🔬 mesoscale physics

Interaction enhanced altermagnet in the Hatsugai-Kohmoto model

This paper investigates the $d_{x^2-y^2}$ altermagnetic Hatsugai-Kohmoto model and reveals that increasing electronic correlations drive a many-body Lifshitz transition into an "interaction-enhanced altermagnet" state characterized by full spin polarization, a correlation-dependent gap in the dynamical susceptibility, and a distinct evolution of signal intensity and static response.

Original authors: Ádám Bácsi, Balázs Dóra

Published 2026-06-29

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Ádám Bácsi, Balázs Dóra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 a crowded dance floor where everyone is paired up, but the music has a strange rule: the dancers are split into two groups based on their "spin" (think of it as their dance style, like "Left-Handed" or "Right-Handed").

In a normal magnetic material, these groups might mix freely. But in this specific material, called an altermagnet, the dance floor is shaped like a weird, twisted oval. Because of this shape, the "Left-Handed" dancers naturally cluster in one part of the room, and the "Right-Handed" dancers cluster in another. They don't mix much, but they aren't fighting each other either; they just occupy different zones. The total number of Left and Right dancers is equal, so the room has no overall "magnetic" pull, even though everyone is dancing in a very specific, organized way.

Now, imagine we turn up the volume on the music to a deafening level. In physics terms, this is increasing the interaction strength between the electrons.

The "No Double-Booking" Rule

In this dance hall, there's a strict rule: No two dancers can occupy the exact same spot at the same time. This is the "Hatsugai-Kohmoto" interaction.

At low volume (weak interaction): Some dancers are brave enough to try to squeeze into the same spot, creating "doubly occupied" zones. The dance floor is a bit messy, with some mixed zones and some single zones.
At high volume (strong interaction): The music gets so loud that the dancers get nervous. They realize that sharing a spot is too expensive energetically. So, they all move out of the shared spots. The "double-occupied" zones vanish completely.

The "Interaction-Enhanced" Moment

Here is the paper's big discovery: As the dancers clear out the shared spots, something amazing happens. The dance floor becomes perfectly organized. Almost every single spot on the floor is now occupied by only one type of dancer (either all Left or all Right).

The authors call this "Interaction-Enhanced Altermagnetism." It's like a paradox: by making the dancers interact more (by turning up the volume), the material becomes more perfectly sorted and polarized than it was before. The "Left" and "Right" zones become larger and more distinct.

The "Jump" (Lifshitz Transition)

There is a specific moment when the volume hits a critical level. This is called a Lifshitz transition.

Before the jump: You can still find a few dancers trying to share a spot.
After the jump: The shared spots are gone forever. The dance floor is now a "Mott insulator" in spirit—it's so organized that the dancers are frozen in their specific zones, and the shape of the dance floor stops changing even if you turn the volume up further.

Listening to the Dance (The Signals)

The researchers also looked at how this system reacts when you poke it with a probe (like a radar or a light beam).

The "Gap" (The Silence): When the interaction is strong, the system develops a "gap." Imagine trying to get the dancers to change their rhythm. If they are too busy protecting their personal space (due to the strong interaction), they won't respond to small nudges. You need a lot of energy to get them to move. This "gap" gets bigger as the interaction gets stronger.
The "Scream" (The Peak): When you do hit them with just the right amount of energy, they don't just move; they scream. The signal gets incredibly loud and sharp.
- As the interaction increases, this "scream" gets louder and louder.
- Right at the moment of the "Jump" (the Lifshitz transition), the scream becomes mathematically infinite (a "log-divergent" peak). It's like the system is screaming at the top of its lungs because the dance floor has just completely reorganized itself.
- After the jump, the scream stays loud but shifts to a higher pitch (higher frequency).

The Static Picture (The Frozen Moment)

Interestingly, if you look at the system without moving it (static view), the interaction doesn't change the basic "weight" of the response. It's like taking a photo of the dance floor: the interaction changes how they move when you poke them, but it doesn't change the total number of dancers or the basic layout of the room. The "altermagnetic" shape of the room is the only thing that matters for this static view.

Summary

In short, the paper shows that if you take a special magnetic material and crank up the internal interactions between its particles, you don't just make it messy. Instead, you force it into a state of extreme order. The particles banish themselves from sharing space, creating a perfectly polarized, "interaction-enhanced" state that screams loudly when probed, revealing a new kind of magnetic behavior driven entirely by the strength of their relationships.

Technical Summary: Interaction Enhanced Altermagnet in the Hatsugai-Kohmoto Model

Problem Statement
Altermagnets are a class of magnetic materials characterized by zero net magnetization yet possessing a spin-dependent band structure typical of ferromagnets, arising from the interplay between collinear antiferromagnetic order and specific crystal symmetries. While the single-particle band features of altermagnets are well-understood, the role of strong electronic correlations on top of altermagnetism remains largely unexplored. Strong interactions in correlated systems often lead to phenomena such as Mott insulating behavior, phase transitions, and non-Fermi liquid physics. The central problem addressed in this work is to investigate how strong electronic correlations modify the band structure and spin response of a two-dimensional $d_{x^2-y^2}$ altermagnet.

