Angular-resolved nonlinear optical response as a probe of Lorentz violation in noncentrosymmetric materials

This paper proposes a method to detect weak Lorentz-violating backgrounds in noncentrosymmetric crystals by measuring a distinct $\pi$-periodic angular modulation in the nonlinear shift photocurrent, which arises from momentum-odd corrections to the Bloch Hamiltonian and offers sensitivity to coupling strengths as low as $\xi\sim10^{-24}\,\mathrm{C\,m}$.

The Gist

The Big Idea: Catching a "Ghost" in the Machine

Imagine the universe has a set of strict rules, like the laws of physics. One of the most famous rules is Lorentz Symmetry. Think of this rule as a "universal fairness policy." It says that the laws of physics should look exactly the same no matter which way you are facing (orientation) or how fast you are moving (uniform motion). It's like saying a game of chess plays the same way whether you are sitting in New York, Tokyo, or on a spaceship moving at a steady speed.

However, some theories suggest that deep down, this "fairness policy" might have tiny cracks in it. These cracks are called Lorentz Violation (LV). They are caused by invisible, fixed "background fields" (like a subtle wind blowing through the universe) that make physics behave slightly differently depending on your direction.

The problem? These cracks are so tiny that we can't see them with our current tools. They are like trying to hear a whisper in a hurricane.

This paper proposes a new, clever way to listen for that whisper.

The Setup: A One-Way Street of Atoms

The researchers suggest using a very specific type of material: a noncentrosymmetric crystal.

• The Analogy: Imagine a row of houses on a street. In a normal, symmetric street, every house looks the same from the front and the back. But in this special material, the houses are different. Maybe House A has a red door and House B has a blue door, and they alternate. This lack of symmetry is crucial.
• The Model: They use a theoretical model called the Rice-Mele model, which describes a chain of atoms that acts like this alternating street.

The Experiment: Spinning the Compass

Here is the experiment they propose:

1. The Chain: You have a long, thin chain of these special atoms.
2. The Light: You shine a laser on it. This light knocks electrons loose, creating an electric current (like wind pushing a sail).
3. The Twist: You apply a static electric field (a steady push) to the side of the chain.
4. The Rotation: This is the key. You slowly rotate the direction of that side-push, like turning a compass needle 360 degrees.

The Prediction: The "Tick-Tock" vs. The "Beep-Beep"

When you rotate the push, you measure the electric current flowing down the chain.

• Scenario A (Normal Physics / No Ghost): If the universe is perfectly fair (Lorentz Symmetric), the current will wiggle as you turn the compass. But it will follow a smooth, predictable pattern that repeats every 360 degrees (one full circle). It's like a clock ticking: Tick... Tock... Tick... Tock.
• Scenario B (The Ghost is Real / Lorentz Violation): If there is that invisible "background wind" (Lorentz Violation), it messes with the quantum mechanics of the electrons. The current will still wiggle, but the pattern will change. Instead of repeating every 360 degrees, it will repeat every 180 degrees.
  • The Metaphor: Imagine a clock that ticks twice as fast. Tick-Tock-Tick-Tock. It hits a peak, drops, hits a peak again, and drops, all within a single full turn of the compass.

Why does this matter?
Detecting a tiny change in the strength of the current is hard because background noise can hide it. But detecting a change in the rhythm (the pattern of repetition) is much easier and more reliable. It's like hearing a different drumbeat rather than trying to hear a quieter drum.

The "Shift Current" Secret Sauce

The paper focuses on a specific type of current called the "Shift Current."

• Analogy: Imagine a crowd of people (electrons) in a hallway. When the light hits them, they don't just run; they take a specific "step" or "shift" to a new spot.
• In normal materials, this step depends on the shape of the hallway.
• The researchers found that the "Lorentz Violation" background changes the shape of the steps themselves. It twists the path the electrons take. This twist creates that special 180-degree rhythm (or $\pi$-periodicity) in the current.

The Results: Can We Actually See It?

The team did the math (using supercomputers) to see if this signal is strong enough to be real.

• The Signal: They predict the current change would be very small—about the size of a picoampere (a trillionth of an amp). That's tiny!
• The Solution: You wouldn't use just one chain of atoms. You would use a bundle of millions of them, all lined up perfectly. It's like having one person whispering; you can't hear them. But if a million people whisper the same thing at the same time, you can hear it clearly.
• Sensitivity: With this setup, they could detect Lorentz Violation strengths as small as $10^{-24}$. This is incredibly sensitive, potentially rivaling the best experiments in particle physics, but using a simple table-top device made of crystals.

Summary

1. The Problem: We want to find proof that the universe's "fairness rules" (Lorentz Symmetry) are slightly broken, but the evidence is too faint to find.
2. The Idea: Use a special, asymmetric chain of atoms and shine light on it while rotating a magnetic/electric field.
3. The Clue: If the universe is "broken" (Lorentz Violation), the electric current will pulse twice as fast as it should when you rotate the field (a 180-degree pattern instead of 360).
4. The Impact: This gives us a new, practical way to hunt for fundamental physics secrets using solid materials, turning a microscopic quantum effect into a measurable electrical rhythm.

In short: They found a way to listen for the universe's "glitch" by watching how electrons dance to a new beat.

