Continuous phase transitions in diagrammatic Monte Carlo

Connor Lenihan, Aaram J. Kim, Fedor Šimkovic +1 more

Diagrammatic Monte Carlo-the technique for the numerically exact summation of all Feynman diagrams to high orders-offers a unique unbiased probe of continuous phase transitions. Being formulated directly in the thermodynamic limit, the diagrammatic series is bound to diverge and is not resummable at the transition due to the nonanalyticity of physical observables. This enables the detection of the transition with controlled error bars from an analysis of the series coefficients alone, avoiding the challenge of evaluating physical observables near the transition. We demonstrate this technique by the example of the Néel transition in the 3D Hubbard model. At half filling and higher temperatures, the method matches the accuracy of state-of-the-art finite-size techniques, but surpasses it at low temperatures and allows us to map the phase diagram in the doped regime, where finite-size techniques struggle from the fermion sign problem. At low temperatures and sufficient doping, the transition to an incommensurate spin density wave state is observed.

Kurt Binder, Erik Luijten, Marcus Müller +2 more

Fedor Šimkovic, Michel Ferrero

The computation of determinants plays a central role in diagrammatic Monte Carlo algorithms for strongly correlated systems. The evaluation of large numbers of determinants can often be the limiting computational factor determining the number of attainable diagrammatic expansion orders. In this work we build on the algorithm presented in [K. Griffin and M. J. Tsatsomeros, Principal minors, part i: A method for computing all the principal minors of a matrix, Linear Algebra Appl. 419, 107 (2006)] which computes all principal minors of a matrix in $O({2}^{n})$ operations. We present multiple generalizations of the algorithm to the efficient evaluation of certain subsets of all principal minors with immediate applications to connected determinant diagrammatic Monte Carlo within the normal and symmetry-broken phases as well as continuous-time quantum Monte Carlo. Additionally, we improve the asymptotic scaling of diagrammatic Monte Carlo formulated in real time to $O({2}^{n})$ and report speedups of up to a factor 25 at computationally realistic expansion orders. We further show that all permanent principal minors, corresponding to sums of bosonic Feynman diagrams, can be computed in $O({3}^{n})$, thus encouraging the investigation of bosonic and mixed systems by means of diagrammatic Monte Carlo.

Nigel B. Wilding

Riccardo Rossi

Kaiyu Yang, Evgeny Kozik, Xin Wang +1 more

We investigate the two-dimensional cooperon-fermion model in the correlated regime with a continuous-time diagrammatic determinant quantum Monte Carlo algorithm. We estimate the transition temperature ${T}_{c}$, examine the effectively reduced band gap and cooperon mass, and find that delocalization of the cooperons enhances the diamagnetism. When applied to diamagnetism of the pseudogap phase in high-${T}_{c}$ cuprates, we obtain results in qualitative agreement with recent torque magnetization measurements.