Magnetic Relaxation in Spin Ice Compounds : Spin Flip Dynamic Driven by a Thermal Bath

Transcription

1 Master Science de la matière Stage École Normale Supérieure de Lyon Jouffrey Victor Université Claude Bernard Lyon I M2 Physique Magnetic Relaxation in Spin Ice Compounds : Spin Flip Dynamic Driven by a Thermal Bath Résumé This report investigates the coupling existing between magnetic and vibrational modes in Spin Ice compounds. The purpose is to understand the temperature independent plateau displayed by the relaxation time of the spins at temperature close to the transition towards the classical spin liquid phase. The exotic physics arising in Spin Ice compounds is introduced before highlighting the unusual behaviour of the spin relaxation time in the temperature region of the phase transition between the classical spin liquid and the thermal paramagnet. In order to shed light on this issue, we investigate how the magnetic excitations in Spin ice materials are connected to the phonon bath. The point charge model is introduced to compute an explicit form of the spin-phonon and give a physical picture of how magneto-elastic coupling may arise in these materials. To make further progress on the connection between spin-phonon coupling and its influence on spin flip dynamics, we reduce the problem to a single spin coupled to a bath. Although no connection to the physics of spin liquid is found to explain the spin relaxation time plateau, we show that the simplest model for single spin flip dynamics in contact with a thermal bath cannot account for the experimental data. Assuming coupling to an ohmic bath, we point out a discrepancy between the transition rate found to model appropriately single spin flip dynamic as a Markov process and the use of a temperature independent transition rate in Monte-Carlo simulation to explore different spin configurations. Mots clefs : Spin Ice, Magnetic Relaxation, Single Spin Flip Dynamic, Spin-Phonon Coupling Stage encadré par : Claudio Castelnovo Cavendish Laboratory 19 J J Thomson Avenue, Cambridge, CB3 0HE, UK http ://

2 Table des matières 1 Introduction 2 2 Spin Ice in a nutshell Single-ion Physics Crystal Electric Field analysis Emergence of the Ice rule and the Classical Spin Liquid Phase Spin-Phonon coupling analysis in Spin Ice compounds Introduction to AC Magnetometry Spin relaxation time in Spin Ice compounds Magneto-elastic coupling in Spin Ice compounds Single Spin Flip Dynamic in a Thermal Bath Linblad Equation for a Two-level System Coupled to a Bath Linblad Master Equation for Purely Dephasing Coupling Linblad Master Equation for Dissipative Coupling Classical Markov Process for Spin Flip Dynamic Comparison between Quantum Linblad Dissipative and Classical Markov Possible extensions of the calculation Conclusion 19 A Magnetic monopole excitations in Spin Ice compounds 21 B Absorbing Transverse Field coupling using a Canonical Transformation 22 1 Introduction Academic and industrial laboratories have been synthesising and analysing new materials for decades. Their diverse properties have led to a broad range of technological applications. However, there is one property that all these materials seem to have in common : When their temperature is lowered below a certain point, order develops out of the high-temperature disordered phase, in a process that involves the (partial or complete) breaking of the original symmetries of the system. In 1973, P.W. Anderson proposed an ansatz, the Resonating Valence Bond state, in an attempt to describe the groundstate of the triangular aniferromagnet [1]. This state had the peculiarity to display no long range order of the spin contrary to the commonly used Néel state which describe to a good approximation the groundstate of numerous unfrustrated antiferromagnets. The ability of the system to remain disordered even at zero temperature is due to strong quantum fluctuations which allow the system to explore a manifold formed by a macroscopic number of degenerate spin configurations related by local operations. This state and other theoretical proposal of groundstates displaying this disordered behaviour down to zero temperature were dubbed "quantum spin liquid". The quest to find experimental realisation of such a spin liquid state has been going on for over 40 years and is still the subject of intense research. In 2001, Bramwell et al. published a paper describing a classical realisation of such a spin liquid state in Ho 2 Ti 2 O 7 [2]. This new class of materials were dubbed Spin Ice due to their residual entropy similar to the one described by Pauli in water ice. A surge of experimental and theoretical investigations of their properties lead to tremendous progress in our understanding of this strongly correlated system. A major step happened in 2008 when Castelnovo et al. [3] managed to identify magnetic monopoles as the effective degrees of freedom of the system within the classical spin liquid phase. This theoretical modelling was very successful in reproducing key feature of the experimental signature in the classical spin liquid phase. The monopole picture allowed subsequent work using Monte-Carlo simulation 2

