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arXiv:cond-mat/9711014 · Physical review. B, Condensed matter

Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures

Kedar Damle, Subir Sachdev A

Published Apr 1, 1998• 175 citations
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Abstract

We discuss the theory of the nonzero temperature $(T)$ spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap $\ensuremath{\Delta}$. For $T\ensuremath{\ll}\ensuremath{\Delta}$, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle $\mathcal{S}$ matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the $O(3)$ $\ensuremath{\sigma}$ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg $S=1/2$ Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an $S=1$, two-particle bound state which leads to the presence of a second peak in the $T=0$ inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate ${1/T}_{1}$ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit $T\ensuremath{\ll}\ensuremath{\Delta}$. The results for ${1/T}_{1}$ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime $\ensuremath{\Delta}<T<$(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.

DOI
10.1103/physrevb.57.8307

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We discuss the theory of the nonzero temperature (T)(T)(T) spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with a gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ. For T\ensuremath≪\ensuremathΔT\ensuremath{\ll}\ensuremath{\Delta}T\ensuremath≪\ensuremathΔ, we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle S\mathcal{S}S matrix which is shown to have a superuniversal form at low momenta. This is established by computations on the O(3)O(3)O(3) \ensuremathσ\ensuremath{\sigma}\ensuremathσ model, and strong- and weak-coupling expansions (the latter using a Majorana fermion representation) for the two-leg S=1/2S=1/2S=1/2 Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals an S=1S=1S=1, two-particle bound state which leads to the presence of a second peak in the T=0T=0T=0 inelastic neutron-scattering (INS) cross section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasiparticle peak in the INS cross section, the spin diffusivity, and for the field dependence of the NMR relaxation rate 1/T1{1/T}_{1}1/T1​ of the effective semiclassical model; these are expected to be asymptotically exact for the quantum antiferromagnets in the limit T\ensuremath≪\ensuremathΔT\ensuremath{\ll}\ensuremath{\Delta}T\ensuremath≪\ensuremathΔ. The results for 1/T1{1/T}_{1}1/T1​ are compared with the experimental findings of Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] and the agreement is quite good. In the regime \ensuremathΔ<T<\ensuremath{\Delta}<T<\ensuremathΔ<T<(a typical microscopic exchange) and we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.
We present a general theory for the intermediate-temperature (T) properties of Heisenberg antiferromagnets of spin-SSS ions on p-leg ladders, valid for 2Sp2Sp2Sp even or odd. Following an earlier proposal for 2Sp2Sp2Sp even [Damle and Sachdev, Phys. Rev. B 57, 8307 (1998)], we argue that an integrable, classical, continuum model of a fixed-length, three-vector applies over an intermediate-temperature range; this range becomes very wide for moderate and large values of 2Sp.2Sp.2Sp. The coupling constants of the effective model are known exactly in terms of the energy gap above the ground state $\ensuremath{\Delta}$ (for 2Sp2Sp2Sp even), or a crossover scale T0{T}_{0}T0​ (for 2Sp2Sp2Sp odd). Analytic and numeric results for dynamic and transport properties are obtained, including some exact results for the spin-wave damping. Numerous quantitative predictions for neutron scattering and NMR experiments are made. A general discussion on the nature of T>0T>0T>0 transport in integrable systems is also presented: an exact solution of a toy model proves that diffusion can exist in integrable systems, provided proper care is taken in approaching the thermodynamic limit.
We use a quantum O(N) nonlinear \ensuremath{\sigma} model to describe one-dimensional quantum Heisenberg antiferromagnets with integer spin S. A large-N analytical treatment of this effective theory leads to definite predictions for thermodynamic quantities in the low-temperature regime. We obtain the Haldane gap as a function of the temperature and note a trend similar to that found in the compound NENP. The specific heat and magnetic susceptibility are also obtained and comparison with exact numerical results for the S=1 spin chain allows us to determine the domain of validity of our approximation. By a calculation at subleading 1/N order we compute the quasiparticle lifetime. Results are given on-shell, for the width observed at the quasiparticle pole as seen in neutron-scattering experiments, and also off-shell, for the NMR relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. The phenomenological interacting boson model is compared with our approach. We discuss the effects of anisotropy and compare our results with present experiments on NENP.
