$$
\newcommand\Tr{\mathrm{Tr}}
\newcommand{\braket}[2]{\langle #1 \mid #2 \rangle}
\newcommand\I{\mathbb{I}}
\newcommand{\avg}[1]{\left< #1 \right>}
\newcommand{\RD}{D}
\newcommand{\ri}{\mathrm{i}}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\Sign}{Sign}
\newcommand{\ii}{\mathrm i}
\newcommand{\vv}{\mathrm v}
\newcommand{\ff}{\mathrm f}
\newcommand{\mm}{\mathrm m}
\newcommand{\ee}{\mathrm e}
\newcommand{\xx}{\mathrm x}
\newcommand{\RR}{\mathrm R}
\newcommand{\dd}{\mathrm d}
\newcommand{\FF}{\mathrm F}
\newcommand{\BB}{\mathrm B}
\newcommand{\vph}{v_{\mathrm{ph}}}
$$