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arXiv:cond-mat/0408016 · Physical Review B

Spin gap in chains with hidden symmetries

Mikhail N. Kiselev, D. N. Aristov, K. Kikoin

Published Mar 24, 2005• 19 citations
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Abstract

We investigate the formation of a spin gap in one-dimensional models characterized by groups with hidden dynamical symmetries. A family of two-parametric models of isotropic and anisotropic spin-rotator chains (SRC's) characterized by $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ and $\mathrm{SO}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{Z}}_{2}$ symmetries is introduced to describe the transition from SU(2) to SO(4) antiferromagnetic Heisenberg chains. The excitation spectrum is studied with the use of the Jordan-Wigner transformation generalized for ${o}_{4}$ algebra and by means of the bosonization approach. Hidden discrete symmetries associated with invariance under various particle-hole transformations are discussed. We show that the spin gap in SRC Hamiltonians is characterized by the scaling dimension $2∕3$, in contrast to dimension 1 in the conventional Haldane problem.

DOI
10.1103/physrevb.71.092404

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We investigate the formation of a spin gap in one-dimensional models characterized by groups with hidden dynamical symmetries. A family of two-parametric models of isotropic and anisotropic spin-rotator chains (SRC's) characterized by SU(2)\ifmmode×\else\texttimes\fiSU(2)\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)SU(2)\ifmmode×\else\texttimes\fiSU(2) and SO(2)\ifmmode×\else\texttimes\fiSO(2)\ifmmode×\else\texttimes\fiZ2\ifmmode×\else\texttimes\fiZ2\mathrm{SO}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{Z}}_{2}SO(2)\ifmmode×\else\texttimes\fiSO(2)\ifmmode×\else\texttimes\fiZ2​\ifmmode×\else\texttimes\fiZ2​ symmetries is introduced to describe the transition from SU(2) to SO(4) antiferromagnetic Heisenberg chains. The excitation spectrum is studied with the use of the Jordan-Wigner transformation generalized for o4{o}_{4}o4​ algebra and by means of the bosonization approach. Hidden discrete symmetries associated with invariance under various particle-hole transformations are discussed. We show that the spin gap in SRC Hamiltonians is characterized by the scaling dimension 2∕32∕32∕3, in contrast to dimension 1 in the conventional Haldane problem.
We introduce a new spin-fermion mapping, for arbitrary spin S generating the SU(2) group algebra, that constitutes a natural generalization of the Jordan-Wigner transformation for S = 1/2. The mapping, valid for regular lattices in any spatial dimension d, serves to unravel hidden symmetries. We illustrate the power of the transformation by finding exact solutions to lattice models previously unsolved by standard techniques. We also show the existence of the Haldane gap in S = 1 bilinear nearest-neighbor Heisenberg spin chains and discuss the relevance of the mapping to models of strongly correlated electrons. Moreover, we present a general spin-anyon mapping for the case d < or = 2.
We study a model of two weakly coupled isotropic spin-1/2 Heisenberg chains with an antiferromagnetic coupling along the chains (spin ladder). It is shown that the system always has a spectral gap and the lower lying excitations are triplets. For the case of identical chains the model in the continuous limit is shown to be equivalent to four decoupled noncritical Ising models with the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}SU(2) symmetry. For this case we obtain the exact expressions for asymptotics of spin-spin correlation functions. It is shown that when the chains have different exchange integrals J1{\mathit{J}}_{1}J1​\ensuremath{\gg}J2{\mathit{J}}_{2}J2​ the spectrum at low energies is described by the O(3)-nonlinear \ensuremath{\sigma} model. We discuss the topological order parameter related to the gap formation and give a detailed description of the dynamical magnetic susceptibility. \textcopyright{} 1996 The American Physical Society.
The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.
In terms of spinless fermions and spin waves, we describe magnetic properties of a spin-1/2 ferromagnetic– antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein– Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool to investigate one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.
In terms of spinless fermions and spin waves, we describe the magnetic properties of a spin-1/2 ferromagnetic-antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On the one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein–Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool for investigating one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.
The composite-spin representation of the spin-1 Heisenberg chain is used to transform it through the Jordan-Wigner transformation to the one-dimensional fermion gas. To properly include the xy couplings between spins, we also consider the bosonized version of the fermion model. Phase diagrams deduced from the two versions of the fermion model are compared against numerical results for finite Heisenberg chains. One of the symmetries of the spin model is lost in the fermionization, and this leads to a topologically incorrect phase diagram in at least one part of the parameter space. There are clear indications of significant coupling of spin and charge degrees of freedom in the fermion model and of strong renormalization of the coupling constants in the continuum limit. The existence of the Haldane gap is discussed in connection with these results.