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.
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.
We study ground state properties of the
S=2 quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden
Z2×Z2 symmetry is broken for
0.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden
Z2×Z2 symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
We use the density matrix renormalization group method to study the ground-state "phase" diagram and some low-energy properties of an antiferromagnetic spin-1 chain with a next-nearest-neighbour exchange J 2 and an alternation δ of the nearest-neighbour exchanges. We find a line running from a gapless point at ( J 2 ,δ) = (0,0.25 ± 0.01) up to a "gapless" point at (0.73 ± 0.005,0) such that the open-chain ground state is fourfold degenerate below the line and is unique above it. A disorder line 2 J 2 + δ = 1 runs from δ = 1 to about δ = 0.136. To the left of this line, the peak in the structure factor S ( q ) is at π, while to the right of the line, it is at less than π.
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$
SiSi+1+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$
SiSi+2, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
We use a tensor network renormalization group method to study random
S=2 antiferromagnetic Heisenberg chains with alternating bond strength distributions. In the absence of randomness, an imposed dimerization with bond alternation induces two quantum critical points between the
S=2 Haldane phase, a partially dimerized phase, and a fully dimerized phase, depending on the strength of dimerization. These three phases, called ($\ensuremath{\sigma},4\ensuremath{-}\ensuremath{\sigma}$) = (2,2), (3,1), and (4,0) phases, are valence-bond solid (VBS) states characterized by $\ensuremath{\sigma}$ valence bonds (effective spin-1/2 singlet pairs) forming across even links and $4\ensuremath{-}\ensuremath{\sigma}$ valence bonds on odd links. Here we study the effects of bond randomness on the ground states of the dimerized spin chain, calculating disorder-averaged twist order parameters and spin correlations. We classify the types of random VBS phases depending on the strength of bond randomness
R and dimerization
D using the twist order parameter, which has a negative/positive sign for a VBS phase with odd/even $\ensuremath{\sigma}$. Our results demonstrate the existence of a multicritical point in the intermediate disorder regime with finite dimerization, where (2,2), (3,1), and (4,0) phases meet. This multicritical point is at the junction of three phase boundaries in the $R\text{\ensuremath{-}}D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundaries that extend to zero randomness, and the (2,2)-(4,0) phase boundary that connects another multicritical point in the undimerized limit. The undimerized multicritical point separates a gapless Haldane phase and an infinite-randomness critical line with the diverging dynamic critical exponent in the large-
R limit at
D=0. Furthermore, we identify the (3,1)-(4,0) phase boundary as an infinite-randomness critical line even at small
R and find the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in the vicinity of the multicritical point.