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=Hubbard Model: 2D Calculations=
=Hubbard Model: 2D Calculations=


Expansion of the Hubbard model Hamiltonian into two dimensions allows us to calculate various properties, including the grand canonical potential and corrections to the chemical potential. In 2D, the Hamiltonian can be written as:
==== Two-dimensional Hubbard Hamiltonian in Momentum Space ====


<math> H = -t \sum_{r=1}^{n_x} \sum_{s=1}^{n_y} \sum_{\sigma = \uparrow,\downarrow} 
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:
(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} </math>


The grand canonical potential, Omega, is best calculated by using coherent state path integral. The grand partition function is defined as:
<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}^{\dagger}(m,n) = \frac{1}{\sqrt{n_{x}n_{y}}}\sum_{k_{x},k_{y}}C_{\sigma}^{\dagger}(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}^{\dagger}(m,n) C_{\sigma}(m',n') + C_{\sigma}^{\dagger}(m',n') C_{\sigma}(m,n)]
+ U \sum_{m,n}^{} C_{\uparrow}^{\dagger}(m,n) C_{\downarrow}^{\dagger}(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}^{\dagger}(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}
\Big [ \sum_{k_{x},k_{y}}C_{\sigma}^{\dagger}(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}^{\dagger}(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} \Big]</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}
\bigg[\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}^{\dagger}(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}^{\dagger}(k_{x},k_{y})C_{\sigma}(p_{x},p_{y})e^{-i p_{x}\lambda}e^{-i p_{y}\lambda'}\bigg]</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>
 
therefore, the hopping term becomes:
 
<math>
\begin{align}
H_{hopping}
&= -t \sum_{k_{x},k_{y}}\sum_{\lambda,\lambda'} \sum_{\sigma\uparrow\downarrow}[C_{\sigma}^{\dagger}(k_{x},k_{y})C_{\sigma}(k_{x},k_{y})e^{i k_{x}\lambda}e^{i k_{y}\lambda'} + C_{\sigma}^{\dagger}(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}^{\dagger}(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})\end{align} </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}^{\dagger}(m,n) C_{\downarrow}^{\dagger}(m,n) C_{\downarrow}(m,n) C_{\uparrow}(m,n) </math>
<math> = \frac{U}{M^{2}} \sum_{m,n}^{} \sum_{k_{1,x},k_{1,y}}C_{\uparrow}^{\dagger}(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}^{\dagger}(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 and defining <math>\bold{k}=(k_{x},k_{y})</math>,
 
<math>\bold{k_{1}}-\bold{k_{4}}=\bold{k_{3}-\bold{k_{2}}}</math>.
 
This relation can be satisfied by defining <math>\bold{q}</math> such that <math> \bold{k_{1}}=\bold{k}, \bold{k_{4}}=\bold{k}-\bold{q}, \bold{k_{2}}=\bold{k}', \bold{k_{3}}=\bold{k}'+\bold{q}</math>.
 
And so,
 
<math> H_{int} = \frac{U}{M}\sum_{\bold{k},\bold{k}',\bold{q}}C_{\uparrow}^{\dagger}(\bold{k})C_{\downarrow}^{\dagger}(\bold{k}')C_{\downarrow}(\bold{k}'+\bold{q})C_{\uparrow}(\bold{k}-\bold{q})</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_{\bold{k},\bold{k}',\bold{q}}C_{\uparrow}^{\dagger}(\bold{k})C_{\downarrow}^{\dagger}(\bold{k}')C_{\downarrow}(\bold{k}'+\bold{q})C_{\uparrow}(\bold{k}-\bold{q})</math>.
 
====Calculation of the Grand Canonical Potential====
 
The grand canonical potential, <math>\Omega</math>, is best calculated by using coherent state path integral. The grand partition function is defined as:


<math> Z = Tr \big[e^{-\beta (H - \mu N)} \big] = e^{-\beta \Omega} </math>
<math> Z = Tr \big[e^{-\beta (H - \mu N)} \big] = e^{-\beta \Omega} </math>
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<math> \Omega_0 = -\frac{2}{\beta} \sum_k ln(1+e^{-\beta(E_k - \mu)}) </math>
<math> \Omega_0 = -\frac{2}{\beta} \sum_k ln(1+e^{-\beta(E_k - \mu)}) </math>


