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Physical review. B, Condensed matter

Ordering and fluctuations in the ground state of the one-dimensional and two-dimensional S =1/2 XXZ antiferromagnets: A study of dynamical properties based on the recursion method

V. S. Viswanath, Shu Zhang, Joachim Stolze, Gerhard H. Muller

Published Apr 1, 1994• 36 citations
Publisher

Abstract

The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ t${sum}_{\mathbf{q}}$${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.

DOI
10.1103/physrevb.49.9702

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This dissertation reports three projects on the zero-temperature dynamics of some prominent one-dimensional (1D) quantum spin systems. A common feature of all three model systems is the presence of competing interactions. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis. The first model investigated is the 1D s = 1 XXZ model with an additional uniaxial single-site term. The results for the in-plane and out-of-plane spin dynamic structure factors $S\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ z$ are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors $S\\sb{zz}(q,\\omega)$ and $S\\sb{DD}(q,\\omega$), which describe (for q = $\\pi$) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of S$\\sb{zz}(\\pi,\\omega$) and $S\\sb{DD}(\\pi,\\omega$) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to $S\\sb{zz}(q,\\omega$) but not to $S\\sb{DD}(q,\\omega$). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors, $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$), are investigated via two different types of continued-fraction analysis. The spectral weight in $S\\sb{ZZ}(q,\\omega$) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in $S\\sb{DD}(q,\\omega$) is dominated by the continuum alone, which shrinks to a single mode at $q = \\pi$.

Journal of Media Literacy Education•Jan 1, 1996•0 citations
The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors S\ensuremathμ\ensuremathμ{\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}S\ensuremathμ\ensuremathμ​(q,\ensuremath{\omega}) and the spin autocorrelation functions S\ensuremathμ\ensuremathμ{\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}S\ensuremathμ\ensuremathμ(\ensuremath{\omega})=N\ensuremath−1{\mathit{N}}^{\mathrm{\ensuremath{-}}1}N\ensuremath−1 t{sum}_{\mathbf{q}}$${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.
We use the recursion method to study the spectral and dynamical properties of the one-dimensional (1D) s=1/2 XXZ method with planar anisotropy at T=0. Distinct methods of continued-fraction analysis have been developed for the weak-coupling and strong-coupling regimes of the corresponding lattice fermion system. The weak-coupling analysis presented here yields detailed information on the spectral-weight distribution in dynamic structure factors and spin autocorrelation functions, notably on the infrared singularities produced by critical fluctuations, and on the bound states for the case of attractive fermion interaction. The same method is then applied to the charge dynamics of the 1D t-J model for strongly correlated electrons. There it yields similar yet distinct results in the regime of weak exchange coupling. The results for renormalized bandwidths of particle-hole excitations are consistent with available results for charge velocities, and the results for the infrared exponent in the charge dynamic structure factor agree with existing results for the exponent of the equal-time charge correlation function.
The dynamical properties at T=0 of the one-dimensional (1D) s=1/2 nearest-neighbor (NN) XXZ model with an additional isotropic next-nearest-neighbor (NNN) coupling are investigated by means of the recursion method in combination with a weak-coupling continued-fraction analysis. The focus is on the dynamic structure factors Szz{\mathit{S}}_{\mathit{zz}}Szz​(q,\ensuremath{\omega}) and SDD{\mathit{S}}_{\mathit{DD}}SDD​(q,\ensuremath{\omega}), which describe (for q=\ensuremath{\pi}) the fluctuations of the N\'eel and dimer order parameters, respectively. We calculate the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distribution in Szz{\mathit{S}}_{\mathit{zz}}Szz​(\ensuremath{\pi},\ensuremath{\omega}) and SDD{\mathit{S}}_{\mathit{DD}}SDD​(\ensuremath{\pi},\ensuremath{\omega}), all in the spin-fluid phase, which is realized for planar NN anisotropy and sufficiently weak NNN coupling. For some parameter values we find a discrete branch of excitations above the spinon continuum. They contribute to Szz{\mathit{S}}_{\mathit{zz}}Szz​(q,\ensuremath{\omega}) but not to SDD{\mathit{S}}_{\mathit{DD}}SDD​(q,\ensuremath{\omega}). \textcopyright{} 1996 The American Physical Society.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
The T=0 dynamics of the one-dimensional s= 1 / 2 planar antiferromagnet is studied by an approach which consists of exact analytic calculations in the Bethe formalism and numerical finite-chain calculations on rings up to 10 spins. The method makes use of well known critical exponents for the correlation functions and of exact sum rules. The authors obtain approximate analytic expressions for both the out-of-plane and the in-plane dynamic structure factors, and for related quantities such as integrated intensities, susceptibilities and autocorrelation functions. The results are discussed in relation to possible experiments on quasi-1D magnetic compounds at low T. The calculations make clear that the T=0 dynamic structure factors are dominated by two-parameter continua of excitations rather than by single branches of spin-waves as predicted by classical spin-wave theory. By varying the planar anisotropy the autocorrelation functions display interesting features in their longtime asymptotic behaviour, such as a crossover from a uniform power-law decay to an oscillatory decay and a crossover between oscillatory decays with different frequencies. They conjecture a possibility of approaching the classical limit s= infinity starting from the quantum limit s= 1 / 2 . This provides a qualitative, and in some aspects even quantitative, understanding of the dynamical behaviour of s> 1 / 2 systems in terms of a quantum approach.
This dissertation reports three projects on the zero-temperature dynamics of some prominent one-dimensional (1D) quantum spin systems. A common feature of all three model systems is the presence of competing interactions. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis. The first model investigated is the 1D s = 1 XXZ model with an additional uniaxial single-site term. The results for the in-plane and out-of-plane spin dynamic structure factors Ssbmumu(q,omega),mu=x,zS\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ zSsbmumu(q,omega),mu=x,z are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors Ssbzz(q,omega)S\\sb{zz}(q,\\omega)Ssbzz(q,omega) and SsbDD(q,omegaS\\sb{DD}(q,\\omegaSsbDD(q,omega), which describe (for q = pi\\pipi) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of Ssbzz(pi,omega\\sb{zz}(\\pi,\\omegasbzz(pi,omega) and SsbDD(pi,omegaS\\sb{DD}(\\pi,\\omegaSsbDD(pi,omega) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to Ssbzz(q,omegaS\\sb{zz}(q,\\omegaSsbzz(q,omega) but not to SsbDD(q,omegaS\\sb{DD}(q,\\omegaSsbDD(q,omega). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of Ssbzz(q,omegaS\\sb{zz}(q,\\omegaSsbzz(q,omega) and SsbDD(q,omegaS\\sb{DD}(q,\\omegaSsbDD(q,omega) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors, Ssbzz(q,omegaS\\sb{zz}(q,\\omegaSsbzz(q,omega) and SsbDD(q,omegaS\\sb{DD}(q,\\omegaSsbDD(q,omega), are investigated via two different types of continued-fraction analysis. The spectral weight in SsbZZ(q,omegaS\\sb{ZZ}(q,\\omegaSsbZZ(q,omega) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in SsbDD(q,omegaS\\sb{DD}(q,\\omegaSsbDD(q,omega) is dominated by the continuum alone, which shrinks to a single mode at q=piq = \\piq=pi.