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Onset of incommensurability in quantum spin chains

K. Nomura

Published Jan 1, 2008• 7 citations
OpenAlex

Abstract

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

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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.
In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS) type points. It was difficult to study the commensurate–incommensurate (C–IC) change. On the one hand field theoretical approaches are not justified because of the short correlation length. On the other hand numerical calculations are not suitable to study the incommensurability since it is needed to treat the large size data. We discuss the relation between the edge state and the incommensurability, partially using the previous our study on the C–IC change.
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{\mathit{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathit{S}}_{\mathit{i}+1}Si+1​+sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ (Si{\mathit{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathit{S}}_{\mathit{i}+1}Si+1​)2{)}^{2})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.
Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic J1J_1J1​-J2J_2J2​ Chain
Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.
The quantum phase transition from a spin-Peierls phase with a small Fermi surface to a paramagnetic Luttinger-liquid phase with a large Fermi surface is studied in the framework of a one-dimensional Kondo-Heisenberg model that consists of an electron gas away from half filling, coupled to a spin-1/2 chain by Kondo interactions. The Kondo spins are further coupled to each other with isotropic nearest-neighbor and next-nearest-neighbor antiferromagnetic Heisenberg interactions which are tuned to the Majumdar-Ghosh point. Focusing on three-eighths filling and using the density-matrix renormalization-group (DMRG) method, we show that the zero-temperature transition between the phases with small and large Fermi momenta appears continuous, and involves a new intermediate phase where the Fermi surface is not well defined. The intermediate phase is spin gapped and has Kondo-spin correlations that show incommensurate modulations. Our results appear incompatible with the local picture for the quantum phase transition in heavy fermion compounds, which predicts an abrupt change in the size of the Fermi momentum.
We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest- and next-nearest-neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins S1>S2,{S}_{1}>{S}_{2},S1​>S2​, using 1/S1/S1/S spin-wave expansions, density-matrix renormalization group, and exact diagonalization techniques. It is argued that the zero-point spin fluctuations completely destroy the classical commensurate-incommensurate continuous transition. Instead, the long-range ferrimagnetic state disappears through a discontinuous transition to a singlet state at a larger value of the frustration parameter. In the ferrimagnetic phase we find a disorder point marking the onset of incommensurate real-space short-range spin-spin correlations.