arXiv:cond-mat/9604066 · Physical review. B, Condensed matter

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

Published Oct 1, 1996• 12 citations

Abstract

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.

DOI: 10.1103/physrevb.54.9242

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arXiv:cond-mat/9604066 · Physical review. B, Condensed matter

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

Published Oct 1, 1996• 12 citations

arXiv HTML PDF

Abstract

{\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.

DOI: 10.1103/physrevb.54.9242

Related papers

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.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.

Physical review. B, Condensed matterApr 1, 199436 citations

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

Yongmin Yu

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 $q = \\pi$ .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. 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Rolando D. Somma, Armando A. Aligia

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arXiv:cond-mat/0105119Physical review. B, Condensed matterJun 19, 200152 citations

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Shu Zhang, Gerhard H. Muller

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Relaxation inS=1/2 quantum spin chains: The role of second neighbor interactions

Surajit Sen, C. N. Hoff, Dennis E. Kuhl +1 more

Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterFeb 1, 19965 citations

{\mathit{S}}_{\mathit{zz}}

(q,\ensuremath{\omega}) and

{\mathit{S}}_{\mathit{DD}}

{\mathit{S}}_{\mathit{zz}}

(\ensuremath{\pi},\ensuremath{\omega}) and

{\mathit{S}}_{\mathit{DD}}

{\mathit{S}}_{\mathit{zz}}

(q,\ensuremath{\omega}) but not to

{\mathit{S}}_{\mathit{DD}}

(q,\ensuremath{\omega}). \textcopyright{} 1996 The American Physical Society.

{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.

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

We have studied the phase diagram of the one-dimensional

S=\frac{1}{2}

XXZ

model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions. We have applied the quantum renormalization group (QRG) approach to obtain stable fixed points and the scaling of coupling constants. The QRG prescription has to be implemented to the second order of interblock interactions to obtain a self-similar Hamiltonian after each step of the QRG. This model shows a rich phase diagram which includes quantum spin-fluid and dimer phases in addition to the classical N\'eel and ferromagnetic ones. We have found the border between different phases by tracing the scaling of coupling constants.

We calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN)

{J}_{1}

and next-NN exchange

{J}_{2}

with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase

{J}_{2}.

For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large

{J}_{2}{/J}_{1},

this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.