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arXiv:2002.08982 · Physical review. B./Physical review. B

Floating, critical, and dimerized phases in a frustrated spin- 32 chain

Natalia Chepiga, Ian K. Affleck, Frédéric Mila

Published May 5, 2020• 17 citations
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Abstract

We study spontaneous dimerization and emergent criticality in a spin-$\frac{3}{2}$ chain with antiferromagnetic nearest-neighbor ${J}_{1}$, next-nearest-neighbor ${J}_{2}$, and three-site ${J}_{3}$ interactions. In the absence of three-site interaction ${J}_{3}$, we provide evidence that the model undergoes a remarkable sequence of three phase transitions as a function of ${J}_{2}/{J}_{1}$, going successively through a critical-commensurate phase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommensurate order, to end up in a fully dimerized phase at very large ${J}_{2}/{J}_{1}$. In the field-theory language, this implies that the coupling constant of the marginal operator responsible for dimerization changes sign three times. For large enough ${J}_{3}$, the fully dimerized phase is stabilized for all ${J}_{2}$, and the phase transitions between the critical phases and this phase are both Wess-Zumino-Witten (WZW) SU(2)${}_{3}$ along part of the boundary and turn first order at some point due to the presence of a marginal operator in the WZW SU(2)${}_{3}$ model. By contrast, the transition between the two-dimerized phase is always first order, and the phase transitions between the partially dimerized phase and the critical phases are Kosterlitz-Thouless. Finally, we discuss the intriguing spin-$\frac{1}{2}$ edge states that emerge in the partially dimerized phase for even chains. Unlike their counterparts in the spin-1 chain, they are not confined and disappear upon increasing ${J}_{2}$ in favor of a reorganization of the dimerization pattern.

DOI
10.1103/physrevb.101.174407

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We study spontaneous dimerization transitions in a Heisenberg spin-1 chain with additional next-nearest-neighbor and three-site interactions using extensive numerical simulations and a conformal field-theory analysis. We show that the transition can be second order in the Wess-Zumino-Witten (WZW) SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ or Ising universality class, or first order. We argue that these features are generic because of a marginal operator in the WZW SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ model and because of two topologically distinct nondimerized phases with or without edge states. We also provide explicit numerical evidence of conformal towers of singlets inside the spin gap at the Ising transition. Implications for other models are briefly discussed.
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We study dimerization transitions in a frustrated spin-3 chain with next-nearest-neighbor and three-site interactions. We show that two independent coupling constants of the model are sufficient to fine-tune the system to the critical point in the Wess-Zumino-Witten SU(2)6\mathrm{SU}{(2)}_{6}SU(2)6​ universality class. This critical point appears as the end point of an extended SU(2)4\mathrm{SU}{(2)}_{4}SU(2)4​ critical line. This implies that the renormalization group flow leads to the critical theory with the largest level kkk such that the number of relevant operators is reduced by one and the parity of the level is preserved. Furthermore, we argue that due to the presence of marginal operator there is only one point in the SU(2)6\mathrm{SU}{(2)}_{6}SU(2)6​ universality class. In addition, we report the appearance of the nonmagnetic Ising transition between the topologically trivial uniform and dimerized phases. This transition takes place within the singlet sector, while the magnetic gap remains open.
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.
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.