Dirac bilinears in condensed matter physics: Relativistic correction for observables and conjugate electromagnetic fields

Inspired by recent developments in electron chirality, this paper bridges condensed matter, quantum chemistry, and particle physics by deriving the non-relativistic limits of Dirac bilinears to identify overlooked microscopic physical quantities and their conjugate electromagnetic fields, thereby enabling the *ab initio* quantification of chirality and axiality in low-symmetry materials for electromagnetic control.

The Gist

Imagine you are looking at a crystal, like a diamond or a piece of salt. To a physicist, this isn't just a pretty rock; it's a bustling city of electrons. For decades, we've studied these electrons using "standard maps" (non-relativistic physics) that work great for slow-moving things. But electrons are tiny, fast, and sometimes act like they are dancing to a different, more complex rhythm dictated by Einstein's relativity.

This paper is like a new, ultra-high-definition dictionary that translates the complex, four-dimensional language of "Relativistic Quantum Mechanics" into the simpler, two-dimensional language we usually use in condensed matter physics.

Here is the breakdown of their discovery using everyday analogies:

1. The "Four-Headed" vs. "Two-Headed" Electron

In the deep theory of particle physics, an electron is described as a four-headed creature (a four-component Dirac field). It has a "head" for the particle, a "head" for the antiparticle (which doesn't really exist in normal materials), and two heads for spin.

In everyday materials (like the copper wire in your phone), we usually ignore the "antiparticle" head because it's too heavy and energetic to show up. We treat the electron as a two-headed creature (a two-component Schrödinger field).

The Problem: When we chop off the "antiparticle" head to make the math easier, we accidentally throw away some subtle information. It's like trying to describe a complex symphony by only listening to the violins and ignoring the cellos. You get the melody, but you miss the deep bass notes that give the music its unique character.

The Solution: The authors figured out exactly how to translate those "missing bass notes" (relativistic effects) back into the language of the two-headed electron. They created a list of new physical quantities that were previously overlooked.

2. The New "Superpowers" of Electrons

By doing this translation, they found that electrons in certain materials have hidden "superpowers" that we didn't know how to measure before. They identified several new types of "currents" and "charges":

• Chirality (The "Handedness" Current): Imagine a screw. It can be right-handed or left-handed. Electrons can also have a "handedness." The authors found a way to measure how many electrons are "right-handed" versus "left-handed" in a material. This is crucial for materials that twist light or electricity in unique ways.
• Axial Current (The "Spin Flow"): Usually, we think of electric current as electrons moving from point A to point B. But these authors found a current where the spin of the electron flows, even if the electron itself isn't moving much. It's like a crowd of people standing still but all turning their heads in a wave.
• Chirality Polarization: Think of this as a "twist" in the material's structure that isn't just about shape, but about how the electrons are spinning and moving together.

3. The "Conjugate Fields" (The Remote Controls)

One of the coolest parts of the paper is identifying the "Remote Controls" for these new superpowers.

In physics, every property has a "partner" field that can control it.

• To control Electric Charge, you use an Electric Field (like a battery).
• To control Magnetism, you use a Magnetic Field (like a magnet).

The authors discovered the remote controls for the new superpowers:

• To control Electron Chirality (handedness), you need a special mix of electric and magnetic fields called "Magnetic Helicity" (imagine a corkscrew-shaped electromagnetic wave).
• To control Axial Current, you need a field that looks like the spin of a photon (light).

Why this matters: This means we might be able to use specific types of light (like circularly polarized laser beams) to "switch on" or "switch off" these hidden properties in materials. It's like finding a new button on a remote control that changes the material from a simple conductor to a super-efficient spin-tronics device.

4. The "No-Go" Theorem (The Traffic Cop)

The paper also revisits a famous rule called the Bloch-Bohm theorem. Think of this as a traffic cop in a city. The rule says: "In a perfectly still, calm city (ground state), no cars (electric current) can be driving around in a loop."

However, the authors found a loophole. There are actually two types of "currents" in the math:

1. The Real Current (what actually flows).
2. The Conjugate Current (what the math says is coupled to the magnetic field).

They proved that while the "Real Current" must be zero in a calm state (the traffic cop is right), the "Conjugate Current" can be non-zero. This distinction is vital for understanding how materials respond to magnetic fields without actually short-circuiting.

