140 problems

Consider the antiferromagnetic Heisenberg model in one spatial dimension with integer spins. Haldane's conjecture. The model is gapped. This conjecture predicts a fundamental disti…

Ferromagnetic ordering conjecture. The spectral gap $\gamma_k$ is independent of $k$; equivalently,

Let a quantum chain have translation-invariant nearest-neighbor interactions and periodic boundary conditions. A conserved charge is independent if it is not generated by the Hamil…

The closed XXX spin-$\frac{1}{2}$ chain with generic complex twist parameters $\{\kappa,\widetilde{\kappa},\kappa_+,\kappa_-\}$ has quantum space…

Let $\Delta_{(N)}$ denote the $N$-fold coproduct of the quantum group, let $E_i^{+}$ be its positive simple-root generators for $i=1,\ldots,n$, and let…

Gaussian level-density conjecture. For $N\gg1$, the level density follows a Gaussian distribution.

Consider the higher-spin eight-vertex model with spin parameter $\ell$, and let the fused transfer matrix corresponding to the spin chain with local interaction determine the physi…

Let $S$ be an $N\times N$ matrix satisfying the cyclic constraint … Let $\mathcal{N}$ denote the recursion operator and let the local evolution equations be the local flows associa…

In the XXZ Bethe-ansatz string description, each string is assigned an integer Bethe number $I$. The equal-length string uniqueness conjecture. No two strings of the same length ca…

Let $M$ rapidities $x_1,\dots,x_M$ satisfy the Bethe constraints … where $\Delta=\cos\theta$. A string of length $n$ is a set of rapidities with a common real part and equidistant…

Global bound. For all $x\in\mathbb{N}$ and all $\xi\geq 1$,

Consider a critical spin chain containing equal numbers of $l$ different spins, and let $\tilde{c}$ denote the effective conformal anomaly obtained from a finite-size computation b…

Let $z_j$ be the spectral parameters in correlation functions of the Heisenberg XXX model, and consider the homogeneous limit … Let the Riemann zeta function be denoted by…

Consider correlation functions of the Heisenberg XXX model and take the homogeneous limit, in which the spectral parameters coincide. Let the Riemann zeta function be denoted by…

Let a correlation function be any correlation function of the Heisenberg XXX spin-chain model, and let the Riemann zeta-function be denoted by $\zeta$. Rational-coefficient conject…

Let $P(n)$ denote the probability of formation of a ferromagnetic string of length $n$ in the antiferromagnetic ground state of the XXX Heisenberg spin-$1/2$ chain. The cases compu…

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…

Transverse matrix-element conjecture. The first-order splitting matrix element satisfies

Let $N=2n+1$ be odd, and let $a_1,\ldots,a_k$ denote the local observables defining the ferromagnetic-string correlations. Ferromagnetic-string probability conjecture. The correlat…

Let $N=2n+1$ be odd, and let $a_j$ denote the local observable used for the spin-chain correlations. Correlation formula conjecture. The next-nearest-neighbor correlation is … The…