Towards an Effective Spin Hamiltonian of the Pyrochlore Spin Liquid TbTiO
Abstract
TbTiO is a pyrochlore antiferromagnet that has dynamical spins and only shortrange correlations even at mK the lowest temperature explored so far which is much smaller than the scale set by the CurieWeiss temperature K. The absence of longrange order in this material is not understood. Recently, virtual crystal field excitations (VCFEs) have been shown to be significant in TbTiO, but their effect on spin correlations has not been fully explored. Building on the work in Phys. Rev. Lett. 98, 157204 (2007), we present details of an effective Hamiltonian that takes into account VCFEs. Previous work found that VCFEsinduced renormalization of the nearest neighbor Ising exchange leads to spin ice correlations on a single tetrahedron. In this paper, we construct an effective spin lowenergy theory for TbTiO on the pyrochlore lattice. We determine semiclassical ground states on a lattice that allow us to see how the physics of spin ice is connected to the possible physics of TbTiO. We observe a shift in the phase boundaries with respect to those of the dipolar spin ice model as the quantum corrections become more significant. In addition to the familiar classical dipolar spin ice model phases, we see a stabilization of a ordered ice phase over a large part of the phase diagram ferromagnetic correlations being preferred by quantum corrections in spite of an antiferromagnetic nearest neighbor exchange in the microscopic model. Frustration is hence seen to arise from virtual crystal field excitations over and above the effect of dipolar interactions in spin ice in inducing icelike correlations. Our findings imply, more generally, that quantum effects could be significant in any material related to spin ices with a crystal field gap of order K or smaller.
pacs:
75.10.Dg, 75.10.Jm, 75.40.Cx, 75.40.GbI Introduction
The problem of finding a low energy effective theory from a microscopic theory or directly from experimental considerations is a ubiquitous one in physics. The purpose is to identify the relevant degrees of freedom at some energy scale in order to capture the important physics at that scale. Often in condensed matter physics, a large separation of energy scales facilitates the process of finding an effective theory: for example in the spin ices Bramwell and Gingras (2001); Gingras ; Bramwell et al. (2004) discussed below. When the separation of scales is not large, virtual (quantum mechanical) processes can become important, as in the Kondo problem in which double occupancy of the impurity in the Anderson model can be treated as a virtual process that generates the wellknown sd exchange interaction. Schrieffer and Wolff (1966) One focus of this paper is the construction of such a low energy effective theory for a highly exotic magnetic material  the TbTiO pyrochlore magnetic material.
A second thread to the present work is frustration, which occurs in magnetism when interactions between spins cannot be minimized simultaneously. This happens, in the case of geometric frustration, as a consequence of the topology of the lattice. As an example, antiferromagnetic isotropic exchange interactions between classical spins on the vertices of the three dimensional pyrochlore lattice of cornersharing tetrahedra are frustrated. Villain (1979); Moessner and Chalker (1998a, b); Reimers (1992); Reimers et al. (1991); Gardner et al. One consequence of this frustration is an extensive (macroscopic) ground state degeneracy and lack of conventional longrange order down to arbitrarily low temperatures. Theoretically, this degeneracy is expected to be lifted, partially, or fully, by other interactions, Palmer and Chalker (2000); Elhajal et al. (2005) perhaps assisted by the presence of thermal or quantum fluctuations. Moessner and Chalker (1998b); Champion et al. (2003); Champion and Holdsworth (2004) These lessons carry over to real pyrochlore magnets in which the frustration of the principal spinspin interaction usually manifests itself in a transition to longrange order Champion et al. (2001); Stewart et al. (2004); Wills et al. (2006a); Champion et al. (2003) or a spin glass transition Gingras et al. (1997); Gardner et al. (1999a) well below the temperature scale set by the interactions the CurieWeiss temperature . In fact, this is a ubiquitous fingerprint of highly frustrated magnets.
