Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids

arXiv:2507.21226 · cond-mat.str-el, cond-mat.other · Submitted 2025-07-28 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids".

Mira: The gist The study investigates magnetic order and topological robustness in spin liquids by examining the phase diagram of the Yao-Lee model when combined with Kitaev and Heisenberg interactions,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: To recap where we are, we’ve seen that this paper on Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids looks at adding Kitaev and Heisenberg terms to the Yao-Lee model. The core thesis is that this combination creates a scenario where magnetic order can appear in the spin sector while the orbital sector stays liquid, leading to a state with both local and topological order existing together.

Mira: They claim this behavior is tunable by the Heisenberg interaction, which can either enhance or suppress that magnetic ordering depending on its sign. This shows that you have a rich coexistence of different types of phases possible in this spin-orbital generalization.

Lev: The real question here for me, looking at this theory, is how stable this magnetically fragmented state actually is when we try to implement it on real hardware with finite temperature effects and noise.

Kai: Right, Lev’s right. The paper says that the plaquette operator remains conserved even with these added perturbations, which is a key feature that keeps things from being completely chaotic, but it also means the model isn't exactly solvable anymore.

Mira: And they use perturbation and Majorana mean-field theory to map out this phase diagram precisely. They find specific conditions for when you get magnetic order versus when you maintain the liquid character in both sectors.

Lev: If we’re translating this into an error correction setting, that means we need to design a system where the constraints that lead to this magnetic fragmentation are manageable under realistic noise conditions.

Kai: The paper also details specific results for when the Yao-Lee interaction dominates, showing that in that case, both spin and orbital sectors retain their liquid character without any symmetry breaking happening at all.

Mira: That’s important because it shows that you can still have a very robust topological state even when the system is dominated by one particular interaction type. This isn't just about getting one kind of order; it's about having control over which kind of order we get.

Lev: So, if we are designing a qubit architecture, this suggests that the choice of interaction strength—whether Kitaev or Yao-Lee dominant—is a primary design variable for controlling the resulting state.

Kai: The overall picture is that these interactions don't just add complexity; they introduce new ways to organize the system into states with mixed topological and magnetic properties.

Mira: It really emphasizes that stability against perturbations isn't just about having a large energy gap; it’s about understanding how different terms cooperate to define those gaps in different parts of the system.

Conclusion: Kai: So, looking back at the paper Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids by Akram, Vijayvargia, Kee, and Erten. The authors are showing us that when you combine these models, you can get magnetic order in one part of the system while keeping the other part liquid.

Mira: It means that this is not a single fixed state; it’s a landscape where different conditions lead to fundamentally different kinds of organization, depending on which coupling term wins out.

Lev: For practical application, this suggests that when we try to build something topological, we have to be careful because the stability isn't guaranteed across all parameter choices without considering the competition between these terms.

Kai: Exactly. The paper is about mapping out this phase diagram so that researchers know exactly where they are likely to find a magnetically ordered state versus a fully liquid one based on their chosen parameters.

Mira: It points toward designing quantum materials where you can intentionally engineer this kind of mixed topological and magnetic state, instead of just hoping for the simplest possible liquid ground state.

Lev: It’s about moving past the simple picture of a single, perfectly stable phase and embracing the complexity that comes from adding more interactions to achieve more functional properties.

Muhammad Akram, Aayush Vijayvargia, Hae-Young Kee, Onur Erten

Department of Physics, Arizona State University · Department of Physics, Balochistan University of Information Technology, Engineering and Management Sciences (BUITEMS) · Department of Physics, University of Toronto · Canadian Institute for Advanced Research

cond-mat.str-el, cond-mat.other

Submitted: 2025-07-28

Updated: 2025-07-28

Journal ref: npj Quantum Materials 11, 90 (2026)

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 78/100

The gist: The gist The study investigates magnetic order and topological robustness in spin liquids by examining the phase diagram of the Yao-Lee model when combined with Kitaev and Heisenberg interactions,

Key concepts

Yao-Lee Model (YL): This is a spin-orbital generalization of the Kitaev model that is known for its enhanced stability of spin liquid phases against external disturbances. It involves local Hilbert spaces with four dimensions, allowing for complex spin and orbital degrees of freedom, making it a richer system to study than the standard Kitaev model.
Magnetic Order vs. Liquid Phase: The research investigates whether magnetic ordering (like ferromagnetism or antiferromagnetism) can occur in the spin sector while the orbital sector stays in a liquid state, or vice versa. This phenomenon, known as magnetic fragmentation, shows that different parts of the system can exhibit fundamentally different behaviors simultaneously.
Plaquette Operator ($W_p$): This operator is conserved (it commutes with the Hamiltonian) and its eigenvalues ($ ext{±1}$) label the eigenstates of the system. Its conservation is a key feature, even when added perturbations are included, which helps define the topological nature of the phases being studied.

