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arXiv:cond-mat/0105284 · Physical review. B, Condensed matter

Haldane-gapped spin chains: Exact low-temperature expansions of correlation functions

Robert Konik

Published Sep 30, 2003• 73 citations
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Abstract

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, ${1/T}_{1}.$ 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, ${\mathrm{AgVP}}_{2}{S}_{6}.$

DOI
10.1103/physrevb.68.104435

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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​.
Spin transport properties of the quantum one-dimensional non-linear sigma model are studied based upon the Bethe ansatz exact solution for the O(3) sigma model and the 1/N-expansion approach for the O(N) sigma model. It is shown that the spin transport at finite temperatures is ballistic and there is no spin diffusion in this system. A possible mechanism for the spin diffusion observed in the NMR experiment of the Haldane gap system AgVP_2S_6 by Takigawa et al. [Phys. Rev. Lett.{\bf 76}, 2173 (1996)] is also discussed. The inclusion of the spin-phonon interaction accounts for the experimental results qualitatively.
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 the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron-scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighboring chains.
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.
We discuss the theory of the nonzero temperature (T)(T)(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 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) $\ensuremath{\sigma}$ 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 $T\ensuremath{\ll}\ensuremath{\Delta}$. 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 $\ensuremath{\Delta}<T<$(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.
We consider the isotropic S=1S=1S=1 Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.