# Entanglement and correlation functions of a recent exactly solvable spin
chain

Research paper by **Ramis Movassagh**

Indexed on: **24 Feb '16**Published on: **24 Feb '16**Published in: **Quantum Physics**

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#### Abstract

We present exact results on the exactly solvable spin chain of Bravyi et al
[Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a
Hamiltonian that is local and translationally invariant in the bulk. It has a
unique (frustration free) ground state with an energy gap that is polynomially
small in the system's size ($2n$). The half-chain entanglement entropy of the
ground state is $\frac{1}{2}\log n+const.$. Here we first write the Hamiltonian
in the standard spin-basis representation. We prove that at zero temperature,
the magnetization is along the $z-$direction i.e., $\langle
s^{x}\rangle=\langle s^{y}\rangle=0$ (everywhere on the chain). We then
analytically calculate $\langle s^{z}\rangle$ and the two-point correlation
functions of $s^{z}$. By analytically diagonalizing the reduced density
matrices, we calculate the Schmidt rank, von Neumann and R\'enyi entanglement
entropies for: 1. Any partition of the chain into two pieces (not necessarily
in the middle) and 2. $L$ consecutive spins centered in the middle. Further, we
identify entanglement Hamiltonians (Eqs. 40 and 50). We prove a small lemma
(Lemma1) on the combinatorics of lattice paths using the reflection principle
to relate and calculate the Motzkin walk 'height' to spin expected values. We
also calculate the, closely related, (scaled) correlation functions of Brownian
excursions. The known features of this model are summarized in a table in
Sec.I.