Calculations in Unified theory of the photovoltaic Hall effect by field- and light-induced Berry curvatures

This paper presents a unified theoretical framework that coherently describes the photovoltaic Hall effect by demonstrating how both light-induced and bias electric field-induced mechanisms arise from Berry curvature engineering, thereby clarifying the distinct yet interconnected roles of field-modified transition parameters and light-dressed states in nonmagnetic materials.

The Gist

Imagine you are trying to understand how a solar panel works, but instead of just generating electricity, it also creates a magnetic-like "push" that moves electrons sideways. This phenomenon is called the Photovoltaic Hall Effect.

For a long time, scientists were confused because they saw two different "pushes" happening at once, and they were using two different rulebooks to explain them. It was like trying to explain a car crash by using one manual for the engine and another for the brakes, without realizing they were part of the same car.

This paper by Murotani, Fujimoto, and Matsunaga acts as the ultimate unified rulebook. They created a single, elegant theory that explains both pushes at the same time, revealing that they are actually two sides of the same geometric coin.

Here is the breakdown using simple analogies:

1. The Two "Pushes" (The Problem)

Imagine a crowd of people (electrons) trying to walk through a hallway (a solid material) while a strong wind (light) blows on them.

• Push A (The Light-Induced Push): If the wind blows in a specific spinning pattern (circularly polarized light), it twists the hallway itself. The floor becomes slippery in a specific direction, forcing people to slide sideways. This is the Light-Induced Anomalous Hall Effect. It's like the light "dresses" the electrons in a new outfit that makes them walk differently.
• Push B (The Bias-Field Push): Now, imagine someone is also pushing the people from behind with a steady shove (a bias electric field). This shove changes how the people react to the wind. It changes the size of their steps and the angle at which they turn. This creates a sideways drift called the Field-Induced Circular Photogalvanic Effect.

The Confusion: Previously, scientists thought Push A and Push B were totally separate things. They used different math to describe them. But in real experiments, both happen at the same time, making it hard to tell which one is doing what.

2. The Unified Theory (The Solution)

The authors realized that both pushes come from the same underlying source: Geometry.

Think of the electrons not as tiny balls, but as dancers on a stage. The "stage" is the material's atomic structure.

• The Berry Curvature: This is like the "twist" or "curvature" of the dance floor. If the floor is curved, a dancer moving straight will naturally drift sideways.
• The Shift Vector: This is like the "step size" or the distance a dancer jumps when they switch partners (moving from one energy level to another).

The paper shows that:

1. The Light changes the shape of the dance floor (creating a new Berry curvature).
2. The Electric Field changes the dancers' step size and how they react to the wind (altering the transition energy and dipole moment).

By treating both the "twist of the floor" and the "change in step size" using the same geometric language, the authors unified the theory. They proved that the electric field doesn't just push the electrons; it actually rewrites the rules of the dance floor itself.

3. The Key Ingredients (The "Secret Sauce")

The paper introduces two special mathematical tools to describe this:

• The "Sensitivity Gauge" (QGT Polarizability): Imagine a dancer who is very sensitive to the wind. If you push them (electric field), they change their spin direction. This tool measures how much the "twist" of the floor changes when you apply a push.
• The "Step-Size Gauge" (Shift Tensor): Imagine a dancer who, when pushed, takes a bigger or smaller leap. This tool measures how the distance of their jump changes under pressure.

The authors found that the "sideways push" (the Hall effect) is a combination of the floor twisting and the dancers changing their step size.

4. Why This Matters

• It Solves a Mystery: In recent experiments, scientists saw a sideways current that they couldn't explain with just the "light-induced" theory. This paper says, "Ah, that's actually the electric field changing the step size!"
• It's a Universal Translator: Whether you are looking at graphene, semiconductors, or exotic quantum materials, this theory works for all of them. It connects the dots between how light interacts with matter and how electric fields steer that interaction.
• Future Tech: Understanding this "geometric dance" helps engineers design better solar cells, ultra-fast switches, and new types of computers that use light and electricity together more efficiently.

The Bottom Line

Think of this paper as the Rosetta Stone for light-matter interaction. It took two confusing, separate languages (Light-Induced vs. Field-Induced effects) and translated them into one beautiful, geometric story. It tells us that when light and electricity meet in a solid, they don't just push electrons around; they reshape the very geometry of the world the electrons live in.

Technical Summary

Here is a detailed technical summary of the paper "Unified theory of photovoltaic Hall effect by field- and light-induced Berry curvatures" by Murotani, Fujimoto, and Matsunaga.

1. Problem Statement

The Photovoltaic Hall Effect (PVHE) is a phenomenon where a transverse photocurrent is generated in a material under light irradiation and a bias electric field. While historically associated with Floquet engineering (light-induced Berry curvature breaking time-reversal symmetry), recent experiments and theories have highlighted a competing mechanism: the field-induced Circular Photogalvanic Effect (CPGE).

The core problem addressed in this work is the lack of a unified theoretical framework that treats these two mechanisms on equal footing.

• Light-induced AHE: Arises from light-dressed states (Floquet states) modifying the band topology.
• Field-induced CPGE: Arises from the bias electric field breaking inversion symmetry, altering transition probabilities and energies.
• Gap: Previous studies often treated these separately or relied on numerical simulations. The relationship between the field-induced CPGE and field-induced Berry curvature remained elusive, and the geometric origins of the field-induced effects were not fully clarified in a single analytical framework.

