🔬 mesoscale physics

Tunable Topological Superconductivity by Fully Compensated Ferrimagnets

This paper proposes that fully compensated ferrimagnet-based heterostructures serve as a versatile platform for realizing and electrically tuning topological superconductivity with Majorana bound states across one, two, and higher dimensions, while preserving superconducting coherence by eliminating the need for external magnetic fields.

Original authors: Yu-Xuan Li, Yicheng Liu, Cheng-Cheng Liu

Published 2026-11-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-Xuan Li, Yicheng Liu, Cheng-Cheng Liu

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

In the quest to build a new kind of computer that can solve problems impossible for today's machines, scientists are looking for a very specific type of particle. They are hunting for something called a Majorana bound state. Imagine a particle that acts as its own antiparticle, a rare and elusive creature that, if found in the right environment, could store information in a way that is incredibly stable against outside noise. This stability is the holy grail for quantum computing, promising machines that do not crash or lose data as easily as current prototypes. To create these particles, researchers usually try to mix two opposing forces: superconductivity, where electricity flows with zero resistance, and magnetism, which usually disrupts that flow. The challenge has always been that the magnetic fields needed to create these particles often destroy the delicate superconducting state, forcing scientists to choose between control and stability.

A team of researchers at the Beijing Institute of Technology has proposed a solution that avoids this trade-off entirely. Instead of using standard magnets that create a net magnetic field, they suggest using a material called a fully compensated ferrimagnet. In this material, the magnetic atoms are arranged so perfectly that their individual magnetic pulls cancel each other out, leaving the material with no overall magnetic field, yet still possessing the internal magnetic structure needed to split electron energy levels. By placing this material next to a superconductor, the researchers show that they can create a platform where these special particles appear and can be controlled without any external magnets.

The team's work, which relies on detailed computer simulations and theoretical modeling, outlines how this material can be used to create these particles in three different shapes: as a line, as a flat surface, and as a corner. In a one-dimensional setup, where a thin wire sits on top of the magnetic material and a superconductor, the simulations show that the special particles appear at the very ends of the wire. Crucially, the researchers found that they could turn these particles on and off, or move them, simply by applying an electric voltage. This voltage changes the internal energy landscape of the magnetic material, effectively switching the system between a normal state and a state where the particles exist. This is a significant departure from previous methods, which often required physically rotating magnets or applying strong external magnetic fields to achieve similar results.

The researchers also explored what happens when this setup is expanded into two dimensions. In a flat sheet of material, the simulations predict that the particles form a stream that flows along the edge of the material, much like a river flowing around a bank. The direction this stream flows is determined by the orientation of the internal magnetic alignment within the material. By changing the direction of this internal alignment, known as the Néel vector, the researchers showed they could reverse the flow of the particles. This ability to control the direction of the flow using only electric fields and the internal magnetic orientation offers a new level of precision for manipulating these quantum states.

Perhaps the most intriguing finding involves a more complex shape known as a higher-order topological phase. In this scenario, the particles do not appear along the edges or at the ends of a wire, but rather get trapped specifically at the corners of the material. The simulations reveal that the researchers can decide exactly which corners host these particles simply by rotating the internal magnetic alignment. If they point the alignment in one direction, the particles gather at one set of corners; if they rotate it, the particles shift to different corners. This spatial control, achieved without any external magnetic fields, suggests a way to route information through a quantum device by simply changing the orientation of the material's internal structure.

The study emphasizes that the absence of a net magnetic field is the key to this success. In traditional setups, the stray magnetic fields from magnets tend to break apart the superconducting pairs of electrons, weakening the very effect the scientists are trying to harness. By using a fully compensated ferrimagnet, the system preserves the superconductivity while still providing the necessary magnetic influence to create the particles. The researchers note that this approach works because the material allows for an electric field to tune the energy splitting of electrons, a feature that is not possible in other types of compensated magnetic materials.

While these results are currently theoretical and based on mathematical models, the paper provides a clear roadmap for how such a system could be built and tested. The authors suggest that scientists could look for these particles using standard tools like scanning tunneling microscopes, which can detect the unique electrical signals these particles produce. They also propose that measuring how heat flows through the material could reveal the presence of the edge streams. The work does not claim to have built a working quantum computer, but rather establishes a versatile and tunable platform where the conditions for these particles can be created and manipulated with unprecedented control. By proving that electric fields and internal magnetic orientations can replace bulky external magnets, the researchers offer a promising path toward scalable and stable quantum devices that could one day power the next generation of computing.

Technical Summary: Tunable Topological Superconductivity by Fully Compensated Ferrimagnets

Problem Statement
The realization of Majorana bound states (MBSs) is critical for non-Abelian quantum operations, yet most engineered platforms rely on magnetization or external magnetic fields to induce the necessary spin splitting. This approach introduces a fundamental trade-off: stray magnetic fields and depairing effects inevitably weaken superconductivity and destabilize the topological gap. While intrinsic topological superconductors remain elusive, current heterostructure proposals (1D nanowires, 2D Rashba systems) struggle to balance tunability with superconducting coherence. The authors propose that fully compensated ferrimagnets (fFIMs)—a class of materials with vanishing net magnetization but electrically tunable spin-splitting bands—can resolve this conflict by enabling topological superconductivity without external magnetic fields.

