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CritPt / Challenge_28_main / Consider a four-dimensional hypercubic lattice with lattice spacing a and…

Problem

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code template

Code

def answer():
    r"""
    Return the values of the exponents of the leading power law dependence on the Fermi momentum
    of the correction along $y$ direction per unit volume, the quasiparticle scattering rate
    and transport scattering rate on the Fermi surface in the zero frequency limit.

    Inputs
    ----------
    None

    Outputs
    ----------
    exponent_correction: float, the exponent of the leading power law dependence on the Fermi momentum
        of the correction along $y$ direction per unit volume in the zero frequency limit.
    exponent_quasiparticle: float, the exponent of the leading power law dependence on the Fermi momentum
        of the quasiparticle scattering rate on the Fermi surface in the zero frequency limit.
    exponent_transport: float, the exponent of the leading power law dependence on the Fermi momentum
        of the transport scattering rate on the Fermi surface in the zero frequency limit.
    """
    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    exponent_correction = ...
    exponent_quasiparticle = ...
    exponent_transport = ...
    # ---------------------------------------------------------------

    return exponent_correction, exponent_quasiparticle, exponent_transport

problem description

# Problem setup: Consider a four-dimensional hypercubic lattice with lattice spacing

a

and nearest-neighbor hopping amplitude

t

. The system includes an on-site Hubbard interaction of strength

U

, and the chemical potential is near the bottom of the conduction band. The Hubbard interaction will give a correction to the real part of the finite-frequency paramagnetic conductivity. # Main problem: At zero temperature, to the second order in

U

, what's the leading power law dependence of the Fermi momentum

k_F

of this correction along

y

direction per unit volume in the zero frequency limit? At zero temperature, to the second order in

U

, what's the leading power law dependence of the Fermi momentum

k_F

of the quasiparticle scattering rate and transport scattering rate on the Fermi surface in the zero frequency limit?

Plain-text mathematical notation (without MathML)

# Problem setup:
Consider a four-dimensional hypercubic lattice with lattice spacing a and nearest-neighbor hopping amplitude t. The system includes an on-site Hubbard interaction of strength U, and the chemical potential is near the bottom of the conduction band.

The Hubbard interaction will give a correction to the real part of the finite-frequency paramagnetic conductivity.

# Main problem:


At zero temperature, to the second order in U, what's the leading power law dependence of the Fermi momentum k_(F) of this correction along y direction per unit volume in the zero frequency limit? At zero temperature, to the second order in U, what's the leading power law dependence of the Fermi momentum k_(F) of the quasiparticle scattering rate and transport scattering rate on the Fermi surface in the zero frequency limit?

Original LaTeX notation

# Problem setup:
Consider a four-dimensional hypercubic lattice with lattice spacing $a$ and nearest-neighbor hopping amplitude $t$. The system includes an on-site Hubbard interaction of strength $U$, and the chemical potential is near the bottom of the conduction band.

The Hubbard interaction will give a correction to the real part of the finite-frequency paramagnetic conductivity.

# Main problem:


At zero temperature, to the second order in $U$, what's the leading power law dependence of the Fermi momentum $k_F$ of this correction along $y$ direction per unit volume in the zero frequency limit? At zero temperature, to the second order in $U$, what's the leading power law dependence of the Fermi momentum $k_F$ of the quasiparticle scattering rate and transport scattering rate on the Fermi surface in the zero frequency limit?

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Source and history

upstream_id / Original problem ID: Challenge_28_main
task_id: 26a20460-a6bc-5854-a421-c9f4df58a2f8
task_key: train--Challenge~5f28~5fmain
task_revision_id: 1
config / split: / train

Official source

initial import