Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State

TL;DR

The article develops a universal, three-parameter description of two-dimensional quantum Heisenberg antiferromagnets near a zero-temperature quantum critical point between Néel order and quantum disorder. Using the O(N) non-linear sigma model and a 1/N expansion complemented by Monte Carlo simulations, it derives scaling functions for the staggered and uniform spin susceptibilities and the specific heat that apply across Néel-ordered, quantum-critical, renormalized-classical, and quantum-disordered regimes. The theory yields explicit predictions for experimental probes such as neutron scattering and NMR, and it coherently explains observations in undoped and lightly doped La_{2−δ}Sr_δCuO_4, including ω/T scaling and finite-temperature crossovers. The work highlights universality in quantum critical spin dynamics and furnishes a quantitatively accurate framework for interpreting experiments in cuprates and related 2D antiferromagnets.

Abstract

We present the general theory of clean, two-dimensional, quantum Heisenberg antiferromagnets which are close to the zero-temperature quantum transition between ground states with and without long-range Néel order. For Néel-ordered states, `nearly-critical' means that the ground state spin-stiffness, $ρ_s$, satisfies $ρ_s \ll J$, where $J$ is the nearest-neighbor exchange constant, while `nearly-critical' quantum-disordered ground states have a energy-gap, $Δ$, towards excitations with spin-1, which satisfies $Δ\ll J$. Under these circumstances, we show that the wavevector/frequency-dependent uniform and staggered spin susceptibilities, and the specific heat, are completely universal functions of just three thermodynamic parameters. Explicit results for the universal scaling functions are obtained by a $1/N$ expansion on the $O(N)$ quantum non-linear sigma model, and by Monte Carlo simulations. These calculations lead to a variety of testable predictions for neutron scattering, NMR, and magnetization measurements. Our results are in good agreement with a number of numerical simulations and experiments on undoped and lightly-doped $La_{2-δ} Sr_δCu O_4$.

Figures (9)

• Figure 1: Phase diagram of ${\cal H}$ (Eqn. (\ref{['calH']})) as a function of $g$ and temperature $T$ (after Ref. CHN). The coupling $g$ measures the strength of the quantum fluctuations. It is inversely proportional to $S$ for large spin and its value also depends on the ratios of the $J_{ij}$. The parameters $x_1 = N k_B T / (2 \pi \rho_s)$, and $x_2 = k_B T / \Delta$ control the scaling properties of the antiferromagnet (here $\rho_s$ is the spin-stiffness of the Néel-ordered ground state, and $\Delta$ is the spin 1 gap in the quantum-disordered ground state).
• Figure 2: Properties of the nearly-critical antiferromagnet as a function of the observation wavevector $k$, or frequency $\omega$ in the three different regions of Fig. \ref{['phasediag']}. The appropriate regime is determined by the larger of $\hbar c k / (k_B T)$ or $\hbar \omega / k_B T$. In the renormalized-classical regime, $\xi$ is the actual correlation length, while $\xi_{J}$ is a Josephson correlation length related to the spin-stiffness by $\hbar c/ \xi_{J} = \rho_s /\Upsilon$, with $\Upsilon$ a universal number. In the quantum disordered region, $\Delta$ is the gap for $S=1$ excitations at $T=0$. The thermodynamic behavior in the various regions is discussed in the text.
• Figure 3: The scaling function $\hbox{Im} \Phi_{1s} ( \overline{k}, \overline{\omega}, \infty )$ for the staggered susceptibility in the quantum-critical region. The results have been computed in a $1/N$ expansion to order $1/N$ and evaluated for $N=3$. The shoulder on the peaks is due to a threshold to three spin-wave decay.
• Figure 4: Scaling function $\Xi_1 ( \overline{k} , \infty )$ for the structure factor in the quantum-critical region.
• Figure 5: Scaling function $F_1 ( \overline{\omega} , \infty)$ for the imaginary part of the local susceptibility in the quantum-critical region. The oscillations at large $\overline{\omega}$ are due to a finite step-size in the momentum integration.