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\usepackage{amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}


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\newcommand{\bC}{\boldsymbol{C}}
\newcommand{\bF}{\boldsymbol{F}}
\newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bHd}{\boldsymbol{H}_\text{d}}
\newcommand{\bHod}{\boldsymbol{H}_\text{od}}
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bEta}{\boldsymbol{\eta}}
\newcommand{\bSig}{\boldsymbol{\Sigma}}
\newcommand{\bpsi}{\boldsymbol{\psi}}
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\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\
\url{https://lcpq.github.io/pterosor}}
\usetheme{pterosor}
\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
\date{14th November 2022}
\title{Similarity Renormalization Group (SRG) Formalism Applied to Green's Function Methods}

\begin{document}

\maketitle

%-----------------------------------------------------
\begin{frame}{First-Quantized Form of SRG}
	\begin{block}{General upfolded many-body perturbation theory (MBPT) problem}
    	\begin{align}
    		\qty[ \bF + \bSig(\omega) ] \bpsi = \omega \bpsi
			& \qq{$\Leftrightarrow$}
    		\bH \bPsi = \omega \bPsi
			\\
			\bSig(\omega) = \bV \qty(\omega \bI - \bC)^{-1} \bV^{\dag} 
			& \qq{$\Leftrightarrow$}
			\bH =
    		\begin{pmatrix}
    			\bF			&	\bV
    			\\
    			\bV^{\dagger}	&	\bC
    		\end{pmatrix}
    	\end{align}	
	\end{block}
	%
	\begin{block}{Perturbative partitioning}
        \begin{equation}
        	\bH \equiv \bH(s=0) =
            \underbrace{
            \begin{pmatrix}
            	\bF & \bO 
				\\
            	\bO		& \bC
            \end{pmatrix}
            }_{\bHd^{(0)}(s=0)}
            + \lambda
			\underbrace{
			\begin{pmatrix}
				\bO				& \bV
				\\
            	\bV^{\dagger}	& \bO
			\end{pmatrix}
			}_{\bHod^{(1)}(s=0)}
			\qq{with}
			\bHd^{(1)}(s=0) = \bHod^{(0)}(s=0) = \bO 
       \end{equation}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Perturbative Expansions}
	%
	\begin{block}{Perturbative partitioning in the SRG framework}
        \begin{equation}
        	\bH(s) =
            \underbrace{
            \begin{pmatrix}
            	\bF(s) & \bO 
				\\
            	\bO		& \bC(s) 
            \end{pmatrix}
            }_{\bHd{}(s)}
            + \lambda
			\underbrace{
			\begin{pmatrix}
				\bO				& \bV(s) 
				\\
            	\bV^{\dagger}(s)	& \bO
			\end{pmatrix}
			}_{\bHod(s)}
       \end{equation}
	\end{block}
	%
	\begin{block}{Components of the Hamiltonian}
        \begin{equation}
			\bH(s) = \bH^{(0)}(s) + \lambda \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots
       \end{equation}	
        \begin{equation}
			\bC(s) = \bC^{(0)}(s) + \lambda \bC^{(1)}(s) + \lambda^2 \bC^{(2)}(s) + \cdots
       \end{equation}	
        \begin{equation}
			\bV(s) = \bV^{(0)}(s) + \lambda \bV^{(1)}(s) + \lambda^2 \bV^{(2)}(s) + \cdots
       \end{equation}	
 	\end{block}
	\begin{block}{Wegner generator}
        \begin{equation}
			\bEta(s) = \bEta^{(0)}(s) + \lambda \bEta^{(1)}(s) + \lambda^2 \bEta^{(2)}(s) + \cdots
       \end{equation}	
 	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Zeroth-Order Terms}
	\begin{block}{Wegner generator}
        \begin{equation}
			\bEta^{(0)}(s) 
			= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} 
			= \bO
       \end{equation}	
 	\end{block}
	%
	\begin{block}{Zeroth-order Hamiltonian}
        \begin{equation}
			\dv{\bH^{(0)}(s)}{s} 
			= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)} 
			= \bO
			\qq{$\Rightarrow$} 
			\bH^{(0)}(s) = \bH^{(0)}{(s=0)}
       \end{equation}	
 	\end{block}
	\alert{NB: we omit the $s$ dependency from hereon}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{First-Order Terms}
	\begin{block}{Wegner generator}
        \begin{equation}
