Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators $C,D$ . For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form $2\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[C,D]{,C}^{\ensuremath{-}1}]$, ${2D}^{2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}CDC}^{\ensuremath{-}1}{D+DC}^{\ensuremath{-}1}\mathrm{DC}$. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators $C,D$. For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form $2\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[C,D]{,C}^{\ensuremath{-}1}]$, ${2D}^{2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}CDC}^{\ensuremath{-}1}{D+DC}^{\ensuremath{-}1}\mathrm{DC}$. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.

Physical Review LettersJul 3, 1995100 citations

Dynamical properties of the S=12 random Heisenberg chain

Yu-Rong Shu, Maxime Dupont, Dao‐Xin Yao +2 more

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.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.

arXiv:1712.01701Physical review. B./Physical review. BMar 28, 201831 citations

Dynamics of the Classical Heisenberg Chain

Z. Slanič, Harvey Gould, Jan Tobochnik

Computers in PhysicsNov 1, 19913 citations

Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators

C,D

. For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form $2\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[C,D]{,C}^{\ensuremath{-}1}]$, ${2D}^{2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}CDC}^{\ensuremath{-}1}{D+DC}^{\ensuremath{-}1}\mathrm{DC}$. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.

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.