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:

$H=-t\sum _{r=1}^{n_{x}}\sum _{s=1}^{n_{y}}\sum _{\sigma =\uparrow ,\downarrow }(c_{r,s,\sigma }^{\dagger }c_{r+1,s,\sigma }+c_{r,s,\sigma }^{\dagger }c_{r,s+1,\sigma }+h.c.)+U\sum _{r=1}^{n_{x}}\sum _{s=1}^{n_{y}}c_{r,s,\uparrow }^{\dagger }c_{r,s,\downarrow }^{\dagger }c_{r,s,\downarrow }c_{r,s,\uparrow }$

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

$Z=Tr{\big [}e^{-\beta (H-\mu N)}{\big ]}=e^{-\beta \Omega }$

which can be expanded as:

$Z=Z_{0}{\big \langle }e^{-S_{int}}{\big \rangle }=Z_{0}e^{-\langle S_{int}\rangle }e^{{\frac {1}{2}}(\langle S_{int}^{2}\rangle -\langle S_{int}\rangle ^{2})}=e^{-\beta \Omega _{0}}e^{-\beta \Omega _{1}}e^{-\beta \Omega _{2}}$

$S_{int}=\int _{0}^{\beta }d\tau H_{int}(\tau )$

which utilizes cumulant expansion. We begin to calculate the grand canonical potential by analyzing the contribution from $Z_{0}$:

$Z_{0}=\prod _{k}(1+e^{-\beta (E_{k}-\mu )})^{2}=e^{2\sum _{k}ln(1+e^{-\beta (E_{k}-\mu )})}=e^{-\beta \Omega _{0}}$

$\Omega _{0}=-{\frac {2}{\beta }}\sum _{k}ln(1+e^{-\beta (E_{k}-\mu )})$

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

$c_{r,s,\sigma }={\frac {1}{\sqrt {M}}}\sum _{k_{x},k_{y}}e^{ik_{x}\cdot r}e^{ik_{y}\cdot s}c_{k_{x},k_{y},\sigma }$

$H_{int}=U\sum _{r=1}^{n_{x}}\sum _{s=1}^{n_{y}}c_{r,s,\uparrow }^{\dagger }c_{r,s,\downarrow }^{\dagger }c_{r,s,\downarrow }c_{r,s,\uparrow }$

$={\frac {U}{M^{2}}}\sum _{k_{x_{1}}...k_{x_{4}}}\sum _{k_{y_{1}}...k_{y_{4}}}e^{-ik_{x_{1}}r}e^{-ik_{x_{2}}r}e^{ik_{x_{3}}r}e^{ik_{x_{4}}r}e^{-ik_{y_{1}}s}e^{-ik_{y_{2}}s}e^{ik_{y_{3}}s}e^{ik_{y_{4}}s}c_{k_{x_{1}},k_{y_{1}},\uparrow }^{\dagger }c_{k_{x_{2}},k_{y_{2}},\downarrow }^{\dagger }c_{k_{x_{3}},k_{y_{3}},\downarrow }c_{k_{x_{4}},k_{y_{4}},\uparrow }$

$={\frac {U}{M}}\sum _{k_{x_{1}}...k_{x_{4}}}\sum _{k_{y_{1}}...k_{y_{4}}}\delta _{k_{x_{1}}+k_{x_{2}},k_{x_{3}}+k_{x_{4}}}\delta _{k_{y_{1}}+k_{y_{2}},k_{y_{3}}+k_{y_{4}}}c_{k_{x_{1}},k_{y_{1}},\uparrow }^{\dagger }c_{k_{x_{2}},k_{y_{2}},\downarrow }^{\dagger }c_{k_{x_{3}},k_{y_{3}},\downarrow }c_{k_{x_{4}},k_{y_{4}},\uparrow }$

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) }