Numerical methods

2.3.1 Monte Carlo simulations for spin systems

The Monte Carlo method is a powerful technique to calculate the thermodynamic properties of a magnetic system. For a canonical ensemble defined with the partition function:

Z=

whereH(φ) is the energy of the configurationφ, the thermal average of an observableOcan be calculated as:

2.3. Numerical methods

Integration in Eq. (2.9) is usually difficult to implement analytically and the Monte Carlo method, which calculates 〈O〉using a large number of samplings, becomes a preferable approximation.

Normally, samples in Monte Carlo simulations are not generated randomly. Instead, only an initial configuration is needed and all the other configurations are generated from it. Different methods exist for this generation. In this thesis, we use the Metropolis method, where the transition probability fromφtoφ⁰can be expressed as:

W(φ7−→φ⁰)=C×







exp[−δH/k_BT] ifδH>0,

1 otherwise.

In this expression,δHis the energy difference caused by an update of the configuration, which can be either a flipping of an Ising spin or a small rotation of a Heisenberg spin. Repeating this update attempt for each of the lattice sites makes up one sweep and gives rise to a new configuration. In this way, a chain of configurations that obeys the Boltzmann distribution is generated. As long as the ergodicity is satisfied, the thermal average of the observableOis then a simple average over all the configurations.

The auto-correlation time is an important quantity in Monte Carlo simulations. It is the number of sweeps required to obtain a configuration that is statistically independent from the initial one. Since the input configuration is usually random and far from being thermal-ized, starting sweeps on the order of the auto-correlation time should be excluded from the measurements. Also, the auto-correlation time is related to the correlation length. As the system approaches a long-range-order transition, more and more sweeps are necessary for both thermalization and measurement. Specifically, for a frustrated system, its configuration might be trapped at a local minimum that is not the global ground state. In this case, special Monte Carlo schemes like the parallel tempering are required to study the ordering behavior.

In this thesis, we used the Monte Carlo method to simulate the quasi-static spin-spin correla-tions〈SiSj〉in MnSc2S4and CdEr2X4(X = Se, S). Our simulations were implemented with the ALPS package [70]. Convergence for the simulations was checked with the built-in binning analysis. To compare with our diffuse neutron scattering results, Eq. (2.3) was implemented for the measurements, which basically calculates the Fourier transform of the real-space spin-spin correlations and then projects out the component that is perpendicular to the scattering vectork.

2.3.2 Linear spin wave theory

Spin excitations in classical ordered magnetic systems, due to the couplings between the spins, are collective modes of spin waves (also called magnons as an analogy to the phonons).

As is discussed in Chapter 1, spin wave excitations renormalize the ground state energy and might completely remove the long-range magnetic order if their zero-point fluctuations are

too strong. For classical systems, quantum fluctuations are relatively weak. Thus the spin wave theory, especially the linear spin wave theory that assumes zero coupling between the magnons, becomes a convenient technique to extract the coupling strengths from the inelastic neutron scattering data.

Technically, the core of the spin wave theory is the Holstein-Primakoff transformation, which effectively treats the∆S=1 excitation as the appearance of a magnon boson:

S⁺_i =p 2S

Ã

1−a_i^†ai

2S

!1/2

a_i≈p 2Sa_i,

S⁻_i =p 2Sa_i^†

Ã

1−a^†_ia_i 2S

!1/2

≈p 2Sa^†_i,

S^z_i =S−a^†_ia_i, (2.10)

wherea^†_i(a_i) is the creation (annihilation) operator and the approximations forS⁺_i andS⁻_i are valid when∆S¿S, which is just the linear spin wave approximation. Using this transfor-mation, the spin Hamiltonian can be expressed by thea^†_i andai operators. Bilinear terms in the Hamiltonian can be diagonalized in reciprocal space, giving rise to the free magnon dispersion relations. The higher-order terms describe the interactions of the magnons and can be treated with the mean-field approach similar to that used in Eq. (1.12), which usually leads to further renormalization of the ground state energy and the dispersion relations. It needs to be mentioned that the boson character of the magnons constrains the diagonalization process and special treatments like the Bogolubov transformation are required [71].

In this thesis, we used linear spin wave theory to analyze the magnon dispersions in MnSc₂S₄ and MgCr2O4. Our calculations were performed with the SpinW package [72].

3 Spin ice state in CdEr ₂ Se ₄ and CdEr ₂ S ₄

Spins on the pyrochlore lattice might realize the spin ice state, where the four spins on each tetrahedra are constrained locally into a 2-in-2-out manifold, similar to the proton positions in water ice. Previous studies on spin ice concentrate on the pyrochlore compounds ofA₂B₂X₇, where both theAandBsublattices are pyrochlore lattices. In this chapter, we will present our study of the spin ice state realized in the spinels of CdEr2Se4and CdEr2S4, where the Cd and Er sublattices constitute the diamond and pyrochlore lattice, respectively.

In the first section, we introduce the spin ice state and its monopole excitations. In the second section, we present our inelastic neutron scattering study of the crystal field transitions, which provides microscopic evidence for the Ising character of the Er³⁺ spins. The third section discusses the quasi-elastic spin-spin correlations in CdEr₂Se₄ and CdEr₂S₄ using neutron diffuse scattering, which reveals the dominance of the dipolar interactions. In the final section, we investigate the monopole dynamics using ac-susceptibility and neutron spin-echo spectroscopy. The very fast low-temperature dynamics in CdEr₂Se₄and CdEr₂S₄ can be explained with the increase of the monopole mobility, and their high-temperature dynamics are found to follow the Orbach mechanism.