The thermal conductivity of a one dimensional magnetic lattice is calculated in the thermal current formalism. It is assumed that the spins are coupled by a nearest-neighbor ferromagnetic Heisenberg exchange interaction. The conductivity is shown to be proportional to a six-spin correlation function which is factored into a product of three two-spin correlation functions. The two-spin functions are evaluated in a semi-classical approximation assuming exponential decay in time. The rate of decay is taken to be proportional to the spin diffusion constant, which is obtained from the non-linear formalism of Bennett and Martin. The resulting expression for the conductivity scales as the product of the spin diffusion constant and the specific heat of a classical chain over the entire temperature range 0≪T≪∞. The significance of this result is discussed with reference to the approximations made in the calculation. An alternative treatment which is appropriate at low temperatures is also given. This latter approach leads to a conductivity which vanishes linearly with temperature in the boundary scattering region.The thermal conductivity of a one dimensional magnetic lattice is calculated in the thermal current formalism. It is assumed that the spins are coupled by a nearest-neighbor ferromagnetic Heisenberg exchange interaction. The conductivity is shown to be proportional to a six-spin correlation function which is factored into a product of three two-spin correlation functions. The two-spin functions are evaluated in a semi-classical approximation assuming exponential decay in time. The rate of decay is taken to be proportional to the spin diffusion constant, which is obtained from the non-linear formalism of Bennett and Martin. The resulting expression for the conductivity scales as the product of the spin diffusion constant and the specific heat of a classical chain over the entire temperature range 0≪T≪∞. The significance of this result is discussed with reference to the approximations made in the calculation. An alternative treatment which is appropriate at low temperatures is also given. This latter approach leads to a conductivity which vanishes linearly with temperature in the boundary scattering region.

Progress of Theoretical PhysicsMay 1, 196816 citations

Thermal conductivity of spin- 12 chains

Efrat Shimshoni, Natan Andrei, Achim Rosch

We study the low-temperature transport properties of clean one-dimensional spin- $\frac{1}{2}$ chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, $\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.$ As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale ${T}^{*}$ turns out to be of order ${\ensuremath{\Theta}}_{D}/2,$ where ${\ensuremath{\Theta}}_{D}$ is the Debye energy, rather than the magnetic exchange interaction $J$ ---in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of $\ensuremath{\kappa}(T,h)$ as a function of h.We study the low-temperature transport properties of clean one-dimensional spin-$\frac{1}{2}$ chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, $\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.$ As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale ${T}^{*}$ turns out to be of order ${\ensuremath{\Theta}}_{D}/2,$ where ${\ensuremath{\Theta}}_{D}$ is the Debye energy, rather than the magnetic exchange interaction $J$---in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of $\ensuremath{\kappa}(T,h)$ as a function of h.

arXiv:cond-mat/0304641Physical review. B, Condensed matterSep 2, 200387 citations

Exact results for the dynamics of the classical nearest-neighbour Heisenberg chain at low temperatures

George Reiter, Alf Sjölander

Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.

Journal of Physics C Solid State PhysicsJun 10, 198056 citations

Spin and thermal conductivity of quantum spin chains and ladders

Christoph Karrasch, Dante Marvin Kennes, Fabian Heidrich‐Meisner

We study the spin and thermal conductivity of spin- $\frac{1}{2}$ ladders and chains at finite temperature, relevant for experiments with quantum magnets. Using a state-of-the-art density matrix renormalization group algorithm, we compute the current autocorrelation functions on the real-time axis and then carry out a Fourier integral to extract the frequency dependence of the corresponding conductivities. The finite-time error is analyzed carefully. We first investigate the limiting case of spin- $\frac{1}{2}$ $XXZ$ chains, for which our analysis suggests nonzero dc conductivities in all interacting cases irrespective of the presence or absence of spin Drude weights. For ladders, we observe that all models studied are normal conductors with no ballistic contribution. Nonetheless, only the high-temperature spin conductivity of $XX$ ladders has a simple diffusive, Drude-like form, while Heisenberg ladders exhibit a more complicated low-frequency behavior. We compute the dc spin conductivity down to temperatures of the order of $T\ensuremath{\sim}0.5J$, where $J$ is the exchange coupling along the legs of the ladder. We further extract mean-free paths and discuss our results in relation to thermal conductivity measurements on quantum magnets.We study the spin and thermal conductivity of spin-$\frac{1}{2}$ ladders and chains at finite temperature, relevant for experiments with quantum magnets. Using a state-of-the-art density matrix renormalization group algorithm, we compute the current autocorrelation functions on the real-time axis and then carry out a Fourier integral to extract the frequency dependence of the corresponding conductivities. The finite-time error is analyzed carefully. We first investigate the limiting case of spin-$\frac{1}{2}$ $XXZ$ chains, for which our analysis suggests nonzero dc conductivities in all interacting cases irrespective of the presence or absence of spin Drude weights. For ladders, we observe that all models studied are normal conductors with no ballistic contribution. Nonetheless, only the high-temperature spin conductivity of $XX$ ladders has a simple diffusive, Drude-like form, while Heisenberg ladders exhibit a more complicated low-frequency behavior. We compute the dc spin conductivity down to temperatures of the order of $T\ensuremath{\sim}0.5J$, where $J$ is the exchange coupling along the legs of the ladder. We further extract mean-free paths and discuss our results in relation to thermal conductivity measurements on quantum magnets.

