Time-reversal positivity

TL;DR

The paper introduces time-reversal positivity as a time-reversal analogue of Majorana reflection positivity and develops a Fock-space version to address time-reversal symmetric many-fermion systems, linking these positivities to sign-free Monte Carlo sampling. It constructs a cone framework with $K_O$, $K$, and $K_{2n}$ to encode positivity properties of time-reversal invariant operators and states, including non-negative traces and overlaps that support sign-free simulations. It then extends Lieb's theorem to time-reversal symmetric non-Hermitian Hubbard models by a Trotter-Suzuki/Hubbard-Stratonovich decomposition, proving ground-state uniqueness in the sector $N=2n$ via $K_{2n}$-non-negativity and irreducibility of $A=1-d\tau H$, with the approach contingent on non-Hermitian hopping matrices that generate the appropriate Lie algebra. The work offers a general framework to analyze sign problems in AFQMC/PQMC for time-reversal symmetric non-Hermitian systems and to guarantee ground-state uniqueness in a broader class of models.

Abstract

We propose a new analytical tool called time-reversal positivity. It is an analogue of the Majorana reflection positivity in time-reversal symmetric case. This new time-reversal positivity can fully explain the relationship between time-reversal symmetry and the sign-free property in quantum Monte Carlo simulations. Using a cone-theoretical method, we show the ground state uniqueness for the time-reversal symmetric non-hermitian Hubbard model.