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arXiv:2211.09560 · Journal of the Physical Society of Japan

Study of Staggered Magnetization in the Spin-S Square-Lattice Heisenberg Model Using Spiral Boundary Conditions

Masahiro Kadosawa, Masaaki Nakamura, Y. Ohta, Satoshi Nishimoto

Published Dec 26, 2022• 7 citations
arXivHTMLPDF

Abstract

We propose an efficient numerical method to obtain local order parameter in two-dimensional systems using spiral boundary conditions. As a benchmark, we first estimate the magnitude of staggered magnetization for the $S=1/2$ XXZ Heisenberg model on a square lattice in the whole range of the XXZ anisotropy by density-matrix renormalization group technique. The validity of our method is confirmed by comparing our results with the previous analytical and numerical studies. Then, as further demonstration, we apply our method to obtain the staggered magnetization for the higher-spin cases from $S=1$ to $S=6$. The accuracy of the obtained results is validated using the series expansion and the spin-wave theory.

DOI
10.7566/jpsj.92.023701

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We propose an efficient numerical method to obtain local order parameter in two-dimensional systems using spiral boundary conditions. As a benchmark, we first estimate the magnitude of staggered magnetization for the S=1/2S=1/2S=1/2 XXZ Heisenberg model on a square lattice in the whole range of the XXZ anisotropy by density-matrix renormalization group technique. The validity of our method is confirmed by comparing our results with the previous analytical and numerical studies. Then, as further demonstration, we apply our method to obtain the staggered magnetization for the higher-spin cases from S=1S=1S=1 to S=6S=6S=6. The accuracy of the obtained results is validated using the series expansion and the spin-wave theory.
We present a detailed investigation of the XXZ Heisenberg model for spin-1/21/21/2 and spin-1 systems on square and honeycomb lattices. Utilizing the density-matrix renormalization group method, complemented by spiral boundary conditions (SBCs) for mapping two-dimensional (2D) clusters onto one-dimensional chains, we meticulously explore the evolution of staggered magnetization and spin gaps across a broad spectrum of easy-axis anisotropies. Our study reveals that despite the lower site coordination number of the honeycomb lattice, which intuitively suggests increased quantum fluctuations in its N\'eel phase compared to the square lattice, the staggered magnetization in the honeycomb structure exhibits only a marginal reduction. Furthermore, our analysis demonstrates that the dependence of staggered magnetization on the XXZ anisotropy $\mathrm{\ensuremath{\Delta}}$, except in close proximity to $\mathrm{\ensuremath{\Delta}}=1$, aligns with series expansion predictions up to the 12th order. Notably, for the S=1/2S=1/2S=1/2 honeycomb lattice, deviations from the 10th-order series expansion predictions near the isotropic Heisenberg limit emphasize the critical influence of quantum fluctuations on the spin excitation in its N\'eel state. Additionally, our findings are numerically consistent with the singular behavior of the spin gap near the isotropic Heisenberg limit as forecasted by spin-wave theory. The successful implementation of SBCs marks a methodological advancement, streamlining the computational complexity involved in analyzing 2D models and paving the way for more precise determinations of physical properties in complex lattice systems.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
We show that the density matrix renormalization group can be used to study magnetic ordering in two-dimensional spin models. Local quantities should be extrapolated with the truncation error, not with its square root. We introduce sequences of clusters, using cylindrical boundary conditions with pinning fields, which provide for rapidly converging finite-size scaling. We determine the magnetization for both the square and triangular Heisenberg lattices with errors comparable to the best alternative approaches.
We develop a formalism for calculating arbitrary expectation values for any extensive lattice Hamiltonian system using an analytic Lanczos expansion, or plaquette expansion, and a recently proved exact theorem for ground-state energies. The ground-state energy, staggered magnetization, and excited state gap of the two-dimensional anisotropic antiferromagnetic Heisenberg model are then calculated using this expansion for a range of anisotropy parameters, and compared to other moment-based techniques, such as the t expansion, and spin-wave theory, and series-expansion methods. We find that far from the isotropic point all moment methods give essentially very similar results, but near the isotropic point the plaquette expansion is generally better than the others.
We study the phase diagram of the spin-1/2 XXZ Heisenberg chain in a staggered magnetic field. An exact solution is obtained along the phase boundary, where the exchange anisotropy and the staggered-field strength satisfy an exact analytical relation, yielding a highly degenerate manifold of ground states with off-diagonal long-range order. We demonstrate that these exact ground states lie on the boundary between the ferromagnetic phase and a gapless Luttinger liquid phase. Upon further decreasing the anisotropy, the gapless phase eventually gives way to a gapped quantum-disordered phase. Large-scale density-matrix renormalization group calculations confirm this phase structure and indicate that the transition across the exact boundary is of first order. Our results provide an exact benchmark for understanding quantum phase transitions driven by staggered magnetic fields in one-dimensional quantum spin systems.