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arXiv:2310.11774 · Journal of Physics Condensed Matter

On the magnetization of the 120° order of the spin-1/2 triangular lattice Heisenberg model: a DMRG revisited

Jiale Huang, Xiangjian Qian, Mingpu Qin

Published Jan 23, 2024• 9 citations
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Abstract

We revisit the issue about the magnetization of the 120° order in the spin-1/2 triangular lattice Heisenberg model with density matrix renormalization group (DMRG). The accurate determination of the magnetization of this model is challenging for numerical methods and its value exhibits substantial disparities across various methods. We perform a large-scale DMRG calculation of this model by employing bond dimension as large asD=24000and by studying the system with width as large asLy=12. With careful extrapolation with truncation error and suitable finite size scaling, we give a conservative estimation of the magnetization asM0=0.208(8). The ground state energy per site we obtain isEg=-0.5503(8). Our results provide valuable benchmark values for the development of new methods in the future.

DOI
10.1088/1361-648x/ad21a8

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We revisit the issue about the magnetization of the 120° order in the spin-1/2 triangular lattice Heisenberg model with density matrix renormalization group (DMRG). The accurate determination of the magnetization of this model is challenging for numerical methods and its value exhibits substantial disparities across various methods. We perform a large-scale DMRG calculation of this model by employing bond dimension as large asD=24000and by studying the system with width as large asLy=12. With careful extrapolation with truncation error and suitable finite size scaling, we give a conservative estimation of the magnetization asM0=0.208(8). The ground state energy per site we obtain isEg=-0.5503(8). Our results provide valuable benchmark values for the development of new methods in the future.
After decades of debate, now there is a rough consensus that at zero temperature the spin-1/21/21/2 Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond Eb{E}_{b}Eb​ and magnetization per spin M0{M}_{0}M0​ in the thermodynamic limit are obtained with high precision. The former is estimated to be ${E}_{b}=\ensuremath{-}0.18334(10)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as M0=0.161(5){M}_{0}=0.161(5)M0​=0.161(5). It is about 3232%32 of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about 2020%20 smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.
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.
A density matrix renormalization group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multichain-type two-dimensional (2D) DMRG approach that utilizes previously determined system and environment blocks at all points. One first builds up an effective quasi-1D system and environment blocks of width L and these quasi-1D blocks are then used to as the initial building blocks of the 2D infinite-lattice algorithm introduced here. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N=20\ifmmode\times\else\texttimes\fi{}20$ lattice using this treatment are given by ${E}_{g}{/N}_{B}=\ensuremath{-}0.3321$ and ${E}_{g}{/N}_{B}=\ensuremath{-}1.1525,$ respectively.
A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z=3Z=3Z=3 and antiferromagnetic exchange between nearest-neighbor spins s=1/2s=1/2s=1/2 or 1 sites in successive generations ggg. The algorithm is accurate for s=1s=1s=1 sites. The ground states are magnetic with spin S(g)=2gsS(g)={2}^{g}sS(g)=2gs, staggered magnetization that persists for large g>20g>20g>20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S>S(g)S>S(g)S>S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for sss === 1/2 and 1 are interpreted in terms of an analytical three-site model.
We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-12\frac{1}{2}21​ Heisenberg model on a honeycomb lattice with first (J1{J}_{1}J1​) and second (J2{J}_{2}J2​) neighbor antiferromagnetic interactions, where a Z2{Z}_{2}Z2​ spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size $N=2\ifmmode\times\else\texttimes\fi{}6\ifmmode\times\else\texttimes\fi{}6$. With increasing J2{J}_{2}J2​, we find a N\'eel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment ms2{m}_{s}^{2}ms2​ on finite-size cylinders to the thermodynamic limit, we find the N\'eel order vanishing at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.22$. For $0.25<{J}_{2}/{J}_{1}\ensuremath{\le}0.35$, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For 0.22<J2/J1<0.250.22<{J}_{2}/{J}_{1}<0.250.22<J2​/J1​<0.25, both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.