24 problems

Consider the one-dimensional long-range Ising model with interaction exponent $\alpha$; a phase transition means the existence of more than one Gibbs measure. Kac–Thompson conjectu…

Frobenius light-cone conjecture. Entanglement is governed by the Frobenius light cone for power-law interactions.

Long-range interacting systems exhibit spontaneous symmetry breaking, with collective fluctuation modes whose frequencies remain nonzero as the wave number tends to zero. The relev…

Noncommutativity conjecture. This noncommutativity of the thermodynamic and infinite-time limits should be an ubiquitous feature in non-extensive systems.

Inverse-temperature conjecture. The lower-bound constant should satisfy $c_\beta\asymp\beta^{-1}$ as $\beta\to\infty$.

Let $(\phi_i)_{i\in\mathbb Z}$ be the height field of the one-dimensional long-range discrete Gaussian chain, with formal Hamiltonian … where $\phi_i\in\mathbb Z$ and…

Effective-temperature conjecture. For $alpha ain(2,3)$,

Delocalization conjecture. When $\alpha>2$, the discrete Gaussian chain is delocalized at all temperatures.

Let $({\mathbb Z}^d,W)$ be the lattice with long-range interactions described above, where the edge weights are not necessarily locally finite but satisfy the stated long-range con…

Let $s\in\mathbb{S}^{d-1}$, let $n$ tend to infinity, let $\beta_{\rm sat}^{*}(s)$ be the saturation threshold for the truncated two-point function, let $\beta_{\rm c}$ be the crit…

Let $J_x$ be the coupling constants, let $u{beta}(s)$ denote the inverse correlation length in direction $s$, and let $beta{rm c}$ be the critical inverse temperature. **Exponenti…

Let $\beta_{\rm sat}(s)$ denote the saturation threshold in direction $s$, and let $\beta_{\rm sat}^{*}(s)$ be the corresponding threshold for the truncated two-point function. Let…

Consider a range-three spin chain with local Hilbert space $V$, and set $W=V\otimes V$. Let $\mathcal{L}_{a_1,a_2,n_1,n_2}(u)$ be a Lax operator acting on $W\otimes W$, where the i…

Let Skyrmions be topologically protected quasiparticles in a two-dimensional spin lattice, and consider their behavior in the presence of dipolar interactions. Dipole-interaction r…

Let $W_\alpha^{\pm}$ be the wave operators, and consider the scattering matrix at non-threshold energies in the stationary scattering theory of an $N$-body long-range system. The s…

Consider a one-dimensional periodic cell containing $n$ balls of each type $A$, $B$, and $C$, with positive radii parameters $\omega_i$ satisfying $\sum_i\omega_i=1$, and let…

Let $A_0$ and $B_r$ be local operators at sites separated by distance $r$ in a one-dimensional spin chain with power-law interactions of exponent $\alpha$, and let…

Consider a spin chain with power-law interactions and let the Frobenius light cone be the lower-bound timescale for the spreading of commutators measured in the Frobenius norm. A v…

Let $t_{\mathrm{F}}(r)$ denote the Frobenius light-cone timescale for local operators separated by distance $r$ in a spin system with power-law interactions decaying with exponent…

Continuum long-range polymer conjecture. Under these conditions, the partition function ${{\mathcal Z}} _{\beta,\mathrm{long}}^{\omega,a}$ converges almost surely to a non-degenera…

Linear decoherence light-cone conjecture. This light cone is linear for

Consider a $D$-dimensional long-range interacting system whose interactions decay with distance as a power law with exponent $\alpha$. The authors have proved that $\alpha>2D+1$ is…

Consider a quantum system with power-law interactions decaying with exponent $\alpha$, and let the scrambling operators in the referenced scrambling expression act on a single latt…

Long-range interactions (LRI) can generate mean-field potentials through the coherent addition of microscopic interactions across the sample. Generic coherent addition conjecture.…