Getting started

Installation

Requirements: Julia 1.11 or later.

To install the latest stable version of SymBasis.jl, you can use the Julia package manager. Either use the Julia REPL package mode (by pressing ]):

pkg> add SymBasis

or open the Julia REPL and run the following command:

julia> import Pkg; Pkg.add("SymBasis")

Basic concepts

Before diving into the usage of SymBasis.jl, it is helpful to understand some basic concepts related to the package. SymBasis.jl is designed to generate bases with certain symmetries for systems with a discrete number of degrees of freedom, which is particularly useful for quantum many-body problems.

Base positional numbering

In SymBasis.jl, the basis states are represented using a base positional numbering system with a corresponding base. This means that each state is assigned a unique integer based on its configuration, and the position of each degree of freedom contributes to the overall number. This allows for efficient storage and manipulation of basis states. This is handled internally by the SymBasis.DigitBase submodule, which is responsible for the base positional numbering system used in the package. Users typically do not need to interact directly with this submodule when using the package to generate bases, but it is an important aspect of how the package represents and manages basis states via the BaseInt type from the SymBasis.DigitBase submodule.

Objects with degrees of freedom a.k.a. DoF-objects

To generate a basis, you first need to define an object with degrees of freedom (DoF-object) that represents the physical system you are interested in. For example, if you are working with a spin-1/2 system, you can define a DoF-object for spin-1/2 using the dof_object function from the SymBasis.DoFObjects submodule. This object will then be used to specify the symmetries and generate the corresponding basis.

Additionally, you can define custom DoF-objects for other types of systems, such as a system containing flag emojis of countries, by specifying the appropriate parameters when creating the DoF-object via the DoFObject constructor from the SymBasis.DoFObjects submodule. This flexibility allows you to use SymBasis.jl for a wide range of systems.

Symmetry groups

Symmetry groups are a crucial Julia struct to generate bases with symmetries. In SymBasis.jl, you can contruct symmetry groups using the sym function from the SymBasis.SymGroups submodule. This function allows you to call various types of predefined symmetries, such as total magnetization, translation, reflection, and more. You can also combine multiple symmetries to generate bases that conserve multiple symmetries simultaneously.

Similar to DoF-objects, you can also define your own custom symmetries by specifying the appropriate parameters when constructing the symmetry group via the SymGroup constructor from the SymBasis.SymGroups submodule. This feature provides great flexibility in generating bases that are tailored to the very specialized symmetries of your system. These symmetry groups can be also combined with each other to generate bases that conserve multiple symmetries simultaneously via the ∘ function from the SymBasis.SymGroups submodule, which eventually returns CombSymGroup types that represent the combined symmetries.

Generating bases

Once you have defined the DoF-object and the symmetry group(s), you can generate the basis using the basis function from the SymBasis.Bases submodule. This function takes the DoF-object, the number of sites, and the symmetry group(s) as input and returns a Basis type that contains the basis states and their normalization factors to define operators like Hamiltonian in the basis. The generated basis will conserve the specified symmetries, allowing you to work with a reduced Hilbert space that is more manageable for quantum many-body problems.