Two-dimensional Irreducible Representations Research Articles

Orbital-active Dirac materials from the symmetry principle

Dirac materials, starting with graphene, have drawn tremendous research interest in the past decade. Instead of focusing on the p_{z} orbital as in graphene, we move a step further and study orbital-active Dirac materials, where the orbital degrees of freedom transform as a two-dimensional irreducible representation of the lattice point group. Examples of orbital-active Dirac materials occur in a broad class of systems, including transition-metal-oxide heterostructures, transition-metal dichalcogenide monolayers, germanene, stanene, and optical lattices. Different systems are unified based on symmetry principles. The band structure of orbital-active Dirac materials features Dirac cones at K(K') and quadratic band touching points at Γ, regardless of the origin of the orbital degrees of freedom. In the strong anisotropy limit, i.e., when the π-bonding can be neglected, flat bands appear due to the destructive interference. These features make orbital-active Dirac materials an even wider playground for searching for exotic states of matter, such as the Dirac semi-metal, ferromagnetism, Wigner crystallization, quantum spin Hall state, and quantum anomalous Hall state.

Superconductivity from repulsive interactions on the kagome lattice

The discovery of superconductivity in layered vanadium-based kagome metals $A$V$_3$Sb$_5$ ($A$: K, Rb, Cs) has added a new family of materials to the growing class of possible unconventional superconductors. However, the nature of the superconducting pairing in these materials remains elusive. We present a microscopic theoretical study of the leading superconducting instabilities on the kagome lattice based on spin- and charge-fluctuation mediated Cooper pairing. The applied methodology includes effects of both on-site and nearest-neighbor repulsive Coulomb interactions. Near the upper van Hove filling -- relevant for the $A$V$_3$Sb$_5$ materials -- we find a rich phase diagram with several pairing symmetries being nearly degenerate. In particular, while a substantial fraction of the phase diagram is occupied by a spin-singlet order parameter transforming as a two-dimensional irreducible representation of the point group, several nodal spin-triplet pairing states remain competitive. We compute the band and interaction parameter-dependence of the hierarchy of the leading superconducting instabilities, and determine the detailed momentum dependence of the resulting preferred gap structures. Crucially, for moderate values of the interaction parameters, the individual pairing states depend strongly on momentum and exhibit multiple nodes on the Fermi surface. We discuss the properties of these superconducting gap structures in light of recent experimental developments of the $A$V$_3$Sb$_5$ materials.

Successive phase transitions of the spin–orbit-coupled metal Cd2Re2O7 probed by high-resolution synchrotron x-ray diffraction

The 5d pyrochlore oxide superconductor Cd2Re2O7 (CRO) has attracted significant interest as a spin–orbit-coupled metal (SOCM) that spontaneously undergoes a phase transition to an odd-parity multipole phase by breaking the spatial inversion symmetry due to the Fermi liquid instability caused by strong spin–orbit coupling. Despite the significance of structural information during the transition, previous experimental results regarding lattice deformation have been elusive. We have conducted ultra-high resolution synchrotron radiation x-ray diffraction experiments on a high-quality CRO single crystal. The temperature-dependent splitting of the 0 0 16 and 0 0 14 reflections, which are allowed and forbidden, respectively, in the high-temperature cubic phase I (space group Fd–3m), has been clearly observed and reveals the following significant facts: inversion symmetry breaking and tetragonal distortion occur simultaneously at T s1 = 201.5(1) K; the previously believed first-order transition between phase II (I–4m2) and phase III (I4122) at T s2 ∼120 K consists of two close second-order transitions at T s2 = 115.4(1) K and T s3 ∼ 100 K; there is a new orthorhombic phase XI (F222) in between. The order parameters (OPs) of these continuous transitions are uniquely represented by a two-dimensional irreducible representation Eu of the Oh point group, and the OPs of phase XI are a linear combination of those of phases II and III. Each phase is believed to correspond to a distinct odd-parity multipole order, and the complex successive transitions observed may be the result of an electronic phase transition that resolves the Fermi liquid instability in the SOCM.

