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


<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>
<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> 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> \Omega_1 = \frac{U}{M} \sum_{k,k'} n_{F}^2(\epsilon_k) </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>
Combining both terms, the grand canonical potential to first order is:


<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>
<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>


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:
====Calculation of the Chemical Potential====


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


The only contraction combination possible, due to orthogonal spins, results in the following set of Green's functions:
<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}(\tau) c_{k',\downarrow}^{\dagger}(\tau) c_{k'+q,\downarrow}(\tau) c_{k-q,\uparrow}(\tau) \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} 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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