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arXiv:cond-mat/0005211 · Physical review. B, Condensed matter

Commensurate and incommensurate correlations in Haldane-gap antiferromagnets

Gábor Fáth, András Sütö

Published Aug 1, 2000• 19 citations
arXivHTMLPDF

Abstract

We analyze the onset of incommensurabilities around the VBS point of the $S=1$ bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point ${\ensuremath{\theta}}_{\mathrm{VBS}}.$ Moreover, the theory predicts another special point ${\ensuremath{\theta}}_{\mathrm{disp}},$ distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense.

DOI
10.1103/physrevb.62.3778

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arXiv:cond-mat/0306365•Jun 13, 2003•2 citations
We analyze the onset of incommensurabilities around the VBS point of the S=1S=1S=1 bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point \ensuremathθVBS.{\ensuremath{\theta}}_{\mathrm{VBS}}.\ensuremathθVBS​. Moreover, the theory predicts another special point \ensuremathθdisp,{\ensuremath{\theta}}_{\mathrm{disp}},\ensuremathθdisp​, distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense.
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.
Commensurate-incommensurate change on the one-dimensional S=1S=1S=1 bilinear-biquadratic model $[\mathcal{H}(\ensuremath{\alpha})={\ensuremath{\sum}}_{i}{{\mathbf{S}}_{i}∙{\mathbf{S}}_{i+1}+\ensuremath{\alpha}{({\mathbf{S}}_{i}∙{\mathbf{S}}_{i+1})}^{2}}]$ is examined. The gapped Haldane phase has two subphases (the commensurate Haldane subphase and the incommensurate Haldane subphase) and the commensurate-incommensurate change point (the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\alpha}=1∕3$). There have been two different analytical predictions about the static structure factor in the neighborhood of this point. By using the S\o{}rensen-Affleck prescription, these static structure factors are related to the Green functions, and also to the energy gap behaviors. Numerical calculations support one of the predictions. Accordingly, the commensurate-incommensurate change is recognized as a motion of a pair of poles in the complex plane.
Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of −1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value −1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.
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
A Monte Carlo simulation study based on the heat-bath algorithm on a checkerboard arrangement is applied to the spin-1 antiferromagnetic chain model with both bilinear and biquadratic exchange interactions. Changing the parameter \ensuremath{\lambda} which denotes strength of the biquadratic interaction against the bilinear one from -1.0 to 2.0, we obtain the following results according to the measurements of energy and two-point correlation function. At \ensuremath{\lambda}=1 and -(1/3 where some physical quantities are derived rigorously, our Monte Carlo simulation results agree well with the rigorous ones. For ${\ensuremath{\lambda}}_{c}$\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}${\ensuremath{\lambda}}_{1}$ with taking 0.25\ensuremath{\le}${\ensuremath{\lambda}}_{c}$\ensuremath{\le}0.5 and ${\ensuremath{\lambda}}_{1}$\ensuremath{\ge}2.0, its model has the gapless ground state accompanied by a power-law decaying correlation function. This implies the existence of a critical line. For -1.0${\ensuremath{\lambda}}_{c}$, on the other hand, its model has the ground state with energy gap accompanied with exponentially decaying correlation function. This indicates that the valence-bond-solid (VBS) state asserted by Affleck et al. is stabilized there. As the point of \ensuremath{\lambda}=0 comes to the standard Heisenberg antiferromagnetic model and is in the region of -1.0${\ensuremath{\lambda}}_{c}$, the Haldane's conjecture is supported that this ground state is described by the VBS state for the case of integer spin, e.g., S=1.
Quantum phase transitions induced by an external magnetic field in the Haldane-gapped spin-1 chains are studied in a fermionic field-theoretic description of the model. In the case with broken axial symmetry, two transitions occurs consecutively as increasing the field: hc=(m1m2)1/2h_c=(m_1 m_2)^{1/2}hc​=(m1​m2​)1/2 (with m1m_1m1​ and m2m_2m2​ the transverse masses) is the critical Ising point and h~c∗(>hc)\tilde{h}_c^* (> h_c)h~c∗​(>hc​) is the commensurate-incommensurate transition point. Although the latter is thermodynamically indiscernible, its property is revealed in the spin correlation functions (both uniform and staggered) which are calculated upon expressing the correlations of coupled Ising pair fields in terms of block-Toeplitz determinants.