Auxiliary-Field Patterns, Random Matrices Theory, and Topology in Determinant Quantum Monte Carlo Simulation

The Hirsch’s Hubbard–Stratonovich (HS) decomposition maps two-body interactions onto single-particle problems coupled to auxiliary fields, forming the foundation of determinant quantum Monte Carlo (DQMC). A well-known relation between auxiliary fields and spin–spin correlations leads to a characteristic staggered pattern in the Mott insulating regime.



In this work, we extend this framework in three directions. First, by applying a polar decomposition to the fermionic single-particle propagator matrix in DQMC, we analyze the singular-value statistics of —equivalently, the eigenvalue statistics of . These quantities capture the symmetry information encoded in the staggered auxiliary-field pattern and successfully identify the interaction-driven Mott insulating phase in two classes of Hubbard models on the honeycomb lattice, insensitive to the presence of the sign problem.



Second, the determinant of may serve as a topological invariant, suggesting a possible link between the sign problem and the topological characteristics of the propagator.



Finally, we generalize Hirsch’s relation to alternative HS decompositions, establishing explicit correspondences between auxiliary fields and correlation functions across various decoupling schemes. This generalized formulation naturally incorporates random-matrix theory (RMT) tools, which help us search for and characterize phase transitions in many-body systems.