The aim of the lattice gallery s to provide an impression
of the results that can be obtained using the HQS lattice library.
It is not intended to provide a detailed description of the methods.
Band structure of a simple 1D chain with hopping parameter \(t\)
and repulsive Hubbard-type interaction of strength \(U/t=5.0\). The band structure
was obtained by means of Cluster Perturbation Theory (CPT) with a cluster size of \(M_C = 18\)
lattice sites.
The band structure features a Mott gap and the separation of the dispersing features into
spin and charge excitations.#
Effective 1D model of a Transpolyacetylene with Hubbard-type interactions#
Band structure of an effective lattice model for Transpolyacetylene molecules.
This effective model is also known as the Su-Schrieffer-Heeger chain.
Here we plot the band structure of the non-interacting model (\(U/t=0\)) derived
from CPT with cluster size \(M_C=8\).
The band structure displays two bands, the binding one, with energies below the Fermi energy
\(E_F=0\) and the non-binding one.#
Here we plot the band structure of the Transpolyacetylene model with interaction strength
\(U/t=1.0\) derived from CPT with cluster size \(M_C=8\).
The impact of the artificial broadening \(\eta\) of the
self-energy has been reduced via an approach employing equations of motion.#
Band structure of the Transpolyacetylene model with interaction strength
\(U/t=5.0\) derived from CPT with cluster size \(M_C=8\). We observe significant
broadening of the spectral weight around the \(\Gamma\) point as well as a shift of the
lower band to energies above the Fermi energy.#
Band structure of a one-dimensional diamond chain calculated with CPT using a cluster of
\(M_C=6\) lattice sites. The chain features three orbitals per single
unit cell. The resulting band structure at interaction strength \(U/t=0.0\) consists of two dispersing bands enclosing a flat band located at
the Fermi surface with a threefold degeneracy at the \(X\) symmetry points.#
The band structure at interaction strength \(U/t=1.0\) still consists of two dispersing bands enclosing a flat band located at
the Fermi surface with a threefold degeneracy at the \(X\) symmetry points, albeit with slightly increased broadening.#
Band structure for interaction strength \(U/t=5.0\). We observe a significant broadening
of the dispersive bands. The flat band appears to have split into two bands of reduced spectral
weight symmetrically above and below the Fermi energy. The degeneracy at the symmetry points
\(X\) has been lifted.#
Band structure of a one-dimensional Lieb chain calculated with CPT using a cluster of
\(M_C=6\) lattice sites. The chain features three orbitals per single
unit cell. The resulting band structure at interaction strengths \(U/t=0\) and \(U/t=1.0\) consists of two dispersing bands enclosing
a flat band located at the Fermi surface.#
The band structure at interaction strength \(U/t=1.0\) still consists of two dispersing bands enclosing a flat band located at
the Fermi surface with a threefold degeneracy at the \(X\) symmetry points, albeit with slightly increased broadening.#
Band structure for interaction strength \(U/t=5.0\). We observe a significant broadening
of the dispersive bands. The flat band appears to have split into two sections of significantly broadened spectral
weight symmetrically above and below the Fermi energy.#
Band structure of the non-interacting dimensional square lattice. The band structure was calculated
with CPT using a \(M_C = 2\times 2\) cluster.
The impact of the artificial broadening of the
self-energy has been reduced via an approach equations of motion approach.
The band structure features cosine bands in \(x\)- and \(y\)-direction.#
Band structure for repulsive interaction \(U/t = 1.0\). We observe slight broadening over
the entire Brillouin zone and extensive broadening around the symmetry points \(X\) and
half-way between the symmetry points \(M\) and \(\Gamma\).#
Band structure for strong repulsive interaction \(U/t = 5.0\).
A Mott gap has developed at the symmetry points \(X\) and on the k-point path between \(M\)
and \(\Gamma\).#
Band structure of the non-interacting honeycomb lattice of Graphene with Graphene specific hopping parameters \(t=2.6\) eV.
At the symmetry points \(K (H)\) we observe the well-known Dirac-cone.#
Band structure of the honeycomb lattice of Graphene with a Hubbard-type interaction of
strength \(U/t=1.o\) and Graphene specific hopping parameters \(t=2.6\) eV.
At the symmetry points \(K (H)\) we observe the well-known Dirac-cone.#
Band structure of the honeycomb lattice of Graphene with a Hubbard-type interaction of
strength \(U/t=4.0\) and Graphene specific hopping parameters \(t=2.6\) eV.
The Dirac-cone has given way to a gap.#
Band structure of the Lieb lattice without interaction and cluster size \(M_C=12\). This lattice features three sites per single unitcell
at \((0,0)\), \((0, 0.5)\), \((0.5, 0)\).
