One-dimensional Irreducible Representations Research Articles

Topological Structure of the Order Parameter of Unconventional Superconductors Based on d- and f- Elements

The superconducting order parameter (SOP) of a triplet superconductor UTe2 was constructed using the topological space group approach, in which, in contrast to phenomenological and topological approaches, the single pair function and phase winding in condensate are different quantities. The connection between them is investigated for the D2h point group and the m′m′m magnetic group. It is shown how a non-unitary pair function of UTe2 can be constructed using one-dimensional real irreducible representations and Ginzburg–Landau phase winding. It is also shown that the total phase winding is non-zero in magnetic symmetry only. Experimental data on the superconducting order parameter of topological superconductors UPt3, Sr2RuO4, LaPt3P, and UTe2 are considered and peculiarities of their nodal structures are connected with the theoretical results of the topological space group approach.

Leading superconducting instabilities in three-dimensional models for Sr2RuO4

The superconductor Sr2RuO4 has been the subject of enormous interest over more than two decades, but until now the form of its order parameter has not been determined. Since groundbreaking NMR experiments revealed that the pairs are of dominant spin-singlet character, attention has focused on time-reversal symmetry breaking linear combinations of $s$-, $d$- and $g$-wave one-dimensional (1D) irreducible representations. However, a state of the form $d_{xz}+id_{yz}$ has also been proposed. We present a systematic study of the stability of various superconducting candidate states, assuming that pairing is driven by the fluctuation exchange mechanism, including a realistic three-dimensional Fermi surface, full treatment of both local and non-local spin-orbit couplings, and a wide range of interaction parameters $U,J,U',J'$. The leading superconducting instabilities are found to exhibit nodal even-parity $A_{1g} (s')$ or $B_{1g} (d_{x^2-y^2})$ symmetries, similar to the findings in two-dimensional models without longer-range Coulomb interaction which tends to favor $d_{xy}$ over $d_{x^2-y^2}$. Within the so-called Hund's coupling mean-field pairing scenario, the $E_g (d_{xz}/d_{yz})$ solution can be stabilized for large $J$ and specific forms of the spin-orbit coupling, but for all cases studied here the eigenvalues of other superconducting solutions are significantly larger when the full fluctuation exchange vertex is included in the pairing kernel. Additionally, we compute the spin susceptibility in relevant superconducting candidate phases and compare to recent neutron scattering and NMR Knight shift measurements.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

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 $D_{6}$.

Lepton flavor mixing and CP violation in the minimal type-(I+II) seesaw model with a modular A4 symmetry

In this paper, we study the implications of the modular A4 flavor symmetry in constructing a supersymmetric minimal type-(I+II) seesaw model, in which only one right-handed neutrino and two Higgs triplets are introduced to account for the tiny neutrino masses, flavor mixing and CP violation. We consider the most economical case where the right-handed neutrino and the Higgs triplets in this model are assigned into the trivial one-dimensional irreducible representation of the modular group A4, and all the modular forms are with the lowest weights they can take. We find that the octant of the mixing angle θ23 strongly depends on the hierarchy of free model parameters in the charged-lepton sector. We also show that the neutrino mass matrix can possess an approximate μ−τ reflection symmetry for some specific values of free parameters. Moreover, our model predicts relatively large masses of three light neutrinos, thus can be easily tested in future neutrino experiments.

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.

Permutation invariant polynomial neural network approach to fitting potential energy surfaces. IV. Coupled diabatic potential energy matrices.

A machine learning method is proposed for representing the elements of diabatic potential energy matrices (PEMs) with high fidelity. This is an extension of the so-called permutation invariant polynomial-neural network (PIP-NN) method for representing adiabatic potential energy surfaces. While for one-dimensional irreducible representations the diagonal elements of a diabatic PEM are invariant under exchange of identical nuclei in a molecular system, the off-diagonal elements require special symmetry consideration, particularly in the presence of a conical intersection. A multiplicative factor is introduced to take into consideration the particular symmetry properties while maintaining the PIP-NN framework. We demonstrate here that the extended PIP-NN approach is accurate in representing diabatic PEMs, as evidenced by small fitting errors and by the reproduction of absorption spectra and product branching ratios in both H2O( ) and NH3( ) non-adiabatic photodissociation.

ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

Theoretical prediction of fragile Mott insulators on plaquette Hubbard lattices

Employing extensive cellular dynamical mean-field theory (CDMFT) calculations with exact diagonalization impurity solver, we investigate the ground state phase diagrams and non-magnetic metal-insulator transitions of the half-filled Hubbard model on two plaquette -- the 1/5 depleted and checkerboard -- square lattices. We identify three different insulators in the phase diagrams: dimer insulator, antiferromagnetic insulator, and plaquette insulator. And we demonstrate that the plaquette insulator is a novel fragile Mott insulator (FMI) which features a nontrivial one-dimensional irreducible representation of the $C_{4v}$ crystalline point-group and cannot be adiabatically connected to any band insulator with time-reversal symmetry. Furthermore, we study the non-magnetic quantum phase transitions from the metal to the FMI and find that this Mott metal-insulator transition is characterized by the splitting of the non-interacting bands due to interaction effects.

Group theory analysis of the rhombohedral magnetic structures in rare earth iron garnets revisited

The representation analysis of Bertaut is used to enumerate all possible magnetic structures in rare earth iron garnets (RE3Fe5O12) for the RE and Fe ions on the 6e, 6e' and 12f, 2b, 6d Wyckoff positions respectively in the highest rhombohedral subgroup R3̄c of the cubic space group Ia3̄d. The basis vectors of the one-dimensional irreducible representation A2g lead to noncollinear magnetic structures of the RE moments which produce a better description than those based on the models of Wolf et al. Some results of the "double umbrella" structure found in our recent neutron powder diffraction studies of terbium iron garnet are reported.

Molecular Symmetry Properties of Conical Intersections and Nonadiabatic Coupling Terms: Theory and Quantum Chemical Demonstration for Cyclopenta-2,4-dienimine (C5H4NH)

This paper discovers molecular symmetry (MS) properties of conical intersections (CIs) and the related nonadiabatic coupling terms (NACTs) in molecules which allow large amplitude motions such as torsion, in the frame of the relevant molecular symmetry group, focusing on groups with one-dimensional (1-d) irreducible representations (IREPs). If one employs corresponding MS-adapted nuclear coordinates, the NACTs can be classified according to those IREPs. The assignment is supported by theorems which relate the IREPs of different NACTs to each other, and by properties of the NACTs related to the CIs. For example, planar contour integrals of the NACTs evaluated along loops around the individual CIs are equal to +pi or -pi, depending on the IREP-adapted signs of the NACTs. The + or - signs for the contour integrals may also be used to define the "charges" and IREPs of the CIs. We derive various general molecular symmetry properties of the related NACTs and CIs. These provide useful applications; e.g., the discovery of an individual CI allows one to generate, by means of all molecular symmetry operations, the complete set of CIs at different symmetry-related locations. Also, we show that the seams of CIs with different IREPs may have different topologies in a specific plane of MS-adapted coordinates. Moreover, the IREPs impose symmetrical nodes of the NACTs, and this may support their calculations by quantum chemical ab initio methods, even far away from the CIs. The general approach is demonstrated by application to an example. Specifically, we investigate the CIs and NACTs of cyclopenta-2,4-dienimine (C(5)H(4)NH) which has C(2v)(M) molecular symmetry with 1-d IREPs. The results are confirmed by quantum chemical calculations, starting from the location of a CI based on the Longuet-Higgins phase change theorem, until a proof of self-consistency, i.e., the related symmetry-adapted NACTs fulfill quantization rules which have been derived in [Baer, M. Beyond Born-Oppenheimer: Electronic non-Adiabatic Coupling Terms and Conical Intersections; Wiley & Sons Inc.: Hoboken, NJ, 2006].

Modular forms for the even modular lattice of signature (2,10)

We consider the principal congruence subgroup Γ [ 2 ] \Gamma [2] of level two inside the orthogonal group of an even and unimodular lattice of signature ( 2 , 10 ) (2,10) . Using Borcherds’ additive lifting construction, we construct a 715 715 -dimensional space of singular modular forms. This space is the direct sum of the one-dimensional trivial and a 714 714 -dimensional irreducible representation of the finite group O ⁡ ( F 2 12 ) \operatorname {O}(\mathbb {F}^{12}_2) (even type). It generates an algebra whose normalization is the ring of all modular forms. We define a certain ideal of quadratic relations. This system appears as a special member of a whole system of ideals of quadratic relations. At least some of them have geometric meaning. For example, we work out the relation to Kondo’s approach to the modular variety of Enriques surfaces.