3 to draw connection between the dynamic within the classical spin liquid and the physics of weak electrolyte [4]. These remarkable achievements in the understanding of Spin Ice materials still leave a few questions unanswered. In particular, these materials offer a unique playground to further our understanding of the transition in and out of a classical spin liquid phase. The work in this report was originally motivated by the unusual behaviour of the spin relaxation time in the temperature region where the Spin Ice physics sets in [5]. A good agreement with the experimental data was obtained using Monte- Carlo simulation within the spin liquid phase[6] but gives no indication as to why the magnetic degrees of freedom seemingly decouple from the temperature bath in the transition region. The following report describes ongoing work to determine whether this unusual behaviour may be due to exotic physics happening in the transition in or out of the classical spin liquid phase. We will show how this question relates to more general consideration regarding how magnetic relaxation can be driven by the coupling to a phonon bath. The report starts by a brief introduction to the physics of Spin Ice compounds. The aim is to discuss the effect of the crystal electric field on the free lanthanide ion groundstate in order to provide a physical picture for the emergence of Ising spins in these compounds. A brief discussion of the Spin Ice groundstate is included as well as a sketch of the mechanism underlying the classical spin liquid behaviour. Although not directly relevant, an annexe presenting the exotic excitation supported by the spin liquid phase, the so-called magnetic monopole, has been included. The second part investigates temperature dependence of the spin relaxation time and coupling mechanism between magnetic and vibrational degrees of freedom. In particular, we explicitly show the existence of a dissipative coupling between the magnetic system and the phonon bath in the non-kramer Spin Ice compound Ho 2 Ti 2 O 7 and compute its explicit form. In order to focus on the effect of this coupling, we turned to model where the dynamic of the magnetic system, a single spin, is entirely driven by its coupling to the bath. In the final part of the report, we study the physics of a single spin in contact with a thermal bath, either purely dephasing or dissipative. We show how the quantum mechanical treatment of the spin flip dynamics using the framework set by Linblad is compatible with a classical Markovian description provided the transition rate is appropriately chosen. We point out that for an ohmic bath, the Markov transition rate do not display any temperature independent asymptotic behaviour. This conclusion is in direct contradiction to the behaviour observed in Spin Ice compounds. It also suggests that the temperature independent transition rate used in Monte-Carlo simulation may not be a valid approximation when the system is coupled to an ohmic thermal bath. The final paragraph discuss possible extension of the calculation based on ongoing works. 2 Spin Ice in a nutshell Introduction to Spin Ice compounds, Ho 2 Ti 2 O 7 (HTO) and Dy 2 Ti 2 O 7 (DTO), are usually focused on the classical spin liquid behaviour which emerges at low temperature. Although a brief explanation of how the Spin Ice physics emerges from magnetic doublet coupled by long range dipolar interactions will be sketched, the emphasis will rather be on the single ion physics and crystal symmetries which gives rise to the Ising spins in the first place. 2.1 Single-ion Physics The magnetic ions in the Spin Ice compounds are Holmium (Ho 3+ ) and Dysprosium (Dy 3+ ). Their electronic configurations are respectively 4f 10 and 4f 9. The emergence of a strongly anisotropic lowenergy magnetic doublet in both compounds result in the subtle interplay between three different mechanisms : Coulomb interaction, Spin-Orbit Coupling (SOC) and Crystal Electric Field (CEF). We will first describe the free ground state of Lanthanide ions before adding the material specific CEF. The largest energy scale is set by the Coulomb repulsion ( ev) between the f electrons. This is due to f-orbitals being notoriously narrow and known to lead to large intra and inter-orbital interactions. At this stage, the electrons wavefunction can be written as a tensor product of the spin and spatial 3

4 part. To minimize the the energy cost associated with the Coulomb repulsion, the system prefers to anti-symmetrize the spatial part of the wavefunction, minimizing the overlap and the cost of Coulomb repulsion, rather than the spin part. Thus, the electrons will have different orbital momenta whenever possible and symmetrize their spin part when the orbital is singly occupied. A schematic orbital occupancy for the magnetic ions can be drawn : Dy Ho ˆL = 3 ˆL = 2 ˆL = 1 ˆL = 0 ˆL = 1 ˆL = 2 ˆL = 3 We therefore have ˆL tot = 5 and Ŝtot = 5/2 for Dy 3+ and ˆL tot = 6 and Ŝtot = 2 for Ho 3+. Spin-orbit coupling is a relativistic correction to the energy of the electrons whose strength scale as the fourth power of the atomic number. It is therefore large in rare-earth ions ( 100 mev) and plays a key role to give an Ising flavour to the groundstate doublet. It mixes the orbital momentum and spin number of electrons effectively leaving only one good quantum number, the total angular momentum Ĵ. When the filling of the f-orbitals is over half, the lowest level is given by Ĵ = ˆL + Ŝ. For Dy3+ and Ho 3+, the total angular momentum is respectively Ĵ = 15/2 which is 16 times degenerate and Ĵ = 8 which is 17 times degenerate. These degenerate states form the groundstate of the free Lanthanide ions [7]. 2.2 Crystal Electric Field analysis Further progress in determining the groundstate of the magnetic ions in Spin ice compounds relies on a perturbative treatment of the electric field created by the surrounding ions in those materials. Rather than carrying out a full perturbative treatment of the crystal electric field, we will use group theory and pictures to understand what happens in this last step. At this point, it is worth highlighting a major difference between the two ions. Dysprosium is a Kramer system (odd number of electrons) and is therefore expected to display a magnetic groundstate protected by time-reversal symmetry. Holmium is a non-kramer system, therefore time-reversal symmetry do not impose a degenerate groundstate and the emergence of magnetic groundstate doublet is only due to the symmetry of the crystal point group. In Spin Ice compounds, the magnetic ions sit at the vertex of a pyrochlore lattice constituted of corner-sharing tetrahedra. This lattice is well-known to give rise to geometrical frustration effects in magnetic compounds leading to new interesting phenomena. This geometrical frustration is key in the emergence of the Spin Ice physics and will be discussed in the next section. We focus here on the symmetry at the position of the magnetic ions. Symmetry analysis of the crystal environment shows that the wyckoff position of the magnetic atom has a D 3d point group. Figure 1 shows the crystal environment for Holmium in the Spin Ice compound HTO. A simple way to visualize the symmetry operations associated with this point group is to picture two equilateral triangles shifted by a π/3 rotation (see Figure 1 ). The first obvious symmetry is the inversion around the position of the magnetic ion. Then one may see that 2π/3 and 4π/3 rotations along the (111) crystal axis also leave the crystal invariant. Third comes improper rotations, composed of a rotation by π/3, π or 5π/3 immediately followed by a reflection about the plan parallel to both triangles. The symmetry operations involving a π rotation is actually equivalent to the inversion symmetry around the magnetic ion position on the (111) axis which is equidistant from both triangles. The fourth generating symmetries are reflections along planes perpendicular to the triangles and intersecting one of their corner. The last type of operations are simply π/2 rotations around three axis 4

