{"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":"2f16bf90-a9e3-5b89-97c9-846da494fe76","task_key":"train--Challenge~5f40~5fmain","task_revision_id":"1","upstream_id":"Challenge_40_main","short_description":"Effective field theory is a powerful tool used to construct phenomenological…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nchi, kappa, sigma, k = sp.symbols('chi kappa sigma k', real=True)\\n\\ndef answer(chi, kappa, sigma, k):\\n    r\\\"\\\"\\\"\\n    Return the expression of hydrodynamic mode spectrum $\\\\omega(k)$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    chi: sympy.Symbol, charge susceptibility, $\\\\chi$\\n    kappa: sympy.Symbol, quadrupole superfluid stiffness, $\\\\kappa$\\n    sigma: sympy.Symbol, coefficient of the leading order dissipative term, $\\\\sigma$\\n    k: sympy.Symbol, momentum, $k$\\n\\n    Outputs\\n    ----------\\n    omega: set[sympy.Expr], a set of hydrodynamic mode dispersion relation(s), $\\\\omega(k)$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    omega = ...  # a set of SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return omega\",\"problem_description\":\"\\n# Problem setup:\\nEffective field theory is a powerful tool used to construct phenomenological models via symmetries. The method has recently been extended to dissipative systems via the Schwinger-Keldysh formalism. In this problem we will study the dissipative effective field theory associated to spontaneous symmetry breaking of multipolar $U(1)$ symmetries.\\n\\n\\nConsider a $1d$ system with a conserved density $N = \\\\int \\\\rho$, conserved dipole moments $D = \\\\int x \\\\rho$, and a conserved quadrupole moment $Q = \\\\int x^2 \\\\rho$. Assume that these are the only conserved quantities in the system.\\n\\nSuppose that the quadrupole $Q$ generator is spontaneously broken, and the charge $N$ and dipole $D$ generators are unbroken.\\n\\n# Main problem:\\n\\nCompute the spectrum of hydrodynamic modes $\\\\omega(k)$.\\n\\nLet $\\\\chi$ be the charge susceptibility, $\\\\kappa$ be the quadrupole superfluid stiffness, and $\\\\sigma$ be the coefficient of the leading order dissipative term. You may assume all other EFT coefficients are zero. Express your answer in terms of $\\\\chi$, $\\\\kappa$, and $\\\\sigma$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}