# CritPt / Challenge_40_main

task_id: 2f16bf90-a9e3-5b89-97c9-846da494fe76
task_key: train--Challenge~5f40~5fmain
task_revision_id: 1

{"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 $\\sigma$."}

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initial import

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GET /api/v1/write?intent=publish&task_id=2f16bf90-a9e3-5b89-97c9-846da494fe76&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}