Topological quantum computation (TQC) based on Majorana zero modes (MZM) has been actively pursued in the past decade. Aside from well-recognized challenges in unambiguously identifying MZM in existing effective one-dimensional p-wave superconductors, the next step of demonstrating braiding of MZM is even more formidable. The most prevalent proposal on realizing braiding operation is based on networks of effective p-wave superconductor wires subject to many tiny electric gates that are distributed along the wires and can be sequentially turned on or off, so that the MZM can be adiabatically moved across different parts of the network. It is then a natural question to ask if there exists an alternative, possibly more practical architecture for braiding MZM, that may even be more friendly to near-term MZM systems.

Motivated by this question, as well as the recent experimental breakthrough on demonstrating MZM in a minimal Kitaev chain consisting of only two quantum dots [Dvir et al., Nature 614, 445 (2023)], we propose in [1] a couple of new structures for braiding MZM based on triangular superconducting islands that can either be constructed in a bottom-up manner as a minimal extension of the two-site Kitaev chain, or appear naturally in epitaxial growth of ultrathin films. For the former case, we have shown that a minimal 3-site “Kitaev triangle” can host MZM at different pairs of vertices that can be controlled by a non-uniform vector potential. We have demonstrated braiding two MZM in this minimal model by explicitly calculating the many-body Berry phase. For the latter case, we have extended the 3-site Kitaev triangle to a finite-size triangular island with a hollow interior. We have shown that using a uniform vector potential, one can induce a pair of MZM at different pairs of vertices of the triangle. Moreover, rotating the uniform vector potential can change the positions of the MZM without closing the bulk band gap, i.e., adiabatically. Finally we have given a scalable design, backed by numerical calculations, for braiding two out of four MZM that corresponds to nontrivial logical gate operations, in a network of corner-sharing hollow triangles.

Our work provides a novel platform in parallel with the coupled-wire network design for braiding MZM, and is practical especially considering near-term MZM devices. The Kitaev triangle offers an alternative strategy towards MZM-based TQC that is not based on bulk-boundary correspondence, which may be easier to realize in near-term quantum-dot=based devices than coupled wires. On the other hand, our proposal of a uniform vector potential coupled to hollow triangles explores the utility of geometry rather than the individual control of superconducting nanowires. It highlights that triangles, as a geometry, are unique compared to other quasi-2D structures such as wires, squares, or circles, since they naturally break 2D inversion symmetry and do not present a straightforward strategy for morphing into either 1D or 2D structures with periodic boundary conditions.

[1] Aidan Winblad and Hua Chen, Superconducting triangular islands as a platform for manipulating Majorana zero modes, Physical Review B 109, 205158 (2024).

The phenomenon of spin-transfer torque in bilayer metallic ferromagnets is well understood in terms of Landau-Lifshitz equation modified by magnetic torques carried by electric currents. However, when each ferromagnet layer is replaced by a single atomic layer of vdW magnetic insulators and electric transport through the bilayer is highly coherent, is the above picture still applicable? In a recent collaborative work [1], we argued that switching between layer-resolved collinear antiferromagnetic and ferromagnetic states in bilayer insulating CrI3 is achieved by tunneling-current-induced spin accumulation of opposite signs at metallic electrodes in contact with the respective CrI3 monolayers, which from a symmetry point of view is a general mechanism for switching between uniform and nonuniform magnetic states. A pedagogical news article summarizing the main findings of the work can be found here.

[1] ZhuangEn Fu, Piumi I. Samarawickrama, John Ackerman, Yanglin Zhu, Zhiqiang Mao, Kenji Watanabe, Takashi Taniguchi, Wenyong Wang, Yuri Dahnovsky, Mingzhong Wu, TeYu Chien, Jinke Tang, Allan H. MacDonald, Hua Chen* & Jifa Tian*, “Tunneling current-controlled spin states in few-layer van der Waals magnets”, Nature Communications 15, 3630 (2024).