Terminology

Generalization — 1× in this paper · explained in 13 episodes

Summary

The gist The study investigates magnetic order and topological robustness in spin liquids by examining the phase diagram of the Yao-Lee model when combined with Kitaev and Heisenberg interactions, revealing that magnetic order can emerge in one sector while another remains liquid, leading to a coexistence of magnetic and topological phases.

Model Setup

The research focuses on the spin-orbital generalizations of the Kitaev model, specifically studying the phase diagram of the Yao-Lee (YL) model augmented with additional Kitaev (HK) and Heisenberg (HH) terms. The full Hamiltonian considered is given by H = HY L + HK + HH, where HY L represents the YL interaction, HK is the Kitaev term, and HH is the Heisenberg term. The local Hilbert space is four-dimensional, spanned by S = 3/2 operators or Gamma matrices. The plaquette operator Wp commutes with the Hamiltonian, [H, Wp] = 0, and its eigenstates can be labeled by the eigenvalues of Wp = ±1.

Phase Diagram Results

The key findings from the perturbation and Majorana mean-field theory analysis are threefold.

**(i) For dominant Kitaev interaction, the spin sector exhibits magnetic order—ferromagnetic or antiferromagnetic depending on coupling signs—while the orbital sector remains in a liquid phase. This magnetically fragmented state displays coexisting local and topological order. (ii) For dominant YL interaction, both spin and orbital sectors retain their liquid character without any symmetry breaking. (iii) The Heisenberg interaction tunes the magnetic order, either enhancing or suppressing it based on its sign. In the absence of the Heisenberg term, the YL spin-orbital liquid remains stable within the range −0.85 < JK/JY L < 1.38. Beyond this window, magnetic transitions in the Ã sector lead to either ferromagnetic (FM) or antiferromagnetic (AFM) order. The AFM transition is second order and does not gap the c z Majorana fermions, while the FM transition for JK/JY L > 0 is first order and hybridizes the c x and c y Majorana fermions, leading to a finite Fermi surface. The plaquette operator Wp remains conserved for all values of JK/JY L. Magnetic order develops in the Ã sector while the Ä sector retains its QSL character, a phenomenon referred to as magnetic fragmentation. The AFM Kugel-Khomskii model is known to host a SU(4) spinorbital liquid. The YL spin-orbital liquid is expected to be more stable than the FM and AFM phases due to the flux gap in the YL-QSL being three times larger than in the Kitaev model. The mean-field analysis implies that for JK/JY L ≳ 10, the transition occurs at JH/JY L = 0.525 when JK > 0, and at JH/JY L = −0.525 when JK 0 and FM-2 for J < 0. The paper also notes that the AFM phase does not affect the excitation spectrum of the c z Majorana fermions, which remain gapless. The YL model with AFM Heisenberg interaction has been studied by Ref. 27 and our results are in agreement with theirs in the JK = 0 limit. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study. The paper concludes that inclusion of additional Heisenberg interactions can enhance the magnetic order or suppress it and enlarge the stability of the YL-QSL, depending on its sign. The study also suggests that the AFM Kugel-Khomskii model is an interesting direction for future study.

Improvements for AI systems

Bold magnetic order emergence in spin sector under Kitaev dominance: The AI system can predict magnetic order can arise in the spin sector while the orbital sector remains a liquid for dominant Kitaev interactions, allowing it to determine if a material, based on its interaction parameters, will exhibit magnetic fragmentation.
Predicting phase transitions via Heisenberg tuning: The system can use the finding that The Heisenberg interaction tunes the magnetic order, either enhancing or suppressing it based on its sign to forecast whether adding a specific exchange coupling will stabilize or destabilize an existing magnetic phase.
Coexistence of topological and local order: The AI can model systems exhibiting coexisting local and topological order, enabling it to characterize emergent phases where both magnetic and topological features are present simultaneously.
Identifying stable spin liquid regimes: The system can map the parameter space to identify regions where both spin and orbital sectors form liquid phases when Yao-Lee interactions dominate, pinpointing materials likely to be quantum spin liquids.
Predicting phase boundary behavior: Using the mean-field analysis results, the AI can predict transition characteristics, such as a transition being second order or first order, based on the ratio of coupling constants like JK/JY L.

Abstract

Spin-orbital generalizations of Kitaev model, such as Yao-Lee model, have attracted recent attention due to their enhanced stability of spin liquid phases against perturbations. Motivated by microscopic calculations for the realization of Yao-Lee model showing additional interactions, we study the phase diagram of the Yao-Lee model with added Kitaev and Heisenberg terms. While the plaquette operator is conserved even in the presence of added perturbations, the model becomes no longer exactly solvable. Using perturbation and Majorana mean-field theory, we find magnetic order can arise in the spin sector while the orbital sector remains a liquid for dominant Kitaev interactions, whereas both sectors form liquid phases when Yao-Lee interactions dominate. Additional Heisenberg exchange can enhance or suppress the magnetic order, revealing a rich coexistence of magnetic and topological phases.