We report a phase diagram of the antiferromagnetic spin-1 chain with nearest-neighbor Heisenberg and three-site interactions in the presence of single-ion anisotropy. We show that the Gaussian and Ising transitions that separate the topological Haldane phase from the two anisotropic phases eventually fuse into a higher symmetry point characterized by the Wess-Zumino-Witten (WZW) SU(2)$_2$ critical theory providing a lattice realization of the conformal embedding. On the other side of the WZW multi-critical point, the Ising critical line reappears together with the eight-vertex transition. This transition is a one-dimensional realization of a deconfined quantum criticality separating the dimerized and Ising antiferromagnetic phases - two ordered phases with incompatible order parameters.
We report a phase diagram of the antiferromagnetic spin-1 chain with nearest-neighbor Heisenberg and three-site interactions in the presence of single-ion anisotropy. We show that the Gaussian and Ising transitions that separate the topological Haldane phase from the two anisotropic phases eventually fuse into a higher symmetry point characterized by the Wess-Zumino-Witten (WZW) SU(2)2 critical theory providing a lattice realization of the conformal embedding. On the other side of the WZW multi-critical point, the Ising critical line reappears together with the eight-vertex transition. This transition is a one-dimensional realization of a deconfined quantum criticality separating the dimerized and Ising antiferromagnetic phases - two ordered phases with incompatible order parameters.
We investigate the phase diagram of a quantum mixed-spin (1/2,1/2,1,1) Heisenberg tetramer chain featuring bond-alternating antiferromagnetic nearest-neighbor exchange interactions and single-ion anisotropy. This model exhibits a rich phase structure comprising three gapped phases: the antiferromagnetic, topological valence-bond-solid (TVBS), and the trivial nonmagnetic phases (TnM). We identify a multicritical Wess-Zumino-Witten SU(2)1 point where two Ising critical lines merge, forming a higher-symmetry point that realizes conformal embedding on the lattice. Beyond this multicritical point, a Gaussian critical line emerges, determining the boundary between the TVBS and the TnM phases. Utilizing the tangential finite-size scaling method, we demonstrate the precision with which critical points and correlation length critical exponents can be extracted along this Gaussian line. Our results reveal rich critical phenomena induced by the interplay of bond-alternating exchange and anisotropic single-ion interactions in quantum many-body systems.
We report an extremely rich ground-state phase diagram of two spin-1 Haldane chains frustrated with a three-site exchange and coupled by the antiferromagnetic Heisenberg interaction on a zigzag ladder. A particular feature of the phase diagram is the extended quantum phase transition in the Ashkin-Teller universality class that separates the plaquette phase, which spontaneously breaks translation symmetry, and the uniform disordered phase. The former is connected to the Haldane phase, stabilized by large interchain coupling, via the topological Gaussian transition. Upon decreasing the interchain interactions, this intermediate disorder phase vanishes, giving place to a dimerized phase separated from the plaquette phase on one side via a nonmagnetic Ising transition and from the Haldane phase on the other side by a topological weak first-order transition. Finally, in the limit of two decoupled chains, we recover a quantum critical point that corresponds to two copies of the Wess-Zumino-Witten SU ( 2 ) 2 criticality with a total central charge c = 3 .
We investigate the cluster Ising chain with an additional nearest-neighbor interaction using tensor-network methods and a weak-coupling field theory. Without the interaction, the Jordan-Wigner transformation decomposes the model into a triplet of identical Majorana chains related by an exact O(3) flavor symmetry. Their common mass vanishes at the SU(2)2 Wess-Zumino-Novikov-Witten critical point with central charge c=3/2 separating the symmetry-protected topological cluster phase from a ferromagnet. The interaction reduces O(3) to its cyclic subgroup C3, splitting the triplet into a singlet and a doublet whose gaps close separately, producing a critical bifurcation into Ising (c=1/2) and Gaussian (c=1) critical lines. A second ferromagnetic phase opens between these critical lines for repulsive interactions and a disordered phase for attractive ones. The Gaussian line then separates two Landau-incompatible ferromagnets, realizing a deconfined quantum critical line with emergent O(2) symmetry, along which our independently extracted exponents vary continuously yet satisfy the parameter-free relation β=(2ν−1)/4 of the eight-vertex weak universality class. At stronger repulsion this line opens into an extended gapless floating phase with incommensurate algebraic correlations, entered through Berezinskii-Kosterlitz-Thouless transitions. All of these phases and the transitions between them are captured by the weak-coupling theory. Beyond its regime of validity, our simulations reveal a translation-symmetry-breaking antiferromagnet reached through first-order transitions.
Exotic criticality in the dimerized spin-1 XXZ chain with single-ion anisotropy
We consider the dimerized spin-1 XXZ chain with single-ion anisotropy D. In absence of an explicit dimerization there are three phases: a large-D, an antiferromagnetically ordered and a Haldane phase. This phase structure persists up to a critical dimerization, above which the Haldane phase disappears. We show that for weak dimerization the phases are separated by Gaussian and Ising quantum phase transitions. One of the Ising transitions terminates in a critical point in the universality class of the dilute Ising model. We comment on the relevance of our results to experiments on quasi-one-dimensional anisotropic spin-1 quantum magnets.
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.
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 universality class. This critical point appears as the end point of an extended SU(2)4 critical line. This implies that the renormalization group flow leads to the critical theory with the largest level k 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 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.