When hearing “pi-phase” and “quantum oscillations”, would you immediately jump to Berry phase? In a recent collaborative work, we have shown that a new kind of “pi-phase shift”, i.e., that between quantum oscillations in longitudinal resistivities along orthogonal directions, in the prototypical spin density wave material Cr, is caused by quantum interference effects between coupled semiclassical orbits. For an in-depth introduction, see this note which was a presentation given at the “Magneticians’ meeting” at Johns Hopkins University in 2023.

Y. Feng, Y. Wang, T. F. Rosenbaum, P. B. Littlewood, and H. Chen, “Quantum interference in superposed lattices”, Proc. Natl. Acad. Sci. 121, e2315787121 (2024).

Since the original proposal by Anderson on spin liquid being a possible mechanism for high-Tc superconductivity, electrical transport in frustrated quantum spin systems has always been one of the most enticing topics in condensed matter physics. However, a majority of quantum spin ice and spin liquid candidates are good insulators, making it challenging for standard electrical transport techniques. In contrast, in the few cases of metallic frustrated spin systems, transport experiments usually lead to surprising discoveries.

In this collaborative work, through electrical transport experiments on a recently identified metallic kagome spin ice compound HoAgGe, we made another surprising discovery: There exists a time-reversal-like degeneracy for certain noncollinear ice-rule states on field-induced magnetic plateaus, which have the same energy and magnetization, but different sizes of the anomalous Hall effect (AHE). Since the AHE transforms as a pseudovector with fixed length under all magnetic group operations, the symmetry operation underlying such a degeneracy is beyond the standard paradigm of magnetic space group for classifying magnetic structures.

Through comprehensive neutron refinement and numerical search in the degenerate ice-rule manifold, we are able to nail down the spin structures of the degenerate states and to come up with a microscopic model for calculating their transport properties. Again surprisingly, we find that the degenerate spin structures have exactly the same band structure, but different Berry curvatures in momentum space, and hence different sizes of the anomalous Hall effect. Although the Berry curvature is more related to the geometric properties of the Bloch wavefunctions than to the eigenenergies, it is hard to imagine without seeing that real physical systems with identical band structures can have different Berry curvatures.

Our systematic experimental and theoretical efforts elucidated the rather subtle symmetry operation that connects the degenerate states, which critically relies on a reversal of lattice distortion in the structure of HoAgGe. Since the Berry connection inherently depends on the atomic location in a unit cell, while the latter is largely invisible to the momentum space Hamiltonian, the distortion reversal in such a quasi-symmetry operation eventually changes the size of the AHE but not the band structure.

Our work is built upon the active research in the topical areas of quantum spin ice systems and anomalous transport in noncollinear antiferromagnets, but advances these fields by cross-disciplinary findings that crucially require ingredients from both: The massive ground state degeneracy of frustrated spin systems provides a reservoir for novel noncollinear spin configurations that host characteristic anomalous transport properties. Our work also points to the special role of lattice distortion and distortion reversal in quantum spin systems. In particular, our theoretical analysis indicates that similar phenomena should generally occur in quantum spin systems with nontrivial lattice distortion and relatively weak spin-orbit coupling.

• Zhao, K., Tokiwa, Y., Chen, H., and Gegenwart, P., Discrete degeneracies distinguished by the anomalous Hall effect in a metallic kagome ice compound. Nat. Phys. (2024).
• News & Views: Enke Liu, Electronic transport probes a hidden state, Nat. Phys. (2024).

Spintronics exploits the idea of using electron spin instead of charge to encode and process information. But can electrons carry more than just spin across diverse materials? Our work answers with a resounding yes, unveiling the transport of spin “shape” in solids. The word “shape” means how spin is distributed spatially about the center of mass of a single electron wave packet. In practice it can be described by the multipole moments of spin distributions. We have isolated the part of such multipole moments that does not depend on how the wave packets are constructed, thus having an absolute meaning. Similar to spin, the well-defined spin or magnetic multipole moments can then be transported by electrons driven by external fields. As one example, we showed that an electric current flowing through phosphorene subjected to a perpendicular electric field can generate magnetic octupole moments. Such octupole moments are manifested by accumulation of spins with alternating signs at square sample corners, as illustrated in the figure below. Our work propels the emergence of “multipoletronics,” a promising new phase beyond spintronics.

