{"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":"e7e65389-6681-589e-a5e9-ed9aa498b6cf","task_key":"train--Challenge~5f46~5fmain","task_revision_id":"2","upstream_id":"Challenge_46_main","short_description":"The PXP model is a one-dimensional spin chain with a Rydberg blockade…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the energies of the scar states and the corresponding base-10 logarithms\\n    of their squared overlaps with the $Z_2$ state.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    energies: list[float]\\n        Energies of the scar states in ascending order (four-decimal precision).\\n    log_overlaps: list[float]\\n        $\\\\log_{10}|\\\\langle Z_2|\\\\psi\\\\rangle|^2$ for each scar state in the same order as `energies` (four-decimal precision).\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    energies = [...]\\n    log_overlaps = [...]\\n    # ---------------------------------------------------------------\\n\\n    return energies, log_overlaps\",\"problem_description\":\"# Problem setup:\\nThe PXP model is a one-dimensional spin chain with a Rydberg blockade constraint. We use $|1\\\\rangle$ to denote the excited state and $|0\\\\rangle$ for the ground state. The Rydberg blockade forbids adjacent $1$s. The system's Hamiltonian is given by\\n\\\\begin{equation}\\nH = \\\\sum_{i=1}^L P_{i-1}X_iP_{i+1},\\n\\\\end{equation}\\nwhere $X_i$ is the Pauli X operator acting on the $i$th spin, and $P_i$ is the projector $|0\\\\rangle\\\\langle 0|$ on site $i$, indicating that excitations are only possible when both neighboring sites are in the $|0\\\\rangle$ state.\\n\\nWe impose periodic boundary conditions, and the Hamiltonian exhibits both translation and reflection symmetries. As a result, we can block diagonalize the Hamiltonian into simultaneous eigenstates of the lattice translation operator $T$ and reflection operator $P$. We consider the subspace $\\\\mathcal{D}_0^+$, which has zero momentum and even reflection parity.\\n\\nThe scar eigenstates have exceptionally larger overlap with the $Z_2$ state $|101010\\\\cdots10\\\\rangle$ compared to the neighboring eigenstates.\\n\\n# Main problem:\\n\\nFor the eigenstates inside $\\\\mathcal{D}_0^+$, list the energies of the scar states and the base-10 logarithm of their overlap with the $Z_2$ state: $\\\\log_{10}|\\\\langle Z_2|\\\\psi\\\\rangle|^2$. All numerical results should be given to four decimal places. Set the system size to $L = 26$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}