The ordinary Heisenberg model that possesses the SU(2) symmetry can easily be generalized to higher symmetry simply by regarding a symbol “S” in the Heisenberg Hamiltonian as a generator of the SU(N) algebra rather than SU(2). This very simple and fundamental model, according to some theoretical predictions, exhibits surprizingly wide variety of ground states depending upon the value of N and the representation, in two dimensions. Here, chaging the representation corresponds to changing the magnitude of spins in the ordinary SU(2) Heisenberg model. While the representation-dependent ground state is well-known as a Haldane phenomenon in one dimension, no systematic numerical study has been done in two dimensions. We employed quantum Monte Carlo method to study this model system, using newly-developed algorithm. We found that the ground state is the Neel state for N is 4 or smaller, whereas for N larger than 4 the lattice translational symmetry is broken to yield a dimerized state as shown schematically in Fig.1. (Darker lines correspond to pairs of nearest-neighbor spins with larger correlation.)
Figure 1
Moreover, the space of the ground states has an approximate U(1) symmetry. Figure 2 shows the distribution function of the dimer order parameters in the x and the y directions. In the case of larger representations, we observed that non-trivial spontaneous spatial structure is suppressed.
Figure 2
（by Naoki KAWASHIMA）
References
Kenji Harada, Naoki Kawashima and Matthias Troyer: “Neel and Spin-Peierls ground states of two-dimensional SU(N) quantum antiferromagnets”, Phys. Rev. Lett. 90 117203-117206 (2003).
Naoki Kawashima and Yuta Tanabe: “Representation-Dependent Ground-States of the SU(N) Heisenberg Model”, Phys. Rev. Lett. 98 057202(1-4) (2007).
NiCl2−4SC(NH2)2 (DTN) is a quantum S=1 chain system with strong easy-pane anisotropy and a new candidate for the Bose-Einstein condensation of the spin degrees of freedom. ESR studies of magnetic excitations in DTN in fields up to 25 T are presented. Based on analysis of the single-magnon excitation mode in the high-field spin-polarized phase and previous experimental results [Phys. Rev. Lett. 96, 077204 (2006)], a revised set of spin-Hamiltonian parameters is obtained. Our results yield D=8.9K, Jc=2.2K, and Ja,b=0.18K for the anisotropy, intrachain, and interchain exchange interactions, respectively. These values are used to calculate the antiferromagnetic phase boundary, magnetization, and the frequency-field dependence of two-magnon bound-state excitations predicted by theory and observed in DTN for the first time. Excellent quantitative agreement with experimental data is obtained. (Abstract of Ref. [1])
(by Mitsuaki Tsukamoto)
Reference
[1] S. A. Zvyagin, J. Wosnitza, C. D. Batista, M. Tsukamoto, N. Kawashima, J. Krzystek, V. S. Zapf, M. Jaime, N. F. Oliveira, Jr., and A. Paduan-Filho: ” Magnetic Excitations in the Spin-1 Anisotropic Heisenberg Antiferromagnetic Chain System NiCl2-4SC(NH2)2″, Phys. Rev. Lett. 98, 047205 (2007).
[2] Mitsuaki Tsukamoto, Cristian Batista and Naoki Kawashima: “Quantum Monte Carlo simulation for S=1 Heisenberg model with uniaxial anisotropy”, J. Magn. Magn. Mater. 310, 1360 (2007).
The spin glass problem was first manifested in experiments on random magnets, and the Edwards-Anderson model was subsequently proposed. Since then, the simplicity of the model and the fundamental nature of the problem have been encouraging a great deal of effort to be invested on its study. While we confirmed the existence of the spin-glass phase transition in three dimensions by means of Monte Carlo simulations [1], the nature of the spin glass phase is a subject of active debates even today. Early theoretical works produced two paradigms that are consistent with experimental findings: the mean-field picture and the droplet picture. Investigations in the last two decades have been focused on the question as to which paradigm is correct for the three-dimensional model. For solving this problem, the principal tool was numerical calculations. It did not take too long, however, for researchers to realize the difficulty of the problem from the computational point of view. Even a mathematically rigorous statement was made: the problem of finding the ground-state of a given sample of the Edwards-Anderson spin glass model is NP-hard if the lattice is three dimensional or higher. Today’s largest system size that can be handled with conventional methods does not exceed L=30 for three-dimensional models, which is not enough to answer the question beyond reasonable doubts.
Figure 1: A contraction process. Circles are vertices (or spins) and lines are edges (or bonds). Solid lines are those which have not been contracted yet and dashed lines are the ones that have been contracted already.
We proposed an approximate solution to the problem [2] based on recursive deformation. One of the most well-known examples for methods of this kind is the Migdal-Kadanoff (MK) real space renormalization, which yields a raasonably accurate result for the ferromagnetic Ising model and was applied later to spin glasses. Our approximation procedure consists of repeated application of an elementary operation (“contraction”) to the model with a given lattice geometry and the system size. As depicted in Fig.1, a contraction is performed as reconnection of bonds and recalculation of the bond strength. Starting from the the finite lattice on which the model is defined, we gradually deform it by contraction until it becomes a tree with no loop. Once the lattice has been deformed into a tree, it is easy to compute the correlation functions because of the absence of loops. In this way, we can treat much larger systems than we can with conventional numerical methods such as the Monte Carlo simulation.
Figure 2: The Binder parameter for the three-dimensional model of system sizes of L=4, 8, 16, 32, 64 and 128 (from top to bottom at high temperature). The arrows indicate the locations of the crossings between two successive system sizes. The inset shows the finite size scaling for the largest three sizes, L= 32, 64 and 128. (The other sizes are not shown in the inset.)
For the three dimensional system, the spin glass susceptibility shows a drastic change near T=J, which is presumably the transition temperature. This is confirmed by Fig.2, which shows the temperature dependence of the Binder parameter and the finite-size-scaling plot based on it. Curves of various system sizes cross each other and the crossing between larger systems takes place at lower temperature. The crossing point in the thermodynamic limit is located around T = 1.0 J and g = 0.9. We can also produce a finite-size-scaling plot that looks reasonable, as shown in the inset. The plot yields the estimates of the critical exponent; \(\nu=1.85\). The agreement of these values with the results of Monte Carlo simulation [3], Tc = 0.98(5) and \(\nu= 2.00 (15)\), are remarkably good considering the crudeness of the present approximation. This is also an improvement on the MK result [4], Tc = 0.39 and \(\nu=2.8\).
（by Naoki Kawashima）
References
[1] N. Kawashima and A. P. Young: “Phase transition in the three-dimensional \(\pm J\) Ising spin glass”, Phys. Rev. B 53 (1996) R484.
[2] N. Kawashima: “Tree approximation for spin glass models”, J. Phys. Soc. Jpn. 25 (2006) 073002.
[3] E. Marinari, G. Parisi and J. J. Ruiz-Lorenzo: “Phase structure of the three-dimensional Edwards-Anderson spin glass”, Phys. Rev. B 58 (1998) 14852.
[4] B. W. Southern and A. P. Young: “Real space rescaling study of spin glass behavior in 3 dimensions”, J. Phys. C 10 (1977) 2179.
