Proposal for realizing unpaired Weyl points in a three-dimensional periodically driven optical Raman lattice

This paper proposes a feasible experimental scheme using periodically driven ultracold atoms in a 3D optical Raman lattice to realize unpaired Weyl points with tunable net chirality, thereby circumventing the Nielsen-Ninomiya theorem and enabling the observation of the chiral magnetic effect through a quantized charge current.

The Gist

Imagine you are a physicist trying to build a tiny, magical highway for particles called Weyl fermions. These particles are like ghosts in the machine of physics: they are incredibly fast, massless, and have a special property called "chirality" (or "handedness"). Think of chirality like a screw: some spin clockwise (right-handed), and some spin counter-clockwise (left-handed).

In the natural world, there's a strict rule called the Nielsen-Ninomiya theorem. It's like a cosmic bouncer that says: "You cannot have a party with only right-handed screws. For every right-handed one, you must have a left-handed partner."

This rule is a problem because it prevents a cool phenomenon called the Chiral Magnetic Effect (CME). The CME is like a one-way street where a magnetic field forces all the particles to flow in a single direction, creating a super-efficient electric current. But because the "bouncer" forces them to pair up and cancel each other out, this one-way street can't exist in normal, static materials.

So, how do the authors of this paper solve the problem?

They propose a way to trick the cosmic bouncer using ultracold atoms (atoms cooled to near absolute zero) and lasers. Here is the story of their proposal, broken down into simple steps:

1. The Playground: A 3D Optical Raman Lattice

Imagine a giant, invisible 3D grid made of light (lasers) floating in a vacuum chamber. This is the Optical Raman Lattice.

• The Atoms: The scientists use Potassium-40 atoms. Think of these atoms as tiny, four-story buildings with different rooms (energy levels).
• The Lasers: They shine multiple laser beams from different directions. These beams create a "dance floor" for the atoms. The lasers don't just hold the atoms in place; they also act like a conductor, telling the atoms how to spin and move.

2. The Trick: The "Floquet" Dance

Normally, if you just set up this light grid and leave it alone, the "bouncer" rule applies, and the particles pair up. But the authors propose a periodic driving scheme.

• The Metaphor: Imagine the light grid isn't static. Instead, it's a dance floor that vibrates and changes rhythm in a very specific, repeating pattern (like a beat in a song).
• By shaking the grid at just the right speed (adiabatic modulation), they change the rules of the game. In this "dancing" world (known as a Floquet system), the cosmic bouncer gets confused. The rule that forces particles to pair up no longer applies to the average behavior of the system.

3. The Result: Unpaired Weyl Points

Because the rules are broken by the dancing lasers, the system creates Unpaired Weyl Points.

• The Analogy: Imagine a crowd of people in a room. Usually, for every person walking left, someone walks right, so the crowd stays still. But in this dancing light room, the authors manage to get 8 people to start walking left, and zero people walking right.
• This creates a chirality imbalance. The "net handedness" is no longer zero. This is the holy grail: a system where the particles are unpaired.

4. The Payoff: The Chiral Magnetic Effect (CME)

Now that they have these unpaired particles, they introduce a synthetic magnetic field.

• How? They use a clever laser trick called "laser-assisted tunneling." It's like giving the atoms a gentle nudge that makes them feel like they are in a magnetic field, even though they are neutral atoms.
• The Effect: Because the particles are unpaired (all "left-handed" or all "right-handed"), the magnetic field pushes them all in the same direction.
• The Outcome: A quantized charge current flows. This is a perfectly measured, one-way flow of particles. It's the "smoking gun" proof that the Chiral Magnetic Effect has been achieved.

5. Is this real?

The authors didn't just dream this up; they checked the math and the physics.

• Feasibility: They used Potassium-40, an atom that experimentalists already know how to handle.
• Timing: They calculated that the "dancing" of the lasers needs to happen fast enough to be smooth (adiabatic) but slow enough to not break the atoms apart. Their calculations show this is well within the capabilities of current technology.
• Detection: They propose a simple way to see if it worked: just watch the cloud of atoms move. If the CME is happening, the whole cloud will drift in a specific direction, like a boat being pushed by a current.

Summary

In short, this paper proposes a recipe to build a 3D light trap for atoms. By vibrating the trap in a specific rhythm, they can break the natural laws that usually force particles to cancel each other out. This allows them to create a one-way traffic jam of particles driven by a magnetic field.

It's like taking a crowded two-way street, putting up a "One Way" sign that the traffic police (the laws of physics) usually ignore, but using a special vibrating road surface (the lasers) to force the cars to obey. This could open the door to new types of electronics and a deeper understanding of the universe's most fundamental particles.

Technical Summary

1. Problem Statement

In static lattice systems, the Nielsen-Ninomiya theorem dictates that Weyl points (topological defects in momentum space) must appear in pairs with opposite chiralities. This pairing enforces a net zero chirality, which fundamentally prohibits the observation of the Chiral Magnetic Effect (CME)—a phenomenon where a net electric current flows parallel to an external magnetic field due to chiral anomaly—in equilibrium.

While theoretical proposals suggest that non-equilibrium settings (specifically, adiabatic periodic driving) can circumvent this constraint to create "unpaired" Weyl points with non-zero net chirality, existing proposals often suffer from:

• Requiring fine-tuning of multiple parameters.
• Involving complex driving sequences that are experimentally difficult to implement.
• Lacking a concrete, realistic protocol within a specific quantum platform (like ultracold atoms) with a detailed analysis of the resulting CME.

