{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt 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"},"task_id":"55985a73-5bf0-5af2-9126-408dfad041b5","task_key":"train--Challenge~5f38~5fmain","task_revision_id":"1","upstream_id":"Challenge_38_main","short_description":"The Hamiltonian for the Hatsugai-Kohmoto (HK) model is","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nT, U = sp.symbols('T U')\\n\\ndef answer(T):\\n    \\\"\\\"\\\"\\n    Return the temperature dependence of the energy phase space integral in SymPy format.\\n\\n    Inputs\\n    ----------\\n    T: sympy.Symbol, temperature $T$\\n    U: sympy.Symbol, on-site interaction strength $U$\\n\\n    Outputs\\n    ----------\\n    I: sympy.Expr, the energy phase space integral, $I(T)$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    I = ... # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return I\",\"problem_description\":\"# Problem setup:\\nThe Hamiltonian for the Hatsugai-Kohmoto (HK) model is\\n$$\\nH = \\\\sum_{k,\\\\sigma} (\\\\varepsilon_k-\\\\mu) n_{k \\\\sigma} + U \\\\sum_{k}n_{k \\\\uparrow} n_{k \\\\downarrow}.\\n$$\\nThe first term represents the non-interacting part, which corresponds to a non-interacting band $0<\\\\varepsilon_k<W$, and the second term represents the interaction.\\nWe further add a perturbation term, which preserves momentum and respects fermion symmetries:\\n$$\\nH' = \\\\sum_{2,3,4}V(1,2,3,4) \\\\delta_{k_1+k_2,k_3+k_4}c^{\\\\dagger}_4 c^{\\\\dagger}_3 c_2 c_1,\\n$$\\nwhere the index $i \\\\equiv (k_i, \\\\sigma_i)$ and the scattering process can be understood as $1 \\\\rightarrow \\\\bar{2} + 3+4$ and $\\\\bar{2} + 3+4 \\\\rightarrow 1$.\\nThe energy phase space integral of the scattering rate of the propagating modes of the HK model is given by\\n\\\\begin{equation}\\n    I(T)=\\\\langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \\\\rangle_{\\\\epsilon_2,\\\\epsilon_3,\\\\epsilon_4}=\\\\int d\\\\epsilon_2 d\\\\epsilon_3 d\\\\epsilon_4 \\\\langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \\\\rangle \\\\delta(\\\\epsilon_1+\\\\epsilon_2-\\\\epsilon_3-\\\\epsilon_4).\\n\\\\end{equation}\\n\\n# Main problem:\\n\\nCompute the temperature dependence of the energy phase space integral in the limit $U\\\\gg W\\\\gg k_BT\\\\gg \\\\epsilon_1$, where the chemical potential $\\\\mu$ only crosses the lower Hubbard band, i.e., $0<\\\\mu<W$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}