Persistent Charge and Spin Currents in a Ferromagnetic Hatano-Nelson Ring

This paper investigates persistent charge and spin currents in a ferromagnetic Hatano-Nelson ring, demonstrating how non-reciprocal hopping induces a non-Hermitian Aharonov-Bohm effect and revealing that disorder can surprisingly amplify spin transport across various topological and parameter regimes.

The Gist

Imagine a tiny, circular racetrack made of atoms. Usually, electrons running around this track behave like normal, predictable cars. But in this paper, the researchers set up a very special, slightly "glitchy" version of this track where the rules of physics are bent. They call this a non-Hermitian system.

Here is the story of what they found, explained simply:

1. The Glitchy Racetrack (The Hatano-Nelson Ring)

In a normal racetrack, if you drive clockwise, it takes the same effort as driving counter-clockwise. In this study, the track is "biased." It's like a one-way street built into a circle. The electrons find it easier to hop in one direction than the other.

• The Analogy: Imagine a conveyor belt that moves slightly faster in one direction. Even without an external wind or magnet pushing them, the electrons start to circulate on their own. This creates a "persistent current"—a flow that keeps going without stopping.
• The "Synthetic" Magnet: The researchers found that this one-way bias acts exactly like a magnetic field would. It tricks the electrons into behaving as if they are in a magnetic storm, even though there isn't one physically there.

2. The Spin Traffic (Charge vs. Spin)

Electrons have two main properties:

1. Charge: Like the weight of the car (electricity).
2. Spin: Like the direction the car's wheels are spinning (up or down).

Usually, scientists study how the weight (charge) moves. This paper asks: "What happens to the spinning wheels (spin) in this glitchy, one-way track?"

They added a ferromagnetic element, which is like a giant magnet lining the track. This magnet forces some electrons to spin "up" and others to spin "down," separating them into two different lanes.

3. The Two Types of Currents (Real vs. Imaginary)

Because the track is "glitchy" (non-Hermitian), the currents they measured have two parts:

• The Real Part: This is the "normal" flow you could actually measure with a meter. It's the actual traffic moving around the ring.
• The Imaginary Part: This sounds like math jargon, but think of it as the "potential" or the "growth/decay" of the flow. It tells you if the traffic is about to speed up, slow down, or vanish due to the weird rules of the track. It's not a flow you can catch in a bucket, but it's a crucial part of how the system behaves dynamically.

4. The Surprising Discovery: Disorder as a Booster

In the normal world, if you throw rocks (disorder) onto a racetrack, cars crash, and traffic stops. This is called "localization."

The paper's big surprise: In this specific glitchy, one-way track, throwing a little bit of disorder actually speeds up the spin traffic!

• The Analogy: Imagine a crowded hallway where people are trying to walk in a specific direction. If you add a few random obstacles (like chairs), it might actually force the people to find a more efficient path or push them harder, making the flow stronger for a moment before too many obstacles cause a total jam.
• The researchers found that for spin currents, a moderate amount of "messiness" (disorder) can amplify the flow, making it stronger than in a perfectly clean track.

5. The Shape of the Track Matters

The track is made of pairs of atoms (dimers). The researchers played with how tightly these pairs were connected compared to the connections between pairs.

• Topological Phase: The track is "knotted" in a specific way. The current is weak and fades away quickly if the track gets too long.
• Trivial Phase: The track is "loose." The current is stronger and lasts longer.
• Critical Point: This is the exact tipping point between the two. Here, the current is the strongest and most stable, even as the track gets longer.

6. Tilting the Magnet

The researchers also tilted the direction of the magnetic "lanes."

• When the lanes were straight up and down, only the "up/down" spin current existed.
• When they tilted the lanes, the electrons started spinning sideways too, creating currents in the "left/right" and "forward/backward" directions. The strength of these sideways currents depended exactly on the angle of the tilt, like a shadow changing length as the sun moves.

Summary

The paper shows that in a quantum racetrack with one-way rules:

1. You can create a self-sustaining flow of electricity and spin without an external battery.
2. The "spin" of the electrons behaves differently than the "charge," creating complex patterns.
3. Most importantly: A little bit of disorder (messiness) can actually make the spin flow stronger, which is the opposite of what happens in normal materials.

This gives scientists a new way to think about controlling tiny magnetic flows in future quantum devices, using the "glitches" in the system rather than trying to eliminate them.

Technical Summary

Technical Summary: Persistent Charge and Spin Currents in a Ferromagnetic Hatano-Nelson Ring

Problem Statement
While persistent charge and spin currents are well-established phenomena in Hermitian mesoscopic systems, their behavior in non-Hermitian settings remains largely unexplored, particularly regarding spin degrees of freedom. Existing studies on non-Hermitian persistent currents have largely focused on charge transport or spinless models. A critical gap exists in understanding how non-reciprocal hopping (a hallmark of non-Hermitian systems) interacts with ferromagnetic ordering to influence spin-resolved transport. Specifically, the interplay between synthetic gauge fields generated by asymmetric hopping and magnetic exchange splitting requires a rigorous theoretical framework that accounts for the biorthogonal nature of non-Hermitian eigenstates.