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


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


<math> 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} </math>
<math> \begin{align} \langle S_{int} \rangle &= \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 \\
&= \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) \\
&= {\beta} \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) = \beta \Omega_1 \end{align} </math>


<math> = \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} </math>
<math> \Omega_1 = \frac{U}{M} \sum_{k,k'} n_{F}^2(\epsilon_k) </math>


<math> = \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} </math>
Combining both terms, the grand canonical potential to first order is:


For simplicity, we will combine the <math> k_x </math> and <math> k_y </math> into a single index as <math> k </math>. Evaluating the Kronecker deltas yields:
<math> \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) </math>


<math> 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} </math>
====Calculation of the Chemical Potential====


The only contraction combination possible, due to orthogonal spins, results in the following set of Green's functions:
The Grand Canonical Potential for a 2-D lattice is defined as
 
<math> \Omega = -\frac{2}{\beta} \sum_{k}^{} ln(1 + e^{\beta (E_{k}-\mu)})+ \frac{U}{M} (\sum_k \frac{1}{e^{\beta (E_{k}-\mu)}+1)})^2 </math>
 
In the grand canonical scheme,  
<math> N_f = - \frac{\partial{\Omega}}{\partial{\mu}} </math>
 
The interaction induced correction to the chemical potential, δµ, can be found in first order U.
 
<math> -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}
</math>
 
Using the definition, <math> \mu=\mu_o + \delta\mu </math>, and expanding <math> \mu </math>
 
<math>
= -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)
</math>
 
<math>
\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)
</math>
 
By definition,
 
<math> N_f = 2 \sum_{k} n_f \left (E_k - \mu_o \right) </math>
 
As a result, solving for <math> \delta\mu </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>
 
Alternativey, consider
 
<math> \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}</math>
 
The condition for half-filling, or one electron per site is given by:
 
<math>\sum_{\sigma} \langle A_{2m,2n,\sigma}^{\dagger}A_{2m,2n,\sigma} \rangle = 1</math>
 
The grand canonical potential is invariant under the particle hole transformation for half filling, that is:
<math> A = d^{\dagger}; B = -f^{\dagger}; A^{\dagger} = d; B^{\dagger} = -f</math>
 
where d and f obey the commutation relations
<math>
[f,f{\dagger}]=1; [d,d{\dagger}]=1</math>
 
A short digression into the nature of f and d reveals that
<math>
 
\sum_{\sigma} \langle d_{2m,2n,\sigma}d_{2m,2n,\sigma}^{\dagger} \rangle = 1
 
</math>
 
Using the commutation relations, this shows the grand canonical potential to be invariant under particle hole transformation
 
<math> 1- \langle d_{2m,2n,\uparrow}^{\dagger}d_{2m,2n,\uparrow} \rangle + 1 - \langle d_{2m,2n,\downarrow}^{\dagger}d_{2m,2n,\downarrow} \rangle = 1 </math>
 
<math> \langle d_{2m,2n,\uparrow}^{\dagger}d_{2m,2n,\uparrow} \rangle + \langle d_{2m,2n,\downarrow}^{\dagger}d_{2m,2n,\downarrow} \rangle = 1</math>
 
This recovers the half filling condition for a particle hole transformation.
 
Substituting d and f into the Hamiltonian and using the commutation relations to create an analogue to the original <math>\hat{H}-\mu\hat{N}</math>,
it is evident that when comparing
 
<math> H\left(t,\mu,U\right)=H\left(t,\mu,-\mu+U\right)</math>
 
Under such a transformation, it is clear that the chemical potential is
 
<math>
-\mu+U=-\mu \Rightarrow \mu=\frac{U}{2}
</math>
 
This recovers the condition for half-filling from above since <math> M = N_{f} </math>
 
==== Isothermal Compressibility ====
 
In the limit of low temperature, the isothermal compressibility,<math>\kappa_{T}</math> which satisfies the relation <math>\kappa_{T}\rho^{2}=\frac{\partial\rho}{\partial\mu}</math>, can be corrected as a function of <math>\mu_{0}</math> to first order in <math>U</math>.
 