			\bEta^{(1)} = \comm{\bHd^{(0)}}{\bHod^{(1)}} 
			= 
			\begin{pmatrix}
				\bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)}
				\\
				\bC^{(0)} \bV^{(1),\dagger} - \bV^{(1),\dagger} \bF^{(0)} & \bO
 			 \end{pmatrix}         
  		\end{equation}	
 	\end{block}
	%
	\begin{block}{First-order Hamiltonian}
		\begin{equation}
			\dv{\bH^{(1)}}{s} = \comm{\bEta^{(1)}}{\bHd^{(0)}} 
			=
			\begin{pmatrix}
				\dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s} 
				\\
				\dv{\bV^{(1),\dagger}}{s} & \dv{\bC^{(1)}}{s}
			\end{pmatrix} 
		\end{equation}
		with
		\begin{gather}
			\dv{\bF^{(1)}}{s} 
			= \dv{\bC^{(1)}}{s} 
			= \bO
			\\
			\dv{\bV^{(1)}}{s} 
			= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
			- \qty[\bF^{(0)}]^2 \bV^{(1)} 
			- \bV^{(1)} \qty[\bC^{(0)}]^2
		\end{gather}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Second-Order Terms}
	\begin{block}{Wegner generator}
        \begin{equation}
			\bEta^{(2)}
			= \comm{\bHd^{(0)}}{\bHod^{(2)}}
			+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{=\bO}  
			= 
			\begin{pmatrix}
				\bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)}
				\\
				\bC^{(0)} \bV^{(2),\dagger} - \bV^{(2),\dagger} \bF^{(0)} & \bO
 			 \end{pmatrix}      
  		\end{equation}	
 	\end{block}
	%
	\begin{block}{Second-order Hamiltonian}
		\begin{equation}
			\dv{\bH^{(2)}}{s} = \comm{\bEta^{(2)}}{\bHd^{(0)}} + \comm{\bEta^{(1)}}{\bHd^{(1)}} 
			=
			\begin{pmatrix}
				\dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s} 
				\\
				\dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s}
			\end{pmatrix} 
		\end{equation}
		with
		\begin{gather}
			\dv{\bF^{(2)}}{s} 
			= \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} 
			+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} 
			- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
			\\
			\dv{\bC^{(2)}}{s} 
			= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} 
			+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} 
			- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
			\\
			\dv{\bV^{(2)}}{s} 
			= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
			- \qty[\bF^{(0)}]^2 \bV^{(2)} 
			- \bV^{(2)} \qty[\bC^{(0)}]^2
		\end{gather}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Integration of the First-Order Terms}
		\begin{equation}
			\dv{\bF^{(1)}}{s} = \bO 
			\land 
			\bF^{(1)}(0) = \bO 
			\Rightarrow 
			\bF^{(1)}(s) = \bO
		\end{equation}
		\begin{equation}
			\dv{\bC^{(1)}}{s} 
			\land 
			\bC^{(1)}(0) = \bO 
			\Rightarrow 
			\bC^{(1)}(s) = \bO
		\end{equation}
		\begin{equation}
			\dv{\bV^{(1)}}{s} 
			= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
			- \qty[\bF^{(0)}]^2 \bV^{(1)} 
			- \bV^{(1)} \qty[\bC^{(0)}]^2
			\land 
			\Rightarrow 
			\bV^{(1)}(s) = ?
		\end{equation}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Integration of the Second-Order Terms}
		\begin{equation}
			\dv{\bF^{(2)}}{s} 
			= \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} 
			+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} 
			- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
			\land 			
			\bF^{(2)}(0)  = ?  
			\Rightarrow 
			\bF^{(2)}(s) = ?
		\end{equation}
		\begin{equation}
			\dv{\bC^{(2)}}{s} 
			= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} 
			+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} 
			- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
			\land 
			\bC^{(2)}(0) = ? 
			\Rightarrow 
			\bC^{(2)}(s) = ?
		\end{equation}
		\begin{equation}
			\dv{\bV^{(2)}}{s} 
			= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
			- \qty[\bF^{(0)}]^2 \bV^{(2)} 
			- \bV^{(2)} \qty[\bC^{(0)}]^2
			\land 
			\Rightarrow 
			\bV^{(2)}(s) = ?
		\end{equation}
\end{frame}
%-----------------------------------------------------

\end{document}