5 Figure 1 Schematic view of the crystal field environment in Spin Ice compounds. Figure reproduced from [8] parallel to the triangles intersecting one another in I with a 2π/3 angle. The easy axis of the magnetic ion is oriented in the (111) direction, along the so-called principal axis of the point group. Although the crystal electric field created by neighbouring Ti 4+ and Ho 3+ or Dy 3+ ions is non-zero, it is essentially the CEF originating from the oxygen cage around the magnetic ions that will split the degeneracy of the free magnetic ion groundstate. There are two inequivalent positions for the oxygen atoms in SI compounds. The first one corresponds to oxygen ions (O 2 ) sitting at the center of the tetrahedra. There are closer to the magnetic ions fixing the principal axis of the point group and the Ising nature of the emergent magnetic doublet. Thus, the spin will either point toward the center of the tetrahedron or outward. The special properties of these four directions (111),(-1-11),(-11-1) and (1-1-1) for a given tetrahedron form another key ingredient in the emergence of Spin Ice physics. The second type of oygen ion forms the corner of the two triangles which were previously drawn to help thinking about the D 3d point group. The full force of Group theory can now be applied to find the maximum splitting of the free ion groundstate for both Dysprosium and Holmium compounds. The degenerate groundstate of Dysprosium splits into 8 doublets protected by time-reversal symmetry. It is therefore expected that the groundstate doublet is magnetic. The splitting of Holmium leads to both singlets and doublets. We emphasize here that the doublet for the Holmium compound are protected by the point group symmetry and not by time-reversal symmetry. This point will have major consequences when computing possible coupling to the phonons for both compounds. In particular, we expect Holmium to be much more sensitive to phonon coupling as they may break the symmetry which protects the groundstate magnetic doublet. Unfortunately, the energy of each modes depend on the microscopic details of the system. Therefore, finding the groundstate and the energy gap to higher excited states requires a more involve calculations. Relying on both microscopic calculations and experimental investigations [9, 10], we draw three important conclusions. First, the groundstate of both Holmium and Dysprosium in Spin Ice compounds are magnetic doublets displaying a strong easy-axis (Ising). Second, the first excited states are very well separated from the groundstate doublet ( > 100K). At temperature small compare to this gap, we expect the magnetic properties of Spin Ice compounds to be well described by interacting Ising spins. At last, the magnetic doublet is essentially composed ( 95%) of the highest angular momentum states (±15/2 for Dy 3+ and ±8 for Ho 3+ ) leading to two different informations to model Spin Ice compounds. Large spins essentially behave like classical objects, therefore quantum fluctuations are expected to be small and can be neglected in first approximation. Although it means that a quantum spin liquid phase is unlikely to arise in this system, it also means that classical Monte-Carlo simulation should capture the essential physics of these materials. The last bit of information that we need to understand Spin Ice also relies on the large magnetic moment. Exchange interactions are especially weak in f-orbital system, therefore, given the magnitude of the spin, we expect dipolar interactions to play a significant role in the spin-spin interaction. To obtain a simplify model of the interaction between the groundstate magnetic doublet of the rareearth atoms, we will partially forget about their origin and treat them as regular classical pseudo- 5

6 Figure 2 The wavefunction of the magnetic doublet is decomposed on the basis of total angular momentum. One may see that the main contribution stems from the highest angular momentum. Figure reproduced from [8] Figure 3 The figure on the left is a neutron spectra recorded in HTO samples. One may clearly distinguish the pinch points characteristic of the Spin Ice phase [11]. On the right, the the figure present the spectra of the magnetic ion after taking into account the CEF perturbation. Figure reproduced from [8] spin 1/2. In the case of Dysprosium, the appropriate pseudospin operators can easily be constructed from the total angular momentum operators (S z Jˆz, S ± Jˆ± ). In the case of Holmium, we cannot completely forget about the non-kramer nature of the doublet. In particular, the usual spin transformation under time-reversal S S valid for the Kramer doublet of Dysprosium does not hold. Construction of the pseudospin operators lead to (S z Jˆz, S x,y Jˆz Jˆx,y ). The z-component is dipolar, while the transverse components are quadrupolar and even under time-reversal. It will have important consequences when we investigate the coupling to the phonon. 2.3 Emergence of the Ice rule and the Classical Spin Liquid Phase In the previous section, we mentioned several key ingredients which appropriately combined give rise to Spin Ice behaviour. Let us summarize these key points : Pyrochlore lattice constituted of corner-sharing tetrahedra leading to geometrical frustration Emergent Ising Spins pointing along the direction linking the center of neighbouring tetrahedra Dipolar interactions between the Ising Spin (ferromagnetism) 6