Local constancy for reductions of two-dimensional crystalline representations

We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a weight $k$. We find an (explicit) local constancy result with respect to $a_p$ using Fontaine's theory of $(\varphi, \Gamma)$-modules and its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to $k$ (for $a_p\not=0$) will follow from a local study of Colmez's rigid analytic space parametrizing trianguline representations. This work extends some results of Berger obtained in the semi-simple residual case.

Double covering of the modular A5 group and lepton flavor mixing in the minimal seesaw model

In this paper, we investigate the double covering of modular $\Gamma^{}_5 \simeq A^{}_5$ group and derive all the modular forms of weight one for the first time. The modular forms of higher weights are also explicitly given by decomposing the direct products of weight-one forms. For the double covering group $\Gamma^\prime_5 \simeq A^\prime_5$, there exist two inequivalent two-dimensional irreducible representations, into which we can assign two right-handed neutrino singlets in the minimal seesaw model. Two concrete models with such a salient feature have been constructed to successfully explain lepton mass spectra and flavor mixing pattern. The allowed parameter space for these two minimal scenarios has been numerically explored, and analytically studied with some reasonable assumptions.

Tunable double Weyl phonons driven by chiral point group symmetry

Different from spin-$\frac{1}{2}$ Weyl points which are robust due to the protection of topology, the unconventional chiral quasiparticles usually require extra crystalline symmetries for their existence, indicating that such quasiparticles are sensitive to perturbation. Herein, we present that the spin-1 Weyl can transform into quadratic Weyl phonons depending on symmetry variation. Specifically, the spin-1 Weyl nodes arisen from three-dimensional (3D) irreducible representations (IRs) of chiral point groups, $O$(432) or $T$(23), are verified to split into quadratic Weyl points if symmetry breaking decomposes 3D IRs into two-dimensional IRs. Symmetry analysis and low-energy effective models are performed to identify the splitting mechanisms. The evolution of Berry curvature and surface states driven by symmetry breaking is obtained in real materials. Our work not only builds the connection between double Weyl phonons but also offers guidance for exploring the transition among unconventional quasiparticles.

Topological and nematic superconductivity mediated by ferro-SU(4) fluctuations in twisted bilayer graphene

We propose an SU(4) spin-valley-fermion model to investigate the superconducting instabilities of twisted bilayer graphene (TBG). In this approach, bosonic fluctuations associated with an emergent SU(4) symmetry, corresponding to combined rotations in valley and spin spaces, couple to the low-energy fermions that comprise the flat bands. These fluctuations are peaked at zero wave-vector, reflecting the "ferromagnetic-like" SU(4) ground state recently found in strong-coupling solutions of microscopic models for TBG. Focusing on electronic states related to symmetry-imposed points of the Fermi surface, dubbed here "valley hot-spots" and "van Hove hot-spots", we find that the coupling to the itinerant electrons partially lifts the huge degeneracy of the ferro-SU(4) ground state manifold, favoring inter-valley order, spin-valley coupled order, ferromagnetic order, spin-current order, and valley-polarized order, depending on details of the band structure. These fluctuations, in turn, promote attractive pairing interactions in a variety of closely competing channels, including a nodeless $f$-wave state, a nodal $i$-wave state, and topological $d+id$ and $p+ip$ states with unusual Chern numbers $2$ and $4$, respectively. Nematic superconductivity, although not realized as a primary instability of the system, still appears as a consequence of the near-degeneracy of superconducting order parameters that transform as one-dimensional and two-dimensional irreducible representations of the point group $D_{6}$.

A fundamental invariant-neural network representation of quasi-diabatic Hamiltonians for the two lowest states of H3.

