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Physical review. B, Condensed matter

Quantum thermodynamics of the easy-plane ferromagnetic chain

Alessandro Cuccoli, V. Tognetti, Paola Verrucchi, Ruggero Vaia

Published Nov 1, 1992• 32 citations
Publisher

Abstract

The thermodynamic properties of the quantum ferromagnetic chain with easy-plane single-ion anisotropy are evaluated over a wide temperature range. The basic theoretical method is a recently proposed one, called the pure-quantum self-consistent harmonic approximation (PQSCHA), which, starting from the original Hamiltonian operator scr\ifmmode \hat{H}\else \^{H}\fi{}, allows the definition of an effective classical Hamiltonian ${\mathit{scrH}}_{\mathrm{eff}}$. We use the Villain transformation to canonical conjugate variables, in order to treat the quantum renormalizations within this framework. In the resulting effective Hamiltonian we restore classical spin variables, according to the classical counterpart of the Villain transformation. Provided that the quantum coupling parameter (which depends on the spin value and on the anisotropy constant) is not too large, this effective Hamiltonian is able to account for the quantum behavior, by means of classical-like calculations. We present a number of results for various quantum thermodynamic quantities, obtained numerically by the classical transfer-matrix method. Moreover, we also calculate static spin correlation functions. Finally, the case of the real ferromagnet ${\mathrm{CsNiF}}_{3}$ is considered; even though the corresponding easy-plane anisotropy is rather weak, the experimental data for this system are shown to be well reproduced.

DOI
10.1103/physrevb.46.11601

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Classical aspects of quantum spin chains (abstract)

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A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.

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

Specific Heat of a Quantum Spin Chain of an Easy-Plane Type under an In-Plane Magnetic Field

Toshio Tsuzuki

Quantum statistical mechanics is developed for an easy-plane ferromagnetic chain under an in-plane magnetic field. A pair product approximation is made for the density matrix. A transfer integral expression for the partition function is formulated by the use of the coherent state of spin. Specific heat is computed as a function of temperature and magnetic field for the case of strong single-ion anisotropy. From our qualitative study it is suggested that, beyond the sine-Gordon approximation, simultaneous consideration of the quantum effect and the instability against the off-plane fluctuation (the Kumar-Magyari-Thomas instability) is necessary to explain specific heat, in particular, and thermodynamic quantities, in general, of the present system.

Progress of Theoretical Physics Supplement•Jan 1, 1984•3 citations
The thermodynamic properties of the quantum ferromagnetic chain with easy-plane single-ion anisotropy are evaluated over a wide temperature range. The basic theoretical method is a recently proposed one, called the pure-quantum self-consistent harmonic approximation (PQSCHA), which, starting from the original Hamiltonian operator scr\ifmmode \hat{H}\else ^{H}\fi{}, allows the definition of an effective classical Hamiltonian scrHeff{\mathit{scrH}}_{\mathrm{eff}}scrHeff​. We use the Villain transformation to canonical conjugate variables, in order to treat the quantum renormalizations within this framework. In the resulting effective Hamiltonian we restore classical spin variables, according to the classical counterpart of the Villain transformation. Provided that the quantum coupling parameter (which depends on the spin value and on the anisotropy constant) is not too large, this effective Hamiltonian is able to account for the quantum behavior, by means of classical-like calculations. We present a number of results for various quantum thermodynamic quantities, obtained numerically by the classical transfer-matrix method. Moreover, we also calculate static spin correlation functions. Finally, the case of the real ferromagnet CsNiF3{\mathrm{CsNiF}}_{3}CsNiF3​ is considered; even though the corresponding easy-plane anisotropy is rather weak, the experimental data for this system are shown to be well reproduced.
The statistical mechanics of the two-dimensional ferromagnetic model with easy-plane anisotropy is approached by the pure-quantum self-consistent harmonic approximation (PQSCHA), that reduces the calculation of thermodynamic averages to effective classical expressions. In the PQSCHA, the quantum corrections to the classical thermodynamics are reduced to suitable (temperature-dependent) renormalizations of the interaction parameters, so that the full role of the nonlinear excitations is preserved. A particular case is the XX0 model (also known as the quantum XY model), which undergoes a Kosterlitz–Thouless phase transition at some finite temperature Tc. Since it is possible to calculate how much the effective exchange interaction is weakened by quantum fluctuations, we can predict, for instance, the corresponding amount of reduction of Tc for any value of the spin. Even in the extreme quantum case of the spin-1/2 model, our result is compatible with the estimates of Tc obtained by other authors.
Using the semiclassical method called pure-quantum self-consistent harmonic approximation (PQSCHA), we evaluate thermodynamic quantities of one-dimensional Heisenberg ferromagnets and antiferromagnets. Since the PQSCHA reduces their evaluation to classical-like calculations, we take advantage of Fisher's exact solution [M. E. Fisher, Am J. Phys. 32, 343 (1964)] to get all results in an almost fully analytical way. Explicitly considered here are the specific heat, the correlation length, and the susceptibility. Good agreement with available numerical data and Monte Carlo simulations is found for S>1S>1S>1 ferromagnets and antiferromagnets; for the latter it is seen that topological terms and the related Haldane gap are relevant only for the lowest spin values and temperatures.
A classical effective Hamiltonian for the quantum easy-plane ferromagnetic chain with an in-plane applied field is introduced by means of a path-integral variational method. It then becomes possible to include quantum effects of the out-of-plane fluctuations in an easily evaluable phase-space integral which approximates the quantum partition function. A comparison with experimental data for the excess specific heat of CsNiF3{\mathrm{CsNiF}}_{3}CsNiF3​ shows much improvement in the description of this system. In addition, new light is shed on the old problem of defining the classical counterpart of a spin Hamiltonian.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.
Quantum statistical mechanics is developed for an easy-plane ferromagnetic chain under an in-plane magnetic field. A pair product approximation is made for the density matrix. A transfer integral expression for the partition function is formulated by the use of the coherent state of spin. Specific heat is computed as a function of temperature and magnetic field for the case of strong single-ion anisotropy. From our qualitative study it is suggested that, beyond the sine-Gordon approximation, simultaneous consideration of the quantum effect and the instability against the off-plane fluctuation (the Kumar-Magyari-Thomas instability) is necessary to explain specific heat, in particular, and thermodynamic quantities, in general, of the present system.