Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.
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.
Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈
Siz〉 and the spin-correlation function 〈
S1zSiz〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.
Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=
Jtsumi[1-(-1
)i\ensuremath{\delta}]
Si\ensuremath{\cdot}
Si+1, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈
S1zSiz〉 and the local magnetic moment 〈
Siz〉 are visualized as a function of \ensuremath{\delta}. The continuous phase transition at \ensuremath{\delta}=0.25, which was indicated by several authors, is confirmed by the \ensuremath{\delta} dependence of the low-lying level structure, the divergence of the correlation length, and also the collapse of the edge states. The critical exponent \ensuremath{\eta}, governing the power-law decay of the spin correlation function, is estimated to be 0.99\ifmmode\pm\else\textpm\fi{}0.05. The present result raises a possibility that the variety of quantum spin chains including not only the half-odd-integer-S ones but also the integer-S ones may be described in terms of the generic critical theory.
A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J
tsumi [1-(-1
)i\ensuremath{\delta}]
Si\ensuremath{\cdot}
Si+1. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(
v2 sin2q+${\mathrm{\ensuremath{\Delta}}}^{2}$
)1/2 with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature
Tpeak\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J${\ensuremath{\sum}}_{\mathrm{i}}$[1-(-1
)i\ensuremath{\delta}]
Si\ensuremath{\cdot}
Si+1, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-\ensuremath{\sigma}-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter,
Ostringz(\ensuremath{\theta})=${\mathrm{lim}}_{\mathrm{L}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ostringz(\ensuremath{\theta};L) with
Ostringz(\ensuremath{\theta};L)=〈
SL/4z${\ensuremath{\prod}}_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{\ensuremath{-}}1}$exp[i\ensuremath{\theta}
Sjz]
S3L/4z〉. It turns out that the transition points are successfully detected by observing the overall behavior of
Ostringz(\ensuremath{\theta}) at various values of \ensuremath{\delta}. Investigating the dependences of
Ostringz(\ensuremath{\theta};L) on \ensuremath{\theta}, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.