When shortrange spin correlations persist down to arbitrarily low temperatures, as in the isotropic exchange pyrochlore antiferromagnet of Refs. Moessner and Chalker, 1998a, b, the system is referred to as a spin liquid or collective paramagnet. Villain (1979) Given the large proportion of geometrically frustrated magnetic materials which have been studied experimentally and which do ultimately exhibit an ordering transition, it does seem that spin liquids are rather rare in two and three dimensions.Lee (2008); Levi (2007); Nakatsuji et al. (2005); Mendels et al. (2007); Simonet et al. (2008); Okamoto et al. (2007); Gardner et al. (1999b) One would expect, on general grounds, this scarcity to be particularly apparent in three dimensional materials where thermal and quantum fluctuations are the most easily quenched. This paper is concerned with the material TbTiO which is one of the very few three dimensional spin liquid candidates. Gardner et al. (1999b) TbTiO is a pyrochlore antiferromagnet that is not magnetically ordered at any temperature above the lowest explored temperature of 50 mK, Gardner et al. (1999b); Gardner et al. (2001, 2003); Gla although the CurieWeiss temperature, , is about K, that is three hundred times larger. Gingras et al. (2000) Despite ten years Gardner et al. (1999b) of experimental and theoretical interest in this system, the low energy magnetic properties of this material are still not currently understood. Mirebeau et al. (2006); Enjalran et al. (2004); Molavian et al. (2007)
In this article, we build on earlier work Molavian et al. (2007) by presenting further evidence that qualitatively new physics, in the form of geometrical frustration, is generated via virtual crystal field excitations (VCFEs) in TbTiO. The frustration of interactions coming from high energies is not without precedent in condensed matter physics: frustrated exchange beyond nearest neighbor and ring exchange terms arise in small effective theories derived from the Hubbard model at halffilling. Chernyshev et al. (2004); Delannoy et al. (2005, 2009) In this problem, the higher order terms in the effective model have only a quantitative effect on the physics which is already captured by the lowest order terms. Delannoy et al. (2005)
In contrast, qualitatively new phenomena have been proposed to arise by integrating out high energies in a recent work on Mott systems, Leigh et al. (2006) and in gauge theories of frustrated magnetic systems Hermele et al. (2004); Neto et al. (2006) (which, interestingly, take as starting points models closely related to the effective model derived in Sections III and IV of this paper). The substantial effect of VCFEs on low energy physics advocated in Ref. Molavian et al., 2007 and in this article is reminiscent of the recent experimentally motivated proposal that PrAuSi is a disorderfree spin glass owing to frustration dynamically arising from excited crystal field levels. Goremychkin et al. (2008) Before launching into the calculations, we first describe some earlier developments relating to TbTiO to motivate our approach to this problem.
i.1 Phenomenology of TbTiO
There is one particular property that may be useful for making progress towards understanding the low energy physics of TbTiO and which is shared by all the compounds in the RMO family of compounds to varying degrees Gardner et al. (here is a rare earth ion with a magnetic crystal field ground state and is nonmagnetic Ti or Sn). It is the smallness of the energy scale due to interactions, , compared with the crystal field splitting, , between the single ion ground state doublet and the first (lowest) excited states. The interactions are typically of the order of K or smaller while the lowest crystal field splitting is of the order of tens or hundreds of Kelvin.Rosenkranz et al. (2000); Gingras et al. (2000); Mirebeau et al. (2007) This means that the ground state wavefunction and low energy excitations mainly “live” in the Hilbert space spanned by the ground state crystal field states on all lattice sites. As we shall see in detail later on, the interactions, V, admix excited crystal field wavefunctions into the ground state doublet and these quantum corrections are weighted by . Vbr For the spin ices, HoTiO and DyTiO, for which is of the order of K, Rosenkranz et al. (2000) the effect of excited crystal field levels can be ignored to a very good approximation and the angular momenta can then be treated as classical Ising spins. Bramwell and Gingras (2001); den Hertog and Gingras (2000); Gingras In common with the spin ices, TbTiO has a crystal field ground state that can be described in terms of Ising spins.Gingras et al. (2000) But, the (classical) dipolar spin ice model (DSIM) which has, through various studies demonstrated its veracity in comparisons to the spin ices, den Hertog and Gingras (2000); Bramwell et al. (2001); Yavors’kii et al. (2008) is not a good model for TbTiO.
