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Journal of Applied Physics

Correlation functions of the X Y model (abstract)

J. C. Bonner, Gerhard H. Muller, John B. Parkinson

Published May 1, 1990• 0 citations

Abstract

Recently an elegant and quite powerful finite-system approach to determine the exponents ηx and ηz from simple spectral properties has been proposed. For critical systems, the two exponents can be expressed in terms of finite-size spectral gaps as follows: η(N)x=2ΔE01(N)/ΔE(N), η(N)z=2ΔE00(N)/ΔE(N). Here ΔE(N) is the finite-size gap between the ground state (SzT=0,k=0) and the lowest excitation at k=2π/N; ΔE01(N) is the gap to the lowest ‖SzT‖=1 excitations (at k=π), and ΔE00(N) is the gap to the next lowest SzT=0 excited state. The η(N) sequence is then extrapolated to N→∞. For XY models, differences between s=1/2 and s≥1 appear. For s=1/2, the excitations which determine ΔE00(N) and ΔE(N) are degenerate, which implies that ηz=1/2, in agreement with the exact analytic result. For spin-1, however, the next lowest SzT=0 state is located at k=2π/N instead of k=π, and is therefore identical to the state which determines the gap ΔE. The resulting equality ΔE=ΔE00 implies ηz=2, as in the spin-1/2 case. In fact, our result corresponds to power-law decay for all s, and hence we differ from Schulz and Ziman, who claim the out-of-plane correlation function decays exponentially for s>1/2. For the in-plane correlation function, the spectral gap method again agrees with the exact result ηx=0.5 for s=1/2. The consensus of this and other numerical methods for s=1 gives a value ηx≂0.20, considerably different from the case of s=1/2. Hence it is tempting to conjecture that ηx is s dependent, implying that XY models belong to different universality classes for different s. However, a finite-size study of the conformal anomaly produces the result that c=1, independent of s. This situation is further discussed.

DOI: 10.1063/1.345843

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Scaling behavior in the S=1 antiferromagnetic X X Z chain (abstract)

J. C. Bonner, Gerhard H. Muller, John B. Parkinson

A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.

Journal of Applied Physics•Apr 15, 1988•0 citations

CORRELATION FUNCTIONS IN QUANTUM SPIN CHAINS AND VERTEX MODELS

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Some correlation functions of critical models, like the anisotropic spin chain with nearest and next-to-nearest neighbor interaction, or the eight vertex model, are computed as a corollary of the study of the XYZ model in [2].

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Spin correlation function of the one-dimensional S=1 XXZ model

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•Jan 1, 1990•0 citations

Spin-1 biquadratic chain: Magnetization curve (abstract)

J. C. Bonner

The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!

Journal of Applied Physics•May 1, 1990•0 citations

Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain

G. Spronken, B. Fourcade, Yves Lépine

We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.

Physical review. B, Condensed matter•Feb 1, 1986•50 citations

Correlation functions of the XXZ spin-1/2 chain: recent progress

Jean Michel Maillet

International Congress on Mathematical Physics (ICMP)•Mar 1, 2006•0 citations

Recently an elegant and quite powerful finite-system approach to determine the exponents ηx and ηz from simple spectral properties has been proposed. For critical systems, the two exponents can be expressed in terms of finite-size spectral gaps as follows: η(N)x=2ΔE01(N)/ΔE(N), η(N)z=2ΔE00(N)/ΔE(N). Here ΔE(N) is the finite-size gap between the ground state (SzT=0,k=0) and the lowest excitation at k=2π/N; ΔE01(N) is the gap to the lowest ‖SzT‖=1 excitations (at k=π), and ΔE00(N) is the gap to the next lowest SzT=0 excited state. The η(N) sequence is then extrapolated to N→∞. For XY models, differences between s=1/2 and s≥1 appear. For s=1/2, the excitations which determine ΔE00(N) and ΔE(N) are degenerate, which implies that ηz=1/2, in agreement with the exact analytic result. For spin-1, however, the next lowest SzT=0 state is located at k=2π/N instead of k=π, and is therefore identical to the state which determines the gap ΔE. The resulting equality ΔE=ΔE00 implies ηz=2, as in the spin-1/2 case. In fact, our result corresponds to power-law decay for all s, and hence we differ from Schulz and Ziman, who claim the out-of-plane correlation function decays exponentially for s>1/2. For the in-plane correlation function, the spectral gap method again agrees with the exact result ηx=0.5 for s=1/2. The consensus of this and other numerical methods for s=1 gives a value ηx≂0.20, considerably different from the case of s=1/2. Hence it is tempting to conjecture that ηx is s dependent, implying that XY models belong to different universality classes for different s. However, a finite-size study of the conformal anomaly produces the result that c=1, independent of s. This situation is further discussed.

A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.

The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!

We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.