Jorn Venderbos

The distinctive symmetry properties of pure altermagnets make them natural candidates for piezomagnetism. Previous work motivated by the piezomagnetic properties of altermagnets has primarily focused on the spin magnetization response to applied strain. In this paper we study orbital piezomagnetic effects-the orbital magnetization response to applied strain-in minimal lattice models of pure insulating altermagnets in two dimensions. We obtain general microscopic expressions for the orbital magnetization in the presence of strain, as well as the orbital piezomagnetic polarizability, i.e., the defining response characteristic of pure altermagnets. We apply these expressions to three specific tetragonal lattice models, two corresponding to d-wave altermagnets and one describing a g-wave altermagnet. Whereas the d-wave altermagnets are associated with a linear piezomagnetic polarizability, the g-wave altermagnet exhibits a nonlinear piezomagnetic effect. Our analysis reveals how the polarizabilities are related to and determined by the Berry curvature of the occupied bands. Connections to materials of current interest are discussed.

In this work, we present a theory to address conflicting experimental claims regarding the charge density wave (CDW) state in TiSe2, including whether there is single or multiple CDW transitions and whether threefold rotation symmetry (𝐶3) is broken. Using a continuum 𝒌·𝒑 model coupled to the CDW order parameter, we show how commonplace conduction band doping induces a nematic transition from a 𝐶3-symmetric 3⁢𝑄 CDW to a 𝐶3-breaking 3⁢𝑄 CDW, which is favored by the large ellipticity of the conduction bands of TiSe2. We also find that a 1⁢𝑄 stripe CDW is generically stabilized for sufficiently high electron doping. We then show how both stripe and nematic CDW states emerge self-consistently from a minimal interacting tight-binding model for both positive and negative initial gaps. Our theory provides a scenario in which, as temperature is lowered, a second 𝐶3-breaking transition may occur or not, depending on the doping level, potentially explaining the experimental variability. These predictions can be further verified with a variety of probes including transport, photoemission, and tunneling.

Magnetic states with zero magnetization but non-relativistic spin splitting are outstanding candidates for the next generation of spintronic devices. Their electronvolt (eV)-scale spin splitting, ultrafast spin dynamics and nearly vanishing stray fields make them particularly promising for several applications1,2. A variety of such magnetic states with non-trivial spin textures have been identified recently, including even-parity d-wave, g-wave or i-wave altermagnets and odd-parity p-wave magnets3, 4, 5, 6-7. Achieving voltage-based control of the non-uniform spin polarization of these magnetic states is of great interest for realizing energy-efficient and compact devices for information storage and processing8,9. Spin-spiral type II multiferroics are optimal candidates for such voltage-based control, as they exhibit an inversion-symmetry-breaking magnetic order that directly induces ferroelectric polarization, allowing for symmetry-protected cross-control between spin chirality and polar order10, 11, 12, 13-14. Here we combine photocurrent measurements, first-principles calculations and group-theory analysis to provide direct evidence that the spin polarization of the spin-spiral type II multiferroic NiI2 exhibits odd-parity character connected to the spiral chirality. The symmetry-protected coupling between chirality and polar order enables electrical control of a primarily non-relativistic spin polarization. Our findings represent an observation of p-wave magnetism in a spin-spiral type II multiferroic, which may lead to the development of voltage-based switching of non-relativistic spin polarization in compensated magnets.

1T-TaS2 is a non-magnetic Mott insulating transition-metal dichalcogenide with an odd number of electrons per unit cell, making it a potential spin-liquid candidate. This behavior arises from miniband reconstructions in the charge density wave state, producing a nearly flat band at half-filling. We revisit its electronic band structure using a nearest-neighbor tight-binding model, emphasizing the importance of often-neglected “spin-flip” terms in the spin-orbit coupling. By comparing with density functional theory calculations, we estimate the strength of these couplings. We also apply our theory to 1T-TaSe2, which is found to be a promising candidate for a topologically non-trivial flat band. Our findings have significant implications for correlated physics in the flat band, including the emergent spin-spin Hamiltonian at half-filling, identified as a J-K-Γ-
model on a triangular lattice, and for tuning electronic properties away from half-filling.

The electronic spectra of altermagnets are a fertile ground for nontrivial topology due to the unique interplay between time-reversal and crystalline symmetries. This is reflected in the unconventional Zeeman splitting between bands of opposite spins, which emerges in the absence of spin-orbit coupling (SOC) and displays nodes along high-symmetry directions. Here, we argue that even for a small SOC, the direction of the magnetic moments in the altermagnetic state has a profound impact on the electronic spectrum, enabling novel topological phenomena to appear. By investigating microscopic models for two-dimensional (2D) and three-dimensional (3D) altermagnets, motivated in part by rutile materials, we demonstrate the emergence of hitherto unexplored Dirac crossings between bands of the same spin but opposite sublattices. The direction of the moments determines the fate of these crossings when the SOC is turned on. We focus on the case of out-of-plane moments, which forbid an anomalous Hall effect and thus ensure that no weak magnetization is triggered in the altermagnetic state. In the 2D model, the SOC gaps out the Dirac crossings, resulting in mirror Chern bands that enable the Quantum Spin Hall Effect and undergo a topological transition to trivial bands upon increasing the magnitude of the magnetic moment. On the other hand, in the 3D model the crossings persist even in the presence of SOC, forming Weyl nodal loops protected by mirror symmetry. Finally, we discuss possible ways to control these effects in altermagnetic material candidates.

Magnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time -reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold ( Z 3 ) crystal rotation symmetry. We show that the transition can occur in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetically disordered. Since the nematic transition is described by the three -state Potts model, the intermediate phase is a Z 3 Potts-nematic phase. Our prediction of the existence of a Potts-nematic phase is based on an analysis of bound states formed from two-magnon excitations in the paramagnet, which become gapless while single-magnon excitations remain gapped. By considering three different lattice models, we demonstrate a generic instability towards two-magnon bound state formation in the Potts-nematic channel. We present both numerical results and a general analytical perturbative formula for the bound state binding energy similar to BCS theory. We further discuss a number of different materials that realize key features of the model considered, and thus provide promising venues for possible experimental observation.

Tin adatoms on a Si(111) substrate with a one-third monolayer coverage form a two-dimensional triangular lattice with one unpaired electron per site. These electrons order into an antiferromagnetic Mott-insulating state, but doping the Sn layer with holes creates a two-dimensional conductor that becomes superconducting at low temperatures. Although the pairing symmetry of the superconducting state is currently unknown, the combination of repulsive interactions and frustration inherent in the triangular adatom lattice opens up the possibility of a chiral order parameter. Here we study the superconducting state of Sn/Si(111) using scanning tunnelling microscopy, scanning tunnelling spectroscopy and quasiparticle interference imaging. We find evidence for a doping-dependent superconducting critical temperature with a fully gapped order parameter, the presence of time-reversal symmetry breaking and a strong enhancement in zero-bias conductance near the edges of the superconducting domains. Although each individual piece of evidence could have a more mundane interpretation, our combined results suggest the possibility that Sn/Si(111) is an unconventional chiral
d
-wave superconductor.
Adatoms on the surface of silicon can create two-dimensional superconductivity, the order parameter symmetry of which is currently not known. Now, evidence suggests it might be a topological chiral
d
-wave state.

Graphene moire superlattices display electronic flat bands. At integer fillings of these flat bands, energy gaps due to strong electron-electron interactions are generally observed. However, the presence of other correlation-driven phases in twisted graphitic systems at non-integer fillings is unclear. Here, we report the existence of three-fold rotational (C-3) symmetry breaking in twisted double bilayer graphene. Using spectroscopic imaging over large and uniform areas to characterize the direction and degree of C-3 symmetry breaking, we find it to be prominent only at energies corresponding to the flat bands and nearly absent in the remote bands. We demonstrate that the magnitude of the rotational symmetry breaking does not depend on the degree of the heterostrain or the displacement field, being instead a manifestation of an interaction-driven electronic nematic phase. We show that the nematic phase is a primary order that arises from the normal metal state over a wide range of doping away from charge neutrality. Our modelling suggests that the nematic instability is not associated with the local scale of the graphene lattice, but is an emergent phenomenon at the scale of the moire lattice.
Observations of an electronic nematic phase in twisted double bilayer graphene expand the number of moire materials where this interaction-driven state exists.

Graphene-based moire systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moire unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moire superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moire scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moire degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moire systems.

We report the optical conductivity in high-quality crystals of the chiral topological semimetal CoSi, which hosts exotic quasiparticles known as multifold fermions. We find that the optical response is separated into several distinct regions as a function of frequency, each dominated by different types of quasiparticles. The low-frequency intraband response is captured by a narrow Drude peak from a high-mobility electron pocket of double Weyl quasiparticles, and the temperature dependence of the spectral weight is consistent with its Fermi velocity. By subtracting the low-frequency sharp Drude and phonon peaks at low temperatures, we reveal two intermediate quasilinear interband contributions separated by a kink at 0.2 eV. Using Wannier tight-binding models based on first-principle calculations, we link the optical conductivity above and below 0.2 eV to interband transitions near the double Weyl fermion and a threefold fermion, respectively. We analyze and determine the chemical potential relative to the energy of the threefold fermion, revealing the importance of transitions between a linearly dispersing band and a flat band. More strikingly, below 0.1 eV our data are best explained if spin-orbit coupling is included, suggesting that at these energies, the optical response is governed by transitions between a previously unobserved fourfold spin-3/2 node and a Weyl node. Our comprehensive combined experimental and theoretical study provides a way to resolve different types of multifold fermions in CoSi at different energy. More broadly, our results provide the necessary basis to interpret the burgeoning set of optical and transport experiments in chiral topological semimetals.