The two-layer Heisenberg antiferromagnet exhibits a zero temperature quantum phase transition from a disordered dimer phase to a collinear Neel phase, with long range order in the ground state. The spin-wave gap vanishes as ? ? ( J ? – J ? c ) &ngr; approaching the transition point. To account for strong correlations, the S = 1 elementary excitations triplets are described as a dilute Bose gas with infinite on-site repulsion. We apply the Brueckner diagram approach which gives the critical index &ngr; ˜ 0 . 5. We demonstrate also that the linearised in density Brueckner equations give the mean field result &ngr; = 1. Finally, an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, gives &ngr; ˜ 0 . 67. This value agrees reasonably with that of the nonlinear O(3) s model. Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear s model. We discuss the conditions for this to occur.
We consider the two-layer Heisenberg antiferromagnet near a zero-temperature quantum phase transition from a disordered dimer phase to a N\'eel state. At approaching the transition point the spin-wave gap vanishes as $\ensuremath{\Delta}\ensuremath{\propto}{(J}_{\ensuremath{\perp}}\ensuremath{-}{J}_{\ensuremath{\perp}c}{)}^{\ensuremath{\nu}}.$ To account for strong correlations between the
S=1 elementary excitations we apply the Brueckner diagram approach which gives the critical index $\ensuremath{\nu}\ensuremath{\approx}0.5.$ We demonstrate also that the linearized in density Brueckner equations give the mean field result $\ensuremath{\nu}=1.$ Finally an expansion of the Brueckner equations in powers of the density, combined with the scaling hypothesis, give $\ensuremath{\nu}\ensuremath{\approx}0.67.$ This value reasonably agrees with that of the nonlinear $O(3) \ensuremath{\sigma} \mathrm{model}.$ Our approach demonstrates that for other quantum spin models the critical index can be different from that in the nonlinear $\ensuremath{\sigma}$ model. We discuss the conditions for this to occur.
We consider quantum spin systems with dimerization, which at strong coupling have singlet ground states. To account for strong correlations, the
S=1 elementary excitations are described as a dilute Bose gas with infinite on-site repulsion. This approach is applied to the two-layer Heisenberg antiferromagnet at
T=0 with general couplings. Our analytic results for the triplet gap, the excitation spectrum, and the location of the quantum critical point are in excellent agreement with numerical results obtained by dimer series expansions.
Quantum phase transition from the N\'eel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter $\ensuremath{\alpha}$ for the exchange coupling that connects the N\'eel $(\ensuremath{\alpha}=0)$ and the dimer $(\ensuremath{\alpha}=1)$ states. We employ the
CP1 representation of the
s=21 spin operator and integrate out the half of the
CP1 variables at odd sites to obtain a
CP1 nonlinear $\ensuremath{\sigma}$ model. The effective coupling constant is a function of $\ensuremath{\alpha}$, and at $\ensuremath{\alpha}=0$, the
CP1 model is in the ordered phase that corresponds to the N\'eel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of ${\ensuremath{\alpha}}_{C}$ as $\ensuremath{\alpha}$ increases. In the N\'eel state, the dynamical composite U(1) gauge field in the
CP1 model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with
s=1 excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and
s=21 spinons appear as low-energy excitations.
We obtain the complete magnetic phase diagram of the two-dimensional antiferromagnetic Heisenberg model, , within the framework of the O ( N ) nonlinear sigma model. We find two magnetically ordered phases, one with Néel order, for , and another with collinear order, for , separated by a nonmagnetic region, for , where a gapped spin liquid is found. The transition at is of the second order while the one at is of the first order and the spin gaps cross at . Our results are exact at N → ∞ and agree with numerical results from different methods.
The quantum phase transition between a spin gap state and an antiferromagnetic phase is investigated. We study S =1/2 antiferromagnetic Heisenberg chains coupled by antiferromagnetic interchain interaction. The intrachain exchanges have alternating strength. The phase boundary between the antiferromagnetically ordered phase and a spin gap phase is also obtained in a parameter space of the amplitude of the interchain coupling and the dimerization. The spin-wave approximation substantially overestimates the antiferromagnetic phase. The competition between the long range order and the spin gap is examined in detail. We estimate a variety of critical exponents at the transition, namely, exponents ν, θ and z defined as the exponent of the correlation length, the magnetization curve and the dynamical exponent, respectively. From the quantum Monte Carlo simulation, the exponents ν, θ and z are estimated to be unity. The exponents ν and θ are different from the estimated values in one dimension. It suggests that the universality changes due to the dimensionality change. In our estimates, the exponent ν does not agree with the prediction from three dimensional classical Heisenberg model. We also discuss the relevance of our result to spin-Peierls systems with lattice distortion.
We study the
S=1/2 Heisenberg antiferromagnet on a square lattice with nearest-neighbor and plaquette four-spin exchanges [introduced by A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).] This model undergoes a quantum phase transition from a spontaneously dimerized phase to N\'eel order at a critical coupling. We show that as the critical point is approached from the dimerized side, the system exhibits strong fluctuations in the dimer background, reflected in the presence of a low-energy singlet mode, with a simultaneous rise in the triplet quasiparticle density. We find that both singlet and triplet modes of high density condense at the transition, signaling restoration of lattice symmetry. In our approach, which goes beyond mean-field theory in terms of the triplet excitations, the transition appears sharp; however since our method breaks down near the critical point, we argue that we cannot make a definite conclusion regarding the order of the transition.