90 problems

Quasilocal commutant conjecture. The commutant is

Fabricius–McCoy conjecture. There exist zero-energy excitations in the form of string operators, manifested in the Bethe ansatz equations as singular solutions.

Logarithmic entanglement-scaling conjecture. The entanglement entropy of any such block, to leading order, scales as

Let the infinite anti-ferromagnetic XXX chain have spin $1/2$, and let its correlation functions be evaluated in the thermodynamic limit. Boos–Korepin conjecture. The correlation f…

A finitely correlated state is a pure state on a quantum spin chain with an associated finite-dimensional memory system. Consider a spin and a subset of other spins, and suppose th…

Let $\omega$ be a translationally invariant state on an infinite quantum spin chain, let $\mathcal{A}_L$ and $\mathcal{A}_R$ denote the left and right half-chain algebras, and let…

Zero-entropy-density conjecture. The entropy density vanishes for every translation-invariant pure state on the spin chain; equivalently, $S_N$ is sublinear.

Let $\Delta=\cosh\eta$ with $\Delta\geq 1$, and let $\gamma_{J,M}$ denote the spectral gap above the ground-state eigenvalue of the spin-$J$ ferromagnetic XXZ chain in the kink sec…

Consider the AKLT state, whose configurations in the Ising basis obey diluted Néel order: between any two $+$ spins there is a $-$ spin and between any two $-$ spins there is a $+$…

Spin-1/2 gap conjecture. Assuming this ordering holds true, for spin $1/2$ and $B\le A$ the gap equals

Continuous-spectrum accumulation conjecture. There is no other continuous spectrum close to $-j|B|$, and therefore $-j|B|$ is purely an accumulation point of eigenvectors.

Large-spin Bose-gas conjecture. There is a function $\gamma_\infty:(1,\infty)\times\mathbb{R}\to\mathbb{R}$ such that

Let $J$ be the spin, $M\in\mathbb{Z}$ the magnetization parameter, and $1<\Delta<\infty$ the anisotropy regime. The lowest excited state means the spectral point immediately above…

Consider the spin-$J$ ferromagnetic XXZ chain in finite volume or infinite volume, whose frustration-free ground states minimize every nearest-neighbor interaction. Gottstein and W…

Let $P(n)$ be the emptiness formation probability in the thermodynamic-limit antiferromagnetic XXX Heisenberg spin chain, defined by … where $P_j=S^z_j+\frac12$ projects the $j$-th…

Let $P(n)$ denote the probability of formation of a ferromagnetic string of length $n$ in the thermodynamic-limit ground state of the antiferromagnetic XXX Heisenberg spin-$1/2$ ch…

Consider a general SU(2)-invariant model with integer spin $S$, and let the AKLT point denote the parameter value corresponding to the Affleck–Kennedy–Lieb–Tasaki valence-bond-soli…

Let the entanglement entropy (EE) be the von Neumann entropy of the reduced density matrix of a subsystem in a quantum spin chain. Consider a large block of spins in either a gappe…

The one-boundary Temperley–Lieb algebra has generators whose representations yield the Hamiltonians $H^{nd}$, $H^d$, and $H^{lp}$; the representation giving $H^d$ is explicitly con…

The spin-$\frac{1}{2}$ XXZ chain has correlation functions represented by multiple integrals, including one-dimensional integrals indexed by an integer $j$. The authors' results fo…

QLEM uniqueness conjecture. Whenever a QLEM exists, it is unique and does not depend on any additional relevant parameter.

Grabowski–Mathieu conjecture. The Hamiltonian $\hat{H}$ is integrable.

Let $J$ be a $7 \times 7$ Jacobi matrix with symmetric spectrum that realizes perfect state transfer (PST). Its early state exclusion (ESE) property means that no state is excluded…

Let $\mathfrak{g}$ be the Lie algebra generated by the raising and lowering operators of the periodic Fredkin chain with $N$ sites. Let $\mathrm{S}^z$ be the third component of the…

Let $\mathfrak{g}$ be the Lie algebra generated by the raising and lowering operators $\Sigma^\pm$ of the periodic Fredkin chain with $N$ sites, and set $n=\lceil N/2\rceil$. B- an…