An estimate of the antiferromagnetic exchange coupling in TbTiO Gingras et al. (2000) puts this compound close to the phase boundary of the DSIM between the paramagnetic spin ice state (or lower temperature longrange ordered spin ice phase) and the four sublattice longrange Néel antiferromagnetic phase (see inset to Fig. 2). den Hertog and Gingras (2000); Melko et al. (2001); Melko and Gingras (2004) None of these states adequately describes TbTiO. The longranged ordered phases can be ruled out on the grounds that no Bragg peaks are observed in the diffuse neutron scattering pattern. Gardner et al. (2001, 2003) A comparison with spin ice phenomenology is a little more subtle. One of the main features of the spin ice state is that it harbors a large residual entropy as deduced by integrating the heat capacity downwards from high temperatures. Ramirez et al. (1999) Whereas, similarly to what has been observed in spin ices, Bramwell and Gingras (2001); Ramirez et al. (1999) there is a broad bump in the specific heat between K and K as the temperature is lowered, at present it remains difficult to determine whether there is a residual entropy in the collective paramagnetic state of TbTiO. Gingras et al. (2000); Hamaguchi et al. (2004) The study in Ref. Hamaguchi et al., 2004 finds a slightly different heat capacity to the one in Ref. Gingras et al., 2000 and claims no evidence of residual entropy in TbTiO owing to almost a complete recovery of the full entropy of the doubletdoublet crystal field levels (see also Ref. Ke et al., 2009 for a similar finding). Instead it reports that there is a sharp feature in the heat capacity at about mK indicating the onset of a glassy state. Glassiness has also been observed in the susceptibility measurements of Ref. Luo et al., 2001. Finally, the diffuse paramagnetic neutron scattering pattern Gardner et al. (1999b); Gardner et al. (2001, 2003); Yasui et al. (2002) of TbTiO differs drastically from the experimental spin ice pattern (which has been reproduced by Monte Carlo simulations of the DSIM Bramwell et al. (2001) and its improvements Yavors’kii et al. (2008)). This strongly suggests that the Ising nature of the localized moments is not an appropriate description for the magnetism in TbTiO, as noted in Ref. Enjalran and Gingras, 2004.
Some important insight into the microscopic nature of TbTiO is provided by a mean field theory for classical spins with only a finite Ising anisotropy. Enjalran and Gingras (2004) Specifically, Ref. Enjalran and Gingras, 2004 finds that a toy model in which spins, subject to a finite anisotropy and interacting via isotropic exchange and dipoledipole interactions, captures the main features of the experimental paramagnetic diffuse neutron scattering pattern in TbTiO. Gardner et al. (1999b) The results of Ref. Enjalran and Gingras, 2004 lead one to suspect that the weaker anisotropy of the spins in TbTiO, in contrast to those in the spin ices, can be attributed to the fact that because the ground to first excited crystal field gap is much smaller in TbTiO, the effect of excited crystal field states cannot be ignored. The effects of VCFEs can be studied, albeit incompletely, within the random phase approximation (RPA). A computation of the RPA diffuse neutron scattering intensity in the paramagnetic regime using the full crystal field level structure and wavefunctions Kao et al. (2003) leads to results that are in good qualitative agreement with experiment, Gardner et al. (2001) adding weight to the idea that one of the effects of VCFEs in TbTiO is to decrease the Ising anisotropy of the spins.
Having identified VCFEs as an important contribution to the physics of TbTiO, we look for a way of examining the effect of VCFEs on the ground state of perhaps the simplest minimal model for TbTiO. An approach that is wellsuited to this problem is an effective Hamiltonian formalism. The low energy theory that is obtained within this formalism inhabits a product of two dimensional Hilbert spaces one for each magnetic site spanned by the ground state crystal field doublet. So, the effective theory can be written in terms of (pseudo) spins onehalf. Neglecting VCFEs, the effective Hamiltonian is simply the theory obtained by projecting onto the ground state crystal field doublet on each magnetic ion which, as we shall see, is the DSIM of interacting (classical) Ising spins i.e. a model in which transverse spin fluctuations are absent. Gingras The separation of energy scales to which we have alluded then allows us to develop a perturbation series in the parameter Vbr where the zeroth order term is the DSIM Gingras and higher order terms explicitly incorporate the effect of VCFEs in terms of operators acting within the projected Hilbert space. The procedure can be written schematically as
where the bare microscopic Hamiltonian , depending on magnetic moments through the crystal field and interactions , is used to derive an effective Hamiltonian in terms of pseudospins , .
