Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Applied Physics

Quantum spin chains and the conformal anomaly

Jill C. Bonner, John B. Parkinson

Published Apr 15, 1988• 5 citations
Publisher

Abstract

The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E0(N)≂E0(∞)−(NΔE/12)c(1/N2), where E0 (N) is the ground-state energy of an N-spin finite system, E0 (∞) is the ground-state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ2 ∼1.15 −1.18 have also been studied.

DOI
10.1063/1.340737

Related papers

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

Numerical studies of the quantum spin chain (abstract)

H. Q. Lin, C. Y. Pan

We report the results of recent numerical studies of the antiferromagnetic Heisenberg chain with S=1/2, 1, 3/2, and 2. The ground-state energy and the energy gap between the ground state and the first excited state are obtained by using exact diagonalization method on various finite lattices and an improved real-space renormalization method. We discuss our numerical methods and results. Some of our results on the S=1 biquadratic antiferromagnetic model will also be discussed.

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

Conformal invariance and the operator content of the XXZ model with arbitrary spin

Francisco Castilho Alcaraz, Marcio Jose Martins

The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.

Journal of Physics A Mathematical and General•Jun 7, 1989•53 citations

Critical Behaviour of Mixed Heisenberg Chains

F. C. Alcaraz, A. L. Malvezzi

The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1

•Jan 1, 1996•28 citations

Quantum spin chains with alternation

Henk W. J. Blöte, Jill C. Bonner

Affleck has made predictions for the critical exponents of a class of integrable spin chains of general spin $s$. The expressions are $s$ dependent. Our interest here is a finite-size scaling analysis of spin-1 systems to determine the validity of the prediction for the alternation critical exponent. We discuss these results in the light of our numerical results for the conformal anomaly $c$, the exponent $\ensuremath{\eta}$, and the mass-gap exponent associated with biquadratic exchange. The numerical results are not generally in good agreement with theory, and hence the possible presence of logarithmic corrections is discussed.

Physical review. B, Condensed matter•Aug 1, 1987•12 citations

T=0 magnetization and susceptibility curves for Heisenberg-like and other spin-1 chains (abstract)

Jill C. Bonner

Recently there has been extensive theoretical interest in the remarkable Haldane prediction that spin-1 antiferromagnetic Heisenberg-like chains should display an energy gap between singlet ground state and excited continua. This behavior has been confirmed numerically, and it has been shown that the energy gap is of a magnitude to be observable experimentally. The phenomenon has captured the interest of experimental groups in neutron scattering and magnetism. A recent experimental study obtains very good experimental agreement with the theoretical T=0 magnetization curve obtained numerically by Parkinson and Bonner.1,2 An explanation will be provided for the unusual phase diagram and other anomalous features observed by the authors of Ref. 2. Numerical susceptibility studies of Heisenberg antiferromagnetic and general XXZ spin-1 chains will be presented, and discussed in terms of a novel quantum-classical cross-over phenomenon.2 Finally, a comparison will be made of T=0 magnetization isotherms for several spin-1 chains, including the Takhtajan–Babujian chain and the chain with pure biquadratic interactions.

Journal of Applied Physics•Apr 15, 1991•0 citations
The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E0(N)≂E0(∞)−(NΔE/12)c(1/N2), where E0 (N) is the ground-state energy of an N-spin finite system, E0 (∞) is the ground-state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ2 ∼1.15 −1.18 have also been studied.
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!
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1
Affleck has made predictions for the critical exponents of a class of integrable spin chains of general spin sss. The expressions are sss dependent. Our interest here is a finite-size scaling analysis of spin-1 systems to determine the validity of the prediction for the alternation critical exponent. We discuss these results in the light of our numerical results for the conformal anomaly ccc, the exponent $\ensuremath{\eta}$, and the mass-gap exponent associated with biquadratic exchange. The numerical results are not generally in good agreement with theory, and hence the possible presence of logarithmic corrections is discussed.
Recently there has been extensive theoretical interest in the remarkable Haldane prediction that spin-1 antiferromagnetic Heisenberg-like chains should display an energy gap between singlet ground state and excited continua. This behavior has been confirmed numerically, and it has been shown that the energy gap is of a magnitude to be observable experimentally. The phenomenon has captured the interest of experimental groups in neutron scattering and magnetism. A recent experimental study obtains very good experimental agreement with the theoretical T=0 magnetization curve obtained numerically by Parkinson and Bonner.1,2 An explanation will be provided for the unusual phase diagram and other anomalous features observed by the authors of Ref. 2. Numerical susceptibility studies of Heisenberg antiferromagnetic and general XXZ spin-1 chains will be presented, and discussed in terms of a novel quantum-classical cross-over phenomenon.2 Finally, a comparison will be made of T=0 magnetization isotherms for several spin-1 chains, including the Takhtajan–Babujian chain and the chain with pure biquadratic interactions.