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.
The phase diagram of the spin-1 chain with bilinear-biquadratic and next-nearest-neighbor interactions, recently investigated by Pixley, Shashi, and Nevidomskyy [Phys. Rev. B 90, 214426 (2014)], has been revisited in the light of results we have recently obtained on a similar model. Combining extensive density-matrix renormalization-group simulations with conformal-field theory arguments, we confirm the presence of the three phases identified by Pixley et al., a Haldane phase, a next-nearest-neighbor (NNN) Haldane phase, and a dimerized phase, but we come to significantly different conclusions regarding the nature of the phase transitions to the dimerized phase: (i) We provide numerical evidence of a continuous Ising transition between the NNN-Haldane phase and the dimerized phase. (ii) We show that the tricritical end point, where the continuous transition between the Haldane phase and the dimerized phase turns into a first-order transition, is distinct from the triple point where the three phases meet. (iii) Finally, we demonstrate that the tricritical end point is in the same Wess-Zumino-Witten
SU(2)2 universality class as the continuous transition line that ends at this point.
The ground-state `phase' diagrams and some low-energy properties of isotropic antiferromagnetic spin- and spin-1 chains with a next-nearest-neighbour exchange and an alternation of the nearest-neighbour exchanges have been compared for the first time using the density matrix renormalization group method. In the spin- chain, the system is gapless for and , and is gapped everywhere else in the plane. At , for small , the gap increases as , where is a disorder line. To the left of this line, the structure factor S(q) peaks at (the Néel `phase'), while to the right, decreases from to (the spiral `phase') as increases. We also discover a ` phase' for large values of both and . In the spin-1 case, we find a line running from a gapless point at up to a `gapless' point at such that the open-chain ground state is fourfold degenerate below the line and is unique above it. There is a disorder line in this case also and it has the same equation as in the spin- case, but the line ends at about . Similarly to the spin- case, to the left of this line, the peak in the structure factor is at (the Néel `phase'), while to the right of the line, it is at less than (the spiral `phase'). For , the system corresponds to a spin ladder and the system is gapped for all values of the interchain coupling for both spin- and spin-1 ladders.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and
0.0≲α≤1.0.
A Monte Carlo simulation study based on the heat-bath algorithm on a checkerboard arrangement is applied to the spin-1 antiferromagnetic chain model with both bilinear and biquadratic exchange interactions. Changing the parameter \ensuremath{\lambda} which denotes strength of the biquadratic interaction against the bilinear one from -1.0 to 2.0, we obtain the following results according to the measurements of energy and two-point correlation function. At \ensuremath{\lambda}=1 and -(1/3 where some physical quantities are derived rigorously, our Monte Carlo simulation results agree well with the rigorous ones. For ${\ensuremath{\lambda}}_{c}$\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}${\ensuremath{\lambda}}_{1}$ with taking 0.25\ensuremath{\le}${\ensuremath{\lambda}}_{c}$\ensuremath{\le}0.5 and ${\ensuremath{\lambda}}_{1}$\ensuremath{\ge}2.0, its model has the gapless ground state accompanied by a power-law decaying correlation function. This implies the existence of a critical line. For -1.0${\ensuremath{\lambda}}_{c}$, on the other hand, its model has the ground state with energy gap accompanied with exponentially decaying correlation function. This indicates that the valence-bond-solid (VBS) state asserted by Affleck et al. is stabilized there. As the point of \ensuremath{\lambda}=0 comes to the standard Heisenberg antiferromagnetic model and is in the region of -1.0${\ensuremath{\lambda}}_{c}$, the Haldane's conjecture is supported that this ground state is described by the VBS state for the case of integer spin, e.g., S=1.
A frustrated spin-
1/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling
J2>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of
J1/J2 and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D
+) phase with the same sign of the
x,y- and
z-component dimer order parameters, two VC dimer (VCD
+/VCD${}_{\ensuremath{-}}$) phases with the sign of the
z-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D
+ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb
2Cu
2Mo
3O
12 is discussed.