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Physical review. B, Condensed matter

Elementary excitations of the quantum sine-Gordon model in the massless regime β2>8π : Derivation from the XXZ model

F. D. M. Haldane

Published Sep 1, 1983• 2 citations
Publisher

Abstract

From the identification of the easy-plane $\mathrm{XXZ}$ spin chain as a ${\ensuremath{\beta}}^{2}>8\ensuremath{\pi}$ quantum sine-Gordon system, the nonlinear elementary excitations of the latter are identified as massless "twists" carrying the topological charge $\ifmmode\pm\else\textpm\fi{}\frac{{\ensuremath{\beta}}^{2}}{8\ensuremath{\pi}}$, and (depending on ${\ensuremath{\beta}}^{2}$) twist-antitwist bound states.

DOI
10.1103/physrevb.28.2505

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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.

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Two-spin-wave spectra of easy-plane ferromagnetic chains

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The easy-plane ferromagnetic chain in an applied magnetic field is known to map onto a sine-Gordon field theory, in the classical continuum limit. The sine-Gordon field theory supports breather excitations which, when quantised, are identifiable as a hierarchy of bound states of several small-amplitude excitations. For increasing values of the interaction parameter, the highest members of the hierarchy become unstable to the emission of a soliton/anti-soliton pair. For a critical value of the interaction parameter, even the ground state of the quantised sine-Gordon theory becomes unstable. For the quantum spin system, the excitations corresponding to the two lowest members of the hierarchy of breathers are the single-spin-wave and the two-spin-wave bound states. The authors examine the two-spin-wave bound-state excitations of the quantum spin system directly, using a large-S approach. Although the method does rely on large S, they recover previously derived results valid for arbitrary S, such as the exact two-spin-wave bound states for the isotropic Heisenberg ferromagnet and the expansion in the anisotropy about this result. In the limit of large anisotropy the two-spin-wave bound state becomes unstable, in a manner suggestive of the instability of the quantum-sine-Gordon field theory.

Journal of Physics Condensed Matter•Oct 9, 1989•3 citations
From the identification of the easy-plane XXZ\mathrm{XXZ}XXZ spin chain as a \ensuremathβ2>8\ensuremathπ{\ensuremath{\beta}}^{2}>8\ensuremath{\pi}\ensuremathβ2>8\ensuremathπ quantum sine-Gordon system, the nonlinear elementary excitations of the latter are identified as massless "twists" carrying the topological charge \ifmmode±\else\textpm\fi\ensuremathβ28\ensuremathπ\ifmmode\pm\else\textpm\fi{}\frac{{\ensuremath{\beta}}^{2}}{8\ensuremath{\pi}}\ifmmode±\else\textpm\fi8\ensuremathπ\ensuremathβ2​, and (depending on \ensuremathβ2{\ensuremath{\beta}}^{2}\ensuremathβ2) twist-antitwist bound states.
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.
The easy-plane ferromagnetic chain in an applied magnetic field is known to map onto a sine-Gordon field theory, in the classical continuum limit. The sine-Gordon field theory supports breather excitations which, when quantised, are identifiable as a hierarchy of bound states of several small-amplitude excitations. For increasing values of the interaction parameter, the highest members of the hierarchy become unstable to the emission of a soliton/anti-soliton pair. For a critical value of the interaction parameter, even the ground state of the quantised sine-Gordon theory becomes unstable. For the quantum spin system, the excitations corresponding to the two lowest members of the hierarchy of breathers are the single-spin-wave and the two-spin-wave bound states. The authors examine the two-spin-wave bound-state excitations of the quantum spin system directly, using a large-S approach. Although the method does rely on large S, they recover previously derived results valid for arbitrary S, such as the exact two-spin-wave bound states for the isotropic Heisenberg ferromagnet and the expansion in the anisotropy about this result. In the limit of large anisotropy the two-spin-wave bound state becomes unstable, in a manner suggestive of the instability of the quantum-sine-Gordon field theory.