The non-Hermitian magnetic moment

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a school of fish swim through a complex, shifting coral reef. In the world of traditional physics (Hermitian physics), we have a perfect map of how these fish swim. We know exactly how they turn, how fast they go, and how they react to currents or magnetic fields. We even know that if you spin a fish, it creates a tiny magnetic field, like a miniature compass needle. This is called the "orbital magnetic moment."

However, in the real world, things aren't always perfect. Sometimes, fish might get sick and die (losing energy), or they might be fed extra food and grow faster (gaining energy). In physics, we call systems that gain or lose energy "Non-Hermitian." Until now, we didn't have a good map for how these "gain-and-loss" fish behave in a magnetic field.

This paper by Bar Alon, Goldstein, and Ilan is like drawing that missing map. They created a new set of rules to describe how electrons (our fish) behave when they are in a crystal (the reef) that is slightly imperfect or changing, and when they are subjected to a magnetic field.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of Fish (Left and Right)

In a normal, perfect world, a fish is just a fish. But in this "Non-Hermitian" world, the authors realized you have to look at the fish from two different angles to understand it:

The Right Fish: This is the fish as it actually exists and moves.
The Left Fish: This is a "shadow" or a "mirror image" of the fish that tells us how the fish responds to the environment.

In normal physics, these two are the same. In this new physics, they are different. To understand the fish's behavior, you have to mix the "Right" view with the "Left" view. If you ignore the "Left" view, your map will be wrong.

2. The Spinning Top with a Ghost

The main goal of the paper was to figure out the "Magnetic Moment." Imagine an electron spinning like a top. In normal physics, this spinning creates a real magnetic field.

The authors found that in this new, imperfect world, the "spinning" creates two things at once:

The Real Spin (The Physical Top): This is the part we are used to. It creates a real magnetic field. The authors showed that if you define "spin" carefully (using their new "Left/Right" rules), this part still works just like in the old physics.
The Imaginary Spin (The Ghost): This is the new discovery. Because the system gains or loses energy, there is a "ghost" version of the spin. This ghost doesn't create a magnetic field in the traditional sense; instead, it creates gain or loss.

The Analogy: Imagine a spinning top that is also a magic trick.

The Real Spin makes the top point North (magnetism).
The Imaginary Spin makes the top either grow bigger (gain energy) or shrink and disappear (lose energy) depending on how it spins.

3. The "Aharonov-Bohm" Ghost Effect

You might have heard of the Aharonov-Bohm effect, where a magnetic field changes the phase (the timing or rhythm) of a particle, even if the particle never touches the magnet.

The authors discovered a "Non-Hermitian" version of this. They found that the "Imaginary Spin" (the ghost) turns that magnetic rhythm into energy gain or loss.

Normal World: A magnetic field changes the direction or phase of the electron.
New World: A magnetic field changes the life or death (gain or loss) of the electron.

It's as if the magnetic field isn't just pushing the electron around; it's acting like a faucet, turning the electron's energy up or down based on how it rotates.

4. Why This Matters

Why do we care about "gain and loss"?

Real-World Applications: This isn't just about electrons. These ideas apply to lasers (which gain light), sound waves in special materials, and even biological systems where things grow or decay.
New Technology: By understanding this "Imaginary Spin," scientists might be able to design new materials that use magnetic fields to control how much energy a device absorbs or emits. Imagine a magnetic switch that doesn't just turn a light on or off, but controls how bright it gets by feeding it energy.

Summary

The authors built a new theory to explain how electrons behave in "imperfect" worlds where energy can be gained or lost. They found that the electron's magnetic spin is actually a combination of two things:

A Real Spin that acts like a normal magnet.
An Imaginary Spin that acts like a magical faucet, turning the magnetic field into a source of energy gain or loss.

This discovery bridges the gap between the perfect world of textbook physics and the messy, real world of lasers, biological systems, and advanced materials.

1. Problem Statement

Semiclassical band theory is a cornerstone for understanding electron dynamics in periodic media, particularly regarding topological effects like the anomalous Hall effect and orbital magnetization. However, existing theories are largely restricted to Hermitian Hamiltonians. With the rising interest in non-Hermitian systems (describing open systems, gain/loss, and non-equilibrium processes in photonics, mechanics, and cold atoms), there is a critical gap in understanding how charged particles behave in magnetic fields within these frameworks.

Specifically, there was no general non-Hermitian semiclassical theory that:

Provides a closed-form expression for the orbital magnetic moment of a Bloch electron wavepacket.
Relates this magnetic moment to a physically meaningful definition of orbital angular momentum in a non-Hermitian setting.
Addresses the unique complexities arising from the non-orthogonality of left and right eigenstates and the gauge dependence of perturbation theory in non-Hermitian systems.

