Ab initio Quantum Monte Carlo Calculations of Spin Superexchange in Cuprates: The Benchmarking Case of ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$

In view of the continuous theoretical efforts aimed at an accurate microscopic description of the strongly correlated transition metal oxides and related materials, we show that with continuum quantum Monte Carlo (QMC) calculations it is possible to obtain the value of the spin superexchange coupling constant of a copper oxide in a quantitatively excellent agreement with experiment. The variational nature of the QMC total energy allows us to identify the best trial wave function out of the available pool of wave functions, which makes the approach essentially free from adjustable parameters and thus truly ab initio. The present results on magnetic interactions suggest that QMC is capable of accurately describing ground-state properties of strongly correlated materials.

Popular Summary

Modern advances in computer technologies favor the development of statistical methods in theoretical physics based on random numbers. These methods can now be pushed to more complex physical systems and finer energy resolutions. In condensed matter physics and quantum chemistry, quantum Monte Carlo methods are known to provide an exceptionally accurate description of an interacting many-body system. However, until recently, quantum Monte Carlo methods have typically been applied to systems composed of light elements because of computational demands. Now, quantum Monte Carlo methods can be applied to problems that have remained unsolved for decades, such as cuprates. These oxide materials exhibit remarkable phenomena, including high-temperature superconductivity, which may originate from magnetic spin excitations. We use quantum Monte Carlo methods to study the magnetic properties of the Mott insulator ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$, an effectively one-dimensional counterpart of the famous high-temperature superconducting cuprates.

We focus on ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$ because of its relatively light atoms, which minimize relativistic effects. We evaluate the strength of the superexchange magnetic interaction between the localized Cu ion spins. We find that the value of the spin superexchange interaction constant obtained by quantum Monte Carlo methods is in very good quantitative agreement with experimental results. Contrary to some alternative theoretical electronic structure methods, such as density-functional theory, our approach can greatly reduce the dependence of the final result on the starting approximation, as it is largely free of adjustable parameters.

Our successful application of quantum Monte Carlo methods to a cuprate material reveals the power of this method at efficiently handling the strong electronic correlations of transition metal oxides.

Figure 1

(a) The conventional “$1\times 1\times 1$” unit cell of ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$ and ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$. For calculations, the unit cell crystallographic parameters reported in Ref. [11] are used. (b) The total energies of the antiferromagnetic and ferromagnetic states of ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$ calculated with FP DMC as a function of the trial wave function, characterized by the $\mathrm{LDA}+U$ parameter $U$, used here as a variational parameter. For convenience of presentation, the FP-DMC energies are shifted by $E_0=34705\mathrm{eV}$. (c) The nearest-neighbor Cu spin superexchange coupling constant $J$ of ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$ calculated with WIEN2k (DFT) and FP DMC as a function of $U$. For comparison, the ranges of experimentally determined $J$ values of ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$ are shown in light gold and dark gold bands. The light gold band represents all reported experimental estimates [6], while the dark gold band represents the magnetic susceptibility measurements [7, 8].

Figure 2

The comparison of the ${\mathrm{Ca}}_2{\mathrm{CuO}}_3$ densities of states (DOS) calculated using PP and AE codes: (a) antiferromagnetic and (b) ferromagnetic states. Energies are measured relative to the Fermi level $E_F$

Figure 3

(a) Hard and (b) soft Ne-core Cu pseudopotentials (PPs). $V_{\mathrm{loc}}$, $V_s$, $V_p$, and $V_d$ represent the local, $s$, $p$, and $d$ channels, respectively. $2Z_{\text{eff}}/r$ represents the Coulomb potential due to the effective charge $Z_{\mathrm{eff}}$. The arrowed lines of respective colors indicate cutoff radii for the three pseudopotential channels. (c),(d) Cu atom eigenfunctions obtained in LDA with the hard (c) and soft (d) Ne-core Cu PPs. The solid (dashed) lines represent wave functions obtained from all-electron (PP) calculations.

Figure 4

Binding energy as a function of the interatomic separation $d$ in a CuO dimer molecule. The vertical dashed line indicates the experimental equilibrium separation. The solid lines are a quadratic fit to the data.