5. The Big Picture: Why Should You Care?

This paper is a bridge between three worlds:

1. Condensed Matter Physics (studying solids like chips and metals).
2. Quantum Chemistry (studying how atoms bond).
3. Particle Physics (studying the fundamental laws of the universe).

The Takeaway:
By using this new "dictionary," scientists can now:

• Quantify the "Twist": They can calculate exactly how "chiral" or "polar" a material is, even if it's a weird, low-symmetry crystal.
• Design New Materials: They can search for materials that use these hidden "handedness" properties to create faster, more efficient electronics (spintronics) or new types of sensors.
• Control Matter with Light: They have a blueprint for using light to manipulate the internal "spin" and "handedness" of electrons, potentially leading to computers that process information using electron spin rather than just charge.

In short, the authors didn't just find new numbers; they found new knobs we can turn on the fabric of matter, using the language of relativity to unlock the secrets of the quantum world.

Technical Summary

1. Problem Statement

Condensed matter physics often relies on non-relativistic (Schrödinger) descriptions of electrons, which can overlook subtle microscopic physical quantities arising from relativistic effects. While phenomena like electric toroidal multipoles, chirality, and axiality in low-symmetry materials are gaining attention, a rigorous microscopic definition of these quantities based on relativistic quantum mechanics is often missing or incomplete.

Specifically, the paper addresses:

• The lack of a systematic dictionary translating four-component Dirac field quantities (standard in particle physics) into two-component Schrödinger field quantities (standard in condensed matter).
• The identification of conjugate electromagnetic (EM) fields for various physical observables (e.g., what field couples to electron chirality?).
• The need to quantify material properties like polarity, chirality, and axiality using ab initio methods, requiring a clear definition of these observables in the non-relativistic limit (NRL) with relativistic corrections.

2. Methodology

The authors employ a rigorous theoretical framework bridging relativistic quantum mechanics and condensed matter physics:

• Dirac Bilinears Analysis: They start with the full Dirac equation and classify all 16 independent Dirac bilinears (scalar, pseudoscalar, vector, axial vector, and tensor) based on their transformation properties under spatial inversion (SI) and time reversal (TR).
• Non-Relativistic Limit (NRL) Derivation: Using the Berestetskii-Landau method (an elimination method for the small component of the spinor), they derive the explicit relationship between the four-component Dirac field ($\Psi$) and the two-component Schrödinger field ($\psi$). This is performed via a $1/m$ expansion (where $m$ is the electron mass) rather than a $1/c$ expansion, ensuring gauge invariance is manifest.
• Gordon Decomposition: They utilize the Gordon decomposition to separate convective currents from magnetization and polarization currents, allowing for the identification of physical quantities like electric polarization ($P$) and magnetization ($M$) within the relativistic framework.
• Higher-Order Field Equations: To identify conjugate fields for quantities like chirality (which do not appear in the first-order Dirac equation), the authors derive higher-order equations (up to fourth order) of the Dirac operator. This reveals couplings to non-linear EM field combinations (e.g., $E \cdot B$, $E \times A$).
• Bloch-Bohm Theorem Application: They apply the Bloch-Bohm theorem to distinguish between the "true" current density ($j$) and the current density conjugate to the vector potential ($\tilde{j}$) in the effective Hamiltonian, resolving ambiguities regarding equilibrium currents.
• Quantum Anomalies: They incorporate quantum anomalies (Chiral and Weyl anomalies) derived via path integrals in statistical mechanics to correct the continuity equations for chirality and scalar densities.

3. Key Contributions

A. The "Dictionary" of Physical Quantities

The paper provides a comprehensive mapping (summarized in Table I of the paper) of Dirac bilinears to their NRL expressions and relativistic corrections. Key mappings include:

• Charge ($\rho$): Maps to $\psi^\dagger \psi$ plus relativistic corrections involving the divergence of spin-derived polarization ($\nabla \cdot P_S$).
• Current ($j$): Maps to the convective current plus spin-orbit terms and magnetization curl ($\nabla \times M_S$).
• Chirality ($A_0 = \bar{\Psi}\gamma^0\gamma^5\Psi$): In the NRL, this is identified with helicity density ($H \propto \mathbf{p} \cdot \boldsymbol{\sigma}$).
• Axial Current ($A$): Identified with spin density ($M_S$) in the leading order.
• Pseudoscalar ($P$): Identified as the divergence of magnetization ($\nabla \cdot M_S$).
• Chirality Polarization ($P_C$): A new quantity defined as the time derivative of axial current, related to $\nbf{\Pi} \times \mathbf{v}$.

