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.
The scaling function for the spin correlation length \ensuremath{\xi} of the two-dimensional (2D) quantum nonlinear \ensuremath{\sigma} model is obtained by Monte Carlo simulations. This is used to extract the temperature dependence of the spin correlation length, which sheds light on the nature of the ground state of low-dimensional quantum antiferromagnets. A crossover from quantum critical to renormalized classical behavior of the correlation length is found. Comparison is made with numerical results on the 2D square lattice Heisenberg model, as well as with neutron-scattering and nuclear magnetic relaxation experiments.
We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (
1002\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point
gc that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing
aH and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and
aH, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below
gc. We obtain a reasonable fit to the neutron scattering data of the insulator
La2CuO4 by taking J=1270 K, a value close to that reported by Raman scattering experiments.
We discuss the renormalization of the quantum antiferromagnet in two dimensions. With an analysis of the spin-wave Hamiltonian of the quantum Heisenberg model or the quantum nonlinear \ensuremath{\sigma} model (NL\ensuremath{\sigma}M), we consider O(n) invariant observable 〈
Sl\ensuremath{\cdot}
Sj〉 and evaluate \ensuremath{\beta} functions in the dimensional regularization method up to two-loop order. We show that quantum and classical NL \ensuremath{\sigma} M are connected by a simple crossover function. It follows from this crossover phenomenon that the critical temperature
Tc for d>2 is reduced due to the quantum effects and that the thermal fluctuations reduce the critical coupling constant
gc. The disappearance of
gc indicates thermal restoration of the asymptotic freedom. Finally we present an exact expression of the correlation length.
The ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model with spatially anisotropic couplings on a square lattice are investigated by a spin-rotation-invariant Green's-function approach and by Lanczos diagonalizations on lattices up to 36 sites supplemented by finite-size scaling. We focus on the anisotropy-driven transition from the N\'eel state to a paramagnetic state with antiferromagnetic short-range order and on the spatial dependence of spin correlation functions. Our principal result is that a rather sharp crossover in the magnetic behavior occurs at the coupling ratio ${R}_{0}\ensuremath{\simeq}0.2 {(R=J}_{y}{/J}_{x}).$
The
S=1/2 and
S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for
S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.