2. Methodology

The authors construct a semiclassical theory for electrons in a non-Hermitian periodic system subject to slowly varying perturbations (space and time). The methodology proceeds through the following steps:

Non-Hermitian Bloch Formalism: They utilize the bi-orthogonal basis of right eigenstates ( $|\psi^R_{nk}\rangle$ $∣ ψ_{nk}^{R} ⟩$ ) and left eigenstates ( $|\psi^L_{nk}\rangle$ $∣ ψ_{nk}^{L} ⟩$ ) of the unperturbed Hamiltonian $H_0$ $H_{0}$ . They distinguish between two types of Berry connections:
- $A^{RR}$ : Involves only right eigenstates (analogous to the Hermitian Berry connection).
- $A^{LR}$ : Involves both left and right eigenstates, governing the system's response to perturbations.
Wavepacket Construction: A narrow wavepacket $|W\rangle$ is constructed as a superposition of Bloch states. The authors assume the packet is confined to a "maximally amplified" band (where the imaginary part of the energy is maximal) to apply a non-Hermitian adiabatic theorem.
Effective Energy Derivation: They derive the effective energy of the wavepacket to first order in the gradients of the perturbations. This involves calculating the expectation value of the full Hamiltonian $E = \langle W|H|W\rangle / \langle W|W\rangle$ using the projected Schrödinger equation.
Quasi-Local Hamiltonian: To handle spatially varying fields (like magnetic vector potentials), they employ a quasi-local Hamiltonian approximation. This linearizes the Hamiltonian around the wavepacket's center of mass ( $r_c$ ), treating the field as constant locally but accounting for its gradient.
Operator Redefinition: Recognizing that standard operators (like velocity $\hat{v} = -i[\hat{r}, H]$ ) are non-Hermitian and unphysical in this context, they derive a physically meaningful velocity operator using the non-Hermitian generalization of the Ehrenfest theorem. This leads to a new definition for the orbital angular momentum operator.

3. Key Contributions and Results

A. General Expression for Effective Energy

The authors derive a general expression for the effective energy of a wavepacket in a slowly perturbed non-Hermitian crystal (Eq. 13). This energy includes:

The unperturbed band energy.
Terms coupling the wavepacket velocity and energy gradients to the difference between Berry connections ( $A^{LR} - A^{RR}$ ).
A term involving derivatives of eigenstates with respect to momentum and position, which survives even in the Hermitian limit and is responsible for the magnetic moment.

B. The Non-Hermitian Orbital Magnetic Moment

Applying the theory to a uniform magnetic field (using the symmetric gauge and Peierls substitution), they derive the orbital magnetic moment energy (Eq. 18). The magnetic moment is found to have two distinct contributions:

Anomalous Coupling: A term coupling the difference of Berry connections ( $A^{LR} - A^{RR}$ ) to the Lorentz force. This is analogous to non-Hermitian corrections in electric fields.
Zeeman-like Term: A term related to the self-rotation of the wavepacket. In the Hermitian limit, this reduces to the standard orbital magnetic moment ( $\mathbf{B} \cdot \mathbf{L}$ ). In the non-Hermitian case, this term becomes complex-valued.

C. Redefinition of Orbital Angular Momentum

A central contribution is the rigorous definition of a physical angular momentum operator in non-Hermitian systems.

The naive operator $\hat{L} \propto (\hat{r} \times \nabla_k H)$ is non-Hermitian and unphysical.
By using the non-Hermitian Ehrenfest theorem, they define a Hermitian velocity operator and subsequently a Hermitian angular momentum operator $\hat{L}_{phys}$ .
Key Finding: The complex magnetic moment operator derived from the Hamiltonian is the sum of the physical (Hermitian) angular momentum (responsible for the real part of the energy shift) and an "imaginary angular momentum" (responsible for the imaginary part of the energy).

D. Physical Interpretation of the Imaginary Part

The "imaginary angular momentum" is interpreted as a non-Hermitian generalization of the Aharonov-Bohm effect.

While the real angular momentum causes a phase shift (energy shift), the imaginary angular momentum causes gain or loss (amplification or decay) of the wavepacket.
This dissipation/gain is proportional to the magnetic flux through the wavepacket, linking the imaginary part of the energy directly to the Aharonov-Bohm phase.

E. Gauge Invariance and Perturbation Theory

The paper addresses the issue of gauge dependence in non-Hermitian perturbation theory.

Unlike Hermitian theory, where first-order energy corrections are gauge-invariant, non-Hermitian first-order corrections can be gauge-dependent due to the singularity in the limit of vanishing perturbations (discontinuity between left and right eigenstates).
The authors demonstrate that using the symmetric gauge regularizes this singularity, ensuring the derived magnetic moment is physically consistent and gauge-invariant.

4. Significance

Theoretical Framework: This work establishes the first comprehensive semiclassical theory for charged particles in magnetic fields within non-Hermitian periodic systems. It bridges the gap between non-Hermitian topological physics and classical electromagnetic response.
New Physical Phenomena: It predicts that magnetic fields in non-Hermitian systems do not just shift energy levels but can induce gain and loss via the "imaginary angular momentum." This suggests new mechanisms for controlling dissipation and amplification in non-Hermitian metamaterials and photonic systems.
Experimental Relevance: The theory provides a roadmap for interpreting experiments in systems where non-Hermitian effects are realized, such as active mechanical metamaterials, electrical circuits, and cold atomic gases with gain/loss.
Foundational Insight: By clarifying the relationship between the magnetic moment and a redefined physical angular momentum, the paper resolves ambiguities regarding observables in non-Hermitian quantum mechanics, showing that physical observables must be constructed carefully to remain Hermitian even when the underlying Hamiltonian is not.

In summary, the paper successfully generalizes the concept of orbital magnetization to non-Hermitian systems, revealing a dual nature of the magnetic moment: a real part governing energy shifts (standard magnetism) and an imaginary part governing dissipation/amplification (non-Hermitian magnetism).