Phy5670/HubbardModel 2DCalculations: Difference between revisions
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<math> \delta\mu = \frac{1}{2} U \frac{N_f}{M} ; \mu = \mu_o +\frac{1}{2} U \frac{N_f}{M} + O(U^2) </math> | <math> \delta\mu = \frac{1}{2} U \frac{N_f}{M} ; \mu = \mu_o +\frac{1}{2} U \frac{N_f}{M} + O(U^2) </math> | ||
==== 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: | |||
<math> 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} </math>. | |||
And likewise, | |||
<math> 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} </math>, | |||
where, <math>m</math> and <math>n</math> are the discrete lattice site positions, <math>n_{x}</math> and <math>n_{y}</math> are the number of sites in the <math>x</math> and <math>y</math> directions, respectively, and <math>k_{x}</math> and <math>k_{y}</math> are the reciprocal space components to <math>m</math> and <math>n</math>. The total number of sites is then <math>M=n_{x}n_{y}</math>. | |||
The Hamiltonian including both the direct hopping and interaction terms is: | |||
<math> 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) </math> | |||
where <math>m'</math> and <math>n'</math> can only be one lattice spacing away from <math>m</math> and <math>n</math>. | |||
Substituting in the Fourier series for <math>C_{\sigma}^{+}(m,n)</math> and <math>C_{\sigma}^{}(m,n)</math>, the resulting Hamiltonian is: | |||
<math> 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^{-i k_{x}m}e^{-i k_{y}n} \sum_{p_{x},p_{y}}C_{\sigma}(p_{x},p_{y})e^{i p_{x}m'}e^{i p_{y}n'} + \sum_{k_{x},k_{y}}C_{\sigma}^{+}(k_{x},k_{y})e^{-i k_{x}m'}e^{-i k_{y}n'} \sum_{k_{x},k_{y}}C_{\sigma}(p_{x},p_{y})e^{i p_{x}m}e^{i p_{y}n}]</math> | |||
Since there is only direct hopping, <math>m'=m+\lambda</math> and <math>n'=n+\lambda'</math> where <math>\lambda=\pm 1</math> when <math>\lambda'=0</math> and vise-versa. | |||
<math> 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^{i p_{x}\lambda}e^{i p_{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^{-i p_{x}\lambda}e^{-i p_{y}\lambda'}]</math> | |||
However, the sums over <math>m</math> and <math>n</math> create delta-functions: | |||
<math>\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}}</math>, so | |||
<math> 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^{i k_{x}\lambda}e^{i k_{y}\lambda'} + C_{\sigma}^{+}(k_{x},k_{y})C_{\sigma}(k_{x},k_{y})e^{-i k_{x}\lambda}e^{-i k_{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})</math> | |||
where <math>N(k_{x},k_{y})</math> is the number operator for the state <math>(k_{x},k{y})</math>. | |||
Likewise, for the interaction term, | |||
<math> 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^{-i k_{1,x}m}e^{-i k_{1,y}n}\sum_{k_{2,x},k_{2,y}}C_{\downarrow}^{+}(k_{2,x},k_{2,y})e^{-i k_{2,x}m}e^{-i k_{2,y}n}\sum_{k_{3,x},k_{3,y}}C_{\downarrow}(k_{3,x},k_{3,y})e^{i k_{3,x}m}e^{i k_{3,y}n}\sum_{k_{4,x},k_{4,y}}C_{\uparrow}(k_{4,x},k_{4,y})e^{i k_{4,x}m}e^{i k_{4,y}n} </math> | |||
Implementing the delta-functions again, | |||
<math> 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})</math> | |||
where <math>k_{1,i}=k_{4,i}=k_{i}</math> and <math>k_{2,i}=k_{3,i}=k_{i}'</math>. | |||
The total Hamiltonian in reciprocal space is then | |||
<math>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})</math>. | |||
Revision as of 02:15, 13 December 2012
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) }
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 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)} and , the resulting Hamiltonian is:
![{\displaystyle 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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f44bb23daf2d651ce5c8e2a84a7819dc92543e3)
Since there is only direct hopping, and where when and vise-versa.
![{\displaystyle 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 '}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79d0c30c2099a3e586985d88248f5cb9367889b)
However, the sums over and create delta-functions:
, so
![{\displaystyle 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37adf3f4929c996a3a144a8697ca65282f89317)
where is the number operator for the state .
Likewise, for the interaction term,
Implementing the delta-functions again,
where and .
The total Hamiltonian in reciprocal space is then
.
