Singular values statistics of finite-temperature determinant quantum Monte Carlo simulations

Introducing the auxiliary bosonic field, the partition function of a quantum many-body system can be written in terms of the determinant of a matrix M which is not positive definite in the finite-temperature determinant quantum Monte Carlo simulations. That is why sign problems occur and become a limitation for studying strongly correlated fermions beyond one-dimensional systems. Under the polar decomposition, the M matrix can be considered as a factorization of a unitary matrix U which includes the information of the sign and a positive semi-definite matrix (M^†M)^1/2. In this work, we use singular values statistics of the M matrix, i.e., eigenvalues statistics of (M^†M)^1/2, to investigate the phase transitions of two types of Hubbard-like models on the honeycomb lattice. With the increase of the interaction strength, the auxiliary fields, taken as random variables, break the symmetric structure of matrices and make the singular values uncorrelated. On the other hand, typical fields extracted in the simulations within the Mott insulator phase induce level repulsion. Compared with the results of random fields, the level statistics of typical fields are truly different, and the information of the phase transition can thus be extracted via the metric often employed in the theory of random matrices.