Methodology
The authors employ the Hatsugai-Kohmoto (HK) model, a theoretically tractable framework that combines analytical solvability with the physics of non-Fermi liquids and can be continuously deformed into the Hubbard model. The system is modeled by a Hamiltonian containing a spin-dependent kinetic energy term (describing the altermagnetic band structure with an asymmetry parameter $\alpha$ ) and a momentum-space contact interaction term ( $U$ ) that penalizes double occupancies.

Key methodological features include:

Many-Body Formalism: Unlike conventional treatments based on single-particle Green's functions, the authors explicitly evaluate the Kubo formula using many-body occupation probabilities ( $P(k, s)$ ) derived from the diagonal density matrix in the momentum basis. This approach naturally incorporates the effects of the interaction on the ground state occupation.
Linear Response Theory: The spin susceptibility tensor is calculated using the Kubo formula, analyzing both dynamical (frequency-dependent) and static ( $\omega=0$ ) responses.
Ground State Analysis: The study examines the evolution of the Fermi surface and occupation probabilities as a function of interaction strength $U$ at zero temperature.

Key Contributions and Results

Interaction Enhanced Altermagnetism and Many-Body Lifshitz Transition:
The study identifies a many-body Lifshitz transition occurring at a critical interaction strength $U_c$ .
- Below $U_c$ : The Fermi surface contains regions of double occupancy (doublons) alongside singly occupied, spin-polarized regions.
- At and Above $U_c$ : Double occupancies are completely excluded from the Fermi surface. Consequently, almost all momentum states become fully spin-polarized (except for a zero-measure set along degeneracy curves).
- This state is termed "interaction enhanced altermagnetism." It is not an emergent phase but a result of the interplay between the intrinsic altermagnetic dispersion and the interaction. For $U > U_c$ , the system exhibits effective Mott insulating features where the Fermi surface shape remains fixed despite increasing $U$ , and optical excitations cannot probe gapless modes.
Dynamical Spin Susceptibility:
The dynamical spin response reveals distinct interaction-induced features:
- Gap Opening: A gap proportional to the interaction strength $U$ opens in the dynamical susceptibility. This gap represents the energy cost of forming a spin-flip pair (doublon) from two single-spin particles.
- Peak Evolution: For $U < U_c$ , the susceptibility displays a sharp peak at a frequency $\omega_m$ that increases with $U$ . As $U$ approaches $U_c$ , this peak shifts to the lower gap edge ( $\omega = U$ ) and becomes logarithmically divergent due to a saddle point in momentum space.
- Post-Transition Behavior: Above the Lifshitz transition ( $U > U_c$ ), the peak remains at the lower gap edge with log-divergent behavior. The overall shape of the response function remains unchanged, but the entire spectrum shifts to higher frequencies as the separation between the lower and upper Hubbard bands increases.
- Signal Intensity: The signal intensity of the dynamical susceptibility increases with interaction up to the Lifshitz transition and then saturates, reflecting the "frozen" nature of the fully spin-polarized Fermi surface without doublons.
- Transverse Nature: The dynamical susceptibility possesses only transverse non-zero components for small wavevectors, with the longitudinal component ( $\chi_{zz}$ ) vanishing identically.
Static Spin Susceptibility:
In contrast to the dynamical response, the static susceptibility ( $\omega = 0$ ) remains largely unaffected by the interaction strength $U$ .
- The longitudinal component ( $\chi_{zz}$ ) corresponds to the conventional Pauli susceptibility and is independent of both $U$ and the altermagnetic asymmetry parameter $\alpha$ .
- The transverse components ( $\chi_{xx}, \chi_{yy}$ ) are reduced by the altermagnetic band structure but remain independent of $U$ . This independence arises because static susceptibility is dominated by intraband processes, which are preserved even as the interaction splits the bands into upper and lower Hubbard bands without altering their shape.

Significance and Claims
The paper claims to provide a natural framework for analyzing the interplay between altermagnetism and strong correlations by utilizing the exact solvability of the HK model. The primary significance lies in the identification of "interaction enhanced altermagnetism," a state where strong correlations drive the system toward a fully spin-polarized Fermi surface by excluding doublons.

The authors assert that their results reveal specific fingerprints of strong correlations on dynamical spin probes, such as the interaction-induced gap and the logarithmic divergence of the response peak at the Lifshitz transition. They emphasize that while the dynamical response is highly sensitive to interaction strength, the static Pauli response is robust against correlations, being influenced only by the underlying altermagnetic band structure. The work suggests that these qualitative features—particularly the interaction-enhanced polarization and the specific behavior of the dynamical susceptibility—would likely persist in more realistic altermagnetic Hubbard models at low fillings, provided momentum mixing is introduced.