Muhammad Tahir and Hua Chen, Transport of Spin Magnetic Multipole Moments Carried by Bloch Quasiparticles, Phys. Rev. Lett. 131, 106701 (2023).

From spintronics to multipoletronics: How CSU research could allow for big developments in data processing (CSU SOURCE)

In the ordinary Hall effect, it is the magnetic field that tells which way the electric currents driven by a voltage bias should be deflected, simply through the Lorentz force. In a ferromagnetic conductor, it is the direction of the magnetization that determines that of the transverse anomalous Hall current flow, despite the more complicated microscopic mechanisms. However, for antiferromagnets when the magnetization vanishes but the anomalous Hall effect does not, how can the electrons know which way they should be deflected? In this work we answered this question by introducing a new quantity named as electronic chiralization (EC), that is determined by the spatial gradients of the microscopic magnetization rather than its mean. EC transforms in exactly the same way as the anomalous Hall vector under all space group operations, including continuous translation, and is free from the difficulties of defining multipole moments of infinite systems. Moreover, EC provides intuitive guidance for the search of new unconventional magnetic systems hosting the AHE as it suggests what types of spin and charge textures are necessary for the AHE. To demonstrate this we provided two novel, experimentally relevant examples: charge-ordered kagome spin ice, and 2D Dirac electrons skew-scattered by a magnetic charge texture. The former may solve the puzzle of the experimentally observed AHE in certain frustrated spin systems that lack long-range magnetic dipole order, while the latter demonstrates another paradigm of noncollinear magnetic textures, magnetic charge or monopole, that can give rise to the AHE, in contrast to the well-known scalar spin chirality and skyrmions.

Electronic chiralization as an indicator of the anomalous Hall effect in unconventional magnetic systems
Hua Chen, Phys. Rev. B 106, 024421 (2022).

Spin ices are exotic phases of matter characterized by frustrated spins obeying local “ice rules,” in analogy with the electric dipoles in water ice. Namely, on a tetrahedron with one spin or electric dipole moment sitting on each vertex, only the two-in-two-out configurations are allowed. Such a local constraint can lead to a macroscopic number of degenerate ground states or an extensive ground state entropy, which was first calculated by Pauling for water ice. In two dimensions, one can similarly define ice rules for in-plane Ising-like spins arranged on a kagome lattice, so that only one-in-two-out or two-in-one-out configurations are allowed. Such ice rules can result from either nearest neighbor ferromagnetic exchange coupling or long-range dipolar coupling between the spins, which can lead to different multi-stage ordering behavior under changing temperature. However, there have been very few materials systems that can be described as kagome spin ice.

In a recent Research Article published in Science [1], a multi-national team involving two CSU Physics faculty members Hua Chen and Kate Ross demonstrated that an intermetallic compound HoAgGe is a realization of the kagome ice state. Through experimental and theoretical approaches including magnetometry, thermodynamic measurements, neutron scattering, and Monte Carlo simulations, they have established that the low-temperature (<15K) spin states of the system can be reasonably well captured by a classical spin model hosting the kagome ice physics. The relatively high temperatures at which the spin ice behavior appears, in comparison with that in conventional 3D spin ices, are mainly due to the metallic nature of the system, which allows strong RKKY interaction between local Ho moments mediated by conduction electrons. The coexisting metallicity and spin ice phenomena may lead to other exotic physics such as interaction between electric currents and effective magnetic monopoles that deserves future study.

[1] Kan Zhao*, Hao Deng*, Hua Chen*, Kate A. Ross, Vaclav Petříček, Gerrit Günther, Margarita Russina, Vladimir Hutanu, and Philipp Gegenwart, “Realization of the kagome spin ice state in a frustrated intermetallic compound”, Science 367, 1218-1223 (2020).