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.
In this thesis I present three publications about the use of valence bonds to gain information about quantum spin systems. Valence bonds are an essential ingredient of low energy states present in many compounds. The first part of this thesis is dedicated to two studies of the antiferromagnetic J1-J2 chain with S=1/2. We show how automated variational calculations based on valence bond states can be performed close to the Majumdar-Ghosh point (MG-point). At this point, the groundstate is a product state of dimers (valence bonds between nearest neighbours). In the dimerized region surrounding the MG-point, we find such variational computations to be reliable. The first publication is about the entanglement properties of an impurity attached to the chain. We show how to use the variational method to calculate the negativity, an entanglement measure between the impurity and a distant part of the chain. We find that increasing the impurity coupling and a minute explicit dimerization, suppress the long-ranged entanglement present in the system for small impurity coupling at the MG-point. The second publication is about a transition from commensurate to incommensurate behaviour and how its characteristics depend on the parity of the length of the chain. The variational technique is used in a parameter regime inaccessible to DMRG. We find that in odd chains, unlike in even chains, a very intricate and interesting pattern of level crossings can be observed. The publication of the second part is about novel worm algorithms for a popular quantum Monte Carlo method called valence bond quantum Monte Carlo (VBQMC). The algorithms are based on the notion of a worm moving through a decision tree. VBQMC is entirely formulated in terms of valence bonds. In this thesis, I explain how the approach of VBQMC can be translated to the Sz-basis. The algorithms explained in the publication can be applied to this Sz-method.
Every singlet state of a quantum spin-
1/2 system can be decomposed into a linear combination of valence bond basis states. The range of valence bonds within this linear combination as well as the correlations between them can reveal the nature of the singlet state and are key ingredients in variational calculations. In this work, we study the bipartite valence bond distributions and their correlations within the ground state of the Heisenberg antiferromagnet on bipartite lattices. In terms of field theory, this problem can be mapped to correlation functions near a boundary. In dimension $d\ensuremath{\ge}2$, a nonlinear $\ensuremath{\sigma}$ model analysis reveals that at long distances the probability distribution
P(r) of valence bond lengths decays as ${|r|}^{\ensuremath{-}d\ensuremath{-}1}$ and that valence bonds are uncorrelated. By a bosonization analysis, we also obtain $P(\mathbf{r})\ensuremath{\propto}{|r|}^{\ensuremath{-}d\ensuremath{-}1}$ in
d=1 despite the different mechanism. On the other hand, we find that correlations between valence bonds are important even at large distances in
d=1, in stark contrast to $d\ensuremath{\ge}2$. The analytical results are confirmed by high-precision quantum Monte Carlo simulations in
d=1, 2, and 3. We develop a single-projection loop variant of the valence bond projection algorithm, which is well designed to compute valence bond probabilities and for which we provide algorithmic details.
Using a loop–cluster algorithm we investigate the spin- Heisenberg antiferromagnet on a square lattice with exchange coupling J and an additional four-spin interaction of strength Q . We confirm the existence of a phase transition separating antiferromagnetism at J / Q > J c / Q from a valence bond solid (VBS) state at J / Q < J c / Q . Although our Monte Carlo data are consistent with those of previous studies, we do not confirm the existence of a deconfined quantum critical point. Instead, using a flowgram method on lattices as large as 80 2 , we find evidence for a weak first-order phase transition. We also present a detailed study of the antiferromagnetic phase. For J / Q > J c / Q the staggered magnetization, the spin stiffness and the spinwave velocity of the antiferromagnet are determined by fitting Monte Carlo data to analytic results from the systematic low-energy effective field theory for magnons. Finally, we also investigate the physics of the VBS state at J / Q < J c / Q and we show that long but finite antiferromagnetic correlations are still present.
The purpose of this thesis is the treatment of the antiferromagnetic Heisenberg model within the valence bond basis, allowing for the description of its low-energy physics. The manuscript is organized in two parts: In the first part we utilize the fidelity concept in order to detect quantum phase transitions. It is notably shown, that this quantity is accessible in a valence bond projector quantum Monte Carlo algorithm, making available the fidelity approach to large scale simulations. The second part is devoted to the generalization of an idea going back to Rokhsar and Kivelson, which aims to map a Heisenberg model onto a numerically less demanding quantum dimer model. Starting from a rigourous derivation, the method is then applied to the kagome lattice and allows to establish the existence of a tricritical point in the vicinity of the original model. The same technique is also used to treat the J1-J2-J3 Heisenberg model on the honeycomb lattice, showing the existence of a plaquette phase in a determined parameter regime.
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.