Attractive Hubbard Model

Hamiltonian

The Hamiltonian of the attractive (negative $U$) Hubbard model reads

\begin{align} \mathcal{H} = -t \sum{\langle i,j \rangle, \sigma} \left( c^\dagger{i\sigma} c{j\sigma} + \text{h.c.} \right) - |U| \sumj \left( n{j\uparrow} - \frac{1}{2} \right) \left( n{j\downarrow} - \frac{1}{2} \right) - \mu\sumj n{j}, \end{align}

where $\sigma$ denotes spin, $t$ is the hopping amplitude, $U$ the on-site repulsive interaction strength, $\mu$ the chemical potential and $\langle i, j \rangle$ indicates that the sum has to be taken over nearest neighbors. Note that (1) is written in particle-hole symmetric form such that $\mu = 0$ corresponds to half-filling.

Constructor

You can create an attractive Hubbard model instance as follows,

model = HubbardModelAttractive(dims=1, L=8)

The following parameters can be set via keyword arguments:

• dims::Int: dimensionality of the cubic lattice (i.e. 1 = chain, 2 = square lattice, etc.)
• L::Int: linear system size
• t::Float64 = 1.0: hopping energy
• U::Float64 = 1.0: onsite interaction strength, "Hubbard $U$"
• mu::Float64 = 0.0: chemical potential

Supported Monte Carlo flavors

• Determinant Quantum Monte Carlo (DQMC), see details below

DQMC formulation

We decouple the onsite electron-electron interaction by performing a Hirsch transformation, i.e. a discrete Hubbard-Stratonovich transformation in the density/charge channel,

\begin{align} e^{|U|\Delta \tau \left( n{i\uparrow} - \frac{1}{2} \right) \left(n{i\downarrow} - \frac{1}{2} \right)} = \frac{1}{2} e^{-|U|\Delta \tau /4} \sum{s=\pm 1} \prod{\sigma=\pm 1} e^{s\lambda (n_{i\sigma}-\frac{1}{2})}. \end{align}

The interaction matrix of the model then reads

\begin{align} V{ij}(l) &= \delta{ij} Vi(l), \\ Vi(l) &= - \frac{1}{\Delta \tau} \lambda s_i(l). \end{align}

For completeness, the hopping matrix is \begin{align} T_{ij} &= \begin{cases} -t & \text{if i and j are nearest neighbors,} \\ -\mu & \text{if i == j,} \\ 0 & \text{otherwise.} \end{cases} \end{align}

As neither $T$ nor $V$ depend on spin, neither does the equal-times Green's function. We can therefore restrict our computations to one spin flavor (flv=1) and benefit from operating with smaller matrices.

Potential extensions

Pull requests are very much welcome!

• Arbitrary lattices (so far only cubic lattices supported)