuum description to unify previous results regarding monotonic vs. oscillatory gap behavior.
  2. Lattice Calculation: They construct the Bloch-Hofstadter Hamiltonian for commensurate magnetic fluxes ($\Phi_B/\Phi_0 = 1/q$). This allows them to define a magnetic Brillouin zone (mBZ) and compute topological invariants (Chern numbers).
  3. Surface States: They analyze slab geometries to track the evolution of Fermi-arc surface states across phase transitions.

3. Key Contributions and Results

A. Continuum Limit Unification

The authors confirm that in the continuum limit, the gap behavior is governed by the interference of tunneling paths between Fermi pockets:

• $\gamma < 0$ (Elliptical): Single tunneling path leads to a monotonic gap opening that increases with the field.
• $\gamma > 0$ (Crescent): Two close encounters allow for two tunneling paths. Destructive interference leads to periodic gap closings (oscillations) as a function of $Q l_B$ or field strength.

B. Lattice-Induced Phenomenology

The lattice model reveals rich physics absent in the continuum:

1. Two Distinct Tunneling Mechanisms:
Due to the periodicity of the BZ, there are two ways WNs can couple:

• Intra-BZ Tunneling ($Q$): Tunneling within the first BZ.
• Inter-BZ Tunneling ($Q'$): Tunneling across the BZ boundary (involving reciprocal lattice vectors).
  This leads to two topologically distinct gapped states.

2. Phase Diagram for $\gamma < 0$ (Elliptical Pockets):

• Phases: As the node separation $Q$ increases, the system transitions from a Normal Insulator (NI) to a Layered Chern Insulator (LCI).
• Transition: These phases are separated by a narrow gapless WSM region.
• Width: The width of the WSM region is exponentially small ($\sim e^{-1/\Phi_B}$), determined by the finite bandwidth of the Landau levels on the lattice.
• Surface States:
  • In the NI phase (small $Q$), Fermi arcs are short and are immediately gapped out.
  • In the LCI phase (large $Q$), Fermi arcs evolve into extended chiral surface states spanning the magnetic BZ.

3. Phase Diagram for $\gamma > 0$ (Crescent Pockets):

• Oscillatory Behavior: For small $Q$, the system alternates between NI and a new phase called LCI'.
• LCI' Phase: This is a symmetry-protected topological phase (relying on mirror symmetry) consisting of two copies of LCI with opposite Chern numbers, resulting in a net zero Chern number but distinct surface states.
• Complex Transitions: Transitions between NI and LCI' are not simple; they involve a sequence of three phases: WSM $\to$ LCI $\to$ WSM.
• Large $Q$ Limit: For large separations, inter-BZ tunneling dominates, suppressing oscillations, and the system settles into a standard LCI phase.

4. Fate of Fermi-Arc States:

• Short Arcs (NI): Disappear immediately upon gap opening; electrons tunnel into the bulk via chiral LLs.
• Long Arcs (LCI/LCI'): Persist and evolve smoothly. In the LCI' phase, two distinct Fermi arcs exist at mirror-symmetric momenta ($k_y=0$ and $k_y=\pi/q$). If mirror symmetry is broken, LCI' becomes indistinguishable from a trivial NI.

4. Significance and Implications

• Beyond Continuum Theory: The work demonstrates that lattice effects are crucial for understanding WSMs in strong magnetic fields, particularly in the Hofstadter regime. It corrects the continuum prediction by showing that gap closings are not strictly periodic and that new topological phases (LCI') emerge.
• Topological Classification: It provides a rigorous framework for classifying field-induced gapped phases (NI, LCI, LCI') based on $k_x$-dependent Chern numbers and surface state topology.
• Experimental Signatures:
  • Hall Conductivity: The transition from WSM to LCI results in a quantized anomalous Hall conductivity ($\sigma_{yz} = e^2/h$ per layer), whereas transitions to NI or LCI' result in vanishing Hall conductivity.
  • Surface Probes: The presence of zero-energy modes in the LCI' phase (due to closed Fermi arcs) could be detected via Scanning Tunneling Microscopy (STM).
  • Tunability: In layered materials, rotating the magnetic field can switch the anisotropy parameter $\gamma$ (by changing the ratio of velocity components), allowing experimentalists to tune between monotonic and oscillatory gap behaviors and access different topological phases.

Conclusion

This paper establishes that the interplay between orbital magnetic fields and lattice periodicity in Weyl semimetals leads to a complex phase diagram featuring layered Chern insulators and symmetry-protected topological phases. The breakdown of the chiral anomaly is not a simple gap opening but a rich sequence of topological phase transitions governed by the anisotropy of the Weyl cones and the discrete nature of the lattice. These findings offer new avenues for controlling topological states in WSMs via magnetic field tuning.