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Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem description

# Problem setup: Consider the following Hamiltonian: H=−12∑a=1N∑i,j,k,l=1NCijaCklaχiχjχkχlH=-\frac{1}{2}\sum_{a=1}^N\sum_{i,j,k,l=1}^N C_{ij}^a C_{kl}^a\chi_i\chi_j\chi_k\chi_l, where χ\chi are Majorana fermions and N2⟨CijaCklb⟩=Jδab(δikδjl−δilδjk)N^2\langle C_{ij}^a C_{kl}^b\rangle=J\delta_{ab}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}). Consider the large NN limit. Define the zero-temperature entropy as S=limT→0∂F∂TS= \lim_{T\to 0}\frac{\partial F}{\partial T}, where FF is the free energy of the system. # Main problem: Calculate the numerical value of S/NS/N to four decimal places.
Plain-text mathematical notation (without MathML)
# Problem setup:
Consider the following Hamiltonian: H=−(1)/(2)∑_(a=1)^(N)∑_(i,j,k,l=1)^(N)C_(ij)^(a)C_(kl)^(a)χ_(i)χ_(j)χ_(k)χ_(l), where χ are Majorana fermions and N²⟨C_(ij)^(a)C_(kl)^(b)⟩=Jδ_(ab)(δ_(ik)δ_(jl)−δ_(il)δ_(jk)). Consider the large N limit. Define the zero-temperature entropy as S=lim_(T→0)(∂F)/(∂T), where F is the free energy of the system.

# Main problem:

Calculate the numerical value of S/N to four decimal places.
Original LaTeX notation
# Problem setup:
Consider the following Hamiltonian: $H=-\frac{1}{2}\sum_{a=1}^N\sum_{i,j,k,l=1}^N C_{ij}^a C_{kl}^a\chi_i\chi_j\chi_k\chi_l$, where $\chi$ are Majorana fermions and $N^2\langle C_{ij}^a C_{kl}^b\rangle=J\delta_{ab}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$. Consider the large $N$ limit. Define the zero-temperature entropy as $S= \lim_{T\to 0}\frac{\partial F}{\partial T}$, where $F$ is the free energy of the system.

# Main problem:

Calculate the numerical value of $S/N$ to four decimal places.

code template

Code

def answer():
    r"""
    Return the numerical value of S/N.

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

    Outputs
    ----------
    S_per_N: float, The zero temperature entropy per particle, $S/N$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    S_per_N = ...
    # ---------------------------------------------------------------

    return S_per_N

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Source and history

upstream_id / Original problem ID
Challenge_17_main
task_id
7367714e-8073-56b0-9cbe-5c2ed657958e
task_key
train--Challenge~5f17~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import