Phy5670/HubbardModel 2DCalculations

Hubbard Model: 2D Calculations

Expansion of the Hubbard model Hamiltonian into two dimensions allows us to calculate various properties. In 2D, the Hamiltonian can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = -t \sum_{r=1}^{n_x} \sum_{s=1}^{n_y} \sum_{\sigma = \uparrow,\downarrow} (c_{r,s,\sigma}^{\dagger} c_{r+1,s,\sigma} + c_{r,s,\sigma}^{\dagger} c_{r,s+1,\sigma} + h.c. ) + U \sum_{r=1}^{n_x} \sum_{s=1}^{n_y} c_{r,s,\uparrow}^{\dagger} c_{r,s,\downarrow}^{\dagger}c_{r,s,\downarrow}c_{r,s,\uparrow} }

The grand canonical potential, Omega, is best calculated by using coherent state path integral. The grand partition function is defined as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = Tr \big[e^{-\beta (H - \mu N)} \big] = e^{-\beta \Omega} }

which can be expanded as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = Z_{0} \big\langle e^{-S_{int}} \big\rangle = Z_{0} e^{-\langle S_{int} \rangle} e^{\frac{1}{2}( \langle S_{int}^2 \rangle - \langle S_{int} \rangle^2)} = e^{-\beta \Omega_0}e^{-\beta \Omega_1}e^{-\beta \Omega_2} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{int} = \int_{0}^{\beta} d\tau H_{int} (\tau) }

which utilizes cumulant expansion. We begin to calculate the grand canonical potential by analyzing the contribution from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0 } :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{0} = \prod_k (1+e^{-\beta(E_k - \mu)})^2 = e^{2 \sum_k ln(1+e^{-\beta(E_k - \mu)})} = e^{-\beta \Omega_0} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_0 = -\frac{2}{\beta} \sum_k ln(1+e^{-\beta(E_k - \mu)}) }

Now we look at the contribution from the first order cumulant expansion. First we'll need to convert Hint to momentum space:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{r,s,\sigma} = \frac{1}{\sqrt{M}} \sum_{k_x,k_y} e^{i k_x \cdot r} e^{i k_y \cdot s} c_{k_x,k_y,\sigma} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = U \sum_{r=1}^{n_x} \sum_{s=1}^{n_y} c_{r,s,\uparrow}^{\dagger} c_{r,s,\downarrow}^{\dagger}c_{r,s,\downarrow}c_{r,s,\uparrow} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{U}{M^2} \sum_{k_{x_1}...k_{x_4}} \sum_{k_{y_1}...k_{y_4}} e^{-ik_{x_1} r}e^{-ik_{x_2} r}e^{ik_{x_3} r}e^{ik_{x_4} r}e^{-ik_{y_1} s}e^{-ik_{y_2} s}e^{ik_{y_3} s}e^{ik_{y_4} s} c_{k_{x_1},k_{y_1},\uparrow}^{\dagger} c_{k_{x_2},k_{y_2},\downarrow}^{\dagger}c_{k_{x_3},k_{y_3},\downarrow}c_{k_{x_4},k_{y_4},\uparrow} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{U}{M} \sum_{k_{x_1}...k_{x_4}} \sum_{k_{y_1}...k_{y_4}} \delta_{k_{x_1}+k_{x_2},k_{x_3}+k_{x_4}} \delta_{k_{y_1}+k_{y_2},k_{y_3}+k_{y_4}} c_{k_{x_1},k_{y_1},\uparrow}^{\dagger} c_{k_{x_2},k_{y_2},\downarrow}^{\dagger}c_{k_{x_3},k_{y_3},\downarrow}c_{k_{x_4},k_{y_4},\uparrow} }

For simplicity, we will combine the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_x } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_y } into a single index as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k } . Evaluating the Kronecker deltas yields:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = \frac{U}{M} \sum_{k,k',q} c_{k,\uparrow}^{\dagger} c_{k',\downarrow}^{\dagger} c_{k'+q,\downarrow} c_{k-q,\uparrow} }