<math>\rho=\frac{N_{F}}{M}</math>
 
where
 
<math>N_{F}=\sum_{\bold{k}}n_{F}(E_{\bold{k}}-\mu)+2\frac{U}{M}\sum_{\bold{k}}n_{F}(E_{\bold{k}}-\mu)\sum_{\bold{k}'}\frac{\partial n_{F}(E_{\bold{k}'}-\mu)}{\partial\mu}</math>
 
and <math>n_{F}</math> is the Fermi-Dirac distribution function.
 
<math>\frac{\partial N_{F}}{\partial \mu}=-2\sum_{\bold{k}}n_{F}'(E_{\bold{k}}-\mu)-2\frac{U}{M}(\sum_{\bold{k}}n_{F}'(E_{\bold{k}}-\mu))^{2}-2\frac{U}{M}\sum_{\bold{k}}n_{F}(E_{\bold{k}}-\mu)\sum_{\bold{k}'}n_{F}''(E_{\bold{k}'}-\mu)</math>
 
where the primes on the distribution functions denote derivatives with respect to <math>\mu</math>.
 
Setting <math>\mu=\mu_{0}+\delta\mu</math> and expanding around <math>\delta\mu</math>,


<math> \langle S_{int} \rangle = \bigg \langle \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k',q} c_{k,\uparrow}^{\dagger} c_{k',\downarrow}^{\dagger} c_{k'+q,\downarrow} c_{k-q,\uparrow} \bigg \rangle </math>
<math>\frac{\partial N_{F}}{\partial \mu}\simeq-2\sum_{\bold{k}}n_{F}'(E_{\bold{k}}-\mu_{0})+2\delta\mu\sum_{\bold{k}}n_{F}''(E_{\bold{k}}-\mu_{0})-2\frac{U}{M}(\sum_{\bold{k}}n_{F}'(E_{\bold{k}}-\mu_{0}))^{2}+\frac{U}{M}N_{F}\sum_{\bold{k}}n_{F}''(E_{\bold{k}}-\mu_{0})+O(U^{2})</math>.


<math> = \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 </math>
Setting <math>\delta\mu=\frac{UN_{F}}{2M}</math>,


<math> = \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_k) n_{F}(\epsilon_k') </math>
<math>\frac{\partial N_{F}}{\partial \mu}\simeq(-2\sum_{\bold{k}}n_{F}'(E_{\bold{k}}-\mu_{0}))(1+\frac{U}{M}\sum_{\bold{k}'}n_{F}'(E_{\bold{k}'}-\mu_{0}))</math>.


<math> = {\beta} \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_k) n_{F}(\epsilon_k') = \beta \Omega_1 </math>
The above expression can be written in terms of the density of states, and taking <math>T=0</math>:


<math> \Omega_1 = \frac{U}{M} \sum_{k,k'} n_{F}^2(\epsilon_k) </math>
<math>\frac{1}{M}\frac{\partial N_{F}}{\partial\mu}=\frac{\partial\rho}{\partial\mu}=2N(\mu_{0})(1-U N(\mu_{0}))=\rho^{2}\kappa_{T}</math>


Combining both terms, the grand canonical potential to first order is:
where <math>N(\mu_{0})</math> is the density of states at <math>\mu</math> and


<math> \Omega = \Omega_0 + \Omega_1 = -\frac{2}{\beta} \sum_k ln(1+e^{-\beta(E_k - \mu)}) + \frac{U}{M} \sum_{k,k'} n_{F}^2(\epsilon_k) </math>
<math>\kappa_{T}=\frac{2N(\mu_{0})(1-U N(\mu_{0}))}{\rho^{-2}}</math>.

Latest revision as of 12:57, 14 December 2012

Hubbard Model: 2D Calculations

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}^{\dagger}(m,n) = \frac{1}{\sqrt{n_{x}n_{y}}}\sum_{k_{x},k_{y}}C_{\sigma}^{\dagger}(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}^{\dagger}(m,n) C_{\sigma}(m',n') + C_{\sigma}^{\dagger}(m',n') C_{\sigma}(m,n)] + U \sum_{m,n}^{} C_{\uparrow}^{\dagger}(m,n) C_{\downarrow}^{\dagger}(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}^{\dagger}(m,n)} 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 C_{\sigma}^{}(m,n)} , the resulting Hamiltonian 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_{hopping} = -t \sum_{m,n}\sum_{m',n'} \sum_{\sigma = \uparrow \downarrow} \Big [ \sum_{k_{x},k_{y}}C_{\sigma}^{\dagger}(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}^{\dagger}(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} \Big]}