One advantage of this approach is that, by decreasing , we can smoothly connect our results to the physics of spin ice. Bramwell and Gingras (2001); Gingras ; Bramwell et al. (2004) A second more practical advantage is that, since the dimensionality of the relevant Hilbert space is reduced, exact diagonalization calculations on finite size clusters (albeit small clusters), series expansion techniques and the linked cluster method may become tractable. Oitmaa et al. (2006)
A comparison has previously been made Molavian et al. (2007) between the effective Hamiltonian to lowest order in quantum corrections, , with the crystal field gap as a free parameter and the “high energy” microscopic (bare) model from which it was obtained. This involved an exact diagonalization of the two models on a single tetrahedron to determine the ground state as a function of and the exchange coupling. Molavian et al. (2007) The result is shown in Fig. 3. The ground state degeneracies largely coincide over the range of parameters explored, which includes the estimated exchange coupling of TbTiO. Most importantly, in the singlet region of the phase diagram, the ground state of the exact bare microscopic model is a nondegenerate superposition of states each satisfying the spin ice constraint. In contrast, for the classical dipolar ice model with the same exchange coupling, on a single tetrahedron and on a lattice, the ground state is a doubly degenerate allin/allout state (see Fig. 14(a)). That the full quantum problem favors spin icelike correlations at the single tetrahedron level was shown to arise from a renormalization of the Ising exchange in the effective anisotropic spin Hamiltonian when VCFEs are included. Molavian et al. (2007) Finally, it was found that the level structure from exact diagonalization of the original model on a single tetrahedron is sufficient to reproduce the main semiquantitative features of the experimental diffuse neutron scattering pattern for TbTiO. Molavian et al. (2007)
The renormalization of the effective nearest neighbor Ising exchange by VCFEs such that spin ice correlations are energetically preferred over a larger range of the bare exchange couplings than would be the case without quantum corrections shows clearly that quantum effects can have a significant effect on the nature of the correlations in TbTiO. However, owing to the presence of a longrange dipoledipole interaction and the fact that VCFEs in themselves generate interactions beyond nearest neighbor, it was not clear on the basis of earlier work Molavian et al. (2007) whether VCFEs would have a significant, or even the same qualitative effect on the TbTiO correlations when considering the full lattice. That is the main problem that we resolve in this work.
i.2 Scope of the paper
In this article, we present a more detailed derivation of the effective Hamiltonian for TbTiO than was possible in the earlier work Molavian et al. (2007) owing to lack of space. We also take some initial steps beyond the single tetrahedron approximation by calculating the ground states of the effective model assuming that the effective spins are classical spins of fixed length (large approximation). Our main result is shown in Fig. 2 which is discussed more fully in Section V.4. The plot shows the semiclassical phase diagram of the effective model on a cubic unit cell with periodic boundary conditions as a function of the gap and the isotropic exchange coupling in the microscopic model. When , all quantum corrections are suppressed and we recover the limit of the dipolar spin ice model (DSIM) with two phases  a state with the spin ice rule satisfied on each tetrahedron and ordering wavevector (LRSI) and a fourin/fourout Ising state (AIAO) for more antiferromagnetic . Compared to the dipolar spin ice model ground states, the effective model contains one other phase a long range ordered spin ice phase (LRSI). Also, the magnetic moments in the LRSI and LRSI phases are canted away from the local Ising directions as decreases. The region over which the LRSI is the ground state forms a wedge, broadening out to lower until it is the only phase found within the explored range of at the expense of the antiferromagnetic AIAO phase. There are two main physical mechanisms (contributions) to the stabilization of the LRSI state across the phase diagram. The first is that the effective nearest neighbor Ising coupling becomes more ferromagnetic in character as decreases. However, it does eventually change sign as increases over the entire range of studied. So the second reason for the spreading of a spin ice state across the phase diagram as decreases is due to beyond nearest neighbor interactions that arise purely from effective VCFEs and which monotonically increase in strength as decreases.
The outline of the paper is as follows. In Section II, we introduce some notation and describe the microscopic (bare) model for TbTiO from which the effective model is derived. With this in hand, we formulate our approach in more detail than in this introduction. Section III discusses the form and properties of the lowest order (classical dipolar spin ice) term in the effective Hamiltonian. In Section IV, the quantum corrections to this model are enumerated to lowest order in and we study how the longitudinal (Ising) exchange coupling in the dipolar spin ice model (DSIM) is renormalized to this order. Having obtained the effective Hamiltonian for TbTiO to lowest order in the , we treat the effective spins as classical spins and present, in Section V, the resulting semiclassical ground states. This study of the ground states allows us to see how the effect of VCFEs is connected to the physics of spin ice and also clearly shows that spin ice correlations are present even though the bare microscopic exchange coupling is antiferromagnetic.
In other words, geometric frustration in the model (Eqs. (1),(2) and (5)) of TbTiO emerges from quantum virtual crystal field excitations (VCFEs) and manybody physics.
This is the main result of our paper. We discuss these results, in Section VI, in the light of experiments on TbTiO and describe some possible further applications of the effective Hamiltonian that we derive for TbTiO. Finally, we provide in Appendix A, details of the effective Hamiltonian method as a background to the main application to TbTiO described in the remainder of the paper. Appendix B contains further details behind the calculations presented in Section IV and Appendix C gives some data used to convert between crystal field parameters for different rare earth pyrochlore titanates using a point charge approximation.
