The i–δ relations and the criticality condition, which are used in recent theoretical calculations of the Ising-model transition temperature and critical exponent η, are tested against exact calculations for the two-dimensional lattices: square, plane triangular, and honeycomb. They are found to be valid to within 2%. They are also tested against series results for the three-dimensional hydrogen peroxide lattice, with which they are shown to be consistent to within 3.3%. A new derivation is given of the i–δ relations.
In their Comment (see cond-mat/0301237), Sandvik, Sengupta, and Campbell present some numerical evidences to support the existence of an extended bond-order-wave (BOW) phase at couplings (U,V) weaker than a tricritical point (U_t,V_t) in the ground state phase diagram of the one-dimensional half-filled U-V Hubbard model. They claim that their results do not agree with the phase diagram proposed in my Letter (cond-mat/0204244), which shows a BOW phase for couplings stronger than the critical point only. However, I argue here that their results are not conclusive and do not refute the phase diagram described in the Letter.
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.