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arXiv:1712.01701 · Physical review. B./Physical review. B

Dynamical properties of the S=12 random Heisenberg chain

Yu-Rong Shu, Maxime Dupont, Dao‐Xin Yao, Sylvain Capponi, Anders W. Sandvik

Published Mar 28, 2018• 31 citations
arXivHTMLPDF

Abstract

We study dynamical properties at finite temperature $(T)$ 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 $1/{T}_{1})$. Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the $q$ 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 $q$ dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to $q=0$ that are not contained within the random-singlet theory but give non-negligible contributions to the mean $1/{T}_{1}$. To compare with NMR experiments, we consider the distribution of the local relaxation rates $1/{T}_{1}$. We show that the local $1/{T}_{1}$ values are broadly distributed, approximately according to a stretched exponential. The mean $1/{T}_{1}$ first decreases with $T$, 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 $1/{T}_{1}$ has never been experimentally observed. Indeed, we show that the divergence of the mean $1/{T}_{1}$ is due to rare events in the disordered chains and is concealed in experiments, where the typical $1/{T}_{1}$ value is accessed.

DOI
10.1103/physrevb.97.104424

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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,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) and its \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→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,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω), extending throughout the qqq space, instead of being restricted to q\ensuremath≈0q\ensuremath{\approx}0q\ensuremath≈0 and q\ensuremath≈\ensuremathπq\ensuremath{\approx}\ensuremath{\pi}q\ensuremath≈\ensuremathπ as found in the uniform system. Close to q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ, 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=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ. 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 \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​. 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.
Low-dimensional spin systems, consisting of arrays of spins arranged in chains or ladders, have been investigated intensively in recent years using both exactly solvable theoretical models as well as a variety of experimental techniques. In case of random variations of the exchange couplings the renormalization group theory predicts the existence of a random-singlet (RS) state, corresponding to spins coupled at all possible distances and energy scales. The scarce availability of suitable random systems, however, has so far prevented the experimental identification of this peculiar magnetic ground state. In a combined effort using nuclear magnetic resonance (NMR), dc magnetometry and numerical simulations, we now find compelling evidence for the formation of a random-singlet state in this class of materials. Randomness seems to generate a broad distribution of local magnetic fields, in turn reflected in stretched exponential NMR relaxations. This distribution exhibits a progressive broadening with decreasing temperature, caused by the growing inequivalence of magnetic sites, as expected from the RS theory. Our findings suggest that NMR is the tool of choice for probing the low-energy physics also in other disordered magnets, where extended-scale excitations are dominant.
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We consider the isotropic S=1S=1S=1 Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.
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 present a 13C^{13}\text{C}13C NMR study of the magnetic field driven transition to complete polarization of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg chain system copper pyrazine dinitrate Cu(C4H4N2)(NO3)2\text{Cu}({\text{C}}_{4}{\text{H}}_{4}{\text{N}}_{2}){({\text{NO}}_{3})}_{2}Cu(C4​H4​N2​)(NO3​)2​ (CuPzN). The static local magnetization as well as the low-frequency spin dynamics, probed via the nuclear spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$, were explored from the low to the high field limit and at temperatures from the quantum regime (kBT⪡J)({k}_{B}T⪡J)(kB​T⪡J) up to the classical regime (kBT⪢J)({k}_{B}T⪢J)(kB​T⪢J). The experimental data show very good agreement with quantum Monte Carlo calculations over the complete range of parameters investigated. Close to the critical field, as derived from static experiments, a pronounced maximum in ${T}_{1}^{\ensuremath{-}1}$ is found which we interpret as the finite-temperature manifestation of a diverging density of zero-energy magnetic excitations at the field-driven quantum critical point.
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.