We note here that while our specific focus is on the TbTiO pyrochlore magnet, the formalism that we employ below could be straightforwardly used to construct effective low energy theories for many other frustrated rare earth systems where the excited crystal field levels have a somewhat larger energy scale than the microscopic interactions.
Ii Effective Hamiltonian
ii.1 Microscopic (Bare) Model
The microscopic or bare Hamiltonian for the magnetic Tb ions in TbTiO is given by
(1) 
where is the crystal field Hamiltonian and are the interactions between the ions. In the remainder of this section we explain the form of both terms in some detail.
The magnetic Tb ions in TbTiO are arranged on the sites of a pyrochlore lattice. The pyrochlore lattice consists of cornershared tetrahedra which can otherwise be thought of as a facecentered cubic (fcc) lattice with primitive translation vectors for and a basis of four ions (). We follow the same labeling of the four sublattice basis vectors as in Ref. Enjalran and Gingras, 2004. It is useful to introduce a coordinate system on each of the four sublattices with local unit vector along the local cubic direction. The sublattice basis vectors and local Cartesian , and directions are given in Table 1. Below, we also make use of rotation matrices (the elements of which are contained in Table 1) which achieve a passive transformation that takes the local sublattice coordinate system for sublattice into the global Cartesian laboratory axes.
Sublattice  

Spinorbit coupling within the relevant localized levels of the Tb ions leaves total angular momentum as a good quantum number with . The local environment about each Tb ion is responsible for breaking the degeneracy. Its effect can be computed from a crystal field Hamiltonian, , which is constrained by symmetry to take the form Gingras et al. (2000); Rosenkranz et al. (2000); Mirebeau et al. (2007)
(2) 
The magnetic ions are labeled by an fcc site and a sublattice index . Expressions for the operators in terms of the local angular momentum components can be found, for example, in Hutchings. Hutchings (1964) The crystal field in TbTiO has been studied in Refs. Gingras et al., 2000 and Mirebeau et al., 2007 resulting in somewhat differing estimates for the parameters . In the following, all quantitative results that we present for TbTiO were obtained using crystal field parameters for HoTiO, obtained from inelastic neutron scattering in Ref. Rosenkranz et al., 2000, which have been rescaled to the TbTiO parameters according to
(3) 
Here, the are Stevens factors. Stevens (1952) These and the radial expectation values for the rare earth ions Freeman and Desclaux (1979) can be found in Appendix C. We have checked that using the crystal field parameters of Ref. Mirebeau et al., 2007 instead leads to results that are in fairly close quantitative agreement with those obtained using the rescaled parameters from Eq. (3).
The crystal field Hamiltonian, , can be diagonalized numerically exactly; the eigenvalues are and the eigenstates for , which we implicitly arrange in order of increasing energy. One finds a level structure that includes a ground state and a first excited state that are both doubly degenerate. Gingras et al. (2000); Mirebeau et al. (2007) The splitting, , between the ground and first excited states is about K, Gingras et al. (2000); Mirebeau et al. (2007) which is much smaller than the corresponding gap in the spin ices (for example, the gap in HoTiO is about K Rosenkranz et al. (2000)). It is the smallness of this value of compared to for TbTiO and the possibility of admixing between the ground state and excited state crystal field levels that are at the root of all the phenomenology that we explore in the rest of this paper. Fig. 4 shows the level structure of the crystal field spectrum for the four lowest levels determined on the basis of an exact diagonalization of Eq. (2).
We emphasize two features of this spectrum that will be important later on. First of all, let us write down the time reversal properties of the eigenstates, . Let be written as a linear combination of the eigenstates of , denoted ,
Time reversal invariance requires that the coefficients are related to one another by . Tim Secondly, it is possible to interpret the noninteracting single ion angular momenta as Isinglike at low energies, as was done in Ref. den Hertog and Gingras, 2000. This is because, at sufficiently low energies, thermal occupation of excited crystal field levels is negligible and one can focus on the ground state doublet. The ground state doublet states, and , have
(4) 
as the only nonvanishing matrix elements, where the tilde indicates that the axis is taken along the local direction appropriate to each magnetic ion (see Table 1). So, this doublet considered on its own has nonzero angular momentum expectation values only along one axis with vanishing transition matrix elements .