The fundamental invariant neural network (FI-NN) approach is developed to represent coupled potential energy surfaces in quasidiabatic representations with two-dimensional irreducible representations of the complete nuclear permutation and inversion (CNPI) group. The particular symmetry properties of the diabatic potential energy matrix of H3 for the 1A' and 2A' electronic states were resolved arising from the E symmetry in the D3h point group. This FI-NN framework with symmetry adaption is used to construct a new quasidiabatic representation of H3, which reproduces accurately the ab initio energies and derivative information with perfect symmetry behaviors and extremely small fitting errors. The quantum dynamics results on the new FI-NN diabatic PESs give rise to accurate oscillation patterns in the product state-resolved differential cross sections. These results strongly support the accuracy and efficiency of the FI-NN approach to construct reliable diabatic representations with complicated symmetry problems.

The minimal seesaw model with a modular S4 symmetry

In this paper, we incorporate the modular S4 flavor symmetry into the supersymmetric version of the minimal type-I seesaw model, in which only two right-handed neutrino singlets are introduced to account for tiny Majorana neutrino masses, and explore its implications for the lepton mass spectra, flavor mixing and CP violation. The basic idea is to assign two right-handed neutrino singlets into the unique two-dimensional irreducible representation of the modular S4 symmetry group. Moreover, we show that the matter-antimatter asymmetry in our Universe can be successfully explained via the resonant leptogenesis mechanism working at a relatively-low seesaw scale ΛSS≈ 107 GeV, with which the potential problem of the gravitino overproduction can be avoided. In this connection, we emphasize that the observed matter-antimatter asymmetry may lead to a stringent constraint on the parameter space and testable predictions for low-energy observables.

Reality-based algebras with a two-dimensional representation

It is shown that a two-dimensional irreducible representation of a reality-based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti-automorphism on the distinguished basis. This generalizes known results for algebras of dimensions 5, 6, 7, and 8. Explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem.

Reinvestigation of quantum dot symmetry: the symmetric group of the 8-band k⋅p theory Hamiltonian

Self-assembled semiconductor quantum dots, usually formed in pyramid or lens shapes, have an intrinsic geometric symmetry. However, the geometric symmetry of a quantum dot is not identical to the symmetry of the associated Hamiltonian. It is a well-accepted conclusion that the symmetric group of the Hamiltonians for both pyramidal and lens-shaped quantum dots is C2v; consequently, the eigenstate of the Hamiltonian is not degenerate because C2v has only one-dimensional irreducible representations. In this paper, we show the above conclusion is wrong. Using the 8-band k · p theory model and considering the action of group elements on both spatial and electron spin parts of the wavefunction, we find the symmetric group of the Hamiltonian is the C2v double group not C2v. C2v is the symmetric group of the spatial part of the conduction band Hamiltonian only when the inter-band coupling is totally ignored. Employing the C2v double group symmetry, we prove that although the C2v double group has both one-dimensional and two-dimensional irreducible representations, the eigenstates of the 8 × 8 Hamiltonian are always two-fold degenerate and that these degenerate states only correspond to the two-dimensional irreducible representation of the C2v double group. The double group symmetry originates from the coupling between spatial potential and electron half spin. This coupling causes a full 2π rotation in the wavefunction space or the Hilbert space not equal to the unity operation. Finally, the connection between the two-fold degeneracy due to the C2v double group symmetry and Kramers’ degeneracy due to the time inversion symmetry is explored.

Group-theoretical analysis of two-dimensional hexagonal materials

Two-dimensional hexagonal materials such as graphene and silicene have highly symmetric crystal structures and Dirac cones at the K point, which induce novel electronic properties. In this report, we calculate their electronic structures by using density functional theory and analyze their band structures on the basis of the group theory. Dirac cones frequently appear when the symmetry at the K point is high; thus, two-dimensional irreducible representations are included. We discuss the relationship between symmetry and the appearance of the Dirac cone.

