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arXiv:1301.5719 · Physical Review B

Realization of higher Wess-Zumino-Witten models in spin chains

Frédéric Michaud, Salvatore R. Manmana, Frédéric Mila

Published Apr 9, 2013• 28 citations
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Abstract

Building on the generalization of the exactly dimerized Majumdar-Ghosh ground state to arbitrary spin $S$ for the Heisenberg chain with a three-site term $({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\text{H.c.}$, we use density-matrix renormalization group simulations and exact diagonalizations to determine the nature of the dimerization transition for $S=1$, $3/2$, and 2. The resulting central charge and critical exponent are in good agreement with the $\text{SU}{(2)}_{k=2S}$ Wess-Zumino-Witten values $c=3k/(2+k)$ and $\ensuremath{\eta}=3/(2+k)$. Since the three-site term that induces dimerization appears naturally if exchange interactions are calculated beyond second order, these results suggest that $\text{SU}{(2)}_{k>1}$ Wess-Zumino-Witten models might finally be realized in actual spin chains.

DOI
10.1103/physrevb.87.140404

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We discuss spin-SSS antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary SSS. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of S=1S=1S=1 and S=3/2S=3/2S=3/2 systems by using exact diagonalization and level spectroscopy.
We show that spin SSS Heisenberg spin chains with an additional three-body interaction of the form $({\mathbf{S}}_{\mathbf{i}\mathbf{\ensuremath{-}}\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}})({\mathbf{S}}_{\mathbf{i}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}+\mathbf{1}})+\mathrm{H}.\mathrm{c}.$ possess fully dimerized ground states if the ratio of the three-body interaction to the bilinear one is equal to $1/[4S(S+1)\ensuremath{-}2]$. This result generalizes the Majumdar-Ghosh point of the ${J}_{1}\ensuremath{-}{J}_{2}$ chain, to which the present model reduces for S=1/2S=1/2S=1/2. For S=1S=1S=1, we use the density matrix renormalization group method to show that the transition between the Haldane and the dimerized phases is continuous with a central charge c=3/2c=3/2c=3/2. Finally, we show that such a three-body interaction appears naturally in a strong-coupling expansion of the Hubbard model, and we discuss the consequences for the dimerization of actual antiferromagnetic chains.
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We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling J1{J}_{1}J1​, next-nearest neighbor coupling J2{J}_{2}J2​, and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where J2{J}_{2}J2​ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where J3{J}_{3}J3​ refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J2{J}_{2}J2​, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J3{J}_{3}J3​. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J2{J}_{2}J2​. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
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