{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"29f111ff-0ac8-522d-b9d6-de2c03f02c87","task_key":"train--Challenge~5f6~5fmain","task_revision_id":"1","upstream_id":"Challenge_6_main","short_description":"For twisted bilayer MoTe$_2$, we can build the following simplified continuum…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of the Chern numbers of the top three bands and the gauge–invariant Wannier spread TrG.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    chern_numbers: tuple[int, int, int]\\n        (C1, C2, C3) – Chern numbers of the first, second and third top bands, respectively, up to an overall sign.\\n    TrG: float\\n        Gauge–invariant part of the Wannier spread for the top electron band, $\\\\mathop{\\\\mathrm{Tr}}\\\\mathcal{G}$, rounded to 2 decimal places.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    chern_numbers = ...             # three integers\\n    TrG = ...                       # float rounded to 2 decimal places\\n    # ---------------------------------------------------------------\\n\\n    return chern_numbers, TrG\",\"problem_description\":\"# Problem setup:\\nFor twisted bilayer MoTe$_2$, we can build the following simplified continuum model at $\\\\text{K}$ valley:\\n\\\\begin{equation}\\n\\\\mathcal{H} = \\\\int d^2 r\\\\ ( c^\\\\dagger_{\\\\boldsymbol{r},b}, c^\\\\dagger_{\\\\boldsymbol{r},t})\\\\left(\\n\\\\begin{array}{cc}\\n\\\\frac{\\\\hbar^2 \\\\nabla^2}{2 m^*} +2 V \\\\sum_{i=1}^3 \\\\cos(\\\\boldsymbol{g}_i\\\\cdot \\\\boldsymbol{r}- \\\\,\\\\psi) &  w \\\\sum_{i=1}^3 \\\\,e^{-i\\\\,\\\\boldsymbol{q}_i\\\\cdot \\\\boldsymbol{r}} \\\\\\\\\\nw \\\\sum_{i=1}^3 \\\\,e^{i\\\\,\\\\boldsymbol{q}_i\\\\cdot \\\\boldsymbol{r}}  & \\\\frac{\\\\hbar^2 \\\\nabla^2}{2 m^*} + 2 V \\\\sum_{i=1}^3 \\\\cos(\\\\boldsymbol{g}_i\\\\cdot \\\\boldsymbol{r} + \\\\,\\\\psi)\\n\\\\end{array}\\n\\\\right) \\\\left(\\\\begin{matrix} c_{\\\\boldsymbol{r},b} \\\\\\\\ c_{\\\\boldsymbol{r},t}\\\\end{matrix}\\\\right)\\\\ ,\\n\\\\end{equation}\\nwhere $\\\\boldsymbol{g}_1 = \\\\frac{4 \\\\pi}{\\\\sqrt{3} a_{M}} (1,0)^T$,  $\\\\boldsymbol{g}_i = C_3^{i-1} \\\\boldsymbol{g}_1$ with $C_3$ the three-fold rotation symmetry, $\\\\boldsymbol{q}_1 = |\\\\boldsymbol{g}_1| (0, 1/\\\\sqrt{3})^T $, $\\\\boldsymbol{q}_i = C_3^{i-1} \\\\boldsymbol{q}_1$,\\n\\\\begin{equation}\\na_M = \\\\frac{a_0}{2 \\\\sin\\\\left( \\\\frac{\\\\theta}{2} \\\\right)} \\\\ ,\\n\\\\end{equation}\\n$\\\\theta$ is the twist angle, and $a_0=3.52 \\\\text{\\\\AA }$ is the lattice constant of the monolayer MoTe$_2$.\\n\\nWe choose $m^* = 0.6 m_e$ with $m_e$ the mass of an electron, $V=16.5$ meV, $\\\\psi = -105.9^\\\\circ$, and $w = -18.8$ meV.\\n\\n\\nConsider a generic isolated set of $N$ bands with projector $P_{\\\\boldsymbol{k}}$ constructed by the periodic part of the Bloch states.\\n\\nWe can define the quantum metric as\\n\\\\begin{equation}\\ng_{ij}(\\\\boldsymbol{k}) = \\\\frac{1}{2}\\\\mathrm{Tr}[\\\\partial_{k_i} P_{\\\\boldsymbol{k}} \\\\partial_{k_j} P_{\\\\boldsymbol{k}}]\\\\ .\\n\\\\end{equation}\\n\\nThe gauge-invariant part of the Wannier spread of the isolated set of bands is proportional to\\n\\\\begin{equation}\\n\\\\mathop{\\\\mathrm{Tr}}\\\\mathcal{G} = \\\\int d^2 k\\\\ \\\\mathop{\\\\mathrm{Tr}}[g(\\\\boldsymbol{k})]\\\\ ,\\n\\\\end{equation}\\nwhere the integration ranges over the first Brillouin zone.\\n\\n# Main problem:\\n\\nFor $\\\\theta = 3.5^\\\\circ$, what are the Chern numbers of the top three bands of the model, respectively?\\n\\nThe answer can have a global sign freedom owing to the definition of Chern number.\\n\\nNumerically evaluate $\\\\mathop{\\\\mathrm{Tr}}\\\\mathcal{G}$ for the top electron band (to two decimal places) using the following conventions.\\n\\nThe set of Bloch momenta $\\\\boldsymbol{k}$ is\\n\\\\begin{equation}\\n\\\\{ (l_1/L-1/2) \\\\boldsymbol{b}_1 + (l_2/L-1/2) \\\\boldsymbol{b}_2 | l_1,l_2 = 0,1,2,...,L-1 \\\\},\\n\\\\end{equation}\\nwith $L=60$, $\\\\boldsymbol{b}_1 = \\\\boldsymbol{g}_1$, and $\\\\boldsymbol{b}_2 = \\\\boldsymbol{g}_1 + \\\\boldsymbol{g}_2$.\\n\\nChoose $\\\\hbar/(2 m_e) = 7619.96423 \\\\text{ meV} \\\\cdot \\\\AA^2$.\\n\\nChoose the following Fourier transformation convention\\n\\\\begin{equation}\\nc^\\\\dagger_{\\\\boldsymbol{r},l} = \\\\frac{1}{\\\\sqrt{ \\\\mathcal{V}}} \\\\sum_{\\\\boldsymbol{k},\\\\boldsymbol{Q}} e^{-\\\\mathrm{i} (\\\\boldsymbol{k}-\\\\boldsymbol{Q})\\\\cdot \\\\boldsymbol{r} } c^\\\\dagger_{\\\\boldsymbol{k}-\\\\boldsymbol{Q},l}\\\\ ,\\n\\\\end{equation}\\nwhere $\\\\mathcal{V}$ is the volume of the whole sample.\\n\\nChoose all $\\\\boldsymbol{Q}$ for $c^\\\\dagger_{\\\\boldsymbol{k}-\\\\boldsymbol{Q},t}$ to satisfy (i) $\\\\boldsymbol{Q}-\\\\boldsymbol{q}_1$ is a reciprocal lattice vector and (ii) $|\\\\boldsymbol{Q}|< 4.1 |\\\\boldsymbol{b}_1|$.\\n\\nChoose all $\\\\boldsymbol{Q}$ for $c^\\\\dagger_{\\\\boldsymbol{k}-\\\\boldsymbol{Q},b}$ to satisfy (i) $\\\\boldsymbol{Q}+\\\\boldsymbol{q}_1$ is a reciprocal lattice vector and (ii) $|\\\\boldsymbol{Q}|< 4.1 |\\\\boldsymbol{b}_1|$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}