diff options
Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 6 |
1 files changed, 4 insertions, 2 deletions
diff --git a/topology.tex b/topology.tex index 9000b12..2a8eb52 100644 --- a/topology.tex +++ b/topology.tex @@ -1069,12 +1069,14 @@ method. The integrand is stationary at $\tilde{\mathbb Q}=(\mathbb Q-\mathbb M\mathbb M^T)^{-1}$, and substituting this into the above expression results in the term $\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from \eqref{eq:post.hubbard-strat}. The saddle point also yields a prefactor of the form -\begin{equation} +\begin{equation} \label{eq:supermatrix.saddle} \left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\frac{\partial^2\frac12\log\operatorname{sdet}\tilde{\mathbb Q}}{\partial\tilde{\mathbb Q}(1,2)\partial\tilde{\mathbb Q}(3,4)}\right)^{-\frac12} =\left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\tilde{\mathbb Q}^{-1}(3,1)\tilde{\mathbb Q}^{-1}(2,4)\right)^{-\frac12} =1 \end{equation} -where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value. +where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value.\footnote{ + The subscript notation in \eqref{eq:supermatrix.saddle} indicates which superindices of the four-index superoperator associated with the Hessian belong to the domain and codomain, analogous to writing $\det A=\det_{ij}A_{ij}$ for a two-index complex-valued matrix. In this case, the domain is indexed by $\{3,4\}$ and the codomain is indexed by $\{1,2\}$. +} The Hubbard--Stratonovich transformation therefore contributes a factor of \begin{equation} \frac12\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12} |