The band structure exhibits the characteristic
flat-band crossing the cone formed by the two dispersing bands at the \(S\) symmetry points.#
Band structure of the Lieb lattice with a repulsive Hubbard-type interaction of strength \(U/t=5.0\).
The crossing of the dispersing bands has been replaced by a Mott gap at the symmetry points \(S\) and the flat band has split
into two significantly broadened regions of spectral weight above and below the Fermi energy.#
Band structure of the non-interacting Kagome lattice. This lattice features three sites per single unitcell at
\((0,0)\), \((0.5, 0.0)\), \((0.25, 0.433)\). The band structure features two crossing
dispersing bands and a flat band above the higher dispersing band.#
Band structure of the Kagome lattice with Hubbard-type interaction of strength \(U/t=1.0\) and cluster size \(M_C=12\).
We observe increased broadening for the flat band and the lower dispersing band.#
Band structure of the Kagome lattice with Hubbard-type interaction of strength \(U/t=5.0\) and cluster size \(M_C=12\).
We observe very strong broadening across the entire spectrum, but particularly for the flat band.#
Band structure of the Dice lattice with Hubbard-type interaction of strength \(U/t=0.0\) and cluster size \(M_C=12\).
We find a flat band at the Fermi energy.#
Band structure of the Dice lattice with Hubbard-type interaction of strength \(U/t=1.0\) and cluster size \(M_C=12\).
The spectral weight of the flat band is now significantly broadened.#
Band structure of the Dice lattice with Hubbard-type interaction of strength \(U/t=5.0\) and cluster size \(M_C=12\).
There is broadening also for the dispersing bands, while the flat band has split into two regions of spectral weight.#
Density of states for a periodic cubic lattice. The system has size 501 x 501 x 501.
The onsite disorder is modeled by a random potential with a uniform distribution of
amplitude W. The results are averaged over the random potential distributions. The DOS is
obtained via stochastic evaluation of the expectation value of the DOS operator expanded
in Chebyshev polynomials.#
The DOS shows a broadening of spectral features with increasing disorder strength.
Spatially resolved LDOS at the Fermi energy \(E_F=0\). The graphene sheet has size
101x101 unit cells, with periodic boundary conditions and with a vacancy density of 4%.
Only a smaller view of the whole sheet is shown here.
Each triangle is the Voronoi cell associated with each site. White areas denote the defects.
The LDOS is obtained via the Green's function, calculated by means of a Chebyshev expansion
of the resolvent.
The colormap is clipped at a maximum value of 1 for plotting purposes.#
The LDOS shows peaks around vacancies and exhibits interference patterns across different
vacancies.
Spatially resolved LDOS at the Fermi energy \(E_F=0\). The graphene sheet has size
151x151 unit cells, with periodic boundary conditions and with few large circular defects.
Only a smaller view of the whole sheet is shown here.
Each triangle is the Voronoi cell associated with each site. White areas denote the defects.
The LDOS is obtained via the Green's function, calculated by means of a Chebyshev expansion
of the resolvent.
The colormap is clipped at a maximum value of 0.25 for plotting purposes.#
The LDOS shows increased spectral weight around the defects with symmetry depending on the
position within the unit cell of the center of the hole. The effect associated with the
structural irregularity is rather weak as there is little overlap, hence interference,
between different defects.
Electrical conductivity in the x direction \(\sigma_x\) as a function of the chemical
potential \(\mu\), for different vacancy densities. The graphene sheet has size
201x201 unit cells, with periodic boundary conditions.
The conductivity is obtained by evaluating the zero temperature Kubo-Formula via expansion
in Chebyshev polynomials.#
Increasing the vacancy density gradually reduces the conductivity of the system, except for
\(\mu=0\), which corresponds to half-filling, for which a delta-like peak appears.
The inset shows a zoom-in around this point.
Band structure of bulk silicon along the standard k-path for diamond structures. The crystal
Hamiltonian is represented on a lattice through a Wannierization procedure. The band
structure is obtained by evaluating the Green's function in the k-vectors basis via
expansion in Chebyshev polynomials.#
Band structure of disordered silicon along the standard k-path for diamond structures. The
system is modeled as mentioned in the pristine case. Disorder is represented by random onsite
potentials with a distribution of width 2 eV and the band structure is averaged over the
this distribution of disorder configurations. The calculations have been carried out on a
supercell of size 15x15x15.#
Onsite disorder introduces scattering induced broadening of the bands with a k-dependent
width.
Band structure of disordered silicon as mentioned above. In this case the onsite disorder
distribution has width 4 eV. The supercell used is of size 20x20x20.#
Similarly with the previous case, onsite disorder induces a k-dependent band broadening, in
this case stronger, as expected.
