Abstract
We describe a controllable and unbiased strong-coupling diagrammatic Monte Carlo technique that is applicable to a wide range of fermionic systems and spin models. Unlike previous strong coupling methods that generally rely on the Grassmannian Hubbard-Stratonovich transformation, our construction is based on Wick's theorem and a recursive procedure to group contractions into effective connected vertices that are non-perturbative in all local physics and can be calculated exactly. The resulting expansion is described by simple diagrammatic rules that make it suitable for systematic treatment via stochastic sampling. Benchmarks against numerical linked cluster expansion display excellent agreement.
Highlights
Correlated electrons and frustrated spin models are among the most challenging problems in condensed matter theory due to a combination of the sign problem, lack of a natural small parameter, and the computational complexity of series expansions. This topic is at the same time absolutely central to understanding the electronic structure of solids, and a wide range of numerical techniques have been devised to overcome these obstacles
Several of the aforementioned techniques are capable of producing states that seem highly relevant for cuprate superconductivity, including antiferromagnetism, stripes, pseudogap physics, and d-wave superconductivity
It has been known for some time that notable discrepancies may appear both when comparing different techniques and when altering details of the implementation of a given method [16]. This sensitivity that correlated-fermion models display to numerical protocol appears to have a physical origin and be rooted in competition between different states situated very closely in terms of free energy [16,17]
Summary
Correlated electrons and frustrated spin models are among the most challenging problems in condensed matter theory due to a combination of the sign problem, lack of a natural small parameter, and the computational complexity of series expansions. Numerical linked cluster expansion (NLCE) [23] has been applied successfully to both spin models [24] and itinerant fermionic theories like the t–J [25] and Hubbard models For the latter, results exist at infinite on-site repulsion [26], and for finite interactions up to U/t = 16 [27], which is far into the strongly correlated regime. Doubly occupied sites can be reintroduced in the form of hardcore bosons, which are subsequently fermionized, allowing generic correlated systems to be addressed within this framework [32] This technique suffered from poor convergence properties at its inception except at large doping, but this problem has since been overcome through spin-charge transformation, which essentially results in a representation involving fermionic carriers that propagate on a spin background [33]. This series expansion is computationally economical, possesses simple diagrammatic rules, and is free of any large expansion parameter, even for arbitrarily strong interactions
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