{"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":"4b32f00a-e2f7-5424-abc5-434a34fe31f3","task_key":"train--Challenge~5f14~5fmain","task_revision_id":"1","upstream_id":"Challenge_14_main","short_description":"Consider the following spin model on a torus","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of J.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    J: float, value of the coupling constant $J$, when n = 3 and y = 0 in a $100 \\\\times 100$-site lattice\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    J = ...\\n    # ---------------------------------------------------------------\\n\\n    return J\",\"problem_description\":\"# Problem setup:\\nConsider the following spin model on a torus\\n\\\\begin{equation}\\nZ^{(n)}_{\\\\text{RM},\\\\ \\\\alpha}\\\\left[J\\\\right]=\\\\sum_{\\\\left\\\\{\\\\eta_{ij}=\\\\pm 1\\\\right\\\\}}P[\\\\eta]\\\\sum_{\\\\left\\\\{ \\\\sigma^{(f)}=\\\\pm1\\\\right\\\\}|_{f=1, \\\\dots n-1} }e^{J\\\\sum_{f=1}^{n-1}\\\\sum_{\\\\langle i,\\\\ j\\\\rangle}\\\\eta_{ij}\\\\sigma^{(f)}_{i}\\\\sigma^{(f)}_{j}},\\\\ \\\\text{with}\\\\ P[\\\\eta]=\\\\prod_{\\\\langle i, j \\\\rangle}\\\\frac{e^{J\\\\eta_{ij}}}{2\\\\cosh J},\\n\\\\end{equation}\\nwhere $\\\\sigma^{(f)}_{i}=\\\\pm1$ are the $f$-flavor Ising spins on a square lattice with\\n$N$ sites, f=1,2,…n, n denotes the number of flavor, $\\\\eta_{ij}$ is a bond variable associated with $ij$, and $J \\\\geq 0$ is a coupling constant. Here, $\\\\alpha=\\\\text{PP},\\\\ \\\\text{AP},\\\\ \\\\text{PA},\\\\ \\\\text{AA}$\\nindicates, for each flavor, an independent choice of boundary conditions (periodic $\\\\text{P}$ or anti-periodic\\n$\\\\text{A}$ along the non-contractable loop). For brevity, we drop the subscript $\\\\alpha$ when all flavors take periodic-periodic boundary conditions. $Z^{(n)}_{\\\\text{RM}}$ is defined as partition function when all flavors take the periodic-periodic boundary conditions. $y$ is the free energy from twisting boundary conditions, defined as\\n\\\\begin{equation}\\ny= -\\\\frac{2}{n-1}\\\\log_2(\\\\frac{\\\\sum_\\\\alpha Z^{(n)}_{\\\\text{RM}, \\\\alpha}}{2^{n-1}Z^{(n)}_{\\\\text{RM}}}).\\n\\\\end{equation}\\n\\n# Main problem:\\n\\nCalculate the value $J$ for $n=3$ where $y=0$ in a $100\\\\times 100$-site lattice to three decimal places.\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}