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arXiv:cond-mat/9811083 · Physical review. B, Condensed matter

Intermediate-temperature dynamics of one-dimensional Heisenberg antiferromagnets

Chiranjeeb Buragohain, Subir Sachdev A

Published Apr 1, 1999• 23 citations
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Abstract

We present a general theory for the intermediate-temperature (T) properties of Heisenberg antiferromagnets of spin-$S$ ions on p-leg ladders, valid for $2Sp$ even or odd. Following an earlier proposal for $2Sp$ 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 $2Sp.$ The coupling constants of the effective model are known exactly in terms of the energy gap above the ground state $\ensuremath{\Delta}$ (for $2Sp$ even), or a crossover scale ${T}_{0}$ (for $2Sp$ 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 $T>0$ 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.

DOI
10.1103/physrevb.59.9285

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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 \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ (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 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 study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values, c0{\mathit{c}}_{0}c0​. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to T0{\mathit{T}}_{0}T0​\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a, and their further analysis leads us to the inference that below T0{\mathit{T}}_{0}T0​ the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
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 thermal transport in a classical one-dimensional Heisenberg model employing a discrete-time odd-even precessional update scheme. This dynamics equilibrates a spin chain for any arbitrary temperature and finite value of the integration time step $\ensuremath{\Delta}t$. We rigorously show that in presence of driving, the system attains local thermal equilibrium, which is a strict requirement of Fourier law. In the thermodynamic limit, heat current for such a system obeys Fourier law for all temperatures, as has been recently shown [A. V. Savin, G. P. Tsironis, and X. Zotos, Phys. Rev. B 72, 140402(R) (2005)]. Finite systems, however, show an apparent ballistic transport which crosses over to a diffusive one as the system size is increased. We provide exact results for current and energy profiles in zero- and infinite-temperature limits.
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.
We present results for the dynamic structure factor of the S=1∕2S=1∕2S=1∕2 bond alternating Heisenberg chain over a large range of frequencies and temperatures. Data are obtained from a numerical evaluation of thermal averages based on the calculation of all eigenvalues and eigenfunctions for chains of up to 20 spins. Interpretation is guided by the exact temperature dependence in the noninteracting dimer limit which remains qualitatively valid up to an interdimer exchange $\ensuremath{\lambda}\ensuremath{\approx}0.5$. The temperature induced central peak around zero frequency is clearly identified and aspects of the crossover to spin diffusion in its variation from low to high temperatures are discussed. The one-magnon peak acquires an asymmetric shape with increasing temperature. The two-magnon peak is dominated by the S=1S=1S=1 bound state which remains well defined up to temperatures of the order of JJJ. The variation with temperature and wave vector of the integrated intensity for one-magnon and two-magnon scattering and of the central peak are discussed.