36 problems

The XXZ model is considered through ground-state averages of local operators in finite, possibly inhomogeneous, chains, with the reduced quantum Knizhnik–Zamolodchikov (rqKZ) equat…

Let $\mathbf{Z}_{n}$ and $\mathbf{Y}_{n}$ be the higher-order Z and Y charges, let $\mathbf{Q}^{0}_{m}$ and $\mathbf{Q}^{-}_{m}$ be the corresponding higher-order Onsager generator…

Zero-location conjecture. For $n<p_0$, $T^{(3)}_{n-1}(u,v)$ has two real zeros $\pm\zeta^{(3)}_{n-1}$ for some $\zeta^{(3)}_{n-1}\in\mathbf R_{>0}$, while for $n=p_0$ it has a doub…

Zero-location conjecture. All the zeros of $T^{(1)}_{n-1}(u,v)$ are located on the almost straight lines

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

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…

Consider a kink Hamiltonian $H^{+-}$ on a chain of length $L$, with $\operatorname{SU}_q(2)$ irreducible representations labelled by $j=\frac{L}{2},\frac{L-2}{2},\dots,\frac{1}{2}$…

Let $S$ be the spin of a one-dimensional ferromagnetic XXZ chain, let $\Delta$ denote its anisotropy parameter, and let the spectral gap be rescaled by its linear-in-$S$ growth. Cl…

Let the spin $1/2$ XXZ Heisenberg model on a three-dimensional lattice be defined by … with $\Delta>1$. A 111 interface Gibbs-state conjecture asserts that the model has a Gibbs st…

For each integer $n\geq 0$, let $V^{(n)}_\zeta$ be the $(n+1)$-dimensional evaluation representation, let $V(\Lambda_i)$ be a level-one integrable highest-weight representation of…

Massless-regime equality conjecture. The equality

For the even-site ground state $|\Psi^{(2n)}\rangle$, define … and set $\alpha_i=(1+\sigma_i^z)/2$. Korepin–Izergin–Baxter ferromagnetic-string formula. The probabilities of format…

For the even-site ground state $|\Psi^{(2n)}\rangle$, define the normalized expectation of an operator $O$ by … and define $\alpha_i=(1+\sigma_i^z)/2$. The twisted XXZ next-nearest…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, let … and let $A_n$ be the number of alternating-sign $n\times n$ matrices…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, and let $A_n$ denote the number of alternating-sign $n\times n$ matrices.…

Let $H^{(N)}$ be the XXZ Hamiltonian with twisted boundary conditions, with $N$ an even number of sites, twisting angle $\phi$, anisotropy $\Delta$, and total spin $z$-axis project…

Consider the XXZ Hamiltonian with twisted boundary conditions … Here $N$ is the number of sites, $\phi$ is the twisting angle, and $\Delta$ is the anisotropy parameter. The twisted…

Bethe-vector basis conjecture. For every $|\mathbf{x}\rangle\in\mathbb{V}$,

Completeness conjecture. The Bethe Ansatz completely determines the spectrum for all but finitely many values of $\Delta$ with $|\Delta|<\varepsilon$.

Consider the open XXZ spin chain with parameters … where $x$ is real, and let $E_0$ be the eigenvalue … At $x=1$, this is the non-degenerate ground-state eigenvalue, and the same p…

Let the spin-$1/2$ XXZ model be evaluated at a root-of-unity anisotropy, and let the Onsager algebra be regarded as a subalgebra of the $\mathfrak{sl}_{2}$ loop algebra. Onsager sy…

Let $t$ be the XXZ anisotropy parameter, let $\zeta$ denote the regime parameter, and let $\alpha(N_1,\zeta)$ be defined by … where $\widetilde{w}_{fg}^{(N_1,\zeta)}$ is the zero o…

Let $w_1<w_2$ define the interval of narrow pairs, and let $\widehat{B}(w)$ and $\widehat{C}(w)$ be the functions appearing in the expression for $X(w)$. In the unstable regime, th…