In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. Besides, the correlation function decays purely exponentially at VBStype points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in other regions with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applicable for more general cases. Frustration is one of the important problem in many-body physics. Recently, many things in frustrated systems become clearer with field theoretical approaches and numerical calculations (density matrix renormalization group (DMRG) etc.). However, relatively little is known about commensurateincommensurate (C-IC) transitions induced by frustration in quantum spin systems. Among them, it is observed the C-IC change in the region with an excitation gap (therefore, it is not a phase transition) in one-dimensional (1D) quantum spin models. For example, the C-IC change begins from the Majumdar-Ghosh point [1] in the S=1/2 next-nearest-neighbor (NNN) chain
In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. Besides, the correlation function decays purely exponentially at VBS-type points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in the other region with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applicable for more general cases.
Most of the present understanding of the S=1 quantum spin chains displaying the Haldane gap is coming from the so-called valence-bond-solid (VBS) Hamiltonian which has an exactly known ground state. We show that this point is characterized by the onset of short-range incommensurate spin correlations in the one-parameter family of Hamiltonians ${\mathit{H}}_{\mathrm{\ensuremath{\theta}}}$= cos\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$
Si\ensuremath{\cdot}
Si+1+sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ (
Si\ensuremath{\cdot}
Si+1)2. This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range N\'eel order and a ground state with incommensurate long-range order. \textcopyright{} 1996 The American Physical Society.
For the antiferromagnetic
J1-
J2 quantum spin chain with an even number of sites, the point
J2d=J1/2 is a disorder point. It marks the onset of incommensurate real space correlations for
J2>J2d. At a distinct larger value of
J2L=0.52036(6)J1, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at
J2d=J1/2 but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at
J2L, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any
J2>0.24J1 and has the ground-state momentum
k=0. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point
J2=J1/2, we expect the variational results to be very precise.
We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.
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.