From 7c7e35214f28f26a98afd096b46cbb95b61e27ea Mon Sep 17 00:00:00 2001 From: Huanchen Zhai Date: Sun, 20 Oct 2024 23:38:09 -0700 Subject: [PATCH] docs: fix coeff --- docs/source/tutorial/custom-hamiltonians.ipynb | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/source/tutorial/custom-hamiltonians.ipynb b/docs/source/tutorial/custom-hamiltonians.ipynb index ddff16f3..3daf6b53 100644 --- a/docs/source/tutorial/custom-hamiltonians.ipynb +++ b/docs/source/tutorial/custom-hamiltonians.ipynb @@ -914,13 +914,13 @@ "Second, we need to transform the local operators so that they have well defined quantum numbers in the $U(1) \\times Z_2$ symmetry group. We first write each operator in the above $|0\\rangle, |1\\rangle, |2\\rangle, |3\\rangle$ basis. Since now each big site has four states, we have two $\\hat{S}^+$ operators denoted as $\\hat{S}^+_L$ and $\\hat{S}^+_R$, acting on the site from the left-half-chain and the right-half-chain, respectively (for example, when $L=8$, in the first big site, $\\hat{S}^+_L = \\hat{S}^+_0$ and $\\hat{S}^+_R = \\hat{S}^+_7$). We have\n", "\n", "$$\n", - "\\hat{S}_L^+ = \\begin{pmatrix} 0 & 1 & -1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix}\n", + "\\hat{S}_L^+ = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & 1 & -1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix}\n", "$$\n", "\n", "As $\\hat{S}_L^+$ do not commute or anticommute with $\\hat{R} = \\mathrm{diag}(1, 1, -1, 1)$, we split it into the odd and even parts:\n", "$$\n", - "\\hat{S}^+_{L,\\mathrm{odd}} = \\begin{pmatrix} 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix},\\quad\n", - "\\hat{S}^+_{L,\\mathrm{even}} = \\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix}\n", + "\\hat{S}^+_{L,\\mathrm{odd}} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix},\\quad\n", + "\\hat{S}^+_{L,\\mathrm{even}} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\\\end{pmatrix}\n", "$$\n", "\n", "such that $[\\hat{S}^+_{L,\\mathrm{odd}}, \\hat{R}]_+ = 0$ and $[\\hat{S}^+_{L,\\mathrm{even}}, \\hat{R}]_- = 0$.\n",