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arXiv:cond-mat/9409037 · Physical review. B, Condensed matter

Scaling regimes, crossovers, and lattice corrections in two-dimensional Heisenberg antiferromagnets

N. Elstner, Rodney L. Glenister, Rajiv R. P. Singh, Alexander Sokol

Published Apr 1, 1995• 20 citations
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Abstract

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, ${\mathit{c}}_{0}$. 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 ${\mathit{T}}_{0}$\ensuremath{\sim}0.6${\mathit{c}}_{0}$/a, and their further analysis leads us to the inference that below ${\mathit{T}}_{0}$ 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${\mathit{c}}_{0}$/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.

DOI
10.1103/physrevb.51.8984

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The large-NNN expansion of the quasi-two-dimensional quantum nonlinear $\ensuremath{\sigma}$ model is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at $N=\ensuremath{\infty}$, the renormalized coordination number introduced by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)] is a universal number in the limit of ${J}^{\ensuremath{'}}∕J\ensuremath{\rightarrow}0$. Moreover, similar scaling relations proposed by Hastings and Mudry [Phys. Rev. Lett. 96, 027215 (2006)] are derived at $N=\ensuremath{\infty}$ for the three-dimensional static spin susceptibility at the wave vector $(\ensuremath{\pi},\ensuremath{\pi},0)$, as well as for the instantaneous structure factor at the same wave vector. We then use 1∕N1∕N1∕N corrections to study the relation between interplane coupling, correlation length, and critical temperature, and show that the universal scaling relations lead to logarithmic corrections to previous mean-field results.
Quantum critical points exist at zero temperature, yet, experimentally their influence seems to extend over a large part of the phase diagram of systems such as heavy-fermion compounds and high-temperature superconductors. Theoretically, however, it is generally not known over what range of parameters the physics is governed by the quantum critical point. We answer this question for the spin-density wave to Fermi-liquid quantum critical point in the two-dimensional Hubbard model. This problem is in the d =2, z =2 universality class. We use the two-particle self-consistent approach, which is accurate from weak to intermediate coupling, and whose critical behavior is the same as for the self-consistent-renormalized approach of Moriya. Despite the presence of logarithmic corrections, numerical results demonstrate that quantum critical scaling for the static magnetic susceptibility can extend up to very high temperatures but that the commensurate to incommensurate crossover leads to deviations to scaling.
We present thermodynamic measurements of various physical observables of the two-dimensional S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 isotropic quantum Heisenberg antiferromagnet on a square lattice, obtained by quantum Monte Carlo methods. The results are in excellent agreement with field-theoretical predictions. The issue of the existence of a crossover from quantum critical to renormalized classical regime is clarified.
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.