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Physical review. B, Condensed matter

Equivalence between the nonlinearσmodel and the spin-(1/2 antiferromagnetic Heisenberg model: Spin correlations in La2 CuO4

Efstratios Manousakis, Román Salvador

Published Aug 1, 1989• 31 citations
Publisher

Abstract

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 (${100}^{2}$\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point ${g}_{c}$ 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 ${a}_{H}$ 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 ${a}_{H}$, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below ${g}_{c}$. We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.

DOI
10.1103/physrevb.40.2205

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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{100}^{2}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{g}_{c}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\ensuremathσ{a}_{\ensuremath{\sigma}}a\ensuremathσ​=f(T/T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​). Even though the lattice spacing vanishes, a finite unit of length a\ensuremathσ{a}_{\ensuremath{\sigma}}a\ensuremathσ​ and a temperature scale T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​ 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\ensuremathσ{a}_{\ensuremath{\sigma}}a\ensuremathσ​ and T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​ to the lattice spacing aH{a}_{H}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\ensuremathσ{a}_{\ensuremath{\sigma}}a\ensuremathσ​,T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​ and aH{a}_{H}aH​, J is obtained, and f(T/T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​) can be accurately approximated by an exponential of T\ensuremathσ{T}_{\ensuremath{\sigma}}T\ensuremathσ​/T below gc{g}_{c}gc​. We obtain a reasonable fit to the neutron scattering data of the insulator {\mathrm{La}}_{2}$${\mathrm{CuO}}_{4} by taking J=1270 K, a value close to that reported by Raman scattering experiments.
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.
The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.
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