We show that disordered flat phases in crystal surfaces are equivalent to valence-bond-type phases in integer and half-integer spin quantum chains. In the quantum spin representation the disordered flat phase represents a fluid-type phase with long-range antiferromagnetic spin order. This order is stabilized dynamically by the hopping of the particles and short-range spin-exchange interactions. The mass of N\'eel solitons is finite. Numerical finite-size-scaling results confirm this. We identify the order parameter of the valence-bond phase. The Haldane conjecture suggests a fundamental difference between half-integer and integer antiferromagnetic Heisenberg spin chains. We find that disordered flat phases are realized in both cases, have exactly the same type of long-range antiferromagnetic spin order, and are stabilized by exactly the same mechanism. They differ only in the mathematical formulation of broken symmetry in the spin representation. We suggest experimental methods of observing disordered flat phases in crystal surfaces.
The ground-state phase diagram of a spin-
21 XXZ chain with competing ferromagnetic nearest-neighbor (
J1<0) and antiferromagnetic second-neighbor (
J2>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
The spin-1/2 Heisenberg octahedral chain with regularly alternating monomeric and square-plaquette sites is investigated using various analytical and numerical methods: variational technique, localized-magnon approach, exact diagonalization (ED) and density-matrix renormalization group (DMRG) method. The model belongs to the class of flat-band systems and it has a rich ground-state phase diagram including phases with spontaneously broken translational symmetry. Moreover, it exhibits an anomalous low-temperature thermodynamics close to continuous or discontinuous field-driven quantum phase transitions between three quantum ferrimagnetic phases, tetramer-hexamer phase, monomer-tetramer phase, localized-magnon phase and two different spin-liquid phases. If the intra-plaquette coupling is at least twice as strong as the monomer-plaquette coupling, the variational method furnishes a rigorous proof of the monomer-tetramer ground state in a low-field region and the localized-magnon approach provides an exact evidence of a single magnon trapped at each square plaquette in a high-field region. In the rest of parameter space we have numerically studied the ground-state phase diagram and magnetization process using DMRG and ED methods. It is shown that the zero-temperature magnetization curve may involve up to four intermediate plateaus at zero, one-fifth, two-fifth and three-fifth of the saturation magnetization, while the specific heat exhibits a striking low-temperature peak in a vicinity of discontinuous quantum phase transitions.
The quantum phases of one-dimensional spin
s=1/2 chains are discussed for models with two parameters, frustrating exchange
g=J2>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at
g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin
S=SA+SB, have singlet GS for any
g, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at
1/g=0 to linear Heisenberg antiferromagnets on sublattices
A and
B of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap
Em to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at
gc=1/4ln2 from a GS with perfect LRO to a decoupled phase with
SA=SB=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.
We investigate the phase diagram of hard-core bosons on a square lattice with competing interactions. The hard-core bosons can also be represented by spin-
1/2 operators and the model can therefore be mapped onto an anisotropic
J1-
J2 Heisenberg model. We find the N\'eel state and a collinear antiferromagnetic state as classical ordered phases to be suppressed by the introduction of ferromagnetic exchange terms in the
x-
y plane that result in a ferromagnetic phase for large interactions. For an intermediate regime, the emergence of new quantum states like valence bond crystals or supersolids is predicted for similar models. We do not observe any signal for long-range order in terms of conventional order or dimer correlations in our model and find an exponential decay in the spin correlations. Hence, all evidence is pointing toward a quantum-disordered ground state for a small region in the phase diagram.
We revisit the phase diagram of the dimerized XXZ spin-
21 chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic
XY couplings which have a uniform value and
ZZ couplings which have a dimerized form, with strengths
Ja and
Jb on alternate bonds. We find a rich phase diagram in the region of positive
Ja,Jb. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge
c=1. In contrast, there are two phase transition lines separating
IPM0 or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by
c=21 CFTs and correspond to quantum Ising transitions. In fact, the
c=1 line bifurcates into the two
c=21 lines at the point
Ja=Jb=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that
IPM0 is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-
21 states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.