docs: fix coeff

docs/source/tutorial/custom-hamiltonians.ipynb

@@ -914,20 +914,20 @@
         "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",
         "\n",
         "The other operators can be symmetry-adapted in a similar way.\n",
         "\n",
-        "Finally, for the Hamiltonian we require $[\\hat{H}, \\hat{R}]_- = 0$. So terms like $S^+_{i}S^-_{i+1}$ will be rewrite as\n",
+        "Finally, for the Hamiltonian we require $[\\hat{H}, \\hat{R}]_- = 0$. So terms like $S^+_{i}S^-_{i+1}$ will be rewritten as\n",
         "\n",
         "$$\n",
         "S^+_{i,\\mathrm{odd}}S^-_{i+1,\\mathrm{odd}} + S^+_{i,\\mathrm{even}}S^-_{i+1,\\mathrm{even}}.\n",