7 It is far from obvious to see how the interplay between these different aspect of the Spin ice compound physics can lead to a classical spin liquid. The last part is especially troubling, as we usually associate antiferromagnetism with frustration. However, one has to remember that the local easy axis for each spin is different (toward the center of the tetrahedron). In particular, the scalar product between the directions of two nearest neighbours spin is negative : e i e j = 1/3. Although the dipolar interaction between the spin is ferromagnetic, the nearest neighbour part is effectively an antiferromagnetic coupling : Ĥ dip = Da 3 [ ei e j r i,j ij 3 3(e ] i r ij )(e j r ij ) r ij 5 S i S j (1) Ĥ dip = D n.n Ŝi z Ŝz j + long range tail (2) i,j where we introduced the nearest neighbour distance a, the distance r ij between two spins i and j, the dipolar coupling D and its effective value for nearest neighbours D n.n. Dipolar interactions are slowly decaying and it is not straightforward to understand why the long-range tail can be neglected in first approximation. A simple calculation comparing the amplitude of the dipolar interaction for nearest and next nearest neighbour leads to D n.n 0.2D n.n.n. Furthermore, the long range dipolar interaction is not the only one contributing to the nearest neighbour interaction. The exchange part of the Hamiltonian, especially the Ising interaction, cannot be entirely neglected. Based on our previous symmetry analysis, we can introduce the most general Hamiltonian allowed by the D 3d point group : Ĥ ex = i,j J zz Ŝ z i Ŝz j + J ± (Ŝ+ i Ŝ j + Ŝ i Ŝ + j ) + J ±± ( γ ij Ŝ + i Ŝ + j + γ ijŝ i Ŝ j ) + J z± (Ŝz i (ζ ij Ŝ + j + ζ ijŝ j ) ( + ζ ij Ŝ i + + ζ ijŝ i ) Ŝz j ) (3) where we introduced the set of exchange coupling {J zz, J ±, J z±, J ±± } and the sum is over nearest neighbour only. In Spin Ice compounds, the first term largely dominates over the others. This fact is a direct consequence of the large angular momentum carried by the magnetic ions. Indeed, the three other terms can be seen as quantum fluctuations around the easy-axis limit. Although these three terms can be neglected when modelling Spin Ice compounds, they are expected to drive the system in a quantum spin liquid phase in recently synthesized materials [12, 13]. It is worth mentioning that, due to the non-kramer nature of the magnetic doublet in the Holmium compound, the last term coupling the easy axis to the quantum fluctuations of the neighbouring spin is odd under time-reversal symmetry and therefore vanishes. The nearest neighbour Ising interaction is antiferromagnetic in the local axis basis, reducing the effective antiferromagnetic coupling due to the dipolar interaction when recast in the crystal basis. In first approximation, we will forget about the long range tail and the quantum fluctuations to consider a minimal model of Ising spins sitting on the pyrochlore lattice coupled to their nearest neighbours via an antiferromagnetic coupling. Although a vast literature discuss the validity of this approximation in Spin Ice compounds, we will adopt a pragmatic approach and show that this model captures the essential physics which gives rise to the classical spin liquid behaviour. Ĥ min = J eff Ŝi z Ŝz j (4) Remembering that the spins sit on the pyrochlore lattice, one can rewrite the Hamiltonian as a sum over the tetrahedra leading to a much clearer understanding of the Hamiltonian groundstate. i,j 7

