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Journal of the Physical Society of Japan

Real-Space Renormalization-Group Analysisof the S=2 Antiferromagnetic Heisenberg Chain

Yoshihiro Nishiyama, Keisuke Totsuka, Naomichi Hatano, Masuo Suzuki

Published Feb 15, 1995• 37 citations
Publisher

Abstract

The S=2 antiferromagnetic Heisenberg open chain is analyzed with the real-space numerical renormalization-group method improved by White. The low-ly ing spectrum and the one-point function are evaluated to show the degeneracy and the boundary effect on the ground state. Generalized string correlation funct ions were calculated for θ=π and θ=π 2. The θ=π 2 string correlation is found to indicate the long-range hidden order of the ground state. These results are quite consistent with the S=2 VBS argument.

DOI
10.1143/jpsj.64.414

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The S=2 antiferromagnetic Heisenberg open chain is analyzed with the real-space numerical renormalization-group method improved by White. The low-ly ing spectrum and the one-point function are evaluated to show the degeneracy and the boundary effect on the ground state. Generalized string correlation funct ions were calculated for θ=π and θ=π 2. The θ=π 2 string correlation is found to indicate the long-range hidden order of the ground state. These results are quite consistent with the S=2 VBS argument.
The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.
We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ S(j)\ensuremath{\cdot}S(j+1)+ Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​(-1)j{)}^{\mathit{j}})j\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.
Antiferromagnetic Heisenberg spin chains for S =1 and S =2 are treated numerically. We investigate a generalized θ -string correlation function O θ ( j )=< S z 1 exp ( i θ ∑ j -1 k =1 S z k ) S j z > which is a generalization of the string correlation function of den Nijs-Rommelse. When the value of θ lies in an appropriate region, the phase of O θ ( j ) is fixed and O θ ( j ) behaves smoothly as a function of j. This means alternative sign changes of the usual spin-spin correlation O θ =0 ( j ) are well canceled in O θ ( j ) for such values of θ . The cancellation is best at θ =π and π /2 for the S=1 and 2 cases, respectively. The generalized string which appears in the θ -string correlation function plays a role similar to that of a fictitious gauge potential in the definition of the hidden order parameter of the fractional quantized Hall state by Girvin and MacDonald.
The successive phase transitions of the Heisenberg spin chain with bond alternation, H = ∑ i =1 L -1 [1-(-1) i δ] S i · S i +1 , for the spin S =3/2 case under the open boundary condition are investigated by the density-matrix renormalization-group method proposed by White. Besides the Heisenberg point, we find that another gapless point exists at δ c = 0.42 ±0.02. This result qualitatively supports the field-theoretical prediction by Affleck and Haldane. However, our estimation of δ c deviates from the field-theoretical prediction, δ c = 2/3. We also find a fourfold degeneracy of the ground state in the region 0 <δ< δ c in the thermodynamic limit. Our results show that the degeneracy is described by the edge excitations, which is essentially similar to the one in the S =1 case. We calculate the string correlation function and find that its behavior is consistent with the valence-bond-solid (VBS) picture.