Methodology
The authors investigate a one-dimensional ferromagnetic Hatano-Nelson (HN) ring modeled within a nearest-neighbor tight-binding framework. The system consists of a dimerized ring with anti-Hermitian intradimer hopping ($t_c$ and $t_a$, where $t_a = -t_c^*$), which generates a synthetic magnetic flux and drives a non-Hermitian Aharonov-Bohm effect. Ferromagnetic ordering is introduced via a uniform spin-dependent scattering (SDS) parameter ($h$), which breaks spin degeneracy while preserving translational symmetry.

Key methodological features include:

• Biorthogonal Formalism: Since the Hamiltonian is non-Hermitian, the authors employ a biorthogonal basis comprising left ($\langle \psi^L_n |$) and right ($| \psi^R_n \rangle$) eigenstates to compute expectation values. This is essential for obtaining meaningful physical observables.
• Current Operator Method: Persistent charge and spin currents are calculated using the current operator formalism rather than the derivative method, as the former requires explicit knowledge of eigenstates. The charge current operator is derived from the velocity operator, while the spin current operator is defined as the symmetrized product of the spin operator and velocity.
• Parameter Space: The study systematically varies the intradimer hopping amplitude ($t_1$) to explore topological ($|t_1| < t_2$), critical ($|t_1| = t_2$), and trivial ($|t_1| > t_2$) phases. Additional parameters include the chemical potential ($\mu$), magnetic moment orientation (polar angle $\theta$, azimuthal angle $\phi$), system size ($N$), and disorder strength ($W$) modeled via the Aubry-André-Harper (AAH) potential.

Key Contributions and Results

1. Complex Band Structure and Spectral Characteristics:

   • The ferromagnetic exchange field ($h$) induces a rigid spin splitting in the real part of the energy spectrum, separating up- and down-spin bands.
   • Crucially, the imaginary part of the spectrum remains spin-degenerate, as the non-Hermitian term (anti-Hermitian hopping) is spin-independent.
   • The system exhibits distinct spectral behaviors across phases: the topological phase features a real energy gap and a gapless imaginary spectrum; the critical point shows gapless real and imaginary spectra; and the trivial phase displays a gapless real spectrum but a gapped imaginary spectrum.

2. Persistent Charge Currents:

   • Both real and imaginary persistent charge currents are observed. The real part corresponds to net circulating charge flow, while the imaginary part reflects non-conservative probability flow inherent to the anti-Hermitian lattice.
   • In the presence of ferromagnetism, the real and imaginary currents are no longer identical at the critical point, distinguishing them from the spinless case.
   • The currents exhibit magneto-oscillations with a period of $2\pi$ relative to the synthetic flux.

3. Persistent Spin Currents:

   • The study computes all three components ($I_x, I_y, I_z$) of the persistent spin current.
   • Symmetry and Orientation: When magnetic moments are aligned along the $z$-axis, only $I_z$ is non-zero. Tilting the moments ($\theta \neq 0, \phi \neq 0$) induces non-zero $I_x$ and $I_y$. The components exhibit specific symmetries: $I_z$ is antisymmetric with respect to the polar angle $\theta$, while $I_x$ and $I_y$ are symmetric. $I_z$ is independent of the azimuthal angle $\phi$, whereas $I_x$ and $I_y$ vary as $\cos \phi$ and $\sin \phi$, respectively.
   • Chemical Potential and Size Dependence: Spin currents exhibit a staircase-like profile as a function of $\mu$ due to discrete energy level occupancy and are antisymmetric about $\mu=0$ due to particle-hole symmetry. The magnitude of spin currents decreases with increasing system size ($N$), with the decay rate being fastest in the topological phase and slowest at the critical point.

4. Disorder-Induced Amplification:

   • A striking finding is that moderate AAH-type disorder can amplify persistent spin currents, contrary to the typical Anderson localization effect.
   • In the topological and trivial phases, currents initially increase with disorder strength before being suppressed at higher $W$. At the critical point, a non-monotonic behavior is observed where currents are amplified near specific disorder strengths.
   • This amplification is linked to the imaginary part of the biorthogonal spin-bond overlap (BSBO), which serves as a microscopic measure of spin-dependent phase coherence.

Significance and Claims
The paper claims to provide the first detailed theoretical formulation for spin-resolved persistent currents in a non-Hermitian ferromagnetic system. The authors emphasize that their work extends the understanding of equilibrium-like responses in non-Hermitian quantum systems beyond the spinless case.

Key significance points highlighted by the authors include:

• Novel Framework: The development of a biorthogonal current operator method specifically for spin currents in non-Hermitian systems.
• Disorder as a Control Mechanism: The demonstration that disorder can act as a powerful tool to manipulate and even amplify spin transport in non-Hermitian settings, offering new routes for controlling spin currents.
• Distinct Physics: The revelation that non-Hermitian spin currents possess unique characteristics (e.g., complex values, distinct real/imaginary behaviors, and specific symmetry responses to magnetic orientation) that differ fundamentally from their Hermitian counterparts.

The authors conclude that while the current model yields purely real spin currents (due to the lack of spin splitting in the imaginary spectrum), the framework established here lays the foundation for future investigations into more general non-Hermitian settings involving spin-selective gain and loss.