8 Ĥ min = J eff 2 (Ŝz 1, + Ŝz 2, + Ŝz 3, + Ŝz 4, ) 2 = J eff 2 ˆQ 2 (5) where we dropped an unimportant constant and introduce the tetrahedra charge operators ˆQ. One can immediately see that if the effective coupling is antiferromagnetic, a lower bound to the energy is obtained by setting the charge of each tetrahedra to zero. Setting the charge of a tetrahedra to zero is equivalent to setting two out of four spins to point inward as the two others point outward. This constrain is known as the 2-in,2-out rule or the ice rule, it gave its name to Spin Ice compounds. At the level of a single tetrahedron, there is therefore six different spin configurations satisfying the 2-in,2-out rule. The question is now whether the constrains imposed by the corner-sharing geometry of the lattice (each spin is shared by a two tetrahedra) are compatible with enforcing the 2-in, 2-out rule for each tetrahedron. It turns out that the number of configurations satisfying both the ice rule and the geometry constrains scales exponentially with the system size. In the thermodynamic limit, the groundstate of the system is infinitely degenerate. Each groundstate configuration is related to a certain number of others by local transformations. This locality is at the heart of the spin liquid behaviour. To better understand the major importance of the last statement, let us draw a brief comparison with a regular 3D Heisenberg ferromagnet. In this system, the ferromagnetic groundstate is also infinitely degenerate. However, in the thermodynamic limit, each groundstate configuration is connected to the others by a global rotation of all the spins. The impossibility for thermal or quantum fluctuations to lead to a global transformation of the system is essential to the concept of spontaneous symmetry breaking. The fact that groundsates are connected by global transformations underlies the entire classification of matter by Landau. On the other hand, when the groundstate configurations are connected by local transformation, thermal or quantum fluctuation allows for tunneling from one configuration to another and Landau theory breaks down. The nature of the fluctuation connecting the states, thermal or quantum, determines whether the phase is called a classical or a quantum spin liquid. Although there is a large number of different theories to model spin liquid physics, the essential physics relies on the local connection between different groundstate configurations allowing for thermal or quantum fluctuation to explore the groundstate manifold. This clear theoretical distinction between spin liquid behaviour and usual spontaneous symmetry breaking in ordered magnets does not prove that spin liquid can be consider as a distinct phase of matter compare to paramagnetic phases. In the specific case of Spin Ice, this distinction can be made experimentally using neutron scattering techniques. Indeed, enforcing the ice rule for each tetrahedron leads to non-trivial correlations between spins. The most salient feature predicted and experimentally observed in neutron spectra is the existence of pinch-points (see left side graph on figure 3) which are directly linked to the ice rule. This non-trivial magnetic correlations can be used to distinguish between the high temperature trivial paramagnet and the low temperature spin liquid phase. The results presented in this part form the basis of our current understanding of SI compounds. As one may see, the dynamic of the magnetic degrees of freedom in these systems has a very rich phenomenology and is quite well understood, especially in the classical spin liquid phase. In order to study magnetic relaxation in ths system, we need to understand how the magnetic system is connected to the phonon bath in DTO and HTO. This topic will be the focus of the next part of this report. 3 Spin-Phonon coupling analysis in Spin Ice compounds In this section, we will focus on the spin relaxation in Spin Ice compounds. We first briefly introduce AC susceptibility measurements and sketch an intuitive idea of how this quantity relates to the spin relaxation time. Experimental data based on this technique and Monte-carlo simulation showing an unusual behaviour of the spin relaxation time in these compounds are then presented. The plateau in the spin relaxation time suggest an apparent decoupling between the spin and the thermal phonon bath in these compounds. In order to investigate further how the emergent Ising moment in Spin Ice 8

9 are coupled to the bath, we introduce the point charge model which is generally used to compute the crystal electric field. This model is extended to help us derive an explicit form for the magneto-elastic coupling in Spin Ice compounds. 3.1 Introduction to AC Magnetometry The spin relaxation time in a material can be measured through AC susceptibility measurement also called AC magnetometry. The idea is to measure the response of a sample to an external timedependant magnetic field. The time-dependence of the field induces a time dependence in the magnetic moment orientation and provides information about the intrinsic dynamic of the system. The purpose of this section is to give an intuitive understanding of this technique and highlight how the AC susceptibility relates to the spin relaxation time. DC magnetic measurements determine the equilibrium value of the magnetization in a sample. The sample is magnetized by a constant magnetic field and the magnetic moment of the sample is measured, producing a DC magnetization curve M(H). The equilibrium value of the magnetization depends on the orientation of the magnetic field and on its strength. In AC magnetic measurements, a small AC field is applied to the sample causing a shift in time of the magnetic moments. The time-dependent field induced by this shift can be detected using pickup coils giving information about the dynamic of the system. In general, the response of the system depends not only on the direction and amplitude of the field but also on its frequency. This frequency dependence allows one to identify the characteristic response time of the system to a magnetic perturbation and is key to extract the spin relaxation time in the sample. To illustrate how the behaviour of the sample may depend on the frequency, let us consider two extreme cases. The first case corresponds to a slowly varying magnetic field. Here, slow means that the AC field is roughly constant on time scale of the order of the characteristic magnetic relaxation time of the system. Intuitively, this means that the magnetic moments in the system follow quasi adiabatically the magnetic field. In other words, the magnetisation of the sample follows the curve M(H) measured in a DC experiment. As long as the perturbation of the AC field is small, we can apply linear response theory and obtain a mathematical expression for the time-dependant magnetisation M AC : M AC = dm dh H AC sin (ωt) = χh AC sin (ωt) (6) where ω and H AC are respectively the frequency and the magnitude of the AC field and χ = dm/dh is the susceptibility and in this case corresponds to the slope of the M(H) curve. In general, χ is a three by three tensor due to the anisotropy of the system and depends on the frequency. In AC magnetometry, the susceptibility χ is the quantity of interest and characterize the response of the system to a magnetic perturbation of a given frequency. In the second case, we are interested in the opposite limit for the AC field frequency. The time scale over which the magnetic field varies is now shorter than the characteristic magnetic relaxation time of the system. The magnetic field varies faster than it is possible for the system to adapt, faster than the dynamic processes taking place within the sample can change its magnetisation. As a result, there is a phase difference φ between the AC driving field and the magnetisation which can be measured in addition of the amplitude of the magnetic response χ. In the literature, these two quantities are often merged into a complex susceptibility : χ = χ cos(φ) χ = χ sin(φ) χ = (χ ) 2 + (χ ) 2 φ = arctan (χ /χ ) (7) In the limit of low frequency, the real component χ reduces to χ and corresponds to the slope of the M (H) curve as discussed above. At higher frequencies, adiabatic driving of the magnetic moments is no longer possible and dissipative processes occur. The imaginary part of the susceptibility χ measures the prevalence of these dissipative processes. 9