We study both the static and dynamic properties of gapped, one-dimensional, Heisenberg, antiferromagnetic, spin chains at finite temperature through an analysis of the O(3) nonlinear sigma model. Exploiting the integrability of this theory, we are able to compute an exact low-temperature expansion of the finite temperature correlators. We do so using a truncated ``form-factor'' expansion and so provide evidence that this technique can be successfully extended to finite temperature. As a direct test, we compute the static zero-field susceptibility and obtain an exact match to the susceptibility derived from the low-temperature expansion of the exact free energy. We also study transport properties, computing both the spin conductance and the NMR-relaxation rate, 1/T1.{1/T}_{1}.1/T1​. We find these quantities to show ballistic behavior. In particular, the computed spin conductance exhibits a nonzero Drude weight at finite temperature and zero applied field. The physics thus described differs from the spin diffusion reported by Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] from experiments on the Haldane gap material, AgVP2S6.{\mathrm{AgVP}}_{2}{S}_{6}.AgVP2​S6​.
We study the nuclear spin-lattice relaxation rate 1/T1{1/T}_{1}1/T1​ in the two-leg antiferromagnetic spin-1/2 Heisenberg ladder. More specifically, we consider the contribution to 1/T1{1/T}_{1}1/T1​ from the processes with momentum transfer $(\ensuremath{\pi},\ensuremath{\pi}).$ In the limit of weak coupling between the two chains, this contribution is of activation type with gap $2\ensuremath{\Delta}$ at low temperatures $(\ensuremath{\Delta}$ is the spin gap), but crosses over to a slowly decaying temperature dependence at the crossover temperature $T\ensuremath{\approx}\ensuremath{\Delta}.$ This crossover possibly explains the recent high-temperature NMR results on ladder-containing cuprates by Imai et al.
The problem of characterizing low-temperature spin dynamics in antiferromagnetic spin chains has so far remained elusive. Here we reinvestigate it by focusing on isotropic antiferromagnetic chains whose low-energy effective field theory is governed by the quantum nonlinear sigma model. Employing an exact nonperturbative theoretical approach, we analyze the low-temperature behavior in the vicinity of nonmagnetized states and obtain exact expressions for the spin diffusion constant and the NMR relaxation rate, which we compare with previous theoretical results in the literature. Surprisingly, in SU(2)-invariant spin chains in the vicinity of half filling we find a crossover from the semiclassical regime to a strongly interacting quantum regime characterized by zero spin Drude weight and diverging spin conductivity, indicating superdiffusive spin dynamics. The dynamical exponent of spin fluctuations is argued to belong to the Kardar-Parisi-Zhang universality class. Furthermore, by employing numerical time-dependent density matrix renormalization group simulations, we find robust evidence that the anomalous spin transport persists also at high temperatures, irrespective of the spectral gap and integrability of the model.
In this thesis, I use a quantum Monte Carlo (QMC) method based on the stochastic series expansion (SSE) to study transport, quantum phase diagrams, and dynamics of several one- and quasi-one-dimensional spin systems, including spin-1/2 and spin-1 chains, a frustrated four-leg spin tube, and N-leg spin-1/2 ladders. I establish a diffusive channel for the finite temperature spin transport of the spin-1/2 antiferromagnetic Heisenberg chain. Extensive finite size studies are detailed, allowing to extract the diffusion kernel in the thermodynamic limit. The diffusion is found to exhibit no anomalous exponent, moreover it is shown to survive for finite values of exchange anisotropy and magnetic fields, however with a strong suppression away from the isotropic point and for increasing magnetic fields. In the case of the spin-1 chain, I obtain the quantum phase diagram and spin dynamics in the combined presence of finite single-ion anisotropies and external magnetic fields. I also prove the existence of magnon modes and multi-particle continua and determine their evolutions as a function of anisotropies and magnetic fields. Moreover, I calculate the magnetic field dependence of the nuclear magnetic relaxation as a function of the single ion anisotropy. For the quasi-one-dimensional spin system of a frustrated four-leg tube, I investigate the regime of large rung coupling, where QMC is applicable. I evaluate the uniform spin susceptibility and the dynamic structure factor and show that in this regime the spin-tube exhibts a crossover from a gapped effective antiferromagnetic spin-2 chain behavior with massive magnons to a gapless Luttinger liquid phase with spinon excitations. Finally, I investigate the spin dynamics of N-leg spin-1/2 ladders with spacially isotropic interactions for N=1,..,5. I obtain the low-lying excitations of the spin ladders as a function of the number of legs and show that they exhibt an even-odd oscillation which is consistent with Haldane's conjecture and other known results for the thermodynamics of spin ladders.