The only contraction combination possible, due to orthogonal spins, results in the following set of Green's functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle S_{int} \rangle = \bigg \langle \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k',q} c_{k,\uparrow}^{\dagger}(\tau) c_{k',\downarrow}^{\dagger}(\tau) c_{k'+q,\downarrow}(\tau) c_{k-q,\uparrow}(\tau) \bigg \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k'} \big \langle \mathcal{G}_{0}(k\tau,k\tau)\mathcal{G}_{0}(k'\tau,k'\tau) \big \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = {\beta} \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) = \beta \Omega_1 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_1 = \frac{U}{M} \sum_{k,k'} n_{F}^2(\epsilon_k) }

Combining both terms, the grand canonical potential to first order is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \Omega_0 + \Omega_1 = -\frac{2}{\beta} \sum_k ln(1+e^{-\beta(E_k - \mu)}) + \frac{U}{M} \sum_{k} n_{F}^2(\epsilon_k) }

Calculation of the Chemical Potential of Spin 1/2 Fermions on a 2-D Lattice

The Grand Canonical Potential for a 2-D lattice is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = -2 \beta \sum_{k}^{} ln(1 + e^{\beta (E_{k}-\mu)})+ \frac{U}{2} (\sum_k \frac{1}{e^{\beta (E_{k}-\mu)}+1)})^2 }

In the grand canonical scheme, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_f = - \frac{\partial{\Omega}}{\partial{\mu}} }

The interaction induced correction to the chemical potential, δµ, can be found in first order U.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -N_f = -\frac{2}{\beta} \sum_{k} \frac{\beta e^{-\beta(E_k-\mu)}}{e^{-\beta(E_k-\mu)}+1}+2 \frac{U}{M} (\sum_{k} \frac{1}{e^{\beta (E_{k}-\mu)}+1}) \sum_{k'} \frac{\beta e^{\beta (E_{k'}-\mu)}}{(e^{\beta (E_{k'}-\mu)}+1)^2} }

Using the definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu=\mu_o + \delta\mu } , and expanding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -2 \sum_{k} n_f(E_k-\mu_o-\delta\mu) -2\frac{U}{M}\sum_{k} n_f(E_k-\mu_o) \sum_{k'} n_f(E_{k'}-\mu_o) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \simeq -2 \sum_{k} n_f(E_k-\mu_o) + 2\delta\mu \sum_{k'} n_f(E_{k'}-\mu_o)-2 \frac{U}{M}\sum_{k} n_f(E_k-\mu_o) \sum_{k'} n_f(E_{k'}-\mu_o) }

By definition,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_f = 2 \sum_{k} n_f (E_k-\mu_o) }

As a result, solving for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\mu }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\mu = \frac{1}{2} U \frac{N_f}{M} ; \mu = \mu_o +\frac{1}{2} U \frac{N_f}{M} + O(U^2) }

Alternativey, consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{H}-\mu \hat{N} = \\ -t \sum_{\sigma}\sum_{m=1}^{\frac{n_x}{2}}\sum_{n=1}^{\frac{n_y}{2}}(A_{2m,2n,\sigma}^{\dagger}B_{2m+1,2n,\sigma}+A_{2m,2n,\sigma}^{\dagger}B_{2m- 1,2n,\sigma}+A_{2m,2n,\sigma}^{\dagger}B_{2m,2n+1,\sigma}+A_{2m,2n,\sigma}^{\dagger}B_{2m,2n-1,\sigma}+h.c)\\ + U\sum_{m=1}^{\frac{n_x}{2}}\sum_{n=1}^{\frac{n_y}{2}} (A_{2m,2n,\uparrow}^{\dagger}A_{2m,2n,\downarrow}^{\dagger}A_{2m,2n,\downarrow}A_{2m,2n,\uparrow} +B_{2m+1,2n+1,\uparrow}^{\dagger}B_{2m+1,2n+1,\downarrow}^{\dagger}B_{2m+1,2n+1,\downarrow}B_{2m+1,2n+1,\uparrow})\\ - \mu\sum_{\sigma}\sum_{m=1}^{\frac{n_x}{2}}\sum_{n=1}^{\frac{n_y}{2}}(A_{2m,2n,\sigma}^{\dagger}A_{2m,2n,\sigma}+B_{2m+1,2n+1,\sigma}^{\dagger}B_{2m+1,2n+1,\sigma}) \end{align}}

