\( %Residues \let\Res\relax \DeclareMathOperator*{\Res}{Res} %Allow us to write angstroms \let\ring\r \renewcommand{\r}{\textbf{r}} \let\Oersted\O \let\O\relax \DeclareMathOperator{\O}{O} %Define useful symbols for commonly used sets in mathematics \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\TT}{\mathbb{T}} %\newcommand{\H}{\mathbb{H}} %Useful symbols for differential geometry \newcommand{\M}{\mathcal{M}} \renewcommand{\d}{\text{d}} \newcommand{\Sphere}{\mathbb{S}} \newcommand{\Torus}{\mathbb{T}} %Useful notations for electromagnetism \newcommand{\E}{\mathbf{E}} \newcommand{\B}{\mathbf{B}} \newcommand{\A}{\mathbf{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\J}{\mathbf{J}} \newcommand{\R}{\boldsymbol{R}} %Useful symbols for vector notation \newcommand{\x}{\boldsymbol{x}} \newcommand{\y}{\boldsymbol{y}} \newcommand{\z}{\boldsymbol{z}} \newcommand{\p}{\boldsymbol{p}} \newcommand{\q}{\boldsymbol{q}} \newcommand{\w}{\boldsymbol{w}} \renewcommand{\k}{\boldsymbol{k}} \newcommand{\K}{\boldsymbol{K}} \newcommand{\Q}{\boldsymbol{Q}} \renewcommand{\r}{\boldsymbol{r}} %Useful symbols for physics \newcommand{\hc}{\text{H.c.}} \newcommand{\Hc}{\text{H.c.}} \newcommand{\BZ}{\text{BZ}} \newcommand{\Kn}{\text{Kn}} \newcommand{\g}{\mathfrak{g}} \newcommand{\ahat}{\hat{a}} \newcommand{\bhat}{\hat{b}} \newcommand{\chat}{\hat{c}} \newcommand{\dhat}{\hat{d}} \newcommand{\ehat}{\hat{e}} \newcommand{\fhat}{\hat{f}} \newcommand{\ghat}{\hat{g}} \newcommand{\hhat}{\hat{h}} \newcommand{\ihat}{\hat{i}} \newcommand{\jhat}{\hat{j}} \newcommand{\khat}{\hat{k}} \newcommand{\lhat}{\hat{l}} \newcommand{\mhat}{\hat{m}} \newcommand{\nhat}{\hat{n}} \newcommand{\ohat}{\hat{o}} \newcommand{\phat}{\hat{p}} \newcommand{\qhat}{\hat{q}} \newcommand{\rhat}{\hat{r}} \newcommand{\shat}{\hat{s}} \newcommand{\that}{\hat{t}} \newcommand{\uhat}{\hat{u}} \newcommand{\vhat}{\hat{v}} \newcommand{\what}{\hat{w}} \newcommand{\xhat}{\hat{x}} \newcommand{\yhat}{\hat{y}} \newcommand{\zhat}{\hat{z}} \newcommand{\Ahat}{\hat{A}} \newcommand{\Bhat}{\hat{B}} \newcommand{\Chat}{\hat{C}} \newcommand{\Dhat}{\hat{D}} \newcommand{\Ehat}{\hat{E}} \newcommand{\Fhat}{\hat{F}} \newcommand{\Ghat}{\hat{G}} \newcommand{\Hhat}{\hat{H}} \newcommand{\Ihat}{\hat{I}} \newcommand{\Jhat}{\hat{J}} \newcommand{\Khat}{\hat{K}} \newcommand{\Lhat}{\hat{L}} \newcommand{\Mhat}{\hat{M}} \newcommand{\Nhat}{\hat{N}} \newcommand{\Ohat}{\hat{O}} \newcommand{\Phat}{\hat{P}} \newcommand{\Qhat}{\hat{Q}} \newcommand{\Rhat}{\hat{R}} \newcommand{\Shat}{\hat{S}} \newcommand{\That}{\hat{T}} \newcommand{\Uhat}{\hat{U}} \newcommand{\Vhat}{\hat{V}} \newcommand{\What}{\hat{W}} \newcommand{\Xhat}{\hat{X}} \newcommand{\Yhat}{\hat{Y}} \newcommand{\Zhat}{\hat{Z}} \newcommand{\HH}{\mathcal{H}} \newcommand{\Pihat}{\hat{\Pi}} \newcommand{\sigmahat}{\hat{\sigma}} \newcommand{\psihat}{\hat{\psi}} \newcommand{\Psihat}{\hat{\Psi}} \newcommand{\phihat}{\hat{\phi}} \newcommand{\thetahat}{\hat{\theta}} \newcommand{\Thetahat}{\hat{\Theta}} \newcommand{\chihat}{\hat{\chi}} \newcommand{\Phihat}{\hat{\Phi}} \newcommand{\Omegahat}{\hat{\Omega}} \newcommand{\Deltahat}{\hat{\Delta}} \newcommand{\pihat}{\hat{\pi}} \newcommand{\varphihat}{\hat{\varphi}} \newcommand{\rhohat}{\hat{\rho}} \newcommand{\gammahat}{\hat{\gamma}} \newcommand{\etahat}{\hat{\eta}} \newcommand{\V}{\mathcal{V}} \renewcommand{\L}{\mathcal{L}} \newcommand{\D}{\mathcal{D}} \newcommand{\id}{\text{id}} \DeclareMathOperator{\Vol}{Vol} \DeclareMathOperator{\atan}{atan} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\preim}{preim} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\ver}{ver} \DeclareMathOperator{\hor}{hor} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\const}{const} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\SU}{SU} \DeclareMathOperator{\Cl}{C\ell} \DeclareMathOperator{\conf}{conf} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\algsl}{\mathfrak{sl}} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\algsu}{\mathfrak{su}} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Unit}{U} \DeclareMathOperator{\algu}{\mathfrak{u}} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\algso}{\mathfrak{so}} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Log}{Log} \DeclareMathOperator{\arccot}{arccot} \DeclareMathOperator{\arccsc}{arccsc} \DeclareMathOperator{\arcsec}{arcsec} \DeclareMathOperator{\artanh}{artanh} \DeclareMathOperator{\arsinh}{arsinh} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\arcoth}{arcoth} \DeclareMathOperator{\arcsch}{arcsch} \DeclareMathOperator{\arsech}{arsech} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\csch}{csch} \DeclareMathOperator{\Alt}{Alt} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Arg}{Arg} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Spin}{Spin} \DeclareMathOperator{\Diff}{Diff} \DeclareMathOperator{\Der}{Der} \DeclareMathOperator{\Witt}{Witt} \DeclareMathOperator{\Vir}{Vir} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Pf}{Pf} \DeclareMathOperator{\Li}{Li} \DeclareMathOperator{\vecspan}{span} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\1}{\mathbb{1}} \DeclareMathOperator{\Ai}{Ai} \DeclareMathOperator{\Bi}{Bi} \let\Re\relax \DeclareMathOperator{\Re}{Re} \let\Im\relax \DeclareMathOperator{\Im}{Im} %Useful notations for thermodynamics \newcommand{\dbar}} \)

\(\require{cancel}\) \(\require{simpler-wick}\) \(\require{braket}\)