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Acta Physica Polonica A

Towards Spin-Wave Like Approach to the J-J' Heisenberg Antiferromagnet in Terms of Composite Spins

Przemysław R. Grzybowski, Ravindra W. Chhajlany

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Conventional spin-wave approach relies on an expansion around a relevant classical (unentangled) ground state or equivalently the ground state in the mean field approximation for the quantum model.However, for systems which may admit valence bond ground states, such as the staggered J-J Heisenberg antiferromagnet, single site mean field approximation is obviously an incorrect zeroth order approximation.In this paper, we introduce a mean field approximation for clusters consisting of two spins connected by a strong bond in the aforementioned model.We identify the quantum critical point and calculate the ground state magnetization within this cluster mean field approximation in one and two dimensions.Finally, we derive an effective dimer Hamiltonian, in the standard basis operator formalism, which may be a basis for further spin-wave like expansions around the cluster mean field approximation ground state.

DOI
10.12693/aphyspola.115.168

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Conventional spin-wave approach relies on an expansion around a relevant classical (unentangled) ground state or equivalently the ground state in the mean field approximation for the quantum model.However, for systems which may admit valence bond ground states, such as the staggered J-J Heisenberg antiferromagnet, single site mean field approximation is obviously an incorrect zeroth order approximation.In this paper, we introduce a mean field approximation for clusters consisting of two spins connected by a strong bond in the aforementioned model.We identify the quantum critical point and calculate the ground state magnetization within this cluster mean field approximation in one and two dimensions.Finally, we derive an effective dimer Hamiltonian, in the standard basis operator formalism, which may be a basis for further spin-wave like expansions around the cluster mean field approximation ground state.
Finding an exact solution for a realistic interacting quantum many-body problem is often challenging. There are only a few problems where an exact solution can be found, usually in a narrow parameter space. Here, we propose a spin-12\frac{1}{2}21​ Heisenberg model on a square lattice with spatial anisotropy and bond depletion for the nearest-neighbor antiferromagnetic interactions but not for the next-nearest-neighbor interactions. This model has an exact and unique dimer ground state at J2/J1=12{J}_{2}/{J}_{1}=\frac{1}{2}J2​/J1​=21​; a dimer state is a product state of spin singlets on dimers (here, staggered nearest-neighbor bonds). We examine this model by employing the bond-operator mean-field theory and exact diagonalization. These analytical and numerical methods precisely affirm the correctness of the dimer ground state at the exact point (J2/J1=12)({J}_{2}/{J}_{1}=\frac{1}{2})(J2​/J1​=21​). As one moves away from the exact point, the dimer order melts and vanishes when the spin gap becomes zero. The mean-field theory with harmonic approximation indicates that the dimer order persists for $\ensuremath{-}0.35\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}1.35$. However, in nonharmonic approximation, the upper critical point lowers by 0.28 to 1.07, but the lower critical point remains intact. The exact diagonalization results suggest that the latter approximation fares better. The model reveals N\'eel order below the lower critical point and stripe magnetic order above the upper critical point. It has a topologically equivalent model on a honeycomb lattice where the nearest-neighbor interactions are still spatial anisotropic, but the bond depletion shifts into the isotropic next-neighbor interactions. Moreover, these models can also be generalized in the three dimensions.
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.
We find the corrections to the conventional 1/S expansion for the spin-wave velocity in the two-dimensional S=1/2 Heisenberg antiferromagnet. We do this by solving exactly for the ground and excited states of small clusters of up to 25 spins. We use open boundary conditions with a staggered Weiss effective field consistent with the correct quantum Monte Carlo staggered magnetization MS{\mathit{M}}_{\mathit{S}}MS​=0.31. The numerical results are directly compared with the 1/S expansion for the same clusters. The exact spin-wave velocity is given by c=Z1.158 \ensuremath{\surd}2 Ja, where Z is within 2% of unity. Our best estimate gives Z\ensuremath{\approxeq}1.006.
Using the coupled-cluster method we investigate the ground-state properties of an easy-plane spin-one antiferromagnet which has a critical point at zero temperature. This spin system represents a nontrivial system with the single-ion anisotropy. Single-ion anisotropies have a fundamental influence on the behavior of a magnetic system, and prevail in almost all physical systems with spin greater than one half. Because of the complexities caused by the single-ion anisotropy term, the mean-field approximation is commonly used in calculations of the thermodynamic quantities. However, in the mean-field approximation both quantum and spin fluctuation correlations have been neglected. Here we apply the coupled-cluster method to obtain the ground-state energy, staggered magnetization and critical point for both two and three dimensional lattices. In the coupled-cluster method the correlations of the quantum and spin fluctuations have been systematically included.
The ground state of an antiferromagnetic Heisenberg model on L X L clusters joined by a single bond and balanced Bethe clusters are investigated with quantum Monte Carlo and modified spin-wave theory. The improved Monte Carlo method of Sandvik and Evertz is used and the observables include valence bond and loop valence bond observables introduced by Lin and Sandvik as well as the valence bond entropy and the second Renyi entropy. For the bisecting of the Bethe cluster, in disagreement with our previous results and in agreement with modified spin-wave theory, the valence loop entropy and the second Renyi entropy scale as the logarithm of the number of sites in the cluster. For bisecting the L X L - L X L clusters, the valence bond entropy scales as L, however, the loop entropy and the entanglement entropy scale as ln(L). For the entanglement entropy, the coefficient of the logarithm, the number of Goldstone modes/2, is universal and is the analogue of the sub leading term of a LXL cluster. This result was substantiated by a modified spin-wave theory calculation for the ferromagnetic X-Y model. Taken together, the calculations suggest that linking high entanglement objects will not generate much more entanglement. As a consequence, simulating the ground state of clusters joined together by a few bonds should not be much more difficult than simulating the ground state of a single cluster.
The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.