Two-dimensional Hubbard Hamiltonian in Momentum Space

The Hamiltonian for the Hubbard model can be expressed in reciprocal space by performing a Fourier transformation of the real space Hamiltonian. This can be done by writing the creation and annihilation operators in terms of their Fourier series:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{\sigma}(m,n) = \frac{1}{\sqrt{n_{x}n_{y}}}\sum_{k_{x},k_{y}}C_{\sigma}(k_{x},k_{y})e^{i k_{x}m}e^{i k_{y}n} } .

And likewise,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{\sigma}^{+}(m,n) = \frac{1}{\sqrt{n_{x}n_{y}}}\sum_{k_{x},k_{y}}C_{\sigma}^{+}(k_{x},k_{y})e^{-i k_{x}m}e^{-i k_{y}n} } ,

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} are the discrete lattice site positions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{x}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{y}} are the number of sites in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} directions, respectively, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{x}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{y}} are the reciprocal space components to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . The total number of sites is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=n_{x}n_{y}} .

The Hamiltonian including both the direct hopping and interaction terms is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = -t \sum_{m,n}\sum_{m',n'} \sum_{\sigma = \uparrow \downarrow} [C_{\sigma}^{+}(m,n) C_{\sigma}(m',n') + C_{\sigma}^{+}(m',n') C_{\sigma}(m,n)] + U \sum_{m,n}^{} C_{\uparrow}^{+}(m,n) C_{\downarrow}^{+}(m,n) C_{\downarrow}(m,n) C_{\uparrow}(m,n) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m'} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n'} can only be one lattice spacing away from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

Substituting in the Fourier series for $C_{\sigma }^{+}(m,n)$ and $C_{\sigma }^{}(m,n)$, the resulting Hamiltonian is:

$H_{hopping}=-t\sum _{m,n}\sum _{m',n'}\sum _{\sigma =\uparrow \downarrow }[\sum _{k_{x},k_{y}}C_{\sigma }^{+}(k_{x},k_{y})e^{-ik_{x}m}e^{-ik_{y}n}\sum _{p_{x},p_{y}}C_{\sigma }(p_{x},p_{y})e^{ip_{x}m'}e^{ip_{y}n'}+\sum _{k_{x},k_{y}}C_{\sigma }^{+}(k_{x},k_{y})e^{-ik_{x}m'}e^{-ik_{y}n'}\sum _{k_{x},k_{y}}C_{\sigma }(p_{x},p_{y})e^{ip_{x}m}e^{ip_{y}n}]$

Since there is only direct hopping, $m'=m+\lambda$ and $n'=n+\lambda '$ where $\lambda =\pm 1$ when $\lambda '=0$ and vise-versa.

$H_{hopping}=-t\sum _{m,n}\sum _{\lambda ,\lambda '}\sum _{\sigma =\uparrow \downarrow }[{\frac {1}{M}}\sum _{k_{x},k_{y}}\sum _{p_{x},p_{y}}e^{i(p_{x}-k_{x})m}e^{i(p_{y}-k_{y})n}C_{\sigma }^{+}(k_{x},k_{y})C_{\sigma }(p_{x},p_{y})e^{ip_{x}\lambda }e^{ip_{y}\lambda '}+{\frac {1}{M}}\sum _{k_{x},k_{y}}\sum _{p_{x},p_{y}}e^{i(p_{y}-k_{x})m}e^{i(p_{y}-k_{y})n}C_{\sigma }^{+}(k_{x},k_{y})C_{\sigma }(p_{x},p_{y})e^{-ip_{x}\lambda }e^{-ip_{y}\lambda '}]$