2. Methodology

The authors develop a general analytical theory within the third-order nonlinear regime using the length gauge and density matrix formalism.

• Perturbative Approach: They start with the Schrödinger equation in the length gauge, introducing a static bias electric field ($E_0$) and an optical field ($E_1$).
• Unitary Transformation: They apply a unitary transformation to diagonalize the Hamiltonian in the presence of $E_0$, effectively treating the bias field as a perturbation that modifies the band structure and Berry connections.
• Key Geometric Quantities: The theory explicitly tracks how $E_0$ modifies three critical quantities:
  1. Interband Transition Dipole Moment: Modified by the Berry Connection Polarizability ($C_{\nu\mu}$).
  2. Transition Energy: Modified by the Shift Vector ($R_{nm}$), representing the spatial shift of the electron cloud during transition.
  3. Intraband Velocity: Modified by the Berry Curvature ($\Omega_\nu$) and its polarizability.
• New Tensors: The authors introduce and utilize two higher-order geometric tensors to describe these modifications:
  • QGT Polarizability ($T_{\nu\mu}^{abc}$): Describes how the Quantum Geometric Tensor (QGT) changes with $E_0$.
  • Shift Tensor ($S_{\nu\mu}^{abc}$): Acts as a precursor to the shift vector, related to the Hermitian connection in Riemannian geometry.
• Comparison with Floquet Theory: They derive expressions for light-dressed electrons ($\delta\epsilon_\nu$ and $\delta\xi_{\nu\nu}$) and prove their equivalence to results obtained via standard Floquet formalism up to the third order.

3. Key Contributions

The paper makes several significant theoretical breakthroughs:

1. Unified Framework: It successfully unifies the field-induced CPGE and the light-induced Anomalous Hall Effect (AHE) within a single perturbative framework. This allows for a direct comparison of their magnitudes and physical origins.
2. Geometric Interpretation of Field-Induced CPGE: The authors reveal that the field-induced CPGE is not a single mechanism but a sum of three distinct geometric contributions:
   • Modification of Transition Dipole: The bias field alters the transition probability asymmetrically in momentum space, effectively creating a field-induced Berry curvature ($\Delta\Omega$).
   • Modification of Transition Energy: The bias field induces an energy shift dependent on the shift vector ($R_{nm}$), leading to an energy-shift-induced current.
   • Anomalous Velocity: The bias field changes the intraband velocity of photoexcited carriers via the Berry curvature.
3. Identification of New Tensors: The theory explicitly defines the roles of the QGT polarizability ($T$) and Shift tensor ($S$) in mediating the interaction between the bias field and optical response.
4. Resolution of Floquet vs. Field-Induced: The paper clarifies that while Floquet theory describes the light-induced AHE, the field-induced CPGE is a distinct, competing mechanism driven by the breaking of inversion symmetry by $E_0$. The theory shows that in resonant conditions, the field-induced CPGE can dominate over the light-induced AHE.

4. Results and Case Studies

The authors applied their theory to two specific material systems to validate the predictions:

• Massive Dirac Electron System:

  • Calculated the photocurrent generation rates for the dipole, energy, and anomalous velocity terms.
  • Found that near the band gap, the contributions scale as $(\hbar\omega - E_g)^{1/2}$ (consistent with the joint density of states), except for the dipole term which scales as $(\hbar\omega - E_g)^{3/2}$.
  • Demonstrated that the shift vector and Berry curvature modifications lead to a sign change in the photocurrent at twice the band gap energy.

• GaAs Semiconductor (Kane Model):

  • Analyzed the heavy-hole (HH) and light-hole (LH) bands, which are degenerate at the $\Gamma$ point.
  • Singularity: The theory predicts a divergent enhancement of the photocurrent near the band gap ($\sim (\hbar\omega - E_g)^{-1/2}$). This arises from the topological singularity of the Berry curvature in the HH/LH bands (monopole charges).
  • Experimental Validation: This theoretical prediction of a resonant enhancement near the band gap aligns with recent experimental observations in insulating bulk GaAs, substantiating the validity of the unified theory.

5. Significance

• Physical Clarity: The work provides a "physically transparent" way to understand complex nonlinear optical processes by linking them to fundamental geometric quantities (Berry curvature, shift vector, QGT).
• Experimental Guidance: By distinguishing the contributions of field-induced CPGE and light-induced AHE, the theory offers a roadmap for interpreting experimental data, particularly in ultrafast terahertz spectroscopy where these effects compete.
• Foundation for Future Work: The unified formalism lays the groundwork for incorporating scattering mechanisms (side-jump, skew scattering) and correlation effects in future studies of the photovoltaic Hall effect.
• Geometric Language: It enriches the geometric language of condensed matter physics by demonstrating how external fields (electric and optical) can be treated as perturbations to the quantum geometric tensor and shift vectors, revealing a deep connection between non-equilibrium transport and Riemannian geometry.

In summary, this paper resolves the competition between different mechanisms of the photovoltaic Hall effect, proving that the bias electric field induces a Berry curvature and energy shifts that are geometrically distinct from, yet comparable to, light-induced topological effects.