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arXiv:1009.4018 · Journal of Mathematical Physics

Spin-spin correlation functions of the q-valence-bond-solid state of an integer spin model

Chikashi Arita, Kohei Motegi

Published Jun 1, 2011• 13 citations
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Abstract

We consider the valence-bond-solid ground state of the q-deformed higher-spin Affleck, Kennedy, Lieb, and Tasaki model (q-VBS state) with q real. We investigate the eigenvalues and eigenvectors of a matrix (G matrix), which is constructed from the matrix product representation of the q-VBS state. We compute the longitudinal and transverse spin-spin correlation functions, and determine the correlation amplitudes and correlation lengths.

DOI
10.1063/1.3598424

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We consider the valence-bond-solid ground state of the q-deformed higher-spin Affleck, Kennedy, Lieb, and Tasaki model (q-VBS state) with q real. We investigate the eigenvalues and eigenvectors of a matrix (G matrix), which is constructed from the matrix product representation of the q-VBS state. We compute the longitudinal and transverse spin-spin correlation functions, and determine the correlation amplitudes and correlation lengths.
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence bond states proposed in the Affleck--Kennedy--Lieb--Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.
We study the ground-state properties of two-dimensional quantum spin systems having the valence-bond-solid (VBS) type ground states. The `product-of-tensors' form of the ground-state wavefunction of the system is utilized to associate it with an equivalent classical lattice statistical model which can be treated by the transfer-matrix method. For diagonalization of the transfer matrix, we employ the product-wavefunction renormalization group method, which is a variant of the density-matrix renormalization group method. We obtain the correlation length and the sublattice magnetization accurately. For the anisotropically `deformed' S = 3/2 VBS model on the honeycomb lattice, we find that the correlation length as a function of the deformation parameter behaves very much as that in the S = 3/2 VBS chain.
A brief note on the G2_22​ Affleck-Kennedy-Lieb-Tasaki chain
We calculate the reduced density matrix of a block of integer spin- S 's in a q -deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.
We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than ln2\mathrm{ln}2ln2. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when 2S2S2S is not an integer multiple of the coordination number zzz. S=2S=2S=2 spins on the triangular lattice with z=6z=6z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.