Hamiltonian

The Hamiltonian for the lattice Schwinger model is

\[\begin{split} H = &\frac{(qg)^2 a}{2}\sum_{n=1}^N \left(L_n + \frac{\theta}{2\pi}\right)^2 - \frac{i}{2a}\sum_{n=1}^N\sum_{\alpha=1}^F \left(\chi^\dagger_{n,\alpha} \chi_{n+1,\alpha} - \chi^\dagger_{n+1,\alpha} \chi_{n,\alpha}\right) \\ &+\underbrace{\left(m - \frac{(qg)^2 F a}{8}\right)}_{m_\text{lat}}\sum_{n=1}^N \sum_{\alpha=1}^F (-1)^n \chi^\dagger_{n,\alpha} \chi_{n,\alpha} + m' \sum_{n=1}^N\sum_{\alpha=1}^F (-1)^{n+1}\left(\chi^\dagger_{n-1,\alpha}\chi_{n,\alpha} + \chi^\dagger_{n+1,\alpha}\chi_{n,\alpha}\right) \end{split}\]

This is supplemented by the Gauss law

\[L_n = L_{n-1} + Q_n, \qquad Q_n \equiv q\left(\sum_{\alpha=1}^F \chi^\dagger_{n,\alpha} \chi_{n,\alpha} - \begin{cases} F & n\text{ odd} \\ 0 & n\text{ even} \end{cases}\right).\]

In Schwinger.jl, the Hamiltonian can be constructed using three computational backends:

• Exact Diagonalization (ED): Uses sparse matrices for small systems
• ITensors: Uses MPO/MPS via ITensors.jl and ITensorMPS.jl
• MPSKit: Uses MPO/MPS via MPSKit.jl

Unified API

The unified API allows you to construct operators using any backend by specifying the backend keyword argument:

# Using symbols
H = Hamiltonian(lattice; backend=:ED)
H = Hamiltonian(lattice; backend=:ITensors)  # Default
H = Hamiltonian(lattice; backend=:MPSKit)

# Using backend types directly
H = Hamiltonian(lattice; backend=EDBackend())
H = Hamiltonian(lattice; backend=ITensorsBackend())
H = Hamiltonian(lattice; backend=MPSKitBackend())

You can also set a default backend programmatically:

set_default_backend(:MPSKit)
H = Hamiltonian(lattice)  # Now uses MPSKit by default

Exact diagonalization

When using exact diagonalization, Schwinger.jl constructs a basis of states that diagonalize the operators $\chi^\dagger_{n,\alpha} \chi_{n,\alpha}$ and $L_0$. It then builds a sparse matrix for the Hamiltonian acting on this basis.

using Schwinger
lat = Lattice(12; F = 1);
ham = Hamiltonian(lat; backend=:ED);

ham.matrix

924×924 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 6468 stored entries:
⎡⢻⣶⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⣙⠻⣦⡙⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠈⠳⣌⠿⣧⡦⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠈⠈⠫⣻⣾⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠠⣄⠀⠀⠙⠻⣦⣄⠀⠀⠙⠦⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⡻⣮⡢⡀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠪⢿⣷⡀⠱⢄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠐⢄⡈⠻⣦⣄⠓⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠙⡻⣮⡢⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣮⣄⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢤⠙⠻⣦⡈⠑⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠑⢆⠈⢿⣷⡢⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠁⠈⠪⡻⣮⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠲⣄⠀⠀⠙⠻⣦⣄⠀⠀⠙⠂⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⡿⣯⣢⡀⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠺⢻⣶⡙⢦⡀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⣌⠻⣦⣍⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠿⣧⎦

You can also use the backend-specific constructor EDHamiltonian for backward compatibility.

When $q > 1$, the universe (i.e., the allowed values of $L_n$ modulo $q$) can be specified; the default value is 0. When the lattice is periodic, the maximum absolute value of $L_0$ can be set using L_max; the default value is 3.

Using Arpack.jl, Schwinger.jl can find the lowest eigenstates of a Hamiltonian.

using Schwinger
lat = Lattice(10; F = 1, periodic = true);
ham = Hamiltonian(lat; backend=:ED);

map(energy, loweststates(ham, 5))

5-element Vector{Float64}:
 -1.9903699416046092
 -1.4257050000651357
 -1.1927317764793883
 -1.1927317764793888
 -0.903954948059417

Matrix product operators

For larger lattice sizes where exact diagonalization is infeasible, Schwinger.jl can construct a matrix product operator representation of the Hamiltonian using two backends:

• ITensors: Uses ITensors.jl and ITensorMPS.jl
• MPSKit: Uses MPSKit.jl

using Schwinger
lat = Lattice(10; F = 1);

# Compare energy gaps across backends
[
    energygap(Hamiltonian(lat; backend=:ED)),
    energygap(Hamiltonian(lat; backend=:ITensors)),
    energygap(Hamiltonian(lat; backend=:MPSKit))
]

3-element Vector{Float64}:
 0.6204350201341655
 0.6204350207929119
 0.6204350201341642

ITensors Backend

MPSKit Backend