Lattice Harmonics Research Articles

Competing instabilities of the extended Hubbard model on the triangular lattice: Truncated-unity functional renormalization group and application to moiré materials

A simple yet paradigmatic model for the interplay of strong electronic correlations and geometric frustration is the triangular lattice Hubbard model. Recently it was proposed that moir\'e structures of transition metal dichalcogenides can be used to simulate extended versions that include non-local density-density interactions. We study competing instabilities of interacting electrons in such an extended Hubbard model on the triangular lattice near a filling where the density of states has a Van Hove singularity. We employ a truncated-unity functional renormalization group approach to investigate two cases: a paradigmatic minimally extended Hubbard model and a specific model with parameters that are applicable to hetero-bilayers of transition metal dichalcogenides. We unravel rich phase diagrams, including tendencies to spin-density-wave order and unconventional pairing, which can give rise to topological superconductivity. We classify the symmetry of the superconducting instabilities according to their irreducible representations and show that higher lattice harmonics are dominant when the nearest-neighbor interaction is sizable indicating pair formation between second-nearest neighbors. The phenomenological consequences can be enhanced spin and thermal quantum Hall responses in a topological superconductor.

Harmonic fingerprint of unconventional superconductivity in twisted bilayer graphene

Microscopic details such as interactions and Fermiology determine the structure of superconducting pairing beyond the spatial symmetry classification along irreducible point group representations. From the effective pairing vertex, the pairing wave function related to superconducting order unfolds in its orbital-resolved Fourier profile which we call the harmonic fingerprint (HFP). The HFP allows to formulate a concise connection between microsopic parameter changes and their impact on superconductivity. From a random phase approximation analysis of twisted bilayer graphene (TBG) involving $d+id$, $s_{\pm}$, and $f$-wave order, we find that nonlocal interactions, which unavoidably enter the low-energy electronic description of TBG, not only increase the weight of higher lattice harmonics but also have a significant effect on the orbital structure of these pairing states. For gapped unconventional superconducting order such as $s_{\pm}$ and $d+id$, a change in HPF induces enhanced gap anisotropies. Experimental implications to distinguish the different gaps and HPFs are also discussed.

Resonance identification studies at the CERN PS

In view of the LHC Injectors Upgrade (LIU) and the challenging high brightness target beam parameters, a broad range of possible working points for the Proton Synchrotron (PS) is being investigated. High order resonances have been identified, both structural resonances driven by space charge due to the lattice harmonics of the PS, and resonances excited by multipolar components in the machine. This paper provides a summary of the performed tune scan studies, covering both experimental and simulation results. Furthermore, non-linear analysis techniques have been used to characterize the resonances and their effect on the beam in presence of space charge.

Expansion into lattice harmonics in cubic symmetries

On the example of a few sets of sampling directions in the Brillouin zone, this work shows how important the choice of the cubic harmonics is on the quality of approximation of some quantities by a series of such harmonics. These studies led to the following questions: (1) In the case that for a given l there are several independent harmonics, can one use in the expansion only one harmonic with a given l?; (2) How should harmonics be ordered: according to l or, after writing them in terms of (x4 + y4 + z4)n (x2y2z2)m, according to their degree q = n + m? To enable practical applications of such harmonics, they are constructed in terms of the associated Legendre polynomials up to l = 26. It is shown that electron momentum densities, reconstructed from experimental data for ErGa3 and InGa3, are described much better by harmonics ordered with q.

Lattice harmonics expansion revisited

The main subject of the work is to provide the most effective way of determining the expansion of some quantities into orthogonal polynomials, when these quantities are known only along some limited number of sampling directions. By comparing the commonly used Houston method with the method based on the orthogonality relation, some relationships, which define the applicability and correctness of these methods, are demonstrated. They are verified for various sets of sampling directions applicable for expanding quantities having the full symmetry of the Brillouin zone of cubic and non-cubic lattices. All results clearly show that the Houston method is always better than the orthogonality-relation one. For the cubic symmetry we present a few sets of special directions (SDs) showing how their construction and, next, a proper application depend on the choice of various sets of lattice harmonics. SDs are important mainly for experimentalists who want to reconstruct anisotropic quantities from their measurements, performed at a limited number of sampling directions.

Ground-State Properties of Superfluid Fermi Gas in Fourier-Synthesized Optical Lattices

By employing the balance condition between the lattice potential and the interatomic interaction, we study the ground state solutions of superfluid Fermi gases in Fourier-synthesized (FS) optical lattices. The average energy of the ground state, the atoms number, and the atom density distribution of the Fermi system are analytically derived along the Bose—Einstein condensation (BEC) side to the Bardeen—Cooper—Schrieffer (BCS) side. We analyze the properties of ground state solutions at both the BEC limit and unitarity in FS optical lattices. It is found that the relative phase α between the two lattice harmonics impacts greatly on the properties of the ground state of the superfluid Fermi gas. Especially in the BCS limit, when α = π/2, the average energy presents an exponential form with the increase of the potential depth of the lattice harmonics υ2. Meanwhile, there exits a minimal value. Moreover, due to the Fermi pressure, the atom density distribution at unitarity is more outstretched than that in the BEC limit. The average energy at unitarity is apparently larger than that in the BEC limit. The properties of the ground state solution exhibit very different behaviors when the system transits from the BEC side to the BCS side.

