Spectral gaps and order in quantum spin systems

5 problems

• Spectral gap of the AKLT model on the square lattice

  Open

  The AKLT model is a specially designed quantum magnet whose ground state is known exactly. Its energy gap is proven in one dimension and on the honeycomb lattice but not on the square lattice.

• Haldane gap in the spin-1 Heisenberg chain

  Open

  Haldane predicted that a chain of spin-1 magnets with the simplest antiparallel coupling has a finite energy gap above its ground state. Computers confirm it, but there is no proof.

• Gaplessness of half-integer spin Heisenberg chains with $S \ge 3/2$

  Open

  Chains of half-integer spins are expected to have no energy gap. A general theorem allows either no gap or several ground states, and only the spin-1/2 case is solved exactly.

• Neel order in the spin-$1/2$ square-lattice Heisenberg antiferromagnet

  Open

  In the ground state of a square grid of spin-$1/2$ magnets with antiparallel coupling, the spins should line up in a checkerboard pattern with finite strength. This is proven for larger spins but not for spin $1/2$.

• Spontaneous magnetization of the 3D quantum Heisenberg ferromagnet

  Open

  A cubic lattice of quantum magnets with parallel-favouring coupling should be magnetized at low temperature, as real iron is. The proof works for classical spins and for the antiparallel quantum case, but not for the quantum ferromagnet.