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Journal of the Physical Society of Japan

Quantum Lattice Liquid Version of theS=1/2 Linear Heisenberg Antiferromagnet

Tosizumi Aoki

Published Sep 15, 1989• 2 citations
Publisher

Abstract

From the point of view of the quantum lattice liquid, an attempt is made to understand the one-dimensional S =1/2 Heisenberg antiferromagnet. Without assuming a long-range order, the spin wave excitation is approximately derived from the motion of a number operator ( z -component spin). With the aid of a sum rule, the spectrum is estimated. The Boson representation of the system is approximately given on the basis of the equation of motion of the number operator. In this framework, the susceptibility, the specific heat and the correlation function are calculated. The results imply the possibility of a quantum lattice liquid description of the system.

DOI
10.1143/jpsj.58.3334

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From the point of view of the quantum lattice liquid, an attempt is made to understand the one-dimensional S =1/2 Heisenberg antiferromagnet. Without assuming a long-range order, the spin wave excitation is approximately derived from the motion of a number operator ( z -component spin). With the aid of a sum rule, the spectrum is estimated. The Boson representation of the system is approximately given on the basis of the equation of motion of the number operator. In this framework, the susceptibility, the specific heat and the correlation function are calculated. The results imply the possibility of a quantum lattice liquid description of the system.
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.
We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.
We use the diagram technique for spin operators to calculate Green's functions and observables of the spin-12\frac{1}{2}21​ quantum Heisenberg antiferromagnet on a square lattice. The first corrections to the self-energy and interaction are taken into account in the chain diagrams. The approximation reproduces main results of Takahashi's modified spin-wave theory [Phys. Rev. B 40, 2494 (1989)] and is applicable in a wider temperature range. The energy per spin calculated in this approximation is in good agreement with the Monte Carlo and small-cluster exact-diagonalization calculations in the range $0<~T\ensuremath{\lesssim}1.2J$ where J is the exchange constant. For the static uniform susceptibility the agreement is good for $T\ensuremath{\lesssim}0.6J$ and becomes somewhat worse for higher temperatures. Nevertheless the approximation is able to reproduce the maximum in the temperature dependence of the susceptibility near T=0.9J.T=0.9J.T=0.9J.
We develop a mean-field theory of the 1D spin-1 antiferromagnetic Heisenberg model, using a two-orbital fermionic representation for the spin-1 operators. Two ansatz with different orbital pairing symmetries are discussed and the projected mean-field ground states with their corresponding parent Hamiltonians have also been obtained. Then we study the U(1) spin liquid state with large spinon Fermi surfaces. The physical conductivity obtained through Ioffe-Larkin composition rule can support surface plasmon modes propagating along the interface between the spin liquid and a linear medium, which can be excited using a Kretschmann-Raether configuration. Next we generalize the linear current response to including nonlinear effects under the perturbation of a time dependent but spatially uniform electric field in the framework of an effective U(1) gauge theory derived through a Fermi-liquid-like approach. The nonlinear susceptibility can support waves with triple frequency inside the spin liquids through the process of third harmonic generation. As a byproduct during the Fermi-liquid approach, we generalize the standard Landau Fermi liquid theory to describe the intrinsic anomalous Hall conductivity, based on some general symmetric considerations. Fitting result of the anomalous Hall conductivity is well consistent with the experimental data, and behaves better in lower frequency range than the simple two-band model.