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arXiv:cond-mat/0208282 · Physical review. B, Condensed matter

Thermal conductivity of anisotropic and frustrated spin- 12 chains

Fabian Heidrich‐Meisner, A. Honecker, D. C. Cabra, W. Brenig

Published Oct 29, 2002• 90 citations
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Abstract

We analyze the thermal conductivity of anisotropic and frustrated spin-1/2 chains using analytical and numerical techniques. This includes mean-field theory based on the Jordan-Wigner transformation, bosonization, and exact diagonalization of systems with $N<~18$ sites. We present results for the temperature dependence of the zero-frequency weight of the conductivity for several values of the anisotropy \ensuremath{\Delta}. In the gapless regime, we show that the mean-field theory compares well to known results and that the low-temperature limit is correctly described by bosonization. In the antiferromagnetic and ferromagnetic gapped regime, we analyze the temperature dependence of the thermal conductivity numerically. The convergence of the finite-size data is remarkably good in the ferromagnetic case. Finally, we apply our numerical method and mean-field theory to the frustrated chain where we find a good agreement of these two approaches on finite systems. Our numerical data do not yield evidence for a diverging thermal conductivity in the thermodynamic limit in case of the antiferromagnetic gapped regime of the frustrated chain.

DOI
10.1103/physrevb.66.140406

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We analyze the thermal conductivity of anisotropic and frustrated spin-1/2 chains using analytical and numerical techniques. This includes mean-field theory based on the Jordan-Wigner transformation, bosonization, and exact diagonalization of systems with N< 18N<~18N< 18 sites. We present results for the temperature dependence of the zero-frequency weight of the conductivity for several values of the anisotropy \ensuremath{\Delta}. In the gapless regime, we show that the mean-field theory compares well to known results and that the low-temperature limit is correctly described by bosonization. In the antiferromagnetic and ferromagnetic gapped regime, we analyze the temperature dependence of the thermal conductivity numerically. The convergence of the finite-size data is remarkably good in the ferromagnetic case. Finally, we apply our numerical method and mean-field theory to the frustrated chain where we find a good agreement of these two approaches on finite systems. Our numerical data do not yield evidence for a diverging thermal conductivity in the thermodynamic limit in case of the antiferromagnetic gapped regime of the frustrated chain.
We report a detailed analysis of the Drude weights for both thermal and spin transport in one-dimensional spin-12\frac{1}{2}21​ systems by means of exact diagonalization and analytic approaches at finite temperatures. Transport properties are studied first for the integrable XXZ\mathrm{XXZ}XXZ model and second for various nonintegrable systems such as the dimerized chain, the frustrated chain, and the spin ladder. We compare our results obtained by exact diagonalization and mean-field theory with those of the Bethe ansatz, bosonization, and other numerical studies in the case of the anisotropic Heisenberg model both in the gapless and gapped regime. In particular, we find indications that the Drude weight for spin transport is finite in the thermodynamic limit for the isotropic chain. For the nonintegrable models, a finite-size analysis of the numerical data for the Drude weights is presented, covering the entire parameter space of the dimerized and frustrated chain. We also discuss which conclusions can be drawn from bosonization regarding the question of whether the Drude weights are finite or not. One of our main results is that the Drude weights vanish in the thermodynamic limit for nonintegrable models.
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.
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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 study a one-dimensional anisotropic XXZ Heisenberg spin-12\frac{1}{2}21​ chain with weak random fields hizSiz{h}_{i}^{z}{S}_{i}^{z}hiz​Siz​ by means of Jordan-Wigner transformation to spinless Luttinger liquid with disorder and bosonization technique. First, we reinvestigate the phase diagram of the system in terms of dimensionless disorder $\ensuremath{\gamma}=\ensuremath{\langle}{h}^{2}\ensuremath{\rangle}/{J}^{2}\ensuremath{\ll}1$ and anisotropy parameter $\mathrm{\ensuremath{\Delta}}={J}_{z}/{J}_{xy}$, we find the range of these parameters where disorder is irrelevant in the infrared limit and spin-spin correlations are described by power laws, and compare it with previously obtained numerical and analytical results. Then we use the diagram technique in terms of plasmon excitations to study the low-temperature $(T\ensuremath{\ll}J)$ behavior of heat conductivity $\ensuremath{\kappa}$ and spin conductivity $\ensuremath{\sigma}$ in this power-law phase. The obtained Lorentz number $L\ensuremath{\equiv}\ensuremath{\kappa}/\ensuremath{\sigma}T$ differs from the value derived earlier by means of the memory function method. We argue also that in the studied region inelastic scattering is strong enough to suppress quantum interference in the low-temperature limit.
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