We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values,
c0. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to
T0\ensuremath{\sim}0.6
c0/a, and their further analysis leads us to the inference that below
T0 the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6
c0/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
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.
We present results for a complementary analysis of the frustrated planar ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ spin-1/2 quantum antiferromagnet (AFM). Using dynamical functional renormalization group, high-order-coupled cluster calculations, and series expansion based on the flow equation method, we have calculated generalized momentum-resolved susceptibilities, the ground-state energy, the magnetic-order parameter, and the elementary excitation gap. From these, we determine a quantum phase diagram that shows a large window of a quantum paramagnetic (QP) phase situated among the N\'eel, spiral, and collinear states, which are present already in the classical ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ AFM. Our findings are consistent with substantial plaquette correlations in the QP phase. The extent of the QP region is found to be in satisfying agreement between the three different approaches we have employed.
The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a
d-dimensional spin-
S planar Heisenberg ferromagnet with long-range interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}>d$) with the distance
r between spins. We adopt the Callen scheme for spin
S and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the $(\ensuremath{\alpha},d)$ plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of $\ensuremath{\alpha}$, we find that also for dimensionalities $d\ensuremath{\leqslant}2$ a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of $(\ensuremath{\alpha},d)$-dependent scaling functions and effective critical exponents.
We consider two-dimensional Fermi liquids in the vicinity of a quantum transition to a phase with commensurate, antiferromagnetic long-range order. Depending upon the Fermi-surface topology, mean-field spin-density-wave theory predicts two different types of such transitions, with mean-field dynamic critical exponents z=1 (when the Fermi surface does not cross the magnetic zone boundary, type A) and z=2 (when the Fermi surface crosses the magnetic zone boundary, type B). The type-A system only displays z=1 behavior at all energies and its scaling properties are similar (though not identical) to those of an insulating Heisenberg antiferromagnet. Under suitable conditions precisely stated in this paper, the type-B system displays a crossover from relaxational behavior at low energies to type-A behavior at high energies. A scaling hypothesis is proposed to describe this crossover: we postulate a universal scaling function which determines the entire, temperature-, wave-vector-, and frequency-dependent, dynamic, staggered spin susceptibility in terms of four experimentally measurable, T=0 parameters. The scaling function contains the full scaling behavior in all regimes for both type-A and -B systems. The crossover behavior of the uniform susceptibility and the specific heat is somewhat more complicated and is also discussed. Explicit computation of the crossover functions is carried out in a large N expansion on a mean-field model. Some new results for the critical properties on the ordered side of the transition are also obtained in a spin-density-wave formalism. The possible relevance of our results to the doped cuprate compounds is briefly discussed.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.