10 Figure 4 On the left, the imaginary part of the AC susceptibility is plotted as a function of frequency at low temperatures in zero applied field. The prominent single peak in the data suggests that they are well-described by a single characteristic relaxation time. Figure reproduced from [5]. On the right, relaxation time scales τ in Dy2Ti2O7 : experiment and simulation. The experimental data ( ) are from Snyder et al. [5] and the Monte-Carlo simulations were performed by Jaubert et al. [6]. The temperature scale is fixed without any free parameters. Figure reproduced from [6] The dissipative processes measured by the imaginary part of the magnetic susceptibility depends strongly on the nature of the system. In conductive samples, they are Foucault currents which are induced by the varying magnetic field according to Faraday s law of induction. Spin Ice compounds are insulators, therefore such processes are forbidden. In the paramagnetic phase of an insulating sample hosting Ising spins, these processes correspond to single spin flip events (or small domain flip as far as a domain is a well define object in a paramagnetic phase). In the Spin Ice groundstate, we have seen that the majority of the tetrahedra display a spin configuration satisfying the 2-in,2-out rule. A dissipative process correspond to the creation/annihilation of an excitation, i.e. a pair of monopole/anti-monopole in SI compounds (cf. annexe A), which is nothing but a single spin flip. Although the physics around the Spin Ice transition is still not fully understood, it seems reasonable to also attribute the dissipative processes responsible for magnetic relaxation to single spin flip events. If we identify single spin flip event with the dissipative processes corresponding to a finite value of the imaginary part of the susceptibility, we can intuitively understand how the spin relaxation time can be extracted. The imaginary part of the susceptibility characterizes the ability of a material to respond to an external magnetic perturbation. Therefore, when the frequency of a perturbation match the frequency of the dissipative process induced in the system, we expect a resonance. The dissipative process being a single spin flip, one can infer that the natural time scale of the process is set by the energy gap between the two-level. Therefore, one can extract the relaxation time of the spin at a given temperature by finding the frequency of the AC susceptibility maximum. 3.2 Spin relaxation time in Spin Ice compounds In the previous section, we have seen how to extract the spin relaxation time of AC susceptibility measurements. Using this technique, the group of Snyder et al. [5] measured spin relaxation time in Spin Ice compounds as a function of temperature. Their results display an unusual plateau just within the temperature region corresponding to the phase transition in and out of the classical spin liquid phase. This section discuss the experimental and Monte-Carlo data obtained in [6] and how these result lead us to investigate spin flip relaxation in Spin Ice compounds. The imaginary part of the AC susceptibility in DTO is displayed in figure 4. One may clearly see, for 10

11 each temperature, a broad single pic in the system response suggesting that the magnetic response is well described by a unique relaxation time. At low temperature (< 2K), the system is in the classical spin liquid phase. One may see that, as the temperature increase, there is a rapid shift of the resonance frequency to higher value (two order of magnitudes for only one Kelvin). This rapid shift of the frequency response of the system can be readily explained within the monopole picture. As explained in the annexe, monopole hopping energy cost is small compare to the temperature. The system relax to a more favourable spin configuration by hopping of monopole which are fast due to their relatively small energy cost. However, the energy cost of creating a pair of monopole/antimonopole is of the order of 2K. Therefore, the number of monopole shrinks exponentially with the temperature leading to a reduce plasticity of the system. As the number of monopole diminishes, the system progressively freezes. The last curve in figure 4 shows the behaviour of the susceptibility out of the classical spin liquid phase. Despite the temperature difference of 3K, one immediately notice that the resonance frequency is barely higher than in the classical spin liquid phase. This result indicates a qualitative change in the behaviour of the relaxation time within the temperature domain corresponding to the transition out of the classical spin liquid phase. Figure 4 is a plot of the spin relaxation time extracted from the AC measurements over a temperature domain which includes the spin liquid phase, the transition to the thermal paramagnet, and the paramagnetic phase itself. The temperature range between 2K and 5K corresponding to the transition clearly shows a plateau in the spin relaxation time. Although not completely flat, this plateau may suggest a decoupling of the spin relaxation mechanism from the temperature bath. In a insulator, the thermal energy is stored in the lattice vibrations, the so-called phonons. Thermally activated phonons (often referred to as the phonon bath) generally interact with the magnetic degrees of freedom of the system through a magneto-elastic coupling. In the rest of this report, we will discuss several model in order to investigate further how the magneto-elastic coupling may affect the spin relaxation time. Before turning to the analysis of these models, we discuss the results of Monte-Carlo simulation which successfully reproduced the experimental data within the classical spin liquid phase. Spin Ice compounds are very well describe by classical Monte-Carlo simulations due to their large Ising spins (15/2 in DTO and 8 in HTO). The size of the spin prevents quantum fluctuations from playing an important role, leaving a fully classical approach adapted to capture the essential physics. Moreover, the two levels which constitute the groundstate doublet of the magnetic ions are well separated in energy from the other levels, which means that their modelisation as an Ising spin is justified in the temperature range accessed in the experiment. In other words, SI system are as close as possible of the idealized playground for classical Monte-Carlo simulation. One of the feature of Monte-Carlo simulation is the treatment of the coupling to a temperature bath in a very abstract way. We will discuss how the temperature is taken into account and how to calculate the spin relaxation time in MC simulation in the last section of this report. For now, it suffices to say that MC simulations assume that the coupling to a bath provides a transition rate between the two states of the spin and that the temperature of the bath fixes the equilibrium properties of the system. This description completely forgets about the microscopic details of the coupling between the magnetic excitations and the thermal bath. Monte-Carlo simulations performed by Jaubert et al. [6] managed to reproduce the low temperature part of the spin relaxation time experimental data, see figure 4. The MC simulation shows a change of behaviour of the spin relaxation time as the temperature approach the phase transition. The spin relaxation time within the classical spin liquid phase increases extremely fast with temperature before saturating near 2K when the system starts to undergo a transition out of the classical spin liquid phase. As explained above, the fast increase in the spin relaxation time is due to the increase number of monopole at higher temperature. The saturation would correspond to a system where the increase number of monopole would not help the system to relax faster. The quantitative agreement with experimental data although exceptionally good, leaves several open questions. The first question comes purely from the misfit between the shape of the saturation plateau predicted by the Arrhenius law and the experimental data out the classical spin liquid phase. Although quantum 11

