27 problems

Localization dichotomy conjecture. The following statements are equivalent: (a) $P$ admits a generalized Wannier basis that is exponentially localized; (b) $P$ admits a generalized…

Real equivariant almost-commuting sphere matrices conjecture. For every $\eta>0$ there is a $\delta>0$, independent of $n$, such that there is a triple $(X_2,Y_2,H_2)$ of commuting…

In the chiral symmetry classes BDI, CII and AIII, consider a one-dimensional parameter space and the associated winding number, obtained by integrating the winding number density o…

Let $\Omega\subset\mathbb{R}^2$ be a domain, and define its width by … Let $H_\pm$ and $H_e$ be Hamiltonians satisfying Assumptions 1 and 2, and let $\mathcal{G}$ denote the bulk s…

Let $n\in\{2,3,4,6\}$, and consider two-band, insulating, $\mathrm{P}n$-symmetric Hamiltonians with trivial Chern class. Their topological invariants include the Hopf invariant and…

Positive-part edge-index conjecture. In the spectral gap regime,

Let $H_{\mathrm{BHZ}}^{\mathrm{disordered}}$ be the disordered Bernevig–Hughes–Zhang Hamiltonian obtained by adding an i.i.d. random potential whose values have probability measure…

Chern marker conjecture. The Chern marker characterizes the existence of a localized generalized Wannier basis for gapped non-periodic systems: vanishing of $C(P)$ should be equiva…

Let $V\in C^\infty(\mathbb{R}^3/\mathbb{Z}^3)$ be a smooth periodic potential, and consider the periodic Schrödinger operator $-\Delta_{\mathbb{R}^3}+V$. A degeneracy of its Bloch…

Consider a Haldane-model calculation with periodic boundary conditions, where $P$ is the Fermi projection and $X$ is the lattice position operator in the $x$ direction. The operato…

Exponentially localized generalized Wannier bases are conjectured to exist more generally when time-reversal symmetry holds. In the setting of non-periodic insulating systems, and…

Wave-packet propagation conjecture. Wave-packets propagating along boundaries between regions of differing spectral localizer index do not lose significant mass whenever the locali…

Consider a family of class A insulators in spatial dimension $D$, with parameter space ${\mathsf M}$, and let $\Omega^{(D+2)}$ be the closed $(D+2)$-form associated to the family.…

For a family of translationally invariant tight-binding free-fermion Hamiltonians in spatial dimension $D$, let ${\mathsf M}$ be the parameter space and let $\Omega^{(D+2)}$ be the…

Let $d=3$ and let $chat{P}_+$ and $chat{P}_-$ be the surface Fermi projections associated with a chiral system described by a Fermi unitary $U$, under the assumption that the surfa…

Let $H_\omega$ be a covariant Hamiltonian and let $\mu$ be the Fermi level. Its Fermi projection is … Consider topological phases in even dimensions whose classification is by…

A topological insulator belongs to one of ten symmetry classes determined by the fundamental symmetries stabilizing topological insulating phases. For each class, consider the spat…

Let $\hat h=(h,\tilde h)\in M_{2N}(\mathbb C)\times\widehat{\mathpzc A}_d$ with $d=3$, suppose BGH and CH apply, and assume ${\rm Ch}_3(u_F)\neq 0$. Then the boundary spectrum is d…

Let $H_k$ be the smooth family of Bloch–Floquet matrices of a periodic finite-range lattice Hamiltonian over the Brillouin torus $\mathbb T^d$. In the unitary symmetry class, topol…

Let $d\leq 3$ and consider the topological phases listed by the non-vanishing entries in the classification table of the ten symmetry classes. A boundary spectrum is delocalized wh…

Let $H$ and $X$ be self-adjoint matrices in a matrix algebra, and suppose that $(X,H)$ belongs to an Altland–Zirnbauer symmetry class. Lin's theorem states that sufficiently almost…

Topological-order conjecture. Our $(3+1)$-TQFTs are the underlying topological orders for $3D$ topological insulators and their generalizations.

Let $\gamma^\alpha$ be the matrices defining a representation of the Clifford algebra $C^{0,d}$, and let $M$ be a real antisymmetric mass term. Assume that…

Topological phases of fermionic systems are organized by symmetry constraints on gapped quadratic fermion Hamiltonians; the relevant symmetries include unitary and antiunitary oper…

Let $U_1$ and $U_2$ be approximately commuting unitary matrices, with $obreakbiglVert[U1,U2]bigrVertleq ecdelta$, and let $H1,H2,H3$ be the associated Hermitian matrices construct…