2. Methodology

The authors propose a realistic experimental scheme using ultracold atoms (specifically $^{40}$K) trapped in a three-dimensional (3D) Optical Raman Lattice (ORL) under continuous periodic driving.

• System Setup:

  • Atomic States: Utilizes four long-lived internal states of an alkali atom (two hyperfine manifolds $g$ and $e$, each with two Zeeman sublevels $\uparrow, \downarrow$), acting as a four-level system.
  • Laser Configuration: A specific arrangement of laser beams creates:
    1. A state-independent 3D optical lattice potential ($V_{latt}$).
    2. Static Raman coupling potentials ($M_x, M_y, M_z$) that generate effective 3D spin-orbit coupling (SOC).
    3. A time-dependent Raman potential ($M_0$) and a Zeeman term ($m_z$) driven synchronously.
  • Symmetry Engineering: The relative symmetries between the lattice and Raman potentials are engineered such that $M_{x,y,z}$ are antisymmetric along their respective axes (generating spin-flipped hopping) while $M_0$ is symmetric (generating on-site spin flips). This configuration realizes a tunable 3D topological insulator.

• Driving Protocol:

  • The system is subjected to adiabatic periodic modulation of the phase $\eta(t)$ of the $M_0$ potential and the Zeeman term $m_z(t)$.
  • The modulation is defined as:
    $$\eta(t) = \eta_0 \sin(\omega t), \quad m_z(t) = m_z^{(1)} - m_z^{(2)} \cos(\omega t)$$
  • The driving frequency $\omega$ is chosen to be much smaller than the instantaneous energy gap, ensuring the adiabatic limit.

• Synthetic Magnetic Field:

  • To probe the CME, a synthetic magnetic field is introduced via laser-assisted tunneling. A linear potential tilt along the $y$-axis suppresses bare hopping, which is resonantly restored by running-wave laser beams, imprinting a site-dependent Peierls phase factor equivalent to a uniform magnetic field.

3. Key Contributions

1. Realization of Unpaired Weyl Points: The proposal demonstrates that the Floquet quasienergy spectrum of the low-energy sector hosts eight Weyl points at high-symmetry momenta in the 3D Brillouin zone.
2. Tunable Net Chirality: Unlike static systems, the total chirality ($\chi_{tot}$) of these Weyl points can be precisely tuned by adjusting the mass parameter $m_z^{(1)}$. A non-zero net chirality ($\chi_{tot} \neq 0$) signifies the emergence of unpaired Weyl points, effectively breaking the Nielsen-Ninomiya constraint.
3. Direct Observation of CME: The authors demonstrate that this chirality imbalance drives a quantized charge current parallel to the synthetic magnetic field in the weak-field regime.
4. Experimental Feasibility: The paper provides a comprehensive analysis showing that all required parameters (lattice depths, Raman couplings, driving frequencies) are within the reach of current ultracold-atom technology, specifically citing existing ORL implementations with $^{40}$K.

4. Results

• Topological Phase Diagram: By varying the mass parameter $m_z^{(1)}$, the system transitions between different topological phases. For specific parameters (e.g., $m_z^{(1)} = 0.13 E_r$), the system exhibits a second Chern number $C_2 = 3$, corresponding to a net chirality of $\chi_{tot} = 6$ (or 4 in the CME section depending on specific tuning).
• Weyl Point Characteristics:
  • Eight Weyl points emerge at momenta $k_j \in \{(k_x, k_y, k_z) | k_i \in \{0, \pi\}\}$.
  • The chirality of individual points is determined by the spin texture of the Floquet eigenstates.
  • The net chirality $\chi_{tot} = \sum \chi_j$ is non-zero, confirming the unpaired nature.
• Adiabaticity Verification:
  • The authors verify the adiabatic condition by calculating the spin pumping along momentum lines passing through the Weyl points.
  • Results show that for a driving period $T \geq 500 E_r^{-1}$, the pumped spin deviates from the quantized value by less than 1%, satisfying the adiabatic requirement. This timescale is well within the lifetime of ultracold $^{40}$K atoms in optical lattices ($\sim 1300 E_r^{-1}$).
• Chiral Magnetic Effect (CME):
  • In the presence of a synthetic magnetic field $B$, the system exhibits a charge current $J_z$.
  • In the weak-field regime, the pumped charge per cycle $\Delta Q$ is quantized and proportional to the total chirality:
    $$\Delta Q = \frac{\chi_{tot} B}{8\pi}$$
  • This quantization breaks down only when $B$ becomes strong enough to close the gap between low- and high-energy bands, violating the adiabatic approximation.
  • The current can be detected experimentally by tracking the center-of-mass displacement of the atomic cloud.

5. Significance

• Overcoming No-Go Theorems: This work provides a concrete, experimentally viable pathway to bypass the Nielsen-Ninomiya theorem, enabling the study of single-chirality Weyl fermions in a controlled environment.
• Direct CME Detection: It offers a direct method to observe the Chiral Magnetic Effect in a neutral atom system, a phenomenon previously difficult to isolate in solid-state materials due to competing effects and disorder.
• Platform for Non-Equilibrium Topology: The proposed 3D ORL scheme establishes a robust platform for exploring chiral-anomaly physics, Floquet engineering, and non-equilibrium topological phenomena.
• Experimental Readiness: By leveraging established techniques (ORLs, laser-assisted tunneling, and Floquet engineering) and specific atomic species ($^{40}$K), the proposal bridges the gap between abstract theoretical concepts and practical laboratory realization.