Band structure and dynamic behaviors of Bose–Einstein condensates in Fourier-Synthesized optical lattices

By employing a nonlinear three-mode model, we study the band structure of Bose–Einstein condensates in Fourier-Synthesized optical lattices, where the nonlinearity comes from the mean field treatment of interaction between atoms. In linear case, we present the band structure of the system. It is demonstrated that the energy band structure is strongly dependent on the value of relative phase of the two lattice harmonics. In the nonlinear case, we show that the eigenenergies as the functions of the quasi-momentum have a novel bowl structure in the middle energy level. It is found that there exist four critical values of interaction strength at which the band structure will undergo interesting changes. Furthermore, the stability of the eigenstate is also investigated.

Band structure and tunnelling dynamics of Bose–Einstein condensates in Fourier-synthesized optical lattices

By employing a nonlinear three-mode model, we study the band structure of Bose–Einstein condensates in Fourier-synthesized optical lattices, where the nonlinearity comes from the mean field treatment of interaction between atoms. In the linear case, we present the band structure of the system. It is observed that the energy band structure is strongly dependent on the relative phase between the two lattice harmonics. In the nonlinear case, we show that the eigenenergy as a function of energy bias has a striking bowl structure in the middle energy level. It is found that there exist four critical values of interaction strength where the band structure undergoes interesting changes. Because of these topological distortions on the energy levels, we expect that the tunnelling dynamics make a dramatic change. The tunnelling probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunnelling probability as a function of the sweeping rate shows an irregular oscillation for the state on the middle level.

Suppression of Chaos in a Bose–Einstein Condensate Loaded into a Moving Optical Superlattice Potential

We have shown that the application of modulating the secondary lattice is an efficient route to suppressing the generation of chaotic traveling waves of a Bose–Einstein Condensate with attractive interatomic interaction loaded into a moving optical superlattice consisting of two lattices. With the Melnikov method, we obtain the optimal value of the relative phase between the two lattice harmonics for the control of chaos. We also find that the regularization route as the potential depth of the secondary lattice is varied and fairly rich, including the period-doubling bifurcations.

Atomic Landau-Zener Tunneling in Fourier-Synthesized Optical Lattices

We report on an experimental study of quantum transport of atoms in variable periodic optical potentials. The band structure of both ratchet-type asymmetric and symmetric lattice potentials is explored. The variable atom potential is realized by superimposing a conventional standing wave potential of lambda/2 spatial periodicity with a fourth-order multiphoton potential of lambda/4 periodicity. We find that the Landau-Zener tunneling rate between the first and the second excited Bloch band depends critically on the relative phase between the two spatial lattice harmonics.

SPATIAL RESONANCE OVERLAP IN BOSE–EINSTEIN CONDENSATES IN SUPERLATTICE POTENTIALS

We employ Chirikov's overlap criterion to investigate interactions between subharmonic resonances of coherent structure solutions of the Gross–Pitaveskii (GP) equation governing the mean-field dynamics of cigar-shaped Bose–Einstein condensates in optical superlattices. We apply a standing wave ansatz to the GP equation to obtain a parametrically forced Duffing equation describing the BEC's spatial dynamics. We then investigate analytically the dependence of spatial resonances on the depth of the superlattice potential, deriving an order-of-magnitude estimate for the critical depth at which spatial resonances with respect to different lattice harmonics first overlap. We also derive a formula for the size of resonance zones and examine changes in our estimates as the relative superlattice amplitudes corresponding to the different harmonics are adjusted. We investigate the onset of global chaos and support our analytical work with numerical simulations.

Calculation of magnetic anisotropy energy inSmCo5

SmCo5 is an important hard magnetic material, due to its large magnetic anisotropy energy (MAE). We have studied the magnetic properties of SmCo5 using density functional theory (DFT) calculations where the Sm f-bands, which are difficult to include in DFT calculations, have been treated within the LDA+U formalism. The large MAE comes mostly from the Sm f-shell anisotropy, stemming from an interplay between the crystal field and the spin-orbit coupling. We found that both are of similar strengths, unlike some other Sm compounds, leading to a partial quenching of the orbital moment (f-states cannot be described as either pure lattice harmonics or pure complex harmonics), an optimal situation for enhanced MAE. A smaller portion of the MAE can be associated with the Co-d band anisotropy, related to the peak in the density of states at the Fermi energy. Our result for the MAE of SmCo5, 21.6 meV/f.u., agrees reasonably with the experimental value of 13-16 meV/f.u., and the calculated magnetic moment (including the orbital component) of 9.4 mu_B agrees with the experimental value of 8.9 mu_B.