However, the sums over $m$ and $n$ create delta-functions:

$\sum _{m,n}e^{i(p_{y}-k_{x})m}e^{i(p_{y}-k_{y})n}=M\delta _{p_{x},k_{x}}\delta _{p_{y},k_{y}}$, so

$H_{hopping}=-t\sum _{k_{x},k_{y}}\sum _{\lambda ,\lambda '}\sum _{\sigma =\uparrow \downarrow }[C_{\sigma }^{+}(k_{x},k_{y})C_{\sigma }(k_{x},k_{y})e^{ik_{x}\lambda }e^{ik_{y}\lambda '}+C_{\sigma }^{+}(k_{x},k_{y})C_{\sigma }(k_{x},k_{y})e^{-ik_{x}\lambda }e^{-ik_{y}\lambda '}]=-2t\sum _{k_{x},k_{y}}\sum _{\sigma =\uparrow \downarrow }(Cos(k_{x})+Cos(k_{y}))C_{\sigma }^{+}(k_{x},k_{y})C_{\sigma }(k_{x},k_{y})=-2t\sum _{\sigma =\uparrow \downarrow }\sum _{k_{x},k_{y}}(Cos(k_{x})+Cos(k_{y}))N_{\sigma }(k_{x},k_{y})=-4t\sum _{k_{x},k_{y}}(Cos(k_{x})+Cos(k_{y}))N(k_{x},k_{y})$

where $N(k_{x},k_{y})$ is the number operator for the state $(k_{x},k{y})$.

Likewise, for the interaction term,

$H_{int}={\frac {U}{M^{2}}}\sum _{m,n}^{}C_{\uparrow }^{+}(m,n)C_{\downarrow }^{+}(m,n)C_{\downarrow }(m,n)C_{\uparrow }(m,n)={\frac {U}{M^{2}}}\sum _{m,n}^{}\sum _{k_{1,x},k_{1,y}}C_{\uparrow }^{+}(k_{1,x},k_{1,y})e^{-ik_{1,x}m}e^{-ik_{1,y}n}\sum _{k_{2,x},k_{2,y}}C_{\downarrow }^{+}(k_{2,x},k_{2,y})e^{-ik_{2,x}m}e^{-ik_{2,y}n}\sum _{k_{3,x},k_{3,y}}C_{\downarrow }(k_{3,x},k_{3,y})e^{ik_{3,x}m}e^{ik_{3,y}n}\sum _{k_{4,x},k_{4,y}}C_{\uparrow }(k_{4,x},k_{4,y})e^{ik_{4,x}m}e^{ik_{4,y}n}$

Implementing the delta-functions again,

$H_{int}={\frac {U}{M}}\sum _{k_{x},k_{y}}\sum _{k_{x}',k_{y}'}C_{\uparrow }^{+}(k_{x},k_{y})C_{\downarrow }^{+}(k_{x}',k_{y}')C_{\downarrow }(k_{x}',k_{y}')C_{\uparrow }(k_{x},k_{y})$

where $k_{1,i}=k_{4,i}=k_{i}$ and $k_{2,i}=k_{3,i}=k_{i}'$.

The total Hamiltonian in reciprocal space is then

$H=H_{hopping}+H_{int}=-4t\sum _{k_{x},k_{y}}(Cos(k_{x})+Cos(k_{y}))N(k_{x},k_{y})+{\frac {U}{M}}\sum _{k_{x},k_{y}}\sum _{k_{x}',k_{y}'}C_{\uparrow }^{+}(k_{x},k_{y})C_{\downarrow }^{+}(k_{x}',k_{y}')C_{\downarrow }(k_{x}',k_{y}')C_{\uparrow }(k_{x},k_{y})$.