Bold Diagrammatic Monte Carlo Method Applied to Fermionized Frustrated Spins

Abstract

We demonstrate, by considering the triangular lattice spin-1/2 Heisenberg model, that Monte Carlo sampling of skeleton Feynman diagrams within the fermionization framework offers a universal first-principles tool for strongly correlated lattice quantum systems. We observe the fermionic sign blessing-cancellation of higher order diagrams leading to a finite convergence radius of the series. We calculate the magnetic susceptibility of the triangular-lattice quantum antiferromagnet in the correlated paramagnet regime and reveal a surprisingly accurate microscopic correspondence with its classical counterpart at all accessible temperatures. The extrapolation of the observed relation to zero temperature suggests the absence of the magnetic order in the ground state. We critically examine the implications of this unusual scenario. The method of bold diagrammatic Monte Carlo simulation (BDMC) [1] allows one to sample contributions from millions of skeleton Feynman diagrams and extrapolate the results to the infinite diagram order, provided the series is convergent (or subject to resummation beyond the convergence radius). Recent experimentally certified application of BDMC to unitary fermions down to the point of the superfluid transition [2] makes a strong case for BDMC method as a generic method for dealing with correlated fermions described by Hamiltonians without small parameters. One intriguing avenue to explore is to apply it to frustrated lattice spin systems, where, on one hand, standard Monte Carlo (MC) simulation fails because of the sign problem [3], and, on the other hand, the system's Hamiltonian can be always written in the fermionic representation [4-6] which contains no large parameters-exactly what is needed for the anticipated convergence of BDMC series with the diagram order.The BDMC approach is based on the sign blessing phenomenon, when, despite the factorial increase in the number of diagrams with expansion order, the series features a finite convergence radius because of dramatic (sign alternation induced) compensation between the diagrams. With the finite convergence radius, the series can be summed either directly, or with resummation techniques that can be potentially applied down to the critical temperature of the phase transition, if any. (At the critical temperature thermodynamic functions become nonanalytic, and the diagrammatic expansion involving explicit symmetry breaking by the finite order parameter is necessary to treat the critical region and the phase with broken symmetry.) In the absence of the sign blessing, the resummation protocols become questionable in view of the known mathematical theorems regarding asymptotic series. At the moment, there is no theory allowing one to prove the existence of a finite convergence radius analytically. The absence of Dyson's collapse [7] in a given fermionic system is merely providing hope that the corresponding diagrammatic series is not asymptotic and cannot be a priori taken as a sufficient condition for the sign blessing. Hence, the applicab...

Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

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Diagrammatic Monte Carlo study of quasi-two-dimensional Fermi polarons

“…It can stochastically evaluate the integrals and the different topologies occurring in higher-order perturbation theory and thus provide an answer to the question posed above. It has previously been applied to various fermionic problems such as the unitary Fermi gas 15 , Anderson localization 16 , the Hubbard model in the Fermi-liquid regime 17 , and frustrated-spin systems 18,19 . For the unitary Fermi gas and the three-dimensional (3D) Fermi-polaron problem it was found that the full many-body answer is very close to the first-order result given by a hole line on top of a T matrix.…”

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

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“…In DiagMC [10,11,18,19] one expresses the perturbative expansion in terms of connected Feynman diagrams, which can be written directly in the TL. One then performs a random walk in the space of topologies and of integration variables of Feynman diagrams.…”