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arXiv:2109.00552 · Frontiers of Physics

Dynamical properties of the Haldane chain with bond disorder

Jing-Kai Fang, Jun-Han Huang, Han-Qing Wu, Dao‐Xin Yao

Published Nov 30, 2021• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s11467-021-1124-3

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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.
Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor S(k,ω)S(k,ω)S(k,ω) of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, S(k,ω)S(k,ω)S(k,ω) develops broad non-spinon excitations as well as the random-singlet peak near (k,ω)=(π,0)(k, ω) = (π, 0)(k,ω)=(π,0), consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu2_22​(Six_xx​Ge1−x_{1-x}1−x​)2_22​O7_77​.
We study dynamical properties at finite temperature (T)(T)(T) of Heisenberg spin chains with random antiferromagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor $S(q,\ensuremath{\omega})$ and its $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate 1/T1)1/{T}_{1})1/T1​). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the qqq space, instead of being restricted to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as found in the uniform system. Close to $q=\ensuremath{\pi}$, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted qqq dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to q=0q=0q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T11/{T}_{1}1/T1​. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T11/{T}_{1}1/T1​. We show that the local 1/T11/{T}_{1}1/T1​ values are broadly distributed, approximately according to a stretched exponential. The mean 1/T11/{T}_{1}1/T1​ first decreases with TTT, but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ${\ensuremath{\omega}}_{0}$. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean 1/T11/{T}_{1}1/T1​ has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T11/{T}_{1}1/T1​ is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T11/{T}_{1}1/T1​ value is accessed.
We calculate the dynamical structural factor of the S=1S=1S=1 bond-alternating Heisenberg chain. In the Haldane phase, the lowest excited states form the lower edge of the multimagnon continuum in 0< q< qc0<~q<~{q}_{c}0< q< qc​ and the one-magnon mode in ${q}_{c}<~q<~\ensuremath{\pi}.$ As the system approaches the gapless point, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}$ and the largest integrated intensity of the one-magnon mode is decreased. In the singlet-dimer phase, the one-magnon mode appears in 0< q< qc.0<~q<~{q}_{c}.0< q< qc​. As the bond-alternation becomes strong, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}.$ In the antiferromagnetic-ferromagnetic bond-alternation region with a strong ferromagnetic coupling, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~0.2\ensuremath{\pi}$ and $0.8\ensuremath{\pi}<~q<~\ensuremath{\pi},$ and the one-magnon mode appears in $0.2\ensuremath{\pi}<q<0.8\ensuremath{\pi}.$ The largest integrated intensity of the one-magnon mode is 9393%,93 which is slightly smaller than that in the S=1S=1S=1 Haldane-gap system. We further discuss the dynamical structural factor in connection with the inelastic neutron-scattering experiments.
Using a numerical implementation of a strong-disorder renormalization group, we study the low-energy, long-distance properties of layers and bilayers of S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnets with different types of disorder: bond randomness, site dilution, and dimer dilution. Generally the systems exhibit an ordered and a disordered phase separated by a phase boundary on which the static critical exponents appear to be independent of bond randomness in the strong-disorder regime, while the dynamical exponent is a continuous function of the bond disorder strength. The low-energy fixed points of the off-critical phases are affected by the actual form of the disorder, and the disorder-induced dynamical exponent depends on the disorder strength. As the strength of the bond disorder is increased, there is a set of crossovers in the properties of the low-energy singularities. For weak disorder quantum fluctuations play the dominant role. For intermediate disorder nonlocalized disorder fluctuations are relevant, which become localized for even stronger bond disorder. We also present some quantum Monte Carlo simulation results to support the strong-disorder renormalization approach.
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