Exact solution of theD3non-Abelian anyon chain

Commuting transfer matrices for linear chains of interacting non-Abelian anyons from the two-dimensional irreducible representation of the dihedral group $D_3$ (or, equivalently, the integer sector of the $su(2)_4$ spin-$1$ chain) are constructed using the spin-anyon correspondence to a $D_3$-symmetric formulation of the XXZ Heisenberg spin chain. The spectral problem is solved using discrete inversion identities satisfied by these transfer matrices and functional Bethe ansatz methods. The resulting spectrum can be related to that of the XXZ spin-$1/2$ Heisenberg chain with boundary conditions depending on the topological sector of the anyon chain. The properties of this model in the critical regime are studied by finite size analysis of the spectrum. In particular, points in the phase diagram where the anyon chain realizes some of the rational $\mathbb{Z}_2$ orbifold theories are identified.

Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3, ℤ)

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

Existence of Dirac cones in the Brillouin zone of diperiodic atomic crystals according to group theory

We have considered non-magnetic materials with weak spin–orbit coupling, that are periodic in two non-collinear directions, and finite in the third, orthogonal direction. In some cases, the combined time-reversal and crystal symmetry of such systems, allows the existence of Dirac cones at certain points in the reciprocal space. We have investigated in a systematic way, all points of the Brillouin zone of all 80 diperiodic groups and have found sufficient conditions for the existence of s = 1/2 Dirac fermions, with symmetry-provided band touching at the vertex of the Dirac cones. Conversely, complete linear dispersion is forbidden for orbital wave functions belonging to two-dimensional (2D) irreducible representations (irreps) of little groups that do not satisfy certain group theoretical conditions given in this paper. Our results are illustrated by a tight-binding example.

On the reduction modulo p of representations of a quaternion division algebra over a p-adic field

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands correspondence and the local Jacquet–Langlands correspondence are not compatible with the reduction, except in the cases considered by Vignéras.

Full propagation-vector star antiferromagnetic order in quantum spin trimer system Ca3CuNi2(PO4)4

We show that the antiferromagnetic structure in the quantum spin trimer system Ca3CuNi2(PO4)4 is based on both arms of propagation vector k star of the paramagnetic space group C2/c. The structure is generated by a symmetric direction of the order parameter of two-dimensional irreducible representation of C2/c with one active magnetic mode and corresponds to the Shubnikov magnetic space group Ca2/c. We reveal the relation between representation analysis in the propagation vector formalism and Shubnikov symmetry. These types of multi-k structures are extremely rarely observed experimentally. To further prove the specific magnetic structure we have performed the calculations of the spin expectation values in the isolated Ni2+–Cu2+–Ni2+ trimer with realistic Hamiltonian. The calculated spin values 〈SNi〉 = 0.9 and 〈SCu〉 = 0.3 are within 10% accuracy in agreement with the experiment, providing a strong complementary argument in favor of a multi-arm magnetic structure.

