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arXiv:1808.02043 · Physical review. B./Physical review. B

Thermodynamic properties of the Shastry-Sutherland model from quantum Monte Carlo simulations

Stefan Weßel, Ido Niesen, Jonas Stapmanns, Bruce Normand, Frédéric Mila, Philippe Corboz, A. Honecker

Published Nov 27, 2018• 47 citations
arXivHTMLPDF

Abstract

We investigate the minus-sign problem that afflicts quantum Monte Carlo (QMC) simulations of frustrated quantum spin systems, focusing on spin $S=1/2$, two spatial dimensions, and the extended Shastry-Sutherland model. We show that formulating the Hamiltonian in the diagonal dimer basis leads to a sign problem that becomes negligible at low temperatures for small and intermediate values of the ratio of the inter- and intradimer couplings. This is a consequence of the fact that the product state of dimer singlets is the exact ground state both of the extended Shastry-Sutherland model and of a corresponding ``sign-problem-free'' model, obtained by changing the signs of all positive off-diagonal matrix elements in the dimer basis. By exploiting this insight, we map the sign problem throughout the extended parameter space from the Shastry-Sutherland to the fully frustrated bilayer model and compare it with the phase diagram computed by tensor-network methods. We use QMC to compute with high accuracy the temperature dependence of the magnetic specific heat and susceptibility of the Shastry-Sutherland model for large systems up to a coupling ratio of 0.526(1) and down to zero temperature. For larger coupling ratios, our QMC results assist us in benchmarking the evolution of the thermodynamic properties by systematic comparison with exact diagonalization calculations and interpolated high-temperature series expansions.

DOI
10.1103/physrevb.98.174432

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We investigate the minus-sign problem that afflicts quantum Monte Carlo (QMC) simulations of frustrated quantum spin systems, focusing on spin S=1/2S=1/2S=1/2, two spatial dimensions, and the extended Shastry-Sutherland model. We show that formulating the Hamiltonian in the diagonal dimer basis leads to a sign problem that becomes negligible at low temperatures for small and intermediate values of the ratio of the inter- and intradimer couplings. This is a consequence of the fact that the product state of dimer singlets is the exact ground state both of the extended Shastry-Sutherland model and of a corresponding ``sign-problem-free'' model, obtained by changing the signs of all positive off-diagonal matrix elements in the dimer basis. By exploiting this insight, we map the sign problem throughout the extended parameter space from the Shastry-Sutherland to the fully frustrated bilayer model and compare it with the phase diagram computed by tensor-network methods. We use QMC to compute with high accuracy the temperature dependence of the magnetic specific heat and susceptibility of the Shastry-Sutherland model for large systems up to a coupling ratio of 0.526(1) and down to zero temperature. For larger coupling ratios, our QMC results assist us in benchmarking the evolution of the thermodynamic properties by systematic comparison with exact diagonalization calculations and interpolated high-temperature series expansions.
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 L = 200 rungs (400 spins 1/2) and down to very low temperatures.
Building on a recent investigation of the Shastry-Sutherland model [S. Wessel et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to eliminate the Monte Carlo sign problem near the zero-temperature limit in frustrated quantum spin models. If the Hamiltonian of interest and the sign-problem-free Hamiltonian, obtained by making all off-diagonal elements negative in a given basis, have the same ground state and this state is a member of the computational basis, then the average sign returns to one as the temperature goes to zero. We illustrate this technique by studying the triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field above saturation, as well as the Heisenberg antiferromagnet on a modified Husimi cactus in the dimer basis. We also provide detailed Appendices on using linear programming techniques to automatically generate efficient directed loop updates in quantum Monte Carlo simulations.
We study the ground state properties of the extended hard-core Bose–Hubbard model in a two-dimensional Shastry–Sutherland lattice by quantum Monte Carlo simulations. The effects of the geometric frustration, described by the ratio t ′ / t between hopping integrals along the nearest-neighbor links and the diagonal links, are studied. We find that for small values of frustration amplitude t ′ / t , a checkerboard solid state emerges at ρ= 1/2; for large values of frustration amplitude t ′ / t , a solid state emerges at ρ= 1/2 with one boson per dimer (dimers on diagonal links); for intermediate values of frustration amplitude t ′ / t , besides the ρ= 1/2 solid state, the ρ= 1/3,2/3 solid states emerge. No stable supersolid state is observed in our calculations. At half filling, we find solid-superfluid-solid phase transition as the frustration amplitude t ′ / t is increased.
We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of ``semifrustrated'' systems [Heisenberg models with ferromagnetic couplings Jz(r)<0{J}_{z}(r)<0Jz​(r)<0 along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the xy\mathrm{xy}xy plane, for arbitrary distances r],r],r], the sign problem present for algorithms operating in the z basis can be solved within a recent ``operator-loop'' formulation of the stochastic series expansion method [a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ to all orders]. The solution relies on the identification of operator loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models [Phys. Rev. Lett. 83, 3116 (1999)]. Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.
Quantum antiferromagnets have proven to be some of the cleanest realizations available for theoretical, numerical, and experimental studies of quantum fluctuation effects. At finite temperatures, however, the additional effects of thermal fluctuations in the restricted phase space of a low-dimensional system have received much less attention, particularly the situation in frustrated quantum magnets, where the excitations may be complex collective (bound or even fractionalized) modes. We investigate this problem by studying the thermodynamic properties of the frustrated two-leg S=12S=\frac{1}{2}S=21​ spin ladder, with particular emphasis on the fully frustrated case. We present numerical results for the magnetic specific heat and susceptibility, obtained from exact diagonalization and quantum Monte Carlo studies, which we show can be rendered free of the sign problem even in a strongly frustrated system and which allow us to reach unprecedented sizes of L=200L=200L=200 ladder rungs. We find that frustration effects cause an unconventional evolution of the thermodynamic response across the full parameter regime of the model. However, close to the first-order transition they cause a highly anomalous reduction in temperature scales with no concomitant changes in the gap; the specific heat shows a very narrow peak at very low energies and the susceptibility rises abruptly at extremely low temperatures. Unusually, the two quantities have different gaps over an extended region of the parameter space. We demonstrate that these results reflect the presence of large numbers of multiparticle bound-state excitations, whose energies fall below the one-triplon gap in the transition region.
We investigate the Shastry-Sutherland model (SSM), i.e., spin-1/2 quantum Heisen- berg model on a Shastry-Sutherland lattice, using a newly emerging approach exploiting well-developed machine learning techniques. We utilize neural networks as variational quantum states in quantum Monte Carlo investigations of the ground state properties. We first focus on SSM without an external magnetic field. For small lattices accessible via exact diagonalization, we compare the precision of various architectures based on re- stricted Boltzmann machines (RBM) or group-convolutional neural networks. The most versatile and precise architecture, namely complex-valued RBM, is then applied for larger lattices. Here we investigate the frustrated regime. We show that the RBM is able to represent all three of the major and fundamentally different phases of SSM. Finally, we apply the complex-valued RBM for SSM in a finite external magnetic field. We find that it cannot capture the intriguing steps in magnetization typical for SSM correctly due to its tendency to prefer more ordered states with higher magnetization. 1