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Based on online seminar from T. Kurumaji
Topological Hall effect in a nutshell
Topological Hall effect in a nutshell
In certain magnetic materials, electrons don't just respond to external magnetic fields—they also sense an invisible, internal one. This emergent field arises not from a traditional magnet but from the twisting patterns of spins within the material itself. These complex spin textures act like tiny magnetic whirls, deflecting electron motion in subtle but profound ways.
This phenomenon, known as the topological Hall effect, opens a fascinating window into the interplay between topology and electronic transport. Before diving into the mathematical description, let’s explore what makes this effect so intriguing.
From Twisting Spins to Emergent Fields: Why Electrons Feel What Isn’t There
Imagine an electron navigating a magnetic landscape—not through an ordinary external field, but through a swirling texture of local spins. As it moves, the electron’s own spin tries to follow the twisting orientation of the background. But in doing so, something remarkable happens: it begins to behave as if it's in a magnetic field, even though none was applied.
This illusion arises from quantum mechanics itself. When the spin background varies smoothly in space, we can define a local spin frame that rotates with it. In this frame, the Hamiltonian takes a new form—one that reveals a hidden structure:
We begin with the standard Hamiltonian describing (1) itinerant electrons hopping between sites, and (2) their coupling to a textured spin background represented by M(r).
We can reformulate this Hamiltonian into more intuitive form by rotating local spin principal axis by appropriate gauge transform. By doing this, we can treat second term as uniform magnetization.
However, this gauge transform does not just make spin texture as uniform magnetization. Here is the point where quantum mechanics enter as virtual vector potential.
Because the gauge field T(r) varies across space, it does not commute with the momentum operator. This non-commutativity is the heart of the emergent field.
Kinetic energy term can be written as follows. At this point, we see clearly how spin texture reshapes the landscape through which the electron moves.
In short, effect of non-trivial spin texture can be reduced into virtual vector potential, which will be appear as emergent magnetic field by skyrmion.
With the algebra cleaned up, the Hamiltonian reveals its true structure: the electron behaves as if it were charged and moving under a vector potential.
But this potential isn’t imposed from outside—it’s born from within, sculpted by the spin texture itself.
Adiabatic motion & spin texture: How geometry becomes magnetic field
For the sake of completeness, we provide here a few details regarding the adiabatic approximation used in this context.
In the strong coupling limit between itinerant electrons and local spin order, the electron spin is expected to align perfectly with the local spin texture—minimizing the exchange energy associated with the M⋅σ term.
Therefore, hopping to a neighboring site necessarily involves a rotation of the electron’s spin to follow the local spin orientation.
In line with the adiabatic approximation, we assume that conduction electron spins remain perfectly (anti-)aligned with the local spin direction throughout their motion.
To estimate the resulting (virtual) magnetic flux, consider the adiabatic motion of a conduction electron as it traverses a closed loop in real space.
This picture simplifies the problem considerably.
This reflects the situation where the spin of itinerant electrons remains perfectly (anti-)aligned with the local spin texture.
Although this formalism arises naturally in the strong coupling limit, a similar conclusion can be drawn under weak coupling conditions as well.
The steps shown on the left illustrate how the underlying spin texture gives rise to an emergent magnetic field that influences electron motion.
The key point, once again, is this:
An electron moving around a background skyrmion experiences an emergent magnetic field—originating from the accumulated Berry phase.
We can infer that the strength of the emergent magnetic field depends on the size of the spin texture—most conveniently, the size of a skyrmion. Since the gauge field appears as a vector potential, we can compute how it manifests as an emergent magnetic field:
As this expression shows, the smaller the skyrmion, the larger the emergent field experienced by the electron—scaling inversely with the square of the skyrmion size.
This quantity, known as the skyrmion density or scalar spin chirality, effectively acts like a magnetic field. It deflects the electron’s path, producing a transverse Hall signal—even when no real magnetic field is present. This is the essence of the topological Hall effect.
In this way, the Kondo lattice model doesn't just provide a playground for spin-charge interaction—it reveals how geometry and topology give rise to emergent gauge structures in the quantum world.
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