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Journal of Yangzhou University

NUMERICAL SIMULATION OF J _1-J_2 CLUSTERS

Gao Li

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

For the quantum spin clusters of tens of sites, this paper calculates the total spin of ground state by exact diagonalization. In the non frustrated case, the ground state total spin takes not the lowest possible total spin but a higher value. It means the existence of Neel order in the ground state even for the clusters of nano scale. If introducing frustrations into clusters, it can induce a quantum phase transition from Neel order to spin disorder. This transition can be described and marked through the change of the ground state total spin. Calculations show that it should be the first order transition since the existence of level crossing at the critical J 2/J 1. From the numerical results of clusters, it can be inferred that for the infinite two dimensional J 1 J 2 model, the critical J 2/J 1 should be equal to 0 376 2 (±0.000 2). This result is more accurate than the previous data by analytical approximations and scaling analysis based on finite systems.

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For the quantum spin clusters of tens of sites, this paper calculates the total spin of ground state by exact diagonalization. In the non frustrated case, the ground state total spin takes not the lowest possible total spin but a higher value. It means the existence of Neel order in the ground state even for the clusters of nano scale. If introducing frustrations into clusters, it can induce a quantum phase transition from Neel order to spin disorder. This transition can be described and marked through the change of the ground state total spin. Calculations show that it should be the first order transition since the existence of level crossing at the critical J 2/J 1. From the numerical results of clusters, it can be inferred that for the infinite two dimensional J 1 J 2 model, the critical J 2/J 1 should be equal to 0 376 2 (±0.000 2). This result is more accurate than the previous data by analytical approximations and scaling analysis based on finite systems.
By exact diagonalization, we investigate several spin-12\frac{1}{2}21​ ${J}_{1}\ensuremath{-}{J}_{2}$ real clusters of molecular scale with different shapes. Our calculations show that when the ratio $\ensuremath{\eta}$ of next-nearest-neighbor to nearest-neighbor bonds is equal to 1, even a cluster of only 25 sites exhibits bulk behaviors and has quantum phase transitions. Two effective critical points are around J2/J1=0.3762{J}_{2}{/J}_{1}=0.3762J2​/J1​=0.3762 and 0.612, respectively. They are very close to those of the infinite ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice. But when $\ensuremath{\eta}<~0.85,$ the quantum phase transition around J2/J1=0.3762{J}_{2}{/J}_{1}=0.3762J2​/J1​=0.3762 disappears. By calculating the distributions of the average values of Siz,{S}_{i}^{z},Siz​, the bulk behaviors are demonstrated graphically. In the intermediate phase, the sites on the corners have distinctly different character from the other sites. The distribution is obviously centralized on the corner sites.
Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.
How to impose boundary conditions is crucial to finite-cluster calculations of quantum spin systems, but only open-boundary clusters have been used in cluster-effective-field approximations up to now. In the present paper, periodic-boundary clusters are also considered for the formulation of cluster-effective-field approximations in frustrated quantum spin systems. Namely, the open-boundary-condition double-cluster approximation (OBC-DCA) and the periodic-boundary-condition double-cluster approximation (PBC-DCA) are applied to the one-dimensional S= 1/2 frustrated XXZ model at the ground state. These two approximations are compared using the coherent-anomaly method (CAM). Within the limitation of cluster sizes in exact-diagonalization calculations, the PBC-DCA can reproduce the true phase boundary of the Neel-dimer transition. On the other hand, the OBC-DCA severely underestimates the existence of magnetic orders as quantum fluctuation is increased. These findings suggest that previous cluster-effective-field studies based on open-boundary clusters should be reconsidered.
The two-dimensional S=1/2S=1/2S=1/2 asymmetric Heisenberg Mattis model is investigated with the exact diagonalization of finite clusters. The Néel order parameter and the spin glass order parameter can be smoothly extrapolated to the thermodynamic limit in the antiferromagnetic region, as in the pure Heisenberg antiferromagnet. The critical concentration of the Néel phase is consistent with that of the two-dimensional Ising Mattis model, and the spin glass order parameter increases monotonously as the ferro-bond concentration increases. These facts suggest that quantum fluctuation does not play an essential role in two-dimensional non-frustrated random spin systems. KEYWORDS: quantum spin system, ground state, randomness, Mattis model, Néel order, spin glass order
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.
By using the coupled cluster method, the numerical exact diagonalization method, and the numerical density matrix renormalization group method, we investigated the properties of the one-dimensional Heisenberg chain with alternating antiferromagnetic and ferromagnetic next nearest neighbor interactions. In the classical limit, the ground state is in the collinear Neel state if a<1/2, while for a>1/2, there is an noncollinear canted state. For the quantum case, we found that, although the classical Neel state is absent, the canted state exists if the frustration parameter a exceeds a critical point ac1. The precise critical point ac1 can be determined by using the coupled cluster method and the numerical exact diagonalization method separately. The results of the coupled cluster method and the exact diagonalization method both disclose that the type of phase transition occurring at ac1 changes from a classical second-order transition to a quantum first-order transition due to quantum fluctuation. Although there is another critical point ac2 in a finite system at which the ground state evolves from the canted state to the collinear Neel plus ferromagnetic state, that state is absent because ac2 tends to infinity in the thermodynamic limit.