12 fluctuations are still unimportant and the temperature is still too low to invalid the Ising description of the spin, there is a clear disagreement. This may be due to a change in the coupling to the environment which prevents an analysis focused solely on the magnetic degrees of freedom dynamic to be valid. Second point, which follow from the first one is the physical origin of this transition rate between the two spin state. In particular, the transition rate is assumed to be temperature independent in classical Monte-Carlo simulation, is this description valid in SI compounds? More generally, MC simulation assumes that single spin flip adequately describes the exploration by the system of its partition function. It would be interesting to investigate to which extents the single spin flip dynamic describes an underlying physical reality. In the rest of this report, we investigate possible answers to these questions. The remaining of the second part will be devoted to a brief introduction to the analysis of the magneto-elastic coupling in Spin Ice compounds. The last part will focus on more general questions regarding the dynamics of single spin when coupled to a thermal bath. 3.3 Magneto-elastic coupling in Spin Ice compounds The first part of the report discussed how the symmetry of the crystal field or the protection by time-reversal symmetry could give rise to an effective description of the magnetic ion groundstate as an Ising spin in SI compounds. When performing MC simulations using this effective model to investigate Spin Ice behaviour, the temperature bath is assumed to provide a mechanism for single spin flip which allows the system to explore the different configurations in its partition function. This approach has the advantage to be independent of the physical details of system-bath coupling and allows one to focus solely on the dynamics of the magnetic degrees of freedom. However, it is possible that the physics actually depends on the details of the relaxation mechanism and may not be fully capture by MC simulations. In SI compounds the role of the thermal bath is played by the thermally activated phonons which couple to the magnetic degrees of freedom in an unusual way. The calculation sketched in this section aims to provide to the reader an intuitive understanding of how vibrational and magnetic degrees of freedom can be coupled via the crystal electric field in these materials. This method builds on the perturbative calculation of the point charge model performed in [9]. In particular, the calculation relies on the same assumptions as the point charge model to write the matrix element of the magneto-elastic Hamiltonian. The limitations of the method are discussed at the end of the section. For the sake of simplicity, we will focus on the linear coupling between the effective pseudo-spin and the vibrational modes which arises in HTO. The existence of a linear coupling heavily relies on the nature of the symmetry protecting the magnetic doublet. In DTO, the Kramer nature of the doublet (protected by time-reversal) enforces the following transformation under time-reversal : Ŝ Ŝ. The phonon operators being even under time-reversal, a linear coupling would necessarily break timereversal and is therefore forbidden. In HTO, the groundstate doublet is protected by point group symmetry which leads to different transformation rule for the doublet under time-reversal : Ŝz Ŝz and Ŝ± Ŝ±. The quadrupolar components orthogonal to the easy-axis may couple linearly to the phonon modes of the system without breaking time-reversal. The point charge model calculation for the CEF splitting of the single-ion groundstate was performed in [9]. This calculation highlights how the electric field felt by the magnetic ion was dependent on the position of the surrounding ionic cage. In particular, the symmetry of the cage in HTO protects the degeneracy of the groundstate doublet. One may now imagine that the surrounding ionic cage is distorted by a phonon mode, breaking the D 3d symmetry. The electric potential created by the cage is changed by the phonon mode leading to the appearance in the CEF Hamiltonian of terms which were previously forbidden. If the distortion is small, these terms may be considered as small perturbations of the CEF Hamiltonian and act as an effective coupling between the CEF eigenmodes. To make this intuitive picture more concrete, we start from the expansion in terms of Stevens operators Ô l m of the CEF Hamiltonian derived in [9]. 12

