QUANTUM LATTICE

Summary

This program allows to perform tight binding calculations with a user friendly interface in a variety of lattices and dimensionalities.

Video examples

Here you can see four simultaneous examples of the usage of Quantum Lattice.

Below you can see videos showing the real-time usage of this program for individual examples

• Confined modes in graphene nanoislands
• Superlattices
• Interaction-induced magnetism
• Artificial Chern insulators
• Landau levels and quantum Hall edge states
• Twisted bilayer graphene

How to install

Linux and Mac

The program runs in Linux and Mac machines.

Clone the GitHub repository

git clone https://github.com/joselado/quantum-lattice

and execute the script install as

python install.py

The script will install all the required dependencies if they are not already present for the python command used. Afterwards, you can run the program by executing in a terminal

quantum-lattice

You can see here a short video demonstrating the installation.

Windows

For using this program in Windows, the easiest solution is to create a virtual machine using Virtual Box, installing a version of Ubuntu in that virtual machine, and following the previous instructions.

FUNCTIONALITIES

Single particle Hamiltonians

• Spinless, spinful and Nambu basis for orbitals
• Full non-collinear electron and Nambu formalism
• Include magnetism, spin-orbit coupling and superconductivity
• Band structures with state-resolved expectation values
• Momentum-resolved spectral functions
• Local and full operator-resolved density of states
• 0d, 1d, 2d and 3d tight binding models

Interacting mean-field Hamiltonians

• Selfconsistent mean-field calculations with local/non-local interactions
• Both collinear and non-collinear formalism
• Anomalous mean-field for non-collinear superconductors
• Full selfconsistency with all Wick terms for non-collinear superconductors
• Automatic identification of order parameters for symmetry broken states

Topological characterization

• Berry phases, Berry curvatures, Chern numbers and Z2 invariants
• Operator-resolved Chern numbers and Berry density

Spectral functions

• Surface spectral functions for semi-infinite systems
• Single impurities in infinite systems
• Operator-resolved spectral functions

Chebyshev kernel polynomial based-algorithms

• Local and full spectral functions
• Operator resolved spectral functions
• Reaching system sizes up to 1000000 atoms on a single-core laptop

Quantum Lattice uses pyqula.

Screenshot examples

Unconventional superconductivity

Electronic band structure, Berry curvature and momentum resolved surface spectral function of a px + ipy spin-triplet topological superconductor with d-vector (0,0,1).

Interaction-driven non-collinear magnetism

Electronic band structure and selfconsistent local magnetization of a square lattice with an applied Zeeman field and local Hubbard interactions.

Superlattices

Electronic band structure, Fermi surface and local density of states of a superlattice built from a defective triangular lattice

Scanning tunnel spectroscopy of nanographene islands

Real space simulation of the STS spectra, using atomic-like orbitals for a nanographene island

Kagome lattice with first and second neighbor hopping

Fermi surface and band structure of a two-dimensional lattice, including both first and second neighbor hoppings. In the absence of second neighbor hopping, the lowest band is flat. Only first neighbor hoppings are shown in the 3D structure plot.

Interaction-induced symmetry breaking in the Lieb lattice

Non-interacting and interacting band structure of a two-dimensional Lieb lattice. When repulsive local Hubbard interactions are included, an spontaneously ferromagnetic state appears in the system, leading to a real-space magnetic distribution.

Artificial Chern insulators

Kagome lattice with Rashba spin-orbit coupling and exchange field, giving rise to a net Chern number and chiral edge states

Two-dimensional quantum Spin Hall state

Honeycomb lattice with Kane-Mele spin-orbit coupling and Rashba spin-orbit coupling, giving rise to a gapped spectra with a non-trivial Z2 invariant and helical edge states https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.226801

Magnetism in graphene zigzag nanoribbons

Self-consistent mean field calculation of a zigzag graphene ribbon, with electronic interactions included as a mean field Hubbard model. Interactions give rise to an edge magnetization in the ribbon, with an antiferromagnetic alignment between edges

Three-dimensional quantum spin Hall insulators

Three-dimensional quantum spin-Hall insulator, engineered by intrinsic spin-orbit coupling in the diamond lattice. the top and bottom of the slab show an emergent helical electron gas.

Scanning tunnel spectroscopy of graphene nanoribbons

Real space simulation of the STS spectra, using atomic-like orbitals for a graphene nanoribbon

Nodal line semimetals

Band structure of a slab of a 3D nodal line semimetal in a diamond lattice, showing the emergence of topological zero energy drumhead states in the surface of the slab https://link.springer.com/article/10.1007%2Fs10909-017-1846-3

Confined modes in quantum dots

Spectra and spatially resolved density of states of square quantum dot, showing the emergence of confined modes

Colossal quantum dots

Density of states and spatially resolved density of states of a big graphene quantum dot. The huge islands module uses special techniques to efficiently solve systems with hundreds of thousands of atoms.

Landau levels

Electronic spectra of a graphene lattice in the presence of an off-plane magnetic field and antiferromagnetic order, giving rise to Landau levels and chiral edge states

Artificial topological superconductors

Bogoliuvov de Gennes band structure of a two-dimensional gas in a square lattice with Rashba spin-orbit coupling, off-plane exchange field and s-wave superconducting proximity effect. When superconductivity is turned on, a gap opens up in the spectra hosting a non-trivial Chern number, giving rise to propagating Majorana modes in the system

Quantum Valley Hall effect

Band structure of Bernal stacked bilayer graphene, showing the emergence of a gap when an interlayer bias is applied. The previous gap hosts a non-trivial valley Chern number, giving rise to the emergence of pseudo-helical states in the edge of the system

Twisted bilayer graphene

Bandstructure and Fermi surface of a twisted graphene bilayer, showing the emergence of nearly flat bands https://journals.aps.org/prb/abstract/10.1103/PhysRevB.82.121407

Twisted trilayer graphene

Structure and band structure of a twisted graphene trilayer at the magic angle.