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

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_{hopping} = -t \sum_{m,n}\sum_{\lambda,\lambda'} \sum_{\sigma = \uparrow \downarrow} \bigg[\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}^{\dagger}(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}^{\dagger}(k_{x},k_{y})C_{\sigma}(p_{x},p_{y})e^{-i p_{x}\lambda}e^{-i p_{y}\lambda'}\bigg]}

However, the sums over 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} create delta-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 \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}}}

therefore, the hopping term becomes:

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} H_{hopping} &= -t \sum_{k_{x},k_{y}}\sum_{\lambda,\lambda'} \sum_{\sigma\uparrow\downarrow}[C_{\sigma}^{\dagger}(k_{x},k_{y})C_{\sigma}(k_{x},k_{y})e^{i k_{x}\lambda}e^{i k_{y}\lambda'} + C_{\sigma}^{\dagger}(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}^{\dagger}(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})\end{align} }

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

Likewise, for the interaction term,

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^{2}} \sum_{m,n}^{} C_{\uparrow}^{\dagger}(m,n) C_{\downarrow}^{\dagger}(m,n) C_{\downarrow}(m,n) C_{\uparrow}(m,n) } 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_{m,n}^{} \sum_{k_{1,x},k_{1,y}}C_{\uparrow}^{\dagger}(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}^{\dagger}(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} }

Implementing the delta-functions again and defining 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 \bold{k}=(k_{x},k_{y})} ,

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 \bold{k_{1}}-\bold{k_{4}}=\bold{k_{3}-\bold{k_{2}}}} .

This relation can be satisfied by defining 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 \bold{q}} such that 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 \bold{k_{1}}=\bold{k}, \bold{k_{4}}=\bold{k}-\bold{q}, \bold{k_{2}}=\bold{k}', \bold{k_{3}}=\bold{k}'+\bold{q}} .

And so,

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_{\bold{k},\bold{k}',\bold{q}}C_{\uparrow}^{\dagger}(\bold{k})C_{\downarrow}^{\dagger}(\bold{k}')C_{\downarrow}(\bold{k}'+\bold{q})C_{\uparrow}(\bold{k}-\bold{q})}

The total Hamiltonian in reciprocal space 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 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_{\bold{k},\bold{k}',\bold{q}}C_{\uparrow}^{\dagger}(\bold{k})C_{\downarrow}^{\dagger}(\bold{k}')C_{\downarrow}(\bold{k}'+\bold{q})C_{\uparrow}(\bold{k}-\bold{q})} .

Calculation of the Grand Canonical Potential

The grand canonical potential, 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} , 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.

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 }

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 \begin{align} \langle S_{int} \rangle &= \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 \\ &= \int_{0}^{\beta} d\tau \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) \\ &= {\beta} \frac{U}{M} \sum_{k,k'} n_{F}(\epsilon_{k}) n_{F}(\epsilon_{k'}) = \beta \Omega_1 \end{align} }

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:

Calculation of the Chemical Potential

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

In the grand canonical scheme,

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

Using the definition, , and expanding

By definition,

As a result, solving for

Alternativey, consider

The condition for half-filling, or one electron per site is given by:

The grand canonical potential is invariant under the particle hole transformation for half filling, that is:

where d and f obey the commutation relations

A short digression into the nature of f and d reveals that

Using the commutation relations, this shows the grand canonical potential to be invariant under particle hole transformation

This recovers the half filling condition for a particle hole transformation.

Substituting d and f into the Hamiltonian and using the commutation relations to create an analogue to the original , it is evident that when comparing

Under such a transformation, it is clear that the chemical potential is

This recovers the condition for half-filling from above since

Isothermal Compressibility

In the limit of low temperature, the isothermal compressibility, which satisfies the relation , can be corrected as a function of to first order in .

where

and is the Fermi-Dirac distribution function.

where the primes on the distribution functions denote derivatives with respect to .

Setting and expanding around ,

.

Setting ,

.

The above expression can be written in terms of the density of states, and taking :

where is the density of states at and

.

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