B. Identification of Conjugate Electromagnetic Fields

The authors identify the specific EM field configurations that couple to these microscopic quantities, enabling potential control mechanisms:

• Charge/Current: Conjugate to Scalar Potential ($\Phi$) and Vector Potential ($\mathbf{A}$).
• Magnetization ($M$): Conjugate to Magnetic Field ($\mathbf{B}$).
• Electric Polarization ($P$): Conjugate to Electric Field ($\mathbf{E}$).
• Chirality Density ($\tau^Z$): Conjugate to Magnetic Helicity ($\mathbf{A} \cdot \mathbf{B}$). This suggests that light with specific helicity can control electron chirality.
• Axial Current: Conjugate to Spin of Photons ($\mathbf{E} \times \mathbf{A}$).
• Pseudoscalar: Conjugate to $E \cdot B$ (the topological term).

C. Resolution of Current Ambiguities (Bloch-Bohm Theorem)

The paper clarifies a critical distinction between:

1. $j$ (True Current): Defined by the Dirac field operator.
2. $\tilde{j}$ (Conjugate Current): Defined as the derivative of the effective Hamiltonian with respect to $\mathbf{A}$.
   The authors prove that while these differ at higher orders of $1/m$, both vanish in the ground state of a system with time-reversal symmetry, consistent with the Bloch-Bohm theorem. This resolves confusion regarding the "chiral magnetic effect" in equilibrium, confirming that equilibrium currents cannot exist without external driving or non-equilibrium conditions.

D. Hydrodynamic Helicity

The authors introduce a spin-independent definition of chirality using hydrodynamic helicity ($H_{hydro} \propto \mathbf{j} \cdot (\nabla \times \mathbf{j})$). They show this quantity can be finite even in spinless fermionic systems (unlike the spin-dependent helicity), providing a robust metric for chirality in systems without strong spin-orbit coupling.

4. Key Results

• Relativistic Corrections: Explicit formulas for relativistic corrections to charge, current, polarization, and magnetization were derived up to order $O(m^{-3})$.
• Control Mechanisms: The study demonstrates that circularly polarized light can induce magnetization and chirality. Specifically:
  • Simple circularly polarized light induces both.
  • Counter-propagating circularly polarized lights can be tuned to induce chirality without magnetization (by canceling the vector potential term).
  • Superpositions of different helicities can induce axial current (magnetization) without chirality.
• Anomaly Corrections: The continuity equations for chirality and scalar densities were corrected to include the Chiral Anomaly ($\propto \mathbf{E} \cdot \mathbf{B}$) and Weyl Anomaly ($\propto E^2 - B^2$), respectively, within the condensed matter context (excluding antiparticle contributions).
• D-H Representation: The Maxwell equations were reformulated in terms of $\mathbf{D}$ and $\mathbf{H}$ fields, identifying magnetic charge ($\rho_m = -\nabla \cdot M$) and magnetic current ($j_m$) as conserved quantities derived from the pseudoscalar and tensor components of the Dirac field.

5. Significance

• Interdisciplinary Bridge: The work successfully bridges particle physics (Dirac theory, anomalies), quantum chemistry (relativistic corrections, picture change errors), and condensed matter physics (multipole moments, spintronics).
• Ab Initio Quantification: By providing explicit NRL expressions, the paper enables the ab initio calculation of chirality, axiality, and polarity in materials. This is crucial for the systematic search for new functional materials (e.g., ferroaxial materials, chiral metals).
• Electromagnetic Control: The identification of conjugate fields (e.g., $\mathbf{A} \cdot \mathbf{B}$ for chirality) opens new avenues for controlling material properties using light, particularly high-frequency or structured light fields, without the need for static magnetic fields.
• Theoretical Clarity: It resolves long-standing ambiguities regarding the definition of current in relativistic limits and the application of the Bloch-Bohm theorem, providing a solid foundation for future theoretical and experimental studies of relativistic effects in solids.

In summary, this paper establishes a rigorous theoretical foundation for understanding and manipulating microscopic physical quantities in low-symmetry materials by translating relativistic quantum field concepts into the language of condensed matter physics.