{"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":"44768ddb-c30f-5e2f-be38-d96d9fefcb1b","task_key":"train--Challenge~5f16~5fmain","task_revision_id":"1","upstream_id":"Challenge_16_main","short_description":"Consider the following Hamiltonian in two dimensions on a square lattice:","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of the critical interaction strength $U_c$\\n    that causes a phase transition at quarter-filling of the system\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    U_c: float, critical interaction strength\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    U_c = ...\\n    # ---------------------------------------------------------------\\n\\n    return U_c\",\"problem_description\":\"# Problem setup:\\nConsider the following Hamiltonian in two dimensions on a square lattice:\\n\\\\begin{equation}\\n\\\\begin{aligned}\\nH &= 2\\\\sum_{{\\\\bf k}\\\\sigma}(\\\\cos{k_x}-\\\\cos{k_y})(c^\\\\dagger_{1{\\\\bf k}\\\\sigma}c_{1{\\\\bf k}\\\\sigma}-c^\\\\dagger_{2{\\\\bf k}\\\\sigma}c_{2{\\\\bf k}\\\\sigma})\\\\\\\\&+\\\\sqrt{2}\\\\sum_{{\\\\bf k}\\\\sigma}[\\\\text{e}^{i\\\\pi/4}(1+\\\\text{e}^{i(k_y-k_x)})+\\\\text{e}^{-i\\\\pi/4}(\\\\text{e}^{-ik_x}+\\\\text{e}^{ik_y})]c^\\\\dagger_{1{\\\\bf k}\\\\sigma}c_{2{\\\\bf k}\\\\sigma}\\\\\\\\&+\\\\sqrt{2}\\\\sum_{{\\\\bf k}\\\\sigma}[\\\\text{e}^{-i\\\\pi/4}(1+\\\\text{e}^{-i(k_y-k_x)})+\\\\text{e}^{i\\\\pi/4}(\\\\text{e}^{ik_x}+\\\\text{e}^{-ik_y})]c^\\\\dagger_{2{\\\\bf k}\\\\sigma}c_{1{\\\\bf k}\\\\sigma}\\n\\\\\\\\&-\\\\mu\\\\sum_{{\\\\bf k}\\\\sigma}(c^\\\\dagger_{1{\\\\bf k}\\\\sigma}c_{1{\\\\bf k}\\\\sigma}+c^\\\\dagger_{2{\\\\bf k}\\\\sigma}c_{2{\\\\bf k}\\\\sigma})\\n\\\\\\\\&+U\\\\sum_{{\\\\bf k}}(c^\\\\dagger_{1{\\\\bf k}\\\\uparrow}c_{1{\\\\bf k}\\\\uparrow}c^\\\\dagger_{1{\\\\bf k}\\\\downarrow}c_{1{\\\\bf k}\\\\downarrow}+c^\\\\dagger_{2{\\\\bf k}\\\\uparrow}c_{2{\\\\bf k}\\\\uparrow}c^\\\\dagger_{2{\\\\bf k}\\\\downarrow}c_{2{\\\\bf k}\\\\downarrow}),\\n\\\\end{aligned}\\n\\\\end{equation}\\nwhere $c^\\\\dagger$ is the creation operator; $c$ is the annihilation operator; labels 1,2 denote sublattices 1 and 2; $\\\\bf{k}$ is momentum;   $\\\\sigma=\\\\uparrow \\\\text{or} \\\\downarrow$ represents the spin; and $\\\\mu$ is the chemical potential with repulsive interaction $U>0$.\\n\\n# Main problem:\\n\\nCalculate the critical interaction strength $U_c$ that causes a phase transition at quarter-filling of the system. Please be as accurate as possible.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}