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CritPt / Challenge_14_main / Consider the following spin model on a torus
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem description
# Problem setup:
Consider the following spin model on a torus
\begin{equation}
Z^{(n)}_{\text{RM},\ \alpha}\left[J\right]=\sum_{\left\{\eta_{ij}=\pm 1\right\}}P[\eta]\sum_{\left\{ \sigma^{(f)}=\pm1\right\}|_{f=1, \dots n-1} }e^{J\sum_{f=1}^{n-1}\sum_{\langle i,\ j\rangle}\eta_{ij}\sigma^{(f)}_{i}\sigma^{(f)}_{j}},\ \text{with}\ P[\eta]=\prod_{\langle i, j \rangle}\frac{e^{J\eta_{ij}}}{2\cosh J},
\end{equation}
where are the -flavor Ising spins on a square lattice with
sites, f=1,2,…n, n denotes the number of flavor, is a bond variable associated with , and is a coupling constant. Here,
indicates, for each flavor, an independent choice of boundary conditions (periodic or anti-periodic
along the non-contractable loop). For brevity, we drop the subscript when all flavors take periodic-periodic boundary conditions. is defined as partition function when all flavors take the periodic-periodic boundary conditions. is the free energy from twisting boundary conditions, defined as
\begin{equation}
y= -\frac{2}{n-1}\log_2(\frac{\sum_\alpha Z^{(n)}_{\text{RM}, \alpha}}{2^{n-1}Z^{(n)}_{\text{RM}}}).
\end{equation}
# Main problem:
Calculate the value for where in a -site lattice to three decimal places.
Plain-text mathematical notation (without MathML)
# Problem setup:
Consider the following spin model on a torus
\begin{equation}
Z^{(n)}_{\text{RM},\ \alpha}\left[J\right]=\sum_{\left\{\eta_{ij}=\pm 1\right\}}P[\eta]\sum_{\left\{ \sigma^{(f)}=\pm1\right\}|_{f=1, \dots n-1} }e^{J\sum_{f=1}^{n-1}\sum_{\langle i,\ j\rangle}\eta_{ij}\sigma^{(f)}_{i}\sigma^{(f)}_{j}},\ \text{with}\ P[\eta]=\prod_{\langle i, j \rangle}\frac{e^{J\eta_{ij}}}{2\cosh J},
\end{equation}
where σ_(i)^((f))=±1 are the f-flavor Ising spins on a square lattice with
N sites, f=1,2,…n, n denotes the number of flavor, η_(ij) is a bond variable associated with ij, and J≥0 is a coupling constant. Here, α=PP, AP, PA, AA
indicates, for each flavor, an independent choice of boundary conditions (periodic P or anti-periodic
A along the non-contractable loop). For brevity, we drop the subscript α when all flavors take periodic-periodic boundary conditions. Z_(RM)^((n)) is defined as partition function when all flavors take the periodic-periodic boundary conditions. y is the free energy from twisting boundary conditions, defined as
\begin{equation}
y= -\frac{2}{n-1}\log_2(\frac{\sum_\alpha Z^{(n)}_{\text{RM}, \alpha}}{2^{n-1}Z^{(n)}_{\text{RM}}}).
\end{equation}
# Main problem:
Calculate the value J for n=3 where y=0 in a 100×100-site lattice to three decimal places.
Original LaTeX notation
# Problem setup:
Consider the following spin model on a torus
\begin{equation}
Z^{(n)}_{\text{RM},\ \alpha}\left[J\right]=\sum_{\left\{\eta_{ij}=\pm 1\right\}}P[\eta]\sum_{\left\{ \sigma^{(f)}=\pm1\right\}|_{f=1, \dots n-1} }e^{J\sum_{f=1}^{n-1}\sum_{\langle i,\ j\rangle}\eta_{ij}\sigma^{(f)}_{i}\sigma^{(f)}_{j}},\ \text{with}\ P[\eta]=\prod_{\langle i, j \rangle}\frac{e^{J\eta_{ij}}}{2\cosh J},
\end{equation}
where $\sigma^{(f)}_{i}=\pm1$ are the $f$-flavor Ising spins on a square lattice with
$N$ sites, f=1,2,…n, n denotes the number of flavor, $\eta_{ij}$ is a bond variable associated with $ij$, and $J \geq 0$ is a coupling constant. Here, $\alpha=\text{PP},\ \text{AP},\ \text{PA},\ \text{AA}$
indicates, for each flavor, an independent choice of boundary conditions (periodic $\text{P}$ or anti-periodic
$\text{A}$ along the non-contractable loop). For brevity, we drop the subscript $\alpha$ when all flavors take periodic-periodic boundary conditions. $Z^{(n)}_{\text{RM}}$ is defined as partition function when all flavors take the periodic-periodic boundary conditions. $y$ is the free energy from twisting boundary conditions, defined as
\begin{equation}
y= -\frac{2}{n-1}\log_2(\frac{\sum_\alpha Z^{(n)}_{\text{RM}, \alpha}}{2^{n-1}Z^{(n)}_{\text{RM}}}).
\end{equation}
# Main problem:
Calculate the value $J$ for $n=3$ where $y=0$ in a $100\times 100$-site lattice to three decimal places.
code template
Code
def answer():
r"""
Return the value of J.
Inputs
----------
None
Outputs
----------
J: float, value of the coupling constant $J$, when n = 3 and y = 0 in a $100 \times 100$-site lattice
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
J = ...
# ---------------------------------------------------------------
return JDiscussion
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