Scattering Properties of Non‐Spherical Muffin‐Tin Potentials of Surface Atoms

AbstractThe calculation of the generalized phase shifts and of the partial wave amplitudes is described. An expansion into lattice harmonics is preferred to include the simplifications due to the symmetry in a direct way. The non‐spherical potential components are considered for a Ni atom in a surface layer of an (111) surface. Results are presented for an electron energy of 4 Ryd. Even at this relatively high energy remarkable deviations from the spherical case can be observed. It is discussed in detail, how the scattering amplitude depends both, on the direction of the incoming wave and on the azimuthal angle, that means cases are considered, in which the scattering amplitude in the spherical case has always the same value.

Sound velocity anisotropy in cubic crystals

Simple analytical expressions may be derived for sound velocities in cubic crystals by using lattice harmonics or functions which are invariant under the crystal symmetry operations. These expressions are in good agreement with the exact results for typical crystals such as metallic iron and potassium fluoride.

A new method of constructing symmetry-adapted linear combinations for a finite group: II. Correspondence theorem and basis operators

A correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry-adapted linear combination belonging to the same irreducible representation of a point group. By means of basis operators, the general expressions are given for the symmetry-adapted linear combinations, the coupling coefficients, the lattice harmonics and the hybrid atomic orbitals. Illustrative examples are discussed to demonstrate the effectiveness of the formalism.

Investigation of the Electronic Structure of Uranium Compounds UX2 (X Pnigogen) and UAsY (Y Chalcogen) by Positron Annihilation

AbstractThe electron momentum density distributions ϱ(p) in USb2, UP2, and UAsS are calculated for various sets of lattice harmonics Fϱ, using Mijnarends's method. The selection of the adequate set of harmonics is made by a comparison of the anisotropic parts of ϱ(p) with “difference” momentum density distributions ϱr(p) obtained on the basis of the difference angular correlation of positron annihilation quanta curves. From “summed up” distributions ϱs(p) = ϱpoly(p)+ ϱr(p) Fermi momenta along different directions are estimated. The character of the Fermi momenta changes suggests an ellipsoidal shape of Fermi surface, which corresponds to extremely large anisotropy of electric conductivity along [001] and [100] directions of the tetragonal unit cell of these compounds.

A study of the extended-ion model for F centers in alkali halides

The extended-ion model in the calculation of the electronic structure of simple defects in ionic crystals is studied with the example of the F center in LiF. An efficient scheme of computation is set up in which orthogonalization of the defect wave function to the core orbitals and the exchange terms between the defect and the core electrons are treated exactly for a large enough number of shells around the defect. This is made practical by a systematic use of lattice harmonics and Gaussian-type orbitals. The effect of using different sets of core orbitals in the extended-ion model calculation is studied next. It is found that the results are quite sensitive to the choice of core orbitals. It is pointed out that as anions undergo contraction and cations expansion in an ionic lattice, this effect should be taken into account in an extended-ion model work by employing appropriate ionic orbitals.

Lattice harmonics for graphite

A general theory for obtaining lattice harmonics of non-symmorphic space groups is presented, and the representation theory for these groups is briefly reviewed, with particular reference to graphite (space groupD6h4[P63/mmc]). The irreducible matrix representations, compatibilities and lattice harmonics for alll are listed for all symmetry points, lines and planes in the representation domain of the Brillouin zone. The extra degeneracies introduced by time reversal are also considered. An appendix gives full details of the angular momentum conventions used in this paper.

Application of cubic harmonic expansion in electronic structure calculation of defect using LCAO method

A new scheme for calculating the electronic structure of a defect in a solid by the method of linear combinations of atomic orbitals (LCAO) is presented. Instead of evaluating the multi-center integrals individually, the energy matrix elements are calculated en bloc by expanding the crystal potential and the LCAO wave functions in lattice harmonics around the defect centre. A novel procedure is described in which the atom-centered s-, p- etc. character of original atomic orbitals is partly lost, which enhance the variational freedom considerably. As an example, results of a preliminary work on F-centre in LiF are reported using a Gaussian atomic orbital basis.

Lattice Harmonics II. Hexagonal Close-Packed Lattice

A general discussion of the treatment of the asymmorphic space groups is given, with special reference to the hexagonal close-packed lattice ($\frac{P{6}_{3}}{\mathrm{mmc}}$), for which the irreducible representations are given in full and the lattice harmonics listed for all $l$. The expansions possess the same properties as those given for the cubic groups in the preceding paper.