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arXiv:cond-mat/0304641 · Physical review. B, Condensed matter

Thermal conductivity of spin- 12 chains

Efrat Shimshoni, Natan Andrei, Achim Rosch

Published Sep 2, 2003• 87 citations
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Abstract

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.

DOI
10.1103/physrevb.68.104401

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We study the low-temperature transport properties of clean one-dimensional spin-12\frac{1}{2}21​ 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, \ensuremathκ\ensuremath∼eT∗/T.\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.\ensuremathκ\ensuremath∼eT∗/T. As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale T∗{T}^{*}T∗ turns out to be of order \ensuremathΘD/2,{\ensuremath{\Theta}}_{D}/2,\ensuremathΘD​/2, where \ensuremathΘD{\ensuremath{\Theta}}_{D}\ensuremathΘD​ is the Debye energy, rather than the magnetic exchange interaction JJJ---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 T\ensuremath→0\stackrel{\ensuremath{\rightarrow}}{T}0T\ensuremath→​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 \ensuremathκ(T,h)\ensuremath{\kappa}(T,h)\ensuremathκ(T,h) as a function of h.
The heat transport in a system of S=1/2 large-J Heisenberg spin chains, describing closely Sr(2)CuO(3) and SrCuO(2) cuprates, is studied theoretically at T≪J by considering interactions of the bosonized spin excitations with optical phonons and defects. Treating rigorously the multiboson processes, we derive a microscopic spin-phonon scattering rate that adheres to an intuitive picture of phonons acting as thermally populated defects for the fast spin excitations. The mean-free path of the latter exhibits a distinctive T dependence reflecting a critical nature of spin chains and gives a close description of experiments. By the naturalness criterion of realistically small spin-phonon interaction, our approach stands out from previous considerations that require large coupling constants to explain the data and thus imply a spin-Peierls transition, absent in real materials.
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.
We study the spin and heat conductivity of dimerized spin-12\frac{1}{2}21​ chains in homogeneous magnetic fields at finite temperatures. At zero temperature, the model undergoes two field-induced quantum phase transitions from a dimerized, into a Luttinger, and finally into a fully polarized phase. We search for signatures of these transitions in the spin and heat conductivities. Using exact diagonalization, we calculate the Drude weights, the frequency dependence of the conductivities, and the corresponding integrated spectral weights. As a main result, we demonstrate that both the spin and heat conductivity are enhanced in the gapless phase and most notably at low frequencies. In the case of the thermal conductivity, however, the field-induced increase seen in the bare transport coefficients is suppressed by magnetothermal effects, caused by the coupling of the heat and spin current in finite magnetic fields. Our results complement recent magnetic transport experiments on spin-ladder materials with sufficiently small exchange couplings allowing access to the field-induced transitions.
We outline a general approach to the computation of transport properties of interacting systems at low temperetures and frequencies. We show that if the fixed point and the irrelevant operators around it are known, then by studying the structure of the softly violated conserved currents chracterizing the fixed point one may set up an effective calculation in terms of a memory matrix formalism. We apply this approach to the computation of thermal conductivity of spin chains embedded in a matter matrix and interacting with its phonons. The results are found to be in very good agreement with experiment.
We study the magnetic field-dependence of the thermal conductivity due to magnetic excitations in frustrated spin-1/2 Heisenberg chains. Near the saturation field, the system is described by a dilute gas of weakly interacting fermions (free-fermion fixed point). We show that in this regime the thermal conductivity exhibits a nonmonotonic behavior as a function of the ratio $\ensuremath{\alpha}={J}_{2}/{J}_{1}$ between second- and first-nearest-neighbor antiferromagnetic exchange interactions. This result is a direct consequence of the splitting of the single-particle dispersion minimum into two minima that takes place at the Lifshitz point $\ensuremath{\alpha}=1/4$. Upon increasing $\ensuremath{\alpha}$ from zero, the inverse mass vanishes at $\ensuremath{\alpha}=1/4$ and it increases monotonically from zero for $\ensuremath{\alpha}\ensuremath{\ge}1/4$. By deriving an effective low-energy theory of the dilute gas of fermions, we demonstrate that the Drude weight Kth{K}_{\mathrm{th}}Kth​ of the thermal conductivity exhibits a similar dependence on $\ensuremath{\alpha}$ near the saturation field. Moreover, this theory predicts a transition between a two-component Tomonaga-Luttinger liquid and a vector-chiral phase at a critical value $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$ that agrees very well with previous density matrix renormalization group results. We also show that the resulting curve ${K}_{\mathrm{th}}(\ensuremath{\alpha})$ is in excellent agreement with exact diagonalization (ED) results. For the low-magnetic field regime, our ED results show that ${K}_{\mathrm{th}}(\ensuremath{\alpha})$ has a pronounced minimum at $\ensuremath{\alpha}\ensuremath{\simeq}0.7$. We also demonstrate that the thermal conductivity is significantly affected by the presence of magnetothermal coupling.