22 problems

Let $n\geqslant 1$ be an integer, and consider the ground-state eigenvector of the periodic XXZ spin chain with anisotropy parameter $\Delta=-\frac12$ and $N=2n+1$ sites. Normalize…

For the $XXZ$ spin-$\frac12$ Heisenberg chain at $\Delta=1/2$, let $\langle Q_\kappa(m)\rangle$ be the generating function considered in the paper, and write its explicit expressio…

The quantum-invariant-chain 1-sum conjecture.

Let $L=2n$ and consider the twisted XXZ chain at $\Delta=-\frac{1}{2}$, with the groundstate wavefunction normalised as in the source. Let $S_{2n}^{(2)}$ be the sum of the squares…

Let $L=2n$ and consider the twisted XXZ chain at $\Delta=-\frac{1}{2}$, with the groundstate wavefunction normalised as in the source. Let $S_{2n}^{(1)}$ denote the sum of its wave…

Razumov–Stroganov's conjecture. The largest element of $\psi_0$ is $A_n$.

Let $0<\zeta<\pi$ be generic, let ${\mathfrak{s}}\in\mathbb Z$, and let $1=r_1<\dots<r_p$ be the allowed thermal string lengths. Consider a solution…

Low-temperature spectrum completeness conjecture. Running over all such values of $k$ should produce every possible choice of the integers $n_p^{(\sigma)}=n_h^{(\sigma)}$,…

Quasi-periodicity conjecture. Typical transversal magnetization components must be quasi-periodic.

Let $q$ be the deformation parameter of the XXZ chain, with anisotropy $\Delta=(q+q^{-1})/2$, and consider the transversal two-point function at separation $km$, where $k$ is an in…

Global velocity-properties conjecture. The conclusions of Proposition 1 hold for all

Velocity-matching conjecture. This property holds for any regime of the model's parameters. The claim is supported by extensive numerical analysis, but no resolution is given here.

Let $N$ be the chain length and $M$ the number of Bethe roots in the root-of-unity XXZ chain with $p=2$. Let ${\cal D}(N,M)$ denote the degeneracy of the transfer-matrix eigenvalue…

Let $N$ be the number of sites, let $M$ be the number of Bethe roots, and let $\{\lambda_1,\ldots,\lambda_M\}$ be an admissible solution of the root-of-unity XXZ Bethe equations. L…

Let $N$ be the number of sites, $M$ the number of Bethe roots, and let $q$ be generic, meaning that $q$ is not a root of unity. An admissible solution is a finite, pairwise-distinc…

Fix the magnetic field in the XXZ chain and let $T_m=\frac{h}{2m\gamma}$ with $m\in\mathbb{N}$. Consider eigenstates in the sector with spin $s$, equivalently with…

Consider the XXZ chain in the antiferromagnetic massive regime at finite magnetic field and low temperature. Let the eigenvalue ratios of the quantum transfer matrix be those obtai…

Let the XXZ chain be in the antiferromagnetic massive regime at finite magnetic field and sufficiently low temperature. The quantum transfer matrix has eigenstates, and the higher-…

Counting-function conjecture. The second hypothesis—that ${\widehat{\xi}}_{\mu}$ is strictly increasing on $[ -\pi/2,\,\pi/2 ]$ and satisfies

Consider the integrable higher spin XXZ chain with odd length $M=2N+1$, spin parameter $L$, and Bethe-root variables $w_1,\ldots,w_p$. Let $E_1^{(p)}=\sum_{j=1}^p w_j$, and let $A$…

Let $M=2N+1$ be the odd number of sites in the spin $3/2$ higher spin XXZ chain, let $p$ denote the number of Bethe roots, and let $w_1,\ldots,w_p$ be the corresponding variables.…

The XXZ chain is considered in the Suzuki–Trotter formalism, with an auxiliary lattice whose extension in the Trotter direction tends to infinity to produce finite temperature. Let…