Methodology
The authors employ symmetry analysis and microscopic modeling to construct effective Hamiltonians for fFIM-based heterostructures across three dimensions:

Effective Model Derivation: Starting from a lattice Hamiltonian with a $d$ -wave crystal potential (adapted from Refs. [35, 44]), the authors derive a low-energy effective model ( $H_{eff}$ ) for intrinsic fFIMs. This model incorporates a momentum-dependent crystal potential and a sublattice-staggered potential ( $V_0$ ) that is tunable by an electric field. Crucially, $V_0$ breaks the equivalence of magnetic sublattices, allowing for electric-field-controlled spin splitting that transitions from $C_4$ -symmetric $d$ -wave to $C_4$ -broken $s$ -wave.
Dimensional Analysis:
- 1D Nanowires: A minimal Bogoliubov-de Gennes (BdG) Hamiltonian is constructed for a semiconductor nanowire proximitized by an $s$ -wave superconductor and placed on an fFIM substrate. The system is analyzed for particle-hole symmetry (PHS) and time-reversal symmetry, classifying it into the BDI symmetry class.
- 2D Heterostructures: A model combining an fFIM layer, a Rashba electron gas, and an $s$ -wave superconductor is analyzed. The system is classified under symmetry class D, characterized by a Chern number.
- Higher-Order Topology: A sandwich structure involving a topological insulator (TI), fFIM, and superconductor is modeled to investigate higher-order topological superconductivity (HOTSC).
Topological Invariants: The authors calculate topological invariants, including the winding number ( $w$ ) for 1D systems, the Chern number ( $C$ ) for 2D systems, and analyze Dirac mass sign changes at boundaries for higher-order phases. They utilize Wilson loop calculations and block-diagonalization techniques (via conserved quantities like $\Pi = s_x \gamma_x$ ) to map phase diagrams.

Key Contributions and Results
The paper demonstrates that fFIM-based heterostructures host three distinct phases of tunable topological superconductivity, controlled by electric fields ( $V_0$ ) and the orientation of the Néel vector ( $\hat{n}$ ):

1D Tunable TSCs with Majorana Zero Modes (MZMs):
- The system hosts MZMs localized at the ends of 1D nanowires.
- The topological phase transition is driven by the electric field parameter $V_0$ , which modulates band splitting.
- At zero chemical potential, the model maps exactly to the Kitaev chain. The system supports a quartet of zero-energy states in specific parameter regimes, with wavefunctions exponentially localized at wire ends.
- The Néel vector orientation also influences the phase: alignment along the $x$ -axis preserves MZMs, while alignment along the $y$ -axis drives the system into a trivial phase.
2D Tunable TSCs with Chiral Majorana Edge States (MESs):
- The heterostructure supports robust chiral MESs along boundaries.
- The Chern number ( $C$ ) ranges from $-2$ to $2$. In regions where the net Chern number vanishes ( $C=0$ ), the authors identify gapless edge modes arising from two counter-propagating chiral modes with opposite windings in decoupled sectors.
- The chirality of the edge states is tunable via both the electric field ( $V_0$ ) and the polar angle ( $\theta$ ) of the Néel vector. Specifically, an out-of-plane Néel vector component is required to drive the system into a topological phase, whereas an in-plane component results in a trivial phase.
Higher-Order TSCs with Tunable Majorana Corner Modes (MCMs):
- The sandwich structure hosts MCMs localized at corners rather than edges.
- The emergence of MCMs is explained via edge theory, where MCMs bind to domain walls where the Dirac mass changes sign across adjacent edges.
- Spatial Control: The position of MCMs can be spatially redistributed by rotating the Néel vector. For example, in a triangular geometry, rotating the Néel vector from the $x$ -axis to the $y$ -axis shifts the MCMs from the termination of one edge to another by altering the domain wall configuration.
- Electric fields control the generation and annihilation of these modes.

Significance and Claims
The authors claim that their work establishes fFIM-based heterostructures as a versatile platform for tunable topological superconductivity that overcomes the limitations of magnetized systems. The primary significance lies in the absence of net magnetization, which preserves superconducting coherence while allowing for:

All-electric control: Topological phase transitions are driven by electric fields rather than external magnetic fields.
Dual control mechanism: Electric fields govern bulk topological properties (phase transitions), while the Néel vector orientation modulates boundary states (chirality in 2D, spatial distribution in higher-order phases).
Scalability: The proposal offers a pathway toward magnetic-field-free topological quantum devices.

The paper suggests that experimental detection can be achieved through dimension-specific probes: zero-bias conductance peaks ( $2e^2/h$ ) in 1D and 2D higher-order systems via scanning tunneling microscopy, and half-quantized thermal Hall conductance or $4\pi$ -periodic Josephson effects in 2D chiral phases. The authors note that while the proposal focuses on $d$ -wave potentials, the mechanism is applicable to other momentum-dependent potentials (e.g., $g$ -wave, $i$ -wave) found in fFIMs.

Technical Summary: Tunable Topological Superconductivity by Fully Compensated Ferrimagnets

More like this