Fermion Lattice Structures Tutorial#
navigation-menu 🠖 Structures 🠖 Fermion Lattice Structures 🠖 +This tutorial provides a quick overview of the Fermion Lattice Configuration page, which allows you to quickly and easyly setup and simulate 1D and 2D fermionic lattices. Fermion Lattice Structures furthermore provides you with an interactive graph for every created lattice structure as well as a bandstructure plot.
For more information about the parameters see the Parameters page. Note: Many parameters in the UI have info buttons besides them that let you quickly check their purposes. The UIs also provides intelligent validation of parameters that lets you know about restrictions in value assignment.
Spinful and Spinless Fermion Lattices#
Fermion lattices can be either configured as spinful or spinless lattices. Choose this configuration kinds via the Site Type dropdown. See the parametrization for the different kinds of lattices in our tutorial_physics.
The LOAD button lets you load your fermion lattice structures and also provides you with some predefined templates.
Our templates are fully functioning structures that can be loaded and adjusted to your needs.
Configure a Fermion Lattice#
The Unit-Cell#
The lattice plot will be provided as soon as possible in your configuration process to give a visual feedback on the state of your configuration. See the Parameters page for more information about the different sets of parameters for spinful and spinless fermion lattices.
- Add a name.
Provide a unique name for your strcuture.
- Choose a site type.
This will provide you with the appropriate set of parameters for the chosen type of lattice.
- Add sites.
Open the Sites panel and press +. Configure your sites in the 'Add Site' dialog. Every created site will be shown in the sites table in the sites tab. For spinful sites it is possible to choose between 'Spin Channel Notation' and 'Magnetic Field Notation'. The conversion between these two input conventions is done automatically. After choosing the 'Spin Channel Notation', the spin channel table is shown.
Note: 'Spin Channel Notation' and 'Magnetic Field Notation' are just a different notation that both describes the same thing, provided for your convenience.
- Add lattice vectors.
Open the Lattice Vectors panel and configure your lattice vectors.
Note: The lattice vectors, together with cluster size and boundary conditions, describe the geometry of your lattice. You only need to provide lattice vectors for directions along which the cluster size is bigger than one.
- Add bonds within the unit-cell.
Open the Bonds within Unit-Cells panel and press the + button. Configure your bond in the 'Add Bond' dialog.
- Add bonds between unit-cells.
Open the Bonds between Unit-Cells panel and press the + button. Configure your bonds in the 'Add Bond' dialog.
Note: Every bond created will be shown in one of the bonds tables, either in the Bonds within Unit-Cells or the Bonds between Unit-Cells panel. If you create a bond with translation [0, 0] in the Bonds between Unit-Cells it will be automatically transfered to the bonds table in Bonds within Unit-Cells.
The System Settings#
Open the System Settings panel.
Specify Cluster Size by setting x and y.
Add Boundary Conditions Vector a and Boundary Conditions Vector b.
Optionally add Sz and mod Sz for spinful fermions.
Optionally add N and mod N.
Validate your Configuration#
After you have configured all required values, the VALIDATE LATTICE button will turn green and you can validate your configuration by pressing it. If your configuration is valid you will be notified that you can save your lattice. In case of an invalid configuration an error message will provide hints about how to adjust your configuration.
Edit your Fermion Lattice#
Edit via the lattice plot#
- Edit sites
On hovering your cursor over a site it will highlight together with its corresponding sites in the neighboring unit-cells. Sites can be edited by performing a left-click. This will open the 'Edit Site' dialog.
- Edit bonds
On hovering your cursor over a bond it will highlight together with its corresponding bonds in the neighboring unit-cells. Bonds can be edited by performing a left-click. This will open the 'Edit Bond' dialog.
Edit or delete via the Table Entries#
- Edit or delete sites
Open your Sites panel. The table in the sites panel contains all your sites. The action buttons on the right side of each table entry let you either edit or delete the respective site.
- Edit or delete bonds
Both of the bonds panels contain a table with their related bonds. The action buttons on the right side of each table entry let you either edit or delete the respective bond.
Read the lattice plot#
Visualization of Magnetic Fields for Spinful Fermions#
For Spinful Fermions the different colors and sizes of the sites in the image on the left visualize the properties of the magnetic field on each site. The stronger the magnetic field, the bigger the image of the site. The color of the sites refers to the directions of the magnetic field. Each direction of the vector [Bx, By, Bz] corresponds to a different color:
Bx: red
By: green
Bz: blue
So the 'mixture' of directions corresponds directly to the mixture of colors for a site, the values of Bx, By, Bz representing the weight of their respective color. The Plot Settings & Legend section below the plot shows the distribution of colors due to the directions of the vectors. To turn off the coloration of the sites un-check the Show Magnetic Field box in the Plot Settings & Legend panel below the lattice plot.
For Spinless Fermions each site (up to a total of ten sites) with a unique name will be given a unique color. If sites have the same name they will be given the same color.
Visualization of interactions for spinful fermions#
For Spinful Fermions we also provide a visualization of the interaction along bonds. To visualize the interaction of the bonds open the Plot Settings & Legend panel below the lattice plot and in the section Bond Interaction select the interaction value you want to be shown.
Lieb Lattice with Jz value set to -1 on all bonds
