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Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
def answer():
r"""
Return the energies of the scar states and the corresponding base-10 logarithms
of their squared overlaps with the $Z_2$ state.
Inputs
----------
None
Outputs
----------
energies: list[float]
Energies of the scar states in ascending order (four-decimal precision).
log_overlaps: list[float]
$\log_{10}|\langle Z_2|\psi\rangle|^2$ for each scar state in the same order as `energies` (four-decimal precision).
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
energies = [...]
log_overlaps = [...]
# ---------------------------------------------------------------
return energies, log_overlapsproblem description
# Problem setup:
The PXP model is a one-dimensional spin chain with a Rydberg blockade constraint. We use to denote the excited state and for the ground state. The Rydberg blockade forbids adjacent s. The system's Hamiltonian is given by
\begin{equation}
H = \sum_{i=1}^L P_{i-1}X_iP_{i+1},
\end{equation}
where is the Pauli X operator acting on the th spin, and is the projector on site , indicating that excitations are only possible when both neighboring sites are in the state.
We impose periodic boundary conditions, and the Hamiltonian exhibits both translation and reflection symmetries. As a result, we can block diagonalize the Hamiltonian into simultaneous eigenstates of the lattice translation operator and reflection operator . We consider the subspace , which has zero momentum and even reflection parity.
The scar eigenstates have exceptionally larger overlap with the state compared to the neighboring eigenstates.
# Main problem:
For the eigenstates inside , list the energies of the scar states and the base-10 logarithm of their overlap with the state: . All numerical results should be given to four decimal places. Set the system size to .
Plain-text mathematical notation (without MathML)
# Problem setup:
The PXP model is a one-dimensional spin chain with a Rydberg blockade constraint. We use |1⟩ to denote the excited state and |0⟩ for the ground state. The Rydberg blockade forbids adjacent 1s. The system's Hamiltonian is given by
\begin{equation}
H = \sum_{i=1}^L P_{i-1}X_iP_{i+1},
\end{equation}
where X_(i) is the Pauli X operator acting on the ith spin, and P_(i) is the projector |0⟩⟨0| on site i, indicating that excitations are only possible when both neighboring sites are in the |0⟩ state.
We impose periodic boundary conditions, and the Hamiltonian exhibits both translation and reflection symmetries. As a result, we can block diagonalize the Hamiltonian into simultaneous eigenstates of the lattice translation operator T and reflection operator P. We consider the subspace D₀^(+), which has zero momentum and even reflection parity.
The scar eigenstates have exceptionally larger overlap with the Z₂ state |101010⋯10⟩ compared to the neighboring eigenstates.
# Main problem:
For the eigenstates inside D₀^(+), list the energies of the scar states and the base-10 logarithm of their overlap with the Z₂ state: log₁₀|⟨Z₂|ψ⟩|². All numerical results should be given to four decimal places. Set the system size to L=26.Original LaTeX notation
# Problem setup:
The PXP model is a one-dimensional spin chain with a Rydberg blockade constraint. We use $|1\rangle$ to denote the excited state and $|0\rangle$ for the ground state. The Rydberg blockade forbids adjacent $1$s. The system's Hamiltonian is given by
\begin{equation}
H = \sum_{i=1}^L P_{i-1}X_iP_{i+1},
\end{equation}
where $X_i$ is the Pauli X operator acting on the $i$th spin, and $P_i$ is the projector $|0\rangle\langle 0|$ on site $i$, indicating that excitations are only possible when both neighboring sites are in the $|0\rangle$ state.
We impose periodic boundary conditions, and the Hamiltonian exhibits both translation and reflection symmetries. As a result, we can block diagonalize the Hamiltonian into simultaneous eigenstates of the lattice translation operator $T$ and reflection operator $P$. We consider the subspace $\mathcal{D}_0^+$, which has zero momentum and even reflection parity.
The scar eigenstates have exceptionally larger overlap with the $Z_2$ state $|101010\cdots10\rangle$ compared to the neighboring eigenstates.
# Main problem:
For the eigenstates inside $\mathcal{D}_0^+$, list the energies of the scar states and the base-10 logarithm of their overlap with the $Z_2$ state: $\log_{10}|\langle Z_2|\psi\rangle|^2$. All numerical results should be given to four decimal places. Set the system size to $L = 26$.Discussion
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