Symmetries and fuzzy symmetries of Carbon nanotubes

Carbon nanotubes (CNTs) possess the fuzzy cylinder group characteristic. Comparing with the linear and planer molecules, there are included the fuzzy symmetry of the cylinder screw rotation (CSR) in relation to some higher ( $$>$$ 2) fold rotation axis. The CSR may be noted as the product of translation (T) and rotation (C). The CSR symmetry will be imperfect owing to the introduction of T. As the extent of whole translation is more than 10-fold than every time, the membership function of CNT in relation to CSR will be more than about 0.9, and such CNT may be seems as provided with the perfect CSR symmetry. For analyse the CNT we may using the cylindrical orthogonal curvilinear coordinate system. The MO ought to be provided with a pure irreducible representation, but the component of symmetry adapted atomic orbital (SA-AO) set may be not sole, and it is difficult to get and analyse the ‘pure’ $$\uppi $$ -MO. There are some various AO (1S-, 2S-, 2Pz-, 2Pr-, 2Pt of carbon and 1S- for hydrogen)-set components in a certain MO. For the CNT with the same diameter and different length, the MO energy and the SA-AO component versus the relative serial number will be with the similar distribution. The MOs of CNT with higher fold C symmetry may be provided with two-dimensional irreducible representation. For the molecular skeleton and the MO which belong to one-dimensional irreduable representation, their membership functions in relation to the CSR with the product of the same T and different C would be equality. However, for the single MO which belong to two-dimensional irreducible representation that may be somewhat difference. The torus carbon nanotube (TCNT) may be provided the symmetry with the torus group and torus screw rotation (TSR), such symmetry would be or near be not rare in nature. Similar as the planer rectangle (called as the MH rectangle) may composed the Huckel- or Mobius-strip band, the more MH rectangles in the cylinder CNT may be composed the more Huckel- or Mobius-strip bands, such strip bands set may be called strip tube, meanwhile the fuzzy CSR symmetry will be transform to the perfect TSR symmetry. The intersecting line (Z-axis) of the MH rectangles will be transform to the common basic circle of these strip bands. When the CNT to form a TCNT, as one of the MH rectangle form a Huckel-strip band or an $$n(t)$$ -twisted Mobius-strip band itself, the other MH rectangle will be form the strip band with the same topological structure synchronously, and the set of these strip bands may be called the strip tube. The boundary closed curves of the strip band may reflect the torus group symmetrical characteristic of the relative strip bands. The closed curve may correspond to a cyclical group or subgroup. The number of carbon atomic pairs on the closed curve denoted the order of such group or subgroup. As the CNT to form the TCNT, it is different as the single MH rectangle, they may be to form the fractal-twisted Mobius-strip tube synchronously, in which the single Mobius-strip band may be formed from more one MH rectangle, however, single MH rectangle may enter into only one Mobius-strip band. As the hetero-CNT with the helical-structure distribution, such hetero-CNT may form the relative torus hetero-CNT, but according to the continuity of CNT tube side, a certain twisted to form Mobius-strip tube may often be required. There is some interaction between the distributional helical-structure and twisting way, such interaction may touch to the degree of tightness of the helical-structure distribution in torus hetero-CNT.

Reducible Galois Representations and the Homology of GL(3, ℤ)

Let Fp be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, Fp) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup Γ of GL(3,Z). In addition, we prove that the coefficient module needed is, in fact, predicted by the main conjecture of [3].

Raman scattering study of delafossite magnetoelectric multiferroic compounds: CuFeO2 and CuCrO2

Ultrasonic velocity measurements on the magnetoelectric multiferroic compound CuFeO2 reveal that the antiferromagnetic transition observed at TN1 = 14 K might be induced by an pseudoproper ferroelastic transition []. In that case, the group theory states that the order parameter associated with the structural transition must belong to a two-dimensional irreducible representation Eg (x2 − y2, xy). Since this type of transition can be driven by a Raman Eg mode, we performed Raman scattering measurements on CuFeO2 between 5 and 290 K. Considering that the isostructural multiferroic compound CuCrO2 might show similar structural deformations at the antiferromagnetic transition TN1 = 24.3 K, Raman measurements have also been performed for comparison. At ambient temperature, the Raman modes in CuFeO2 are observed at ωEg = 352 cm−1 and ωA1g = 692 cm−1, while these modes are detected at ωEg = 457 cm−1 and ωA1g = 709 cm−1 in CuCrO2. The analysis of the temperature dependence of the modes in both compounds shows that the frequencies of all modes increase with decreasing temperature. This typical behavior is attributed to anharmonic phonon–phonon interactions. These results clearly indicate that none of the Raman active modes observed in CuFeO2 and CuCrO2 drive the pseudoproper ferroelastic transitions observed at the Néel temperature TN1. Finally, a broad band at about 550 cm−1 observed in the magnetoelectric phase of CuCrO2 below TN2 could be associated with magnons.