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 Physics C Solid State Physics

Specific heat calculations for quantum spin chains with Heisenberg-biquadratic exchange in a magnetic field

H -U Desgranges, John B. Parkinson

Published Jan 10, 1988• 2 citations
Publisher

Abstract

The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.

DOI
10.1088/0022-3719/21/1/014

Related papers

Magnetic field properties of the integrable S=1 isotropic quantum spin chain

John B. Parkinson, J. C. Bonner

The integrable S=1 isotropic quantum spin chain of N atoms, introduced by Takhtajan (1982) and Bubujian (1983), is studied as a function of magnetisation and hence of magnetic field. The methods used are (i) direct numerical diagonalisation for N<or=12, (ii) numerical solution of the Bethe ansatz equations for N=6, 12, 16, 20, 40, (iii) analytic solution of the Bethe ansatz equations for N to infinity . The authors obtain results for the multiple string structure using the first two of these methods, showing the dominance of two-strings in the ground state and elementary excitations. Based on this they calculate the magnetisation as a function of field in the thermodynamic limit, over the whole spin-flop regime. They observe a tracking soft mode in the finite N results and also an energy gap which tends to zero as N to infinity . Their conclusion is that the system behaves much more like the integrable S= 1 / 2 Heisenberg model than the nonintegrable S=1 Heisenberg model.

Journal of Physics C Solid State Physics•Oct 30, 1986•4 citations

The S=1 quantum spin chain with equal Heisenberg and biquadratic exchange in a magnetic field

John B. Parkinson

The S=1 quantum spin chain with equal Heisenberg and biquadratic exchange was shown by Sutherland to be integrable. In this paper the equations obtained by Sutherland are solved numerically to obtain the behaviour at zero temperature in a magnetic field. It is found that there is an additional phase change at a lower field than the spin flop field. This new phase change marks the boundary between a regime in which the ground state contains only atoms with S z =1, 0 and one in which 1, 0 and -1 are present. The form of the elementary excitations in the two regimes is discussed and the possibility of a similar phase change in the pure Heisenberg model is considered.

Journal of Physics Condensed Matter•Sep 18, 1989•15 citations

The S = 1 quantum spin chain: some exact and numerical results

D Paczek, John B. Parkinson

Abstract There has been great interest recently in quantum spin chains with S = 1. Much of this work was stimulated by Haldane’s (1983a, b) discussion of the isotropic Heisenberg model for which he predicted the existence of a disordered ground state (i.e. one in which the correlations decay exponentially) and a gap to the first excited state at k = n. We consider a system with isotropic Heisenberg and biquadratic exchange, and a magnetic field, described by the Hamiltonian.

•Mar 21, 1991•0 citations

The statistical mechanics of the related S=1/2anisotropic and S=1 pure biquadratic quantum spin chains

I. D. Paczek, John B. Parkinson

The authors consider a quantum spin chain with S= 1 / 2 and anisotropic exchange (XXZ model) with Delta =- 3 / 2 . This integrable system has a partial mapping to an S=1 chain with pure biquadratic exchange. They calculate the magnetisation curves at zero temperature which are identical for the two systems. They also use the integral equations obtained by Gaudin to study the specific heat as a function of temperature and magnetic field. The results are compared with numerical calculations for short chains of the S=1 system. The maximum occurs at approximately the same position for the two systems, but the shapes of the curves differ considerably.

Journal of Physics Condensed Matter•Jun 18, 1990•1 citation

Generalized Heisenberg quantum spin chains (invited)

Jill C. Bonner

Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.

Journal of Applied Physics•Apr 15, 1987•40 citations

Thermodynamics of a spin-S' impurity in a spin-S antiferromagnetic Heisenberg chain

Pedro D Sacramento

The numerical solution of the Bethe ansatz equations of an integrable SU(2)-invariant generalization of the spin-S antiferromagnetic Heisenberg chain (Takhtajian-Babujian model) with a spin-S' impurity in zero magnetic field is presented. The entropy, specific heat and susceptibility of an impurity of spin S' are obtained numerically as a function of the spin S of the chain. Three situations have to be distinguished: (i) if S'=S the impurity just corresponds to one more site in the chain; (ii) if S'>S the impurity spin is only partially compensated (undercompensated) at T=0, leaving an effective spin of (S'-S); (iii) if S'

Journal of Physics Condensed Matter•Sep 13, 1993•11 citations
The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.
The integrable S=1 isotropic quantum spin chain of N atoms, introduced by Takhtajan (1982) and Bubujian (1983), is studied as a function of magnetisation and hence of magnetic field. The methods used are (i) direct numerical diagonalisation for N<or=12, (ii) numerical solution of the Bethe ansatz equations for N=6, 12, 16, 20, 40, (iii) analytic solution of the Bethe ansatz equations for N to infinity . The authors obtain results for the multiple string structure using the first two of these methods, showing the dominance of two-strings in the ground state and elementary excitations. Based on this they calculate the magnetisation as a function of field in the thermodynamic limit, over the whole spin-flop regime. They observe a tracking soft mode in the finite N results and also an energy gap which tends to zero as N to infinity . Their conclusion is that the system behaves much more like the integrable S= 1 / 2 Heisenberg model than the nonintegrable S=1 Heisenberg model.
The S=1 quantum spin chain with equal Heisenberg and biquadratic exchange was shown by Sutherland to be integrable. In this paper the equations obtained by Sutherland are solved numerically to obtain the behaviour at zero temperature in a magnetic field. It is found that there is an additional phase change at a lower field than the spin flop field. This new phase change marks the boundary between a regime in which the ground state contains only atoms with S z =1, 0 and one in which 1, 0 and -1 are present. The form of the elementary excitations in the two regimes is discussed and the possibility of a similar phase change in the pure Heisenberg model is considered.
The authors consider a quantum spin chain with S= 1 / 2 and anisotropic exchange (XXZ model) with Delta =- 3 / 2 . This integrable system has a partial mapping to an S=1 chain with pure biquadratic exchange. They calculate the magnetisation curves at zero temperature which are identical for the two systems. They also use the integral equations obtained by Gaudin to study the specific heat as a function of temperature and magnetic field. The results are compared with numerical calculations for short chains of the S=1 system. The maximum occurs at approximately the same position for the two systems, but the shapes of the curves differ considerably.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
The numerical solution of the Bethe ansatz equations of an integrable SU(2)-invariant generalization of the spin-S antiferromagnetic Heisenberg chain (Takhtajian-Babujian model) with a spin-S' impurity in zero magnetic field is presented. The entropy, specific heat and susceptibility of an impurity of spin S' are obtained numerically as a function of the spin S of the chain. Three situations have to be distinguished: (i) if S'=S the impurity just corresponds to one more site in the chain; (ii) if S'>S the impurity spin is only partially compensated (undercompensated) at T=0, leaving an effective spin of (S'-S); (iii) if S'