13 H CEF = l,m B l m ({R j }) Ôl m (8) where B l m are coefficients set by the electric field which stems from the surrounding electrons which depends on the equilibrium position of the ionic cage {R j }. This expression can be considerably simplify using group theory when computing matrix elements of the CEF Hamiltonian. However, the purpose here is to calculate the form of the magneto-elastic coupling, therefore we will need to take into account the effects of phonon modes which break the D 3d symmetry of the ionic cage forcing us to work with the full expression. When phonon modes are taken into account, the position of the ions varies around their equilibrium position {R j }. If the oscillations are small, one may Taylor expand the coefficients in front of the Stevens operators in terms of this small parameter to obtain the expression of the magneto-elastic coupling : Ĥ mag-el = g m,iôl l m δˆr i (9) l,m,i where gm,i l = ( rbm(r)) l r=rj are the magneto-elastic coefficients and δr i is the displacement operator due to the phonon modes. We can now apply the full force of group theory to simplify this expression as is done in [9] for the CEF Hamiltonian. The first step is to substitute phonon operators ε to the displacement operators and to distinguish the phonon modes according to their irreducible representations. Ĥ mag-el = l,γ g l ΓÔl (Γ) ε (Γ) (10) where Γ runs over the different irreducible representations. One may notice that because irreducible representations are orthogonal to one another, only terms which involves product of the same irreducible representations for the Stevens operators and the phonon operators are allowed. In SI compounds, the energy gap to the first excited level of the crystal field is of the order of 100 K. This energy gap being large compare to the energy scale of the magneto-elastic coupling [14], we expect a relatively small mixing of the groundstate magnetic doublet with the other excited states of the CEF due to this effect. Thus, the main contribution is likely to stem from the interaction of the two groundstate level which allows us to project the full magneto-elastic coupling onto the groundstate doublet subspace {σ, σ }. Only the Stevens operators are affected by this operation. Ĥ Eg mag-el = l,γ gγ l σ,σ σ σ Ôl (Γ) σ σ ε (Γ) (11) This expression allows for further simplifications. Only irreducible representations of the Stevens operators which have a non-zero expectation value in the magnetic groundstate subspace will contribute. Another simplification also come from the fact that coupling to the number representation of the Stevens operators will only lead to renormalisation of the crystal field parameters and only lead to a global energy shift of the two doublets. Imposing time-reversal invariance will prevent any perturbation that would be proportional to the pseudospin dipolar component Ŝz. The only terms left include Stevens operator which have non-zero off-diagonal term when projected onto the groundstate doublet. One of the terms in this expansion comes from the coupling to a phonon mode doublet (ε +, ε ) which transform according to the E g representation. Stevens operators which also transform according to this representation include : 13

14 Ô1 2 = 1 (Ĵ z Ĵ y + 2 Ĵ y Ĵ z) Ô 1 2 = 1 (Ĵ z Ĵ x + 2 Ĵ x Ĵ z) (12) These operators can be identified with the quadrupolar pseudo-spin operators Ŝx,y [Ĵ z, Ĵ x,y]. Therefore the effective coupling between the E g phonon and the Ising spin used to describe the magnetic properties of SI reads : Ĥ ph-σ = g 2 E g (Ŝ+ ε + E g + Ŝ ε E g ) (13) One may clearly see that this coupling induces spin flip dynamics in the magnetic system. This coupling is only one of the possible coupling which exist in SI compounds and although we believe it to be the most relevant one, its effect are expected to be small [14]. These two Stevens operators were originally not allowed by the symmetry in the CEF Hamiltonian expansion as they transform according to the E g representation and not according to the number representation. The addition of phonon modes which have different symmetries allows a new set of Stevens operator which lead to coupling and splitting between eigenmodes which were originally protected by the crystal symmetries. Although analytically fairly straightforward, numerical implementation of this calculation is fairly tedious compare to the calculation of the CEF. This is due partly to the distortion of the ionic cage surrounding the magnetic ion induced by phonon modes which breaks the D 3d symmetry. When the symmetry is broken all the Stevens operators coefficients must be calculated to be able to derive magneto-elastic coupling. The second problem stems from the large number of phonon modes (26) in SI compounds [15]. Two conclusions can be drawn from this calculation. First, the magnetic degrees of freedom in HTO are connected to the thermal phonon bath through a dissipative coupling (spin flip mechanism). Second, it seems clear that the coupling mechanism to the environment in HTO and DTO must be different due to the nature of the groundstate doublet. Since these two compounds display a similar plateau in their spin relaxation time, it leads us to think that this plateau may be due to a mechanism which does not depend on the nature of the coupling to the bath. In order to make further progress in the understanding of this plateau, we turned to model of a single spin connected to a bath. This simplification will prevent us from analysing any specific effects due to the classical spin liquid phase properties but may allow us to gain a better understanding of the usual behaviour of a spin connected to a bath. This subject will be the focus of the last part of this report. 4 Single Spin Flip Dynamic in a Thermal Bath This part summarizes a simple calculation investigating the effect of a bath on single spin flip dynamic. The aim is to compare the results obtained by a full quantum mechanical treatment of the spin within the framework set up by Linblad and those derived when treating spin flip dynamics as a classical Markov process. In the limit where the temperature of the bath is much higher than the two-level splitting, we expect quantum mechanical effects to play a minor rôle (tunneling and coherence) and the classical Markov process to capture the spin flip dynamic accurately. 4.1 Linblad Equation for a Two-level System Coupled to a Bath The system of interest is a qubit (Ising spin half) coupled to a thermal bath modelized as a set of harmonic oscillator with an ohmic spectrum : Ĥ qb = ω 0 2 ˆσz (14) Ĥ bath = 0 dω ω ˆb (ω)ˆb(ω) (15) 14