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Introduction

A quantum lattice system can be completely described based on its unitcell. Essentially, this description requires three types of information:

  1. The spatial information, such as the coordinates of the points contained in the unitcell;
  2. The internal degrees of freedom, such as the local algebra acting on the local Hilbert space at each point;
  3. The couplings among different degrees of freedom, such as the terms present in the Hamiltonian.

In theory, given the above information, one can easily write down the operator representation of the Hamiltonian of the system. For example, in the phrase "the single orbital electronic Hubbard model with only nearest neighbor hopping on a one dimensional lattice with only two sites", "one dimensional lattice with only two sites" provides the spatial information, "single orbital electronic" defines the local Hilbert space and thus the local algebra, and "Hubbard model with only nearest neighbor hopping" describes the terms present in the Hamiltonian. From this description, we can derive that the Hamiltonian of the system is

\[H=tc^\dagger_{1\uparrow}c_{2\uparrow}+tc^\dagger_{2\uparrow}c_{1\uparrow}+tc^\dagger_{1\downarrow}c_{2\downarrow}+tc^\dagger_{2\downarrow}c_{1\downarrow}+Uc^\dagger_{1\uparrow}c_{1\uparrow}c^\dagger_{1\downarrow}c_{1\downarrow}+Uc^\dagger_{2\uparrow}c_{2\uparrow}c^\dagger_{2\downarrow}c_{2\downarrow}\]

where $t$ is the hopping amplitude, $U$ is the Hubbard interaction strength, and the electronic creation/annihilation operator $c^\dagger_{i\sigma}/c_{i\sigma}$ carries a site index $i$ ($i=1, 2$) and a spin index $\sigma$ ($\sigma=\uparrow, \downarrow$). The unitcell description framework follows exactly this train of thought. For example, the aforementioned system can be constructed with the following code:

using QuantumLattices
using SymPy: Sym, symbols

# define the unitcell
lattice = Lattice([zero(Sym)], [one(Sym)])

# define the internal degrees of freedom, i.e., the single-orbital spin-1/2 fermionic algebra
hilbert = Hilbert(site=>Fock{:f}(1, 2) for site in eachindex(lattice))

# define the terms
t = Hopping(:t, symbols("t", real=true), 1)
U = Hubbard(:U, symbols("U", real=true))

# get the Hamiltonian
operators = expand(OperatorGenerator(bonds(lattice, 1), hilbert, (t, U)))

\[tc^{\dagger}_{2,\,1,\,\downarrow}c^{}_{1,\,1,\,\downarrow}+tc^{\dagger}_{1,\,1,\,\downarrow}c^{}_{2,\,1,\,\downarrow}+tc^{\dagger}_{2,\,1,\,\uparrow}c^{}_{1,\,1,\,\uparrow}+tc^{\dagger}_{1,\,1,\,\uparrow}c^{}_{2,\,1,\,\uparrow}+Uc^{\dagger}_{1,\,1,\,\uparrow}c^{}_{1,\,1,\,\uparrow}c^{\dagger}_{1,\,1,\,\downarrow}c^{}_{1,\,1,\,\downarrow}+Uc^{\dagger}_{2,\,1,\,\uparrow}c^{}_{2,\,1,\,\uparrow}c^{\dagger}_{2,\,1,\,\downarrow}c^{}_{2,\,1,\,\downarrow}\]

The last line displays all the generated operators in the Hamiltonian in LaTeX format. Here, in the subscript of the electronic annihilation/creation operator, an extra orbital index is also displayed.

In the following pages, we will explain in detail how these codes work:

For more detailed information, please refer to the manual of this package.