arXiv:1412.6047Physical Review BMar 16, 201548 citations

Computer studies of spin and energy transport in one-dimensional Heisenberg magnets

N.A. Lurie, David L. Huber, Martin Blume

We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of ${D}_{S}$ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures $\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5$ . Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of ${D}_{S}$ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures $\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5$. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.

Physical review. B, Solid stateMar 1, 197499 citations

Magnetic Properties of the Random Mixture of the Classical Heisenberg Spins

Shigetoshi Katsura

The specific heat, the susceptibility, and the correlation function at zero field above the critical temperature of the random mixture (quenched site and bond problems) of the classical Heisenberg spins with nearest neighbor interaction were obtained exactly for the linear chain and for an infinite Bethe lattice (Bethe approximation of the two and three dimensional lattices) above the critical temperature. The results are simply expressed by the replacements of 2 cosh K → 4π (sinh K)/K and tanh K → L(K) (L(K) = Langevin function) for K = KAA, KAB, KBA, and KBB in the corresponding expressions of the random mixture of the Ising spins, and qualitative properties of both models are similar.The specific heat, the susceptibility, and the correlation function at zero field above the critical temperature of the random mixture (quenched site and bond problems) of the classical Heisenberg spins with nearest neighbor interaction were obtained exactly for the linear chain and for an infinite Bethe lattice (Bethe approximation of the two and three dimensional lattices) above the critical temperature. The results are simply expressed by the replacements of 2 cosh K → 4π (sinh K)/K and tanh K → L(K) (L(K) = Langevin function) for K = KAA, KAB, KBA, and KBB in the corresponding expressions of the random mixture of the Ising spins, and qualitative properties of both models are similar.

Canadian Journal of PhysicsMay 1, 197516 citations

We study the low-temperature transport properties of clean one-dimensional spin-

\frac{1}{2}

chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, $\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.$ As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale

{T}^{*}

turns out to be of order ${\ensuremath{\Theta}}_{D}/2,$ where ${\ensuremath{\Theta}}_{D}$ is the Debye energy, rather than the magnetic exchange interaction

J

---in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of $\ensuremath{\kappa}(T,h)$ as a function of h.

We study the spin and thermal conductivity of spin-

\frac{1}{2}

ladders and chains at finite temperature, relevant for experiments with quantum magnets. Using a state-of-the-art density matrix renormalization group algorithm, we compute the current autocorrelation functions on the real-time axis and then carry out a Fourier integral to extract the frequency dependence of the corresponding conductivities. The finite-time error is analyzed carefully. We first investigate the limiting case of spin-

\frac{1}{2}

XXZ

chains, for which our analysis suggests nonzero dc conductivities in all interacting cases irrespective of the presence or absence of spin Drude weights. For ladders, we observe that all models studied are normal conductors with no ballistic contribution. Nonetheless, only the high-temperature spin conductivity of

XX

ladders has a simple diffusive, Drude-like form, while Heisenberg ladders exhibit a more complicated low-frequency behavior. We compute the dc spin conductivity down to temperatures of the order of $T\ensuremath{\sim}0.5J$, where

J

is the exchange coupling along the legs of the ladder. We further extract mean-free paths and discuss our results in relation to thermal conductivity measurements on quantum magnets.

{D}_{S}

have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures

\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5

. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.