Spontaneous Multipole Ordering by Local Parity Mixing

Abstract

Broken spatial inversion symmetry in spin-orbital coupled systems leads to a mixing between orbitals with different parity, which results in unusual electronic structures and transport properties. We theoretically investigate the possibility of multipole ordering induced by a parity mixing. In particular, we focus on the system in which the parity mixing appears in a sublattice-dependent form. Starting from the periodic Anderson model with such a local parity mixing, we derive an extended Kondo lattice model with sublattice-dependent antisymmetric exchange couplings between itinerant electrons and localized spins. By the variational calculation, simulated annealing, and Monte Carlo simulation, we show that the model on a quasi-one-dimensional zig-zag lattice exhibits an odd-parity multipole order composed of magnetic toroidal and quadrupole components at and near half filling. The multipole order causes a band deformation with the band bottom shift and a magnetoelectric response. The results suggest that unusual odd-parity multipole orders will be widely observed in multi-orbital systems with local parity mixing.

Emergent spin-valley-orbital physics by spontaneous parity breaking

“…This leads to the different type of the asymmetric band deformation in the class #5, as demonstrated in figures 9(b) and 9(c): the bands are modulated with a band bottom shift, with retaining the spin degeneracy as f E 1 (k)σ z τ z is invariant under PT . The band deformation is similar to the ferroic toroidal ordered cases discussed in [18,16,24]. In the case of the zy-SOO phase in the same class #5, a similar band deformation takes place with the band bottom shift to the k x direction because of the induced ASOC, g u zz (k) ∝ k x .…”

“…Such odd-parity multipoles bring about a peculiar modulation of the electronic structures and unconventional off-diagonal responses [26,27,28]. For example, a staggered antiferromagnetic (AFM) order on the zigzag chain, which accompanies a ferroic toroidal order (see section 2), modifies the electronic band structure in an asymmetric way in the momentum space, and results in unusual off-diagonal responses including the magnetoelectric effect [18,16,24] and asymmetric modulation of collective spin-wave excitations [29]. This indicates that the odd-parity multipoles open the further possibility of enriching the spin-orbital entangled phenomena.…”

Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization

“…The preference of the in-plane magnetic states rather than the out-of-plane magnetic states is presumably understood from the effective inter-orbital Rashba-type ASOI, which consists of the odd-parity CEF V intra and the spin-dependent hopping V inter . 35) This tendency to stabilize the in-plane magnetic states is different from the intra-orbital Rashba-type ASOI neglected in the present study, which favors the out-of-plane magnetic anisotropy. In fact, we confirmed that the out-ofplane magnetic state is stabilized by introducing the intraorbital Rashba-type ASOI.…”

Electric Octupole Order in Bilayer Rashba System

“…In this way, the sublattice-dependent ASOC gives rise to unconventional properties in the odd-parity multipole state. [5][6][7][8][9] Note that the spin degeneracy is preserved along lines |k x | = |k y |, as indicated by the red dashed lines in Fig. 1(c).…”

“…In the same way, various odd-parity multipoles may be constructed by staggered evenparity multipoles in locally noncentrosymmetric systems. [5][6][7][8] Among the various mechanisms of multipole order, we consider the forward scattering that leads to the spontaneous deformation of the Fermi surface. [16][17][18][19] This situation is relevant to the two-dimensional (2D) Hubbard model when the Fermi surface is in the vicinity of the van Hove singularity; renormalization group theories have shown the d-wave Pomeranchuk instability (dPI).…”