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Physical review. B, Condensed matter

Zero-temperature dynamics of the one-dimensionalXXZandt-Jmodels: A weak-coupling continued-fraction analysis

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

Published Jan 1, 1995• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.51.368

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Dynamics of one-dimensional quantum spin systems with competing interactions at zero temperature

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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

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 +1 more

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.

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Dynamics of strong-coupling models for cuprate superconductors: Exact results on finite lattices

H. B. Schuettler, C.-X. Chen, A. J. Fedro

We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.

•Jan 1, 1989•0 citations

Dimer and Néel order-parameter fluctuations in the spin-fluid phase of thes=1/2 spin chain with first- and second-neighbor couplings

Yongmin Yu, Gerhard H. Muller, V. S. Viswanath

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 ${\mathit{S}}_{\mathit{zz}}$(q,\ensuremath{\omega}) and ${\mathit{S}}_{\mathit{DD}}$(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 ${\mathit{S}}_{\mathit{zz}}$(\ensuremath{\pi},\ensuremath{\omega}) and ${\mathit{S}}_{\mathit{DD}}$(\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 ${\mathit{S}}_{\mathit{zz}}$(q,\ensuremath{\omega}) but not to ${\mathit{S}}_{\mathit{DD}}$(q,\ensuremath{\omega}). \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9604066•Physical review. B, Condensed matter•Oct 1, 1996•12 citations
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.
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.
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})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ tsumq{sum}_{\mathbf{q}}sumq​${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(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.
The spectral densities of dynamical spin autocorrelation functions at infinite temperature are studied for the S=1/2XXZ model (with exchange couplings Jx{\mathit{J}}_{\mathit{x}}Jx​=Jy{\mathit{J}}_{\mathit{y}}Jy​\ensuremath{\equiv}J,Jz{\mathit{J}}_{\mathit{z}}Jz​) on the linear chain, the square lattice, and the simple cubic lattice. The low-frequency behavior of a given spectral density is inferred from certain characteristic properties of its continued-fraction coefficients as determined from computed frequency moments. The analysis yields estimates for the Jz{\mathit{J}}_{\mathit{z}}Jz​/J dependence of the infrared-singularity exponent. In the d=1 case, the exponent for spin fluctuations perpendicular to the O(2) symmetry axis responds sensitively as the anisotropy parameter sweeps across the O(3) symmetry point Jz{\mathit{J}}_{\mathit{z}}Jz​/J=1, while the exponent for the parallel fluctuations shows little variation. In the cases d=2 and d=3 the same observations are made for autocorrelation functions of aggregate spins in chains and lattice planes, respectively.
We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.
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.