Spectra of Conic Carbon Radicals

Conic carbon radicals with non-trivial automorphism groups are topologically possible when the curvature originates from 1, 3 or 5 pentagons in the otherwise hexagonal graphene sheet. By splitting determinants of quotient graphs, we determine the bonding nature of the last occupied and first unoccupied Huckel orbitals of the radicals constituting the three infinite series with the most plausible topologies. Each member of the series with three pentagons at the tip has one electron in the first anti-bonding orbital, each member of the series with one or five pentagons at the tip has one vacancy in the last bonding orbital, and none of the radicals have any un-bonding orbital. Within the limits of the Huckel model, this implies, respectively, stable conic cations and anions. The quotient graphs also give the collected Huckel energy for each of the one-dimensional irreducible representations of the point groups.

The Fuzzy D2h-symmetries of Ethylene Tetra-halide Molecules and their Molecular Orbitals

Based on our study in relation to the fuzzy symmetry characterization and the application to linear molecule, the fuzzy symmetry of the planar molecules have been analyzed. The prototypical planer molecules we have chosen to study are the C2F3X (X = Cl, Br, and I) and three kinds of C2F2Cl2 isomers. These molecules relate to the fuzzy symmetry in connection with the D2h point group. As we known, the D2h point group includes an identity transformation and seven twofold symmetry transformations but without higher-fold ones. Meanwhile, it is related only to some one-dimensional irreducible representations, but there is not to multi-dimensional irreducible representation. In this paper, the fuzzy symmetries of these molecules and their molecular orbital(MO)s have been studied, such as the membership functions, the representation compositions, the fuzzy correlation diagrams and so on have been analyzed. These analysis methods can be used to analyze the molecular fuzzy symmetries of some other molecule systems, no difficulty.

Opechowski's magic relations

AbstractTransformation properties of tensors under the action of the full magnetic groupSO(3) × Eoare specified by the intrinsic symmetry which defines how the tensor transforms under the action of the proper rotation groupSO(3) and by their parity which defines how the tensor transforms under the action of the group of inversionsEo= {e,i,e′,i′}, whereidenotes the space inversion,e′ the magnetic inversion andi′ =i.e′ their combination. There exist therefore four types of tensors to each intrinsic symmetry whose transformation properties under the groupSO(3) are identical while their transformation under the groupEodetermines their parities with respect to the three inversions. As a result of this separation, transformation properties of tensors of the same intrinsic symmetry under the action of magnetic groups of the same oriented Laue class are strongly correlated by relations for which we propose the name Opechowski's magic relations in homage to late Prof. Opechowski. The origin of these relations is explained and tables which describe them are presented. It is found, in particular, that the number of different forms of tensor decompositions under the action of groups of oriented Laue class of magnetic point groups equals the number of one-dimensional real irreducible representations of the proper rotation group which generates the Laue class. Accordingly, there exists also the same number of allowed forms of tensors which do not vanish.

Néel probability and spin correlations in some nonmagnetic and nondegenerate states of the hexanuclear antiferromagnetic ringFe6:Application of algebraic combinatorics to finite Heisenberg spin systems

