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| -rw-r--r-- | bezout.tex | 17 |
1 files changed, 11 insertions, 6 deletions
@@ -28,13 +28,18 @@ \maketitle -\begin{equation} \label{eq:hamiltonian} - H = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p} +\begin{equation} \label{eq:bare.hamiltonian} + H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, \end{equation} -where the $z$ are constrained by $z\cdot z=N$ and $J$ is a symmetric -tensor whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ -and $\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter -$|\kappa|<1$. +where the $z$ are constrained by $z\cdot z=N$ and $J$ is a symmetric tensor +whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and +$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter +$|\kappa|<1$. The constraint is enforced using the method of Lagrange +multipliers: introducing the $\epsilon\in\mathbb C$, this gives +\begin{equation} \label{eq:constrained.hamiltonian} + H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). +\end{equation} +At any critical point $\epsilon=H/N$, the average energy. \bibliographystyle{apsrev4-2} \bibliography{bezout} |