{"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":"e2fa5ff8-70f0-5ace-a4aa-4ab527840495","task_key":"train--Challenge~5f12~5fmain","task_revision_id":"2","upstream_id":"Challenge_12_main","short_description":"Consider $Z_N$ ($N$ is any integer) parafermion zero-mode operators on four…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nt, k_12, k_13, k_23, k_34, N, q = sp.symbols('t k_{12} k_{13} k_{23} k_{34} N q')\\n\\ndef answer(t, k_12, k_13, k_23, k_34, N, q):\\n    r\\\"\\\"\\\"\\n    Return the expression of the phase between $|\\\\psi^i(q)\\\\rangle$ and $|\\\\psi^f(q)\\\\rangle$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    t: sympy.Symbol, tunneling amplitude $t$\\n    k_12, k_13, k_23, k_34: sympy.Symbol, ground-state fusion channels\\n          $k_{ij}\\\\in Z_N$ with $k_{ij}<-\\\\frac{\\\\phi_{ij}}{2\\\\pi}<k_{ij}+1$\\n    N: sympy.Symbol, integer N defining $Z_N$\\n    q: sympy.Symbol, fusion channel between the unpaired zero modes\\n\\n    Outputs\\n    ----------\\n    phase: sympy.Expr, the expression of the phase between the initial ground state $|\\\\psi^i(q)\\\\rangle$\\n          and final ground state $|\\\\psi^f(q)\\\\rangle$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    phase = ...  # a SymPy expression of the inputs\\n    # ---------------------------------------------------------------\\n\\n    return phase\",\"problem_description\":\"\\n\\n# Problem setup:\\nConsider $Z_N$ ($N$ is any integer) parafermion zero-mode operators on four sites $\\\\alpha_i$ with $i=1,2,3,4$ and a four-stage tunneling process described by\\n\\n$\\nH_{34}\\\\rightarrow H_{23} \\\\rightarrow H_{12}\\\\rightarrow H_{13}\\\\rightarrow H_{34},\\n$\\n\\nwhere\\n$\\nH_{ij}=t\\\\left(e^{-i\\\\phi_{ij}/N}\\\\alpha_i^\\\\dagger\\\\alpha_j+H.c.\\\\right)\\n$, $\\\\phi_{ij}$ is a phase between sites $i$ and $j$, and $H.c.$ stands for Hermitian conjugate. This tunneling process is followed by a permutation between zero modes $(\\\\alpha_1,\\\\alpha_2)$ and $(\\\\alpha_3,\\\\alpha_4)$, respectively.\\n\\n# Main problem:\\nFind the phase between the initial ground state of the process $|\\\\psi^i(q)\\\\rangle$ and final ground state of the process $|\\\\psi^f(q)\\\\rangle$, where $q$ is the fusion channel between the unpaired zero modes. Your answer should depend on $k_{ij}\\\\in Z_N$, where $k_{ij}<-\\\\frac{\\\\phi_{ij}}{2\\\\pi}<k_{ij}+1$. Note that, for a Josephson phase between sites i and j, k_ij is a ground-state fusion channel.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}