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Monte Carlo simulation of the one-dimensional t-J-J' model

Matthias Troyer

Published Jul 5, 1991• 0 citations
Publisher

Abstract

I present a new and efficient disconnected cluster decomposition to perform Monte Carlo Simulations based on the Trotter-Suzuki method for 1-d quantum systems. As opposed to the normal checkerboard world line algorithm this mapping allows one to simulate models with interactions up to next nearest neighbours or even one-dimensional double layers. I have used this cluster decomposition to simulate the one-dimensional t-J-J' model at α=J'/J=0.5 and have investigated the sign problem in this model. At half band filling the system is dimerised into singlet spin pairs when α=0.5 due to frustration. I believe that the spin gap is no longer present around quarter band filling

DOI
10.5169/seals-116319

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I present a new and efficient disconnected cluster decomposition to perform Monte Carlo Simulations based on the Trotter-Suzuki method for 1-d quantum systems. As opposed to the normal checkerboard world line algorithm this mapping allows one to simulate models with interactions up to next nearest neighbours or even one-dimensional double layers. I have used this cluster decomposition to simulate the one-dimensional t-J-J' model at α=J'/J=0.5 and have investigated the sign problem in this model. At half band filling the system is dimerised into singlet spin pairs when α=0.5 due to frustration. I believe that the spin gap is no longer present around quarter band filling
Ground-state properties of the one-dimensional Kondo lattice model are investigated using the quantum Monte Carlo (QMC) simulation method. To reduce the difficulty of the negative sign problem, we employ the variationally optimized singlet wave function. Since the improvement of Monte Carlo sampling efficiency is achieved in the half-filling case, we can perform the simulations on large systems (up to 26-site system). The QMC data on the local spin correlations are closely compared with those of the variational Monte Carlo (VMC) calculations as well as with the exact diagonalization data of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) lattice model. We find that the local spin correlation decays exponentially with the correlation length ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}2.54 in the QMC data, while ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}1.08 in the VMC results (J/t=1). This implies the absence of the long-range order (the existence of the spin gap) and the importance of intersite correlation effects which are not taken into account in the VMC results. On the other hand, the system size dependence is almost absent in the Kondo lattice model results, which contrasts the behavior in the RKKY data.
We extend a Quantum Monte Carlo (QMC) algorithm to the Hubbard model containing bond-dependent off-diagonal hoppings, which describes the manganites and other similar systems, such as systems with spin-flip hoppings. To show correctness of this extension, we detect two simple examples of one-dimensional chains. The first one is half-filled and the hopping matrixes between arbitrary nearest neighboring pairs are uniform, i.e., ti,i+1=ti+1,i+2{t}_{i,i+1}{=t}_{i+1,i+2}ti,i+1​=ti+1,i+2​ (i=1,2,3,…).(i=1,2,3,\dots{}).(i=1,2,3,…). The other one is quarter-filled and the hopping matrixes alternate every four lattice constants with ${t}_{i,i+1}{=t}_{i+1,i+2}\ensuremath{\ne}{t}_{i+2,i+3}{=t}_{i+3,i+4}$ (i=1,5,9,…).(i=1,5,9,\dots{}).(i=1,5,9,…). Results of the two examples are reasonable.
The Suzuki-Trotter formula has been used to get the mth approximant to the classical representation of the partition function of the one-dimensional N-spin S=½ quantum spin systems. The equivalent two-dimensional (N×2m) Ising model with four-spin interactions has been studied in detail by using the numerically exact transfer-matrix method for T≥0.05 and m ≤8. The convergence properties have been examined in two different representations; checkerboard decomposition (CBD) and real-space decomposition (RSD). The spin correlation functions in RSD converge much faster than those in CBD. The limiting m→∞ behavior has been estimated from the extrapolation formula of the form: E(m)=E( ∞)+ α/ m2. The limiting values of the energy derived from the nearest-neighbor correlation agree with the correct values excellently.
Results of a numerical experiment on one-dimensional quantum spin chains are reported. The Heisenberg model with a nearest-neighbor interaction is investigated for spins 1/2\ensuremath{\le}S\ensuremath{\le}2. The procedure of measurement in the conventional checkerboard world-line algorithm is modified to give lower systematic errors. In addition to spatial-correlation properties, the magnon spectral density is extracted from the magnon Green function via the fast maximum-entropy algorithm. In particular, the existence of the Haldane gap in S=1 and S=2 cases is explicitly demonstrated. Hidden antiferromagnetic order peculiar to the Haldane phase is measured for S=1 as a function of one-site anisotropy. The mapping onto the restricted solid-on-solid model is analyzed. The difference between systems with even and odd integer spins is discussed.
Abstract By precisely writing down the matrix element of the local Boltzmann operator (e-th, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With current formula, the Hubbard–Stratonovich transformation is not necessary, and is not based on the determinant approach, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principle calculation. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm can be used in various studies.