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arXiv:2003.07857 · Physical review. B./Physical review. B

Dynamical spin excitations of the topological Haldane gapped phase in the S=1 Heisenberg antiferromagnetic chain with single-ion anisotropy

Jun-Han Huang, Guang-Ming Zhang, Dao‐Xin Yao

Published Jan 4, 2021• 9 citations
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Abstract

We study the dynamical spin excitations of the one-dimensional $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy by using quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation function. Using the transverse dynamic spin structure factor, we observe the quantum phase transition with a critical point between the topological Haldane gapped phase and the trivial phase. At the quantum critical point, we find a broad continuum characterized by the Tomonaga-Luttinger liquid, and its dynamical signatures are similar to a $S=1/2$ Heisenberg antiferromagnetic chain. The broad continuum can be seen as a dynamical signature of spinons. We further identify that the elementary excitations are fractionalized spinons in the critical state.

DOI
10.1103/physrevb.103.024403

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We study the dynamical spin excitations of the one-dimensional S=1S=1S=1 Heisenberg antiferromagnetic chain with single-ion anisotropy by using quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation function. Using the transverse dynamic spin structure factor, we observe the quantum phase transition with a critical point between the topological Haldane gapped phase and the trivial phase. At the quantum critical point, we find a broad continuum characterized by the Tomonaga-Luttinger liquid, and its dynamical signatures are similar to a S=1/2S=1/2S=1/2 Heisenberg antiferromagnetic chain. The broad continuum can be seen as a dynamical signature of spinons. We further identify that the elementary excitations are fractionalized spinons in the critical state.
A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy Dtsumj{\mathit{Dtsum}}_{\mathit{j}}Dtsumj​(Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​)2{)}^{2})2 is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value Dc{\mathit{D}}_{\mathit{c}}Dc​=1 is shown in the lower Haldane gap and the correlation functions.
We present results of a study of the antiferromagnetic spin-1 chain, subject to the simultaneous presence of single-ion anisotropy and external magnetic fields. Using a quantum Monte Carlo calculation based on the stochastic series expansion method, we first uncover a rich quantum phase diagram comprising N\'eel, Haldane, Luttinger-liquid, and large-anisotropy phases. Second, we scan across this phase diagram over a wide range of parameters, evaluating the transverse dynamic structure factor, which we show to exhibit sharp massive modes as well as multiparticle continua. For vanishing anisotropy and fields, comparison with existing results from other analytic and numerical approaches shows convincing consistency.
We use linked-cluster expansions to analyze the quantum phase transitions between symmetry-unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy, while the second one is the dimerized spin-one Heisenberg chain. For both models, we determine the ground-state energy and the one-particle gap inside the nontopological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point and to extract the associated critical exponent. We find that this approach works unsatisfactorily for the anisotropic chain, since the quality of the extrapolation appears insufficient due to the large correlation length exponent. In contrast, extrapolation schemes display very good convergence for the gap closing in the case of the dimerized spin-one Heisenberg chain.
The interplay between competing interactions in low-dimensional quantum many-body systems gives rise to a wide variety of quantum phases and critical phenomena. Motivated by this scenario, we investigate the quantum phase diagram of the mixed-spin 1 2 , 1 antiferromagnetic Heisenberg chain with modulated single-ion anisotropy by means of the density matrix renormalization group (DMRG) method. The resulting phase diagram comprises four distinct quantum phases: three gapped phases, namely the 1 / 3 magnetization plateau, the trivial nonmagnetic (TnM) phase, and the topological valence-bond-solid (TVBS) phase, together with a gapless Luttinger liquid (LL) phase. The different phases are characterized through the analysis of magnetization, spin correlation functions, nonlocal order parameters, and entanglement-based observables. The TVBS phase is identified by the presence of exponentially localized edge states, finite values of the nonlocal string order parameter and twist operator, and the characteristic degeneracy pattern of the entanglement spectrum. The boundaries of the LL phase are determined from the transverse spin correlation function and the corresponding Luttinger parameter, revealing two Kosterlitz–Thouless transition lines characterized by the critical value K c = 2.0 , in agreement with finite-size scaling analyses of the spin gap. In addition, we identify a Gaussian critical line associated with the topological quantum phase transition separating the TVBS and TnM phases. By employing the tangential finite-size scaling approach, we accurately determine this critical line by extracting the correlation-length critical exponent ν and the conformal field theory central charge c . Our results demonstrate that the competition between exchange interactions and modulated single-ion anisotropy gives rise to a rich interplay between topology, quantum criticality, and magnetic ordering, highlighting the fundamental role of these interactions in determining the ground-state properties of low-dimensional quantum spin systems.
By using Lanczos exact diagonalization and quantum Monte Carlo combined with stochastic analytic continuation, we study the dynamical properties of the S = 1 antiferromagnetic Heisenberg chain with different strengths of bond disorder. In the weak disorder region, we find weakly coupled bonds which can induce additional low-energy excitation below the one-magnon mode. As the disorder increases, the average Haldane gap closes at δ Δ ∼ 0.5 with more and more low-energy excitations coming out. After the critical disorder strength δ c ∼ 1, the system reaches a random-singlet phase with prominent sharp peak at ω = 0 and broad continuum at ω > 0 of the dynamic spin structure factor. In addition, we analyze the distribution of random spin domains and numerically find three kinds of domains hosting effective spin-1/2 quanta or spin-1 sites in between. These “spins” can form the weakly coupled long-range singlets due to quantum fluctuation which contribute to the sharp peak at ω = 0.
Scaling aspects of Gaussian topological phase transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy DDD. This model presents a critical line separating the gapped Haldane and large-DDD phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted, in contrast to the standard scaling holding in the Ising-like transition between the gapless antiferromagnetic N\'eel and gapped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.