Fermion sign problem and controlled many-body numerics

5 problems

• Resolution limits of analytic continuation from imaginary-time data

  Open

  Most electron simulations work in imaginary time, and converting their output into measurable spectra is mathematically unstable: tiny noise allows wildly different answers. The best possible resolution for a given noise level, and a way to certify an answer, are not established.

• Certified two-sided error bars for 2D Hubbard ground states

  Open

  For the two-dimensional Hubbard model, a basic model of electrons in a crystal, different simulation methods give competing ground states whose energies differ by less than one percent. Methods that give guaranteed upper and lower bounds tight enough to decide between them do not exist.

• Polynomial-cost real-time simulation of interacting fermion lattices

  Partially resolved

  Simulating how electrons evolve in time requires summing complex phases that cancel more and more as time grows, so the cost rises exponentially with simulated time. For a single magnetic impurity, new methods reduced this to polynomial cost; whether this is possible for whole lattices is open.

• Which phases of matter carry an intrinsic sign problem

  Partially resolved

  Many phases of matter with one-way edge currents provably cannot be simulated sign-free by any model built from local terms. Which other phases share this obstruction is open.

• Complete criteria for sign-free fermion quantum Monte Carlo

  Partially resolved

  Some special electron models can be simulated without the sign problem thanks to hidden symmetries. A complete list of which models allow this, and whether clever changes of variables can remove the problem in others, is not known.