Technical Summary

1. Problem Statement

Lorentz symmetry is a cornerstone of modern physics, yet the Standard-Model Extension (SME) allows for controlled violations (LV) via coupling to fixed background tensors. While high-precision experiments (e.g., atomic spectroscopy, cavity resonators) have placed tight bounds on LV, the influence of actual LV backgrounds on measurable solid-state observables remains largely unexplored.

• The Gap: Existing condensed matter studies often mimic LV effects (e.g., tilted dispersions in Weyl semimetals) but do not directly probe actual LV backgrounds.
• The Challenge: Detecting weak LV signals requires a system with high sensitivity to symmetry-breaking perturbations that modify band geometry and phase structures, distinguishing them from standard Lorentz-symmetric responses.

2. Methodology

The authors propose a theoretical framework to detect LV backgrounds using nonlinear shift photocurrents in noncentrosymmetric (polar) crystals.

• Model System: A one-dimensional, spinful Rice–Mele model representing a polar atomic chain. This model includes alternating hopping amplitudes ($t_1, t_2$) and a staggered on-site potential ($\Delta$).
• LV Perturbation: The system is coupled to a fixed background four-vector $v^\mu$ via a non-minimal coupling in the Dirac equation.
  • A Foldy–Wouthuysen reduction is performed to derive the non-relativistic Hamiltonian.
  • This yields an effective LV Hamiltonian ($H_{LV}$) containing both spin and orbital terms proportional to the background field.
  • Crucially, $H_{LV}$ introduces a momentum-odd ($k$-odd) correction to the intercell hopping channel (via the $\tau_y$ Pauli matrix in sublattice space).
• Experimental Setup:
  • A static electric field ($E_{dc}$) is applied in the transverse ($yz$) plane and rotated by an angle $\theta$.
  • The photocurrent is measured along the chain direction ($\hat{x}$).
  • The system is driven by an optical field, and the second-order shift conductivity ($\sigma^{(2)}$) is calculated.
• Theoretical Tool: The shift conductivity is computed using the length-gauge formalism, which depends on the interband dipole matrix elements and the shift vector (derived from the Berry connection and the phase of the Bloch spinors).

3. Key Contributions

1. Mechanism Identification: The paper demonstrates that an LV background induces a $k$-odd correction to the Bloch Hamiltonian. This correction reshapes the complex phase of the interband dipole matrix elements.
2. Angular Signature Discovery: The authors identify a distinct change in angular periodicity as the primary signature of LV:
   • Lorentz-Symmetric Case: The shift conductivity exhibits a weak $2\pi$-periodic dependence on the field angle $\theta$.
   • LV Case: The perturbation induces a robust $\pi$-periodic modulation in $\sigma^{(2)}(\theta)$.
3. Detection Scheme: They propose that this periodicity shift can be isolated via Fourier analysis of the photocurrent $I(\theta)$, distinguishing the LV signal from the background static-field response without relying on absolute current magnitude.

4. Results

• Spectral Response: Numerical simulations (using $N_k=5001$ k-points) show that near the band-edge resonance ($\hbar\omega = E_{gap}$), the inclusion of the LV perturbation causes a significant deviation in the shift conductivity compared to the zero-field or pure static-field cases.
• Angular Modulation:
  • Without LV, the difference in conductivity between angles $\theta$ and $\theta + \pi/2$ is negligible (weak $2\pi$ dependence).
  • With LV, a pronounced $\pi$-periodic modulation emerges. The extrema of the signal occur at specific angles (e.g., $\approx 55^\circ$ and $55^\circ + \pi$) determined by the orientation of the background vector $v$.
• Signal Magnitude:
  • For realistic optical intensities ($I_{opt} \sim 10^4 \text{ W/cm}^2$) and static fields ($E_0 = 10^5 \text{ V/m}$), the predicted LV-induced current contrast is in the picoampere to nanoampere range ($\delta I \sim 0.6\text{--}0.8 \text{ nA}$ for a single chain).
  • While small for a single chain, the signal scales linearly with the number of chains in a parallel bundle, making it experimentally accessible.
• Sensitivity: The proposed method can detect LV coupling strengths of the order of $\xi \sim 10^{-24} \text{ C m}$.

5. Significance

• New Probe for Fundamental Physics: This work establishes angular-resolved nonlinear optical transport as a viable and robust probe for emergent Lorentz-violating effects in solid-state systems.
• Robustness: The detection relies on the symmetry of the angular response (periodicity) rather than the absolute magnitude of the current. This makes the method less susceptible to noise and calibration errors, as the $\pi$-periodic component is a unique fingerprint of the LV perturbation.
• Experimental Feasibility: The required parameters (optical intensities, static fields) are well within the range of current experimental capabilities in low-dimensional semiconductors (e.g., transition-metal dichalcogenide nanoribbons, ferroelectric layers).
• Theoretical Bridge: It connects high-energy physics concepts (SME, LV backgrounds) with condensed matter phenomena (shift current, Berry geometry), offering a new pathway to test fundamental symmetries using tabletop experiments.

In conclusion, the paper provides a concrete theoretical roadmap to detect Lorentz violation by exploiting the unique angular periodicity of the shift photocurrent in noncentrosymmetric materials, transforming a subtle symmetry-breaking effect into a measurable, distinct experimental signature.