Efficient Quantum Monte Carlo simulations of highly frustrated magnets: the frustrated spin-1/2 ladder

Abstract

Quantum Monte Carlo simulations provide one of the more powerful and versatile numerical approaches to condensed matter systems. However, their application to frustrated quantum spin models, in all relevant temperature regimes, is hamstrung by the infamous "sign problem." Here we exploit the fact that the sign problem is basis-dependent. Recent studies have shown that passing to a dimer (two-site) basis eliminates the sign problem completely for a fully frustrated spin model on the two-leg ladder. We generalize this result to all partially frustrated two-leg spin-1/2 ladders, meaning those where the diagonal and leg couplings take any antiferromagnetic values. We find that, although the sign problem does reappear, it remains remarkably mild throughout the entire phase diagram. We explain this result and apply it to perform efficient quantum Monte Carlo simulations of frustrated ladders, obtaining accurate results for thermodynamic quantities such as the magnetic specific heat and susceptibility of ladders up to L = 200 rungs (400 spins 1/2) and down to very low temperatures.

Easing the Monte Carlo sign problem

“…We initialized the algorithm at the identity matrix (which was randomly perturbed by a small amount). The phase diagram of the non-stoquasticity qualitatively agrees with the findings of (14), where the stochastic series expansion QMC method was studied. There, it was found that the sign problem can be completely eliminated for a completely frustrated arrangement where J × = J ∥ , while the sign problem remains present for partially frustrated couplings J × ≠ J ∥ .…”

“…In an attempt to ease the sign problem of a given Hamiltonian, it is therefore natural to try and improve the average sign. For a few specific models, these improvements have been achieved by different means: For example, one can exploit known physics to find bases with improved average sign (14,22) that are often induced by sparse representations (17,23,24). For particular observables, one can also exploit clever decompositions of the Monte Carlo estimator into clusters with nonnegative sign (25)(26)(27)(28)(29)(30)(31).…”

“…Ladder geometries are interesting not only for the reasons described above but also because, in spite of frustration effects, they often admit sign-problem free QMC methods (11,13,14). For both the J 0 -J 1 -J 2 -J 3 model studied in (11) and the frustrated Heisenberg ladder studied in (13,14), we find a rich optimization landscape in which a relative improvement of the non-stoquasticity by a factor of 2 to 5 can be achieved depending on the region in the phase diagram. Importantly and in spite of those seemingly moderate improvements of non-stoquasticity, we find that the sample complexity of QMC as governed by the inverse average sign is greatly diminished to approximate unity in large regions of the parameter space for the frustrated ladder model (see Fig.…”

“…For specific models, sign-problem free bases can be found analytically, involving nonlocal bases, for example, by using so-called auxiliary-field ( 7 ), Jordan-Wigner ( 8 ), or Majorana ( 9 , 10 ) transformations. One can also exploit specific known properties of the system such as that the system dimerizes ( 11 – 14 ). These findings motivate the quest for a more broadly applicable systematic search for basis changes that avoid the sign problem, in a way that does not depend on the specific physics of the problem at hand.…”

Wave-function positivization via automatic differentiation

“…The most notable example is the Marshall sign rule [2], eliminating the sign structure from the ground states of quantum antiferromagnets on bipartite lattices. The resulting theoretical insight means that new bases that simplify the sign structure for a specific frustrated magnet or fermion model are routinely discovered [3][4][5][6]. In turn, in a few instances it has also been rigorously proven that efficient transformations do not exist [7,8], rendering the sign structure "intrinsic."…”