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arXiv:1404.4898 · Physical Review B

Frustration and multicriticality in the antiferromagnetic spin-1 chain

J. H. Pixley, Aditya Shashi, Andriy H. Nevidomskyy

Published Dec 15, 2014• 37 citations
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Abstract

The antiferromagnetic spin-1 chain has a venerable history and has been thought to be well understood. Here, we show that inclusion of both next-nearest-neighbor $(\ensuremath{\alpha})$ and biquadratic $(\ensuremath{\beta})$ interactions results in a rich phase diagram with a multicritical point that has not been observed before. We study the problem using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wave function, and conformal field theory. For negative $\ensuremath{\beta}<{\ensuremath{\beta}}^{*}$, we establish the existence of a spontaneously dimerized phase, separated from the Haldane phase by the critical line ${\ensuremath{\alpha}}_{c}(\ensuremath{\beta})$ of second-order phase transitions. In the opposite regime, $\ensuremath{\beta}>{\ensuremath{\beta}}^{*}$, the transition from the Haldane phase becomes first order into the next-nearest-neighbor (NNN) AKLT phase. Based on the field theoretical arguments and DMRG calculations, we find that these two regimes are separated by a multicritical point $({\ensuremath{\beta}}^{*},{\ensuremath{\alpha}}^{*})$ of a different universality class, described by the level-4 SU(2) Wess-Zumino-Witten conformal theory. From the DMRG calculations, we estimate this multicritical point to lie in the range $\ensuremath{-}0.2<{\ensuremath{\beta}}^{*}<\ensuremath{-}0.15$ and $0.47<{\ensuremath{\alpha}}^{*}<0.53$. We further find that the dimerized and NNN-AKLT phases are separated from each other by a line of first-order phase transitions that terminates at the multicritical point. We establish that transitions out of the Haldane phase into the dimer or NNN-AKLT phases are topological in nature and occur either with or without closing of the bulk gap, respectively. We also study short-range incommensurate-to-commensurate transitions in the resulting phase diagram. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. Notably, these crossover lines stretch across the entire $(\ensuremath{\beta},\ensuremath{\alpha})$ phase diagram, merging into a single incommensurate-to-commensurate transition line for negative $\ensuremath{\beta}\ensuremath{\lesssim}{\ensuremath{\beta}}^{*}$ inside the dimer phase. This behavior is qualitatively similar to that seen in classical frustrated two-dimensional spin models, by way of the quantum (1+1)D to classical 2D correspondence.

DOI
10.1103/physrevb.90.214426

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The antiferromagnetic spin-1 chain has a venerable history and has been thought to be well understood. Here, we show that inclusion of both next-nearest-neighbor (\ensuremathα)(\ensuremath{\alpha})(\ensuremathα) and biquadratic (\ensuremathβ)(\ensuremath{\beta})(\ensuremathβ) interactions results in a rich phase diagram with a multicritical point that has not been observed before. We study the problem using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wave function, and conformal field theory. For negative \ensuremathβ<\ensuremathβ∗\ensuremath{\beta}<{\ensuremath{\beta}}^{*}\ensuremathβ<\ensuremathβ∗, we establish the existence of a spontaneously dimerized phase, separated from the Haldane phase by the critical line \ensuremathαc(\ensuremathβ){\ensuremath{\alpha}}_{c}(\ensuremath{\beta})\ensuremathαc​(\ensuremathβ) of second-order phase transitions. In the opposite regime, \ensuremathβ>\ensuremathβ∗\ensuremath{\beta}>{\ensuremath{\beta}}^{*}\ensuremathβ>\ensuremathβ∗, the transition from the Haldane phase becomes first order into the next-nearest-neighbor (NNN) AKLT phase. Based on the field theoretical arguments and DMRG calculations, we find that these two regimes are separated by a multicritical point (\ensuremathβ∗,\ensuremathα∗)({\ensuremath{\beta}}^{*},{\ensuremath{\alpha}}^{*})(\ensuremathβ∗,\ensuremathα∗) of a different universality class, described by the level-4 SU(2) Wess-Zumino-Witten conformal theory. From the DMRG calculations, we estimate this multicritical point to lie in the range \ensuremath−0.2<\ensuremathβ∗<\ensuremath−0.15\ensuremath{-}0.2<{\ensuremath{\beta}}^{*}<\ensuremath{-}0.15\ensuremath−0.2<\ensuremathβ∗<\ensuremath−0.15 and 0.47<\ensuremathα∗<0.530.47<{\ensuremath{\alpha}}^{*}<0.530.47<\ensuremathα∗<0.53. We further find that the dimerized and NNN-AKLT phases are separated from each other by a line of first-order phase transitions that terminates at the multicritical point. We establish that transitions out of the Haldane phase into the dimer or NNN-AKLT phases are topological in nature and occur either with or without closing of the bulk gap, respectively. We also study short-range incommensurate-to-commensurate transitions in the resulting phase diagram. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. Notably, these crossover lines stretch across the entire (\ensuremathβ,\ensuremathα)(\ensuremath{\beta},\ensuremath{\alpha})(\ensuremathβ,\ensuremathα) phase diagram, merging into a single incommensurate-to-commensurate transition line for negative \ensuremathβ\ensuremath≲\ensuremathβ∗\ensuremath{\beta}\ensuremath{\lesssim}{\ensuremath{\beta}}^{*}\ensuremathβ\ensuremath≲\ensuremathβ∗ inside the dimer phase. This behavior is qualitatively similar to that seen in classical frustrated two-dimensional spin models, by way of the quantum (1+1)D to classical 2D correspondence.
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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.
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The authors have identified critical points in the Wess-Zumino-Witten (WZW) SU(2)5{}_{5}5​ universality class in the phase diagram of a spin-5/2 chain with frustrated next-nearest-neighbor and 3-site interactions, providing a concrete example of this universality class in a realistic, nonintegrable model. These points lie on a transition line to a dimerized phase and occur when the transition turns from first order into WZW SU(2)3{}_{3}3​ in agreement with the generic possibilities allowed by field theory arguments.