{"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":"b819ac66-b52b-54fb-90be-4ffbb3b09e78","task_key":"train--Challenge~5f37~5fmain","task_revision_id":"2","upstream_id":"Challenge_37_main","short_description":"Consider a single-particle Hamiltonian on a lattice.","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return a tuple of answers (isolated, gap, TrG_over_2pi, Z2_topology, wannier_expressible).\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    result: a tuple whose entries are\\n        isolated            – \\\"Yes\\\", \\\"No\\\", or \\\"N/A\\\"\\n        gap                 – float (4 decimal places) or \\\"N/A\\\"\\n        TrG_over_2pi        – float (4 decimal places) or \\\"N/A\\\"\\n        Z2_topology         – \\\"Trivial\\\", \\\"Nontrivial\\\", or \\\"N/A\\\"\\n        wannier_expressible – \\\"Yes\\\", \\\"No\\\", or \\\"N/A\\\"\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    result = ...  # a SymPy tuple of the five requested quantities\\n    # ---------------------------------------------------------------\\n\\n    return result\",\"problem_description\":\"# Problem setup:\\nConsider a single-particle Hamiltonian on a lattice.\\n\\nGiven a generic isolated set of $N$ bands with projector $P_{\\\\boldsymbol{k}}$ constructed by the perodic part of the Bloch states,\\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\\nGiven the following 2D Hamiltonian,\\n\\\\begin{align}\\\\begin{split}\\nH & = \\\\int d^2 r\\\\ c^\\\\dagger_{\\\\boldsymbol{r}} \\\\left[ - \\\\frac{1}{2 m} \\\\nabla^2 + \\\\lambda (- \\\\mathrm{i} \\\\partial_y \\\\sigma_x +  \\\\mathrm{i} \\\\partial_x \\\\sigma_y)  + \\\\Delta_1 \\\\sum_{i=1}^{3} \\\\sum_{s=\\\\pm} e^{ s  \\\\mathrm{i} \\\\boldsymbol{g}_i^{(1)}\\\\cdot\\\\boldsymbol{r}}  \\\\right] c_{\\\\boldsymbol{r}}  \\\\\\\\\\n& \\\\qquad + \\\\int d^2 r\\\\ c^\\\\dagger_{\\\\boldsymbol{r}} \\\\left[  \\\\mathrm{i} \\\\Delta_2 \\\\sum_{i=1}^{3} \\\\sum_{s=\\\\pm} s e^{ s  \\\\mathrm{i} \\\\boldsymbol{g}_i^{(1)}\\\\cdot\\\\boldsymbol{r}}   +  \\\\sum_{i=1}^{3} \\\\sum_{s=\\\\pm} (\\\\Delta_3 + s  \\\\mathrm{i} \\\\Delta_4)e^{s  \\\\mathrm{i} \\\\boldsymbol{g}_i^{(2)}\\\\cdot\\\\boldsymbol{r}}  \\\\right] c_{\\\\boldsymbol{r}} \\\\ ,\\n\\\\end{split}\\\\end{align}\\nwhere $c^\\\\dagger_{\\\\boldsymbol{r}} = (c^\\\\dagger_{\\\\boldsymbol{r},\\\\uparrow}, c^\\\\dagger_{\\\\boldsymbol{r},\\\\downarrow})$, $c^\\\\dagger_{\\\\boldsymbol{r},s}$ creates an electron at 2D coordinate $\\\\boldsymbol{r}$ and with spin $s$, the primitive reciprocal lattice vectors reads $\\\\boldsymbol{b}_{M,1} = (0,1)$ and $\\\\boldsymbol{b}_{M,2} = C_6 \\\\boldsymbol{b}_{M,1}$, and $\\\\boldsymbol{g}_i^{(1)} = C_3^{i-1}\\\\boldsymbol{b}_{M,1}  $, and $\\\\boldsymbol{g}_i^{(2)} = C_3^{i-1}(\\\\boldsymbol{b}_{M,1} + \\\\boldsymbol{b}_{M,2})  $.\\n\\nWe choose\\n\\\\begin{equation}\\n2 m = 1,\\\\ \\\\lambda = 1.9,\\\\ \\\\Delta_1 = 0.12,\\\\ \\\\Delta_2 = 0.005,\\\\ \\\\Delta_3 = 0.05,\\\\ \\\\Delta_4 = 0.01\\\\ .\\n\\\\end{equation}\\n\\n\\n# Main problem:\\n\\nIs the set of the lowest two bands isolated?\\nWhat is the direct energy gap between the lowest two bands?\\nWhat is $\\\\frac{1}{2\\\\pi}\\\\mathop{\\\\mathrm{Tr}}\\\\mathcal{G}$ for the set of the lowest two bands?\\nIs the Kane-Mele time-reversal $Z_2$ topology of the set of the lowest two bands trivial or nontrivial?\\nCan the set of the lowest two bands be expressed in terms of two exponentially localized Wannier functions?\\n\\nFor all numerical calculations, please use the 43 shortest reciprocal lattice vectors, use a $60\\\\times 60$ equal-spacing hexagonal-symmetric momentum mesh in the Brillouin zone, and keep four decimal places for all numerical answers.\\nIf any quantity is ill-defined, use N/A as the answer.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}