Abstract
The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of gapped, chiral, topological phases of matter. Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or 'qudits'), and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge, and the topological spins of anyons. We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and non-chiral topological matter.
Highlights
Utilizing a random sampling of phase-space according to the Boltzmann probability distribution, Monte Carlo simulations are arguably the most powerful tools for numerically evaluating thermal averages in classical many-body physics [1]
At energies small compared with the bulk gap, the boundary can be described by a chiral conformal field theory (CFT) [84,85], while the bulk reduces to a chiral topological field theory (TFT) [46,47]; see Fig. 1(a)
A notable example for chiral topological phases is given by Chern insulators [86,87,88]: symmetry-protected topological phases (SPTs) protected by the U(1) fermion number symmetry, which admit free-fermion Hamiltonians
Summary
Utilizing a random sampling of phase-space according to the Boltzmann probability distribution, Monte Carlo simulations are arguably the most powerful tools for numerically evaluating thermal averages in classical many-body physics [1]. In 98.5% of first 103 In 96.7% of first 103 For ν ∈/ 12Z Yes Yes In 91.6% of first 103 For n ∈/ 3N Yes Yes Yes Yes open problems in quantum many-body physics continue to defy solution and remain inaccessible for QMC These include the nature of high-temperature superconductivity and the associated repulsive Hubbard model [20,25,26,27], dense nuclear matter and the associated lattice QCD at finite baryon density [28,29,30], and the enigmatic fractional quantum Hall state at filling 5/2 and its associated Coulomb Hamiltonian [31,32,33,34,35,36], all of which are fermionic. VIII we discuss our results and provide an outlook for future work
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