103 problems

Consider the Sachdev–Ye–Kitaev model of $N$ interacting Majorana fermions, with Majorana operators satisfying … Fast scrambling conjecture. The time required for an operator to "gr…

Consider a dilute homogeneous Bose gas with particle density $\rho$, interaction potential $v$, and positive scattering length $a>0$. Let the leading-order ground-state energy asym…

Consider a power-law interacting system on a $d$-dimensional lattice, with exponent $\alpha$, and the transition from an algebraic to a linear Frobenius light cone. Critical-expone…

Let $E_I^D$ denote the distillable entanglement of a cubic region $I=[1,\ldots,m]^{\times D}$, and write the area law as … Here criticality is characterized by two-point correlatio…

Let $\chi_{\alpha}^{(\otimes)}(\beta)$ denote the quantity in equation (GPTK), with $\alpha$ determined by the interaction potential, and let the Gross–Pitaevskii formula denote th…

Let $V$ be the integral operator with kernel $v$, and let $\chi_N^{(\otimes)}$ be the Hartree variational quantity obtained by minimising over normalized functions…

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…

The model is defined by the nodal and antinodal Hamiltonians discussed above, with the approximations introduced in Section 5 used to obtain its effective description. Low-energy a…

Let $v$ be an interaction potential with positive scattering length $a>0$, and suppose that the corresponding system has no $N$-body bound states for any $N$. Let … should hold whe…

Let a high-dimensional lattice have a particle density per site and an optimal magnetic flux per plaquette, namely a flux minimizing the ground-state energy. Flux phase conjecture.…

Propagation of quantum molecular chaos conjecture. The sequence of Heisenberg maps $\phi_{n,t}(A)=e^{iH_nt/\hbar}Ae^{-iH_nt/\hbar}$ propagates chaos: if $\{\rho_n\}$ is $\rho$-chao…

Low-density Bose-gas conjecture. The formula above requires only a positive scattering length and the absence of any many-body, negative-energy bound state.

Polynomial structure conjecture. The displayed polynomial structure of the small-$cL$ expansion of $g_N(x,0)$ continues for all $N$.

Polynomial structure conjecture. This pattern holds for all $N\geq 2$: for every $p\geq 2$, the coefficient of $(N-1)(cL)^p$ is a polynomial of degree $p-2$ in $N-1$.

Let $U$ be the repulsive coupling, $t$ the nearest-neighbor hopping strength, and let $u$ denote the filling, with half-filling corresponding to $u=1/2$. **Strong-coupling ferrom…

Let a Hubbard model have a localized domain wall structure, and consider the band-bending perturbation $H_{\rm hop} '$. The variational analysis compares the energy shifts of domai…

Rational-filling gap conjecture. One may conjecture that

Similarity-transformation conjecture. The similarity transformation can be implemented by an invertible operator $U_j$ corresponding to the invertible anyon $j$, or equivalently by…

Let a $W^p$ droplet have particle density $\rho=p/M$ in its initial domain, and compare its dynamics under extensive-local interacting quantum many-body scar Hamiltonians of type I…

Let $\ket{W^p}$ be a $W^p$ droplet containing $p$ particles in an initial domain of size $M$, and let $\rho=p/M$. Low-density BEC conjecture. The treatment of the $W^p$ state as a…

Free-fermion solvability conjecture. The circuits defined by $\mathcal{V}^{({\rm III})}_M$ are solvable by free-fermion decomposition (FFD).

Let $\mathfrak L_0$ be the space of local observables, let $o,o'\in\mathfrak L_0$, and let $\mathcal T_{\vec\lambda}$ and $\b{\mathcal T}_{\vec\lambda}$ be a twist field and its co…

Spin-1/2 Fermi energy conjecture. The ground-state energy is conjectured to have the expansion

Let $N$ particles form a dilute one-dimensional spin-$1/2$ Fermi gas of density $\rho$, with even-wave and odd-wave scattering lengths $a_e$ and $a_o$, respectively. The free Fermi…

Let $b_n$ be the Lanczos coefficients for a chaotic local Hamiltonian in spatial dimension $d$, and let $b1$ and $b3$ denote constants. Operator Growth Hypothesis. The leading…