48 problems

Consider the AKLT models on the two-dimensional honeycomb lattice and the two-dimensional square lattice, defined by nearest-neighbor AKLT interactions with the corresponding latti…

Let $\gamma(S)$ denote the spectral gap of the higher-spin XXZ chain with spin magnitude $S$. Large-spin spectral-gap limit conjecture. The limit … exists and is strictly positive.…

Anyon–superselection sector correspondence conjecture. In these models, anyons should be in one-to-one correspondence with the superselection sectors.

High-temperature exponential clustering conjecture. There exist $\beta^*$, $C_\mathrm{Cov}$, $c_\mathrm{Cov}>0$, and $n\in\mathbb{N}$ such that, for every $\beta<\beta^*$ and every…

Let $G$ be the relevant symmetry group, and consider symmetry protected topological phases in $u$-dimensional quantum spin systems. An SPT phase is a gapped, short-range-entangled…

Let $H_{QSOS}=P_{QSOS}H_{XXZ}P_{QSOS}$ be the Hamiltonian of the QSOS model, where $P_{QSOS}$ is the orthogonal projection onto the QSOS Hilbert space, and let $d$ be the dimension…

Ferromagnetic Ordering of Energy Levels conjecture. All ferromagnetic Heisenberg models have the FOEL property: if $S'<S$, then

FOEL conjecture. Every ferromagnetic XXX model on any finite graph has the FOEL property.

Let $L$ be the sector label and let $\varepsilon_m$ denote the quantity introduced in the paper for sector $m$. Ground-state energy ordering conjecture. … for all $m$ such that…

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 ${\cal H}$ be the Hamiltonian, $d$ the spatial dimension, $L$ the linear system size, and let $T$ be the twist operator. For inverse temperature $\beta$, define the thermal exp…

Let $L$ denote the linear system size, $V$ the volume, $d$ the spatial dimension, and $V/L$ the system width. Assume that the width $V/L$ is of order $L^{d-1}$ and that $d>1$. Even…

Affleck–Kennedy–Lieb–Tasaki conjecture. These models should have a unique ground state in the thermodynamic limit and a spectral gap.

The local topological order axioms are formulated for a quantum spin system, including the quantum error-correction axiom and the injectivity axiom. The sufficiency conjecture. The…

Cohomological classification conjecture. Such states should be classified by the cohomology group

The AKLT model is a quantum spin model whose ground states are the lowest-energy states of its Hamiltonian. A state has antiferromagnetic long-range order when spin correlations ex…

The two-dimensional invertible-phase space is $\mathit{IP}_2$, and its component group is $\pi_0(\mathit{IP}_2)=\pi_{-2}(\mathit{IP})$. Kitaev's chiral-central-charge conjecture. O…

Let $\{\mathit{IP}_d\}_{d\in\mathbb{Z}_{\geq 0}}$ be the spaces of $d$-dimensional invertible gapped systems, and let…

A quantum spin system is integrable when it possesses local conserved quantities. In a one-dimensional system with translational symmetry and nearest-neighbor interactions, a conse…

Let $V$ be a metric space, let $\beta>0$, and let $\boldsymbol{b}:\mathcal{P}_{f}(V)\to[0,\infty)$ be a subbaditive function. Let $\Phi$ be a local interaction satisfying … For eve…

Let $h_x=\lambda \xi_x$, where $(\xi_x)_{x\in\mathbb Z}$ is a sequence of independent identically distributed random variables whose density is positive at $0$. Consider the one-di…

A gapped boundary theory is associated with a bulk theory describing a $2+1$-dimensional topological phase, and its Drinfeld center is the categorical construction encoding bulk ex…

Let $\Gamma$ be a lattice with $\deg(x)=d$ at each site $x\in\Gamma$, with a spin-$s=d/2$ particle at each site. For nearest-neighbor sites, define the AKLT interaction by … where…

Let $T_L$ be an even-length spin ring with spin $j$, and let $E_{\min}(S;T_L)$ denote the minimum energy in the total-spin-$S$ sector. Singlet-sector energy ordering conjecture. Fo…