The spin correlations ${\ensuremath{\omega}}_{r}^{z},$ $r=1,2,3,$ and the probability ${p}_{\mathrm{N}}$ of finding a system in the N\'eel state for the antiferromagnetic ring ${\mathrm{Fe}}_{6}^{\mathrm{III}}$ (the so-called ``small ferric wheel'') are calculated. States with magnetization $M=0$ and total spin $0<~S<~15,$ labeled by two (out of four) one-dimensional irreducible representations (irreps) of the point symmetry group ${D}_{6},$ are taken into account. This choice follows from importance of these irreps in analyzing low-lying states in each S multiplet. Taking into account the Clebsch-Gordan coefficients for coupling total spins of sublattices ${(S}_{A}{=S}_{B}=\frac{15}{2})$ the global N\'eel probability ${p}_{\mathrm{N}}^{*}$ can be determined. Dependences of these quantities on state energy (per bond and in the units of exchange integral $J)$ and the total spin S are analyzed. Providing we have determined ${p}_{\mathrm{N}}(S),$ etc., for other antiferromagnetic rings $({\mathrm{Fe}}_{10},$ for instance) we could try to approximate results for the largest synthesized ferric wheel ${\mathrm{Fe}}_{18}.$ Since thermodynamic properties of ${\mathrm{Fe}}_{6}$ have been investigated recently, in the present considerations they are not discussed, but only used to verify obtained values of eigenenergies. Numerical results are calculated with high precision using two main tools: (i) thorough analysis of symmetry properties including methods of algebraic combinatorics and (ii) multiple precision arithmetic library GMP. The system considered yields more than 45 000 basic states (the so-called Ising configurations), but application of the method proposed reduces this problem to 20-dimensional eigenproblem for the ground state $(S=0).$ The largest eigenproblem has to be solved for $S=4;$ its dimension is 60. These two facts (high precision and small resultant eigenproblems) confirm the efficiency and usefulness of such an approach, so it is briefly discussed here.

Symmetry constraints and variational principles in diffusion quantum Monte Carlo calculations of excited-state energies

Fixed-node diffusion Monte Carlo (DMC) is a stochastic algorithm for finding the lowest energy many-fermion wave function with the same nodal surface as a chosen trial function. It has proved itself among the most accurate methods available for calculating many-electron ground states, and is one of the few approaches that can be applied to systems large enough to act as realistic models of solids. In attempts to use fixed-node DMC for excited-state calculations, it has often been assumed that the DMC energy must be greater than or equal to the energy of the lowest exact eigenfunction with the same symmetry as the trial function. We show that this assumption is not justified unless the trial function transforms according to a one-dimensional irreducible representation of the symmetry group of the Hamiltonian. If the trial function transforms according to a multidimensional irreducible representation, corresponding to a degenerate energy level, the DMC energy may lie below the energy of the lowest eigenstate of that symmetry. Weaker variational bounds may then be obtained by choosing trial functions transforming according to one-dimensional irreducible representations of subgroups of the full symmetry group.

Superconducting impurity terms in the Ginzburg-Landau equations and supercurrent:A microscopic theory

Finding the correct way to include small defects in the Ginzburg-Landau (GL) theory of superconductivity requires a microscopic analysis. In the presence of a single impurity, we compute terms which must be added to the GL free energy, supercurrent, and the GL differential equation for the order parameter. Our calculation is very general, covering any {bold {cflx k}}-dependent order parameter {Delta}({bold {cflx k}}) which transforms according to a one-dimensional irreducible representation of the crystalline point group. {copyright} {ital 1997} {ital The American Physical Society}

Microscopic theory of vortex pinning: Impurity terms in the Ginzburg-Landau free energy.

In the presence of a single impurity, the Ginzburg-Landau free-energy functional for a superconductor acquires extra terms. Using microscopic theory, we determine the structure of these terms and their coefficients. Our calculation is very general: we assume a k\ifmmode \hat{}\else \^{}\fi{}-dependent order parameter $\ensuremath{\Delta}(\stackrel{^}{\mathrm{k}})$, which transforms according to any one-dimensional irreducible representation of the crystalline point group. This representation may be conventional or unconventional, as appropriate to the current models of high-${T}_{c}$ superconductors. We treat an arbitrary Fermi surface. The physical significance of the theory is discussed, with emphasis on vortex pinning. We estimate the pinning energy of a single vortex.

Ginzburg-landau impurity terms for unconventional superconductors

the Ginzburg-Landau theory of superconductivity provides expressions for the free energy and the supercurrent in terms of a spatially varying order parameter. In the presence of a single impurity, these expressions and the differential equation for the order parameter are supplemented by additional terms. We compute these extra terms using microscopic theory. Our calculation is very general, covering any k-dependent order parameter Δ(k) which transforms according to a one-dimensional irreducible representation of the crystalline point group.