The interactions between the angular momenta, , are taken to be nearest neighbor isotropic exchange and dipoledipole interactions, :
(5) 
The notation is short for with and (where is the edge length of the conventional cubic unit cell) is the distance between neighboring magnetic ions. Gingras et al. (2000) Here, we employ the convention that is antiferromagnetic and is ferromagnetic. This is the simplest Hamiltonian consistent with the nonvanishing Tb dipoledipole coupling, K with the Landé factor, and with the negative CurieWeiss temperature K. Gingras et al. (2000) The exchange coupling has been estimated from for TbTiO and for the diluted compound (YTb)TiO Gingras et al. (2000) to be about K, while a fit in Ref. Mirebeau et al., 2007 gives a value for K that is significantly less antiferromagnetic. Est
In summary, our bare microscopic model for TbTiO consists of three terms: the crystal field Hamiltonian , an isotropic exchange with an antiferromagnetic coupling and a dipoledipole interaction, . Int An extension of the present work could include (i) bare exchange couplings beyond nearest neighbors, (ii) anisotropic nearest neighbor exchange as described in Ref. McClarty et al., 2009 and (iii) direct or virtual (phononmediated) multipolar interactions. Santini et al. (2009)
ii.2 Route to an effective Hamiltonian
If we were able to ignore the excited crystal field levels in TbTiO, the angular momenta could be treated as classical Ising spins den Hertog and Gingras (2000); Gingras because the only nonvanishing matrix elements of the angular momentum are those in Eq. (4). Gingras However, for reasons outlined in the Introduction, this is not a good approximation for this material. The interactions between the angular momenta induce VCFEs that admix excited crystal field wavefunctions into the space spanned by the noninteracting crystal field doublets with the consequence that the magnetic moments behave much less anisotropically than one would expect on the basis of the Isinglike ground state crystal field doublet. These quantum fluctuations can be treated perturbatively because there is a small dimensionless parameter , where . To lowest order in such a perturbation theory, and in a low energy effective model, the spins should be perfectly Isinglike and hence we recover the DSIM. We now proceed to make these ideas more concrete.
Because we seek a Hamiltonian operating within a low energy subspace, we need a projection operator onto the noninteracting single ion crystal field ground states. For a single ion at the site specified by indices , the projection is accomplished by
This operator satisfies the conditions and Hermiticity. With moments on all the sites of the lattice, the projector is . The subspace of the full Hilbert space selected by the projector will be called the model space, , from now on. The Hilbert space is defined as the space spanned by states and on site .
The spinspin interaction
(6) 
is to be treated as a perturbation. Because the perturbation is “small” compared to the difference between the ground and first excited crystal field energies , , we expect that on a crystal of sites, the lowest energy eigenstates of lie mainly within because the admixing of excited crystal field wavefunctions into the model space is a small effect. Our effective Hamiltonian will be defined in such a way that its eigenstates live entirely within while its eigenvalues exactly correspond to the lowest energy eigenvalues of the exact Hamiltonian, . The lowest energy eigenstates of mainly lie within in the sense that the rotation of exact states out of the model space is determined by the relatively small perturbation .
In practice, the exact eigenvalues can be approximated by carrying out perturbation theory in the construction of the effective Hamiltonian . After some work, that is briefly laid out in Appendix A, one finds that the effective Hamiltonian can be written as Lindgren and Morrison (1982)
(7) 
The operator the resolvent operator is given by
(8) 
where, for a finite crystal of sites, is times the energy of the degenerate ground state crystal field levels . The numerator of each term in the resolvent is a projector onto a space orthogonal to a product of crystal field operators where the product is taken over all sites of the lattice with at least one such operator having (i.e. belonging to the group of excited crystal field states); this is the meaning of the notation in the summation index of Eq. (8). The third term on the righthandside of Eq. (7) is the lowest order term in the perturbation series to include the effects of crystal field states outside the model space. This term is therefore the lowest order contribution of the VCFEs that we have referred to above.
Equation (7) makes no reference to a particular model. In Sections III and IV, we develop the terms in the effective Hamiltonian for the model of TbTiO described in Section II.1. Section III is devoted to the lowest order, or classical, term . Section IV enumerates the lowest order terms generated by VCFEs, relating each underlying class of terms that originate from to specific virtual excitation channels. Higher order corrections than are computationally difficult to determine mainly because of the presence of the longrange dipole interactions . See Ref. Bergman et al., 2007 for a model on a pyrochlore for which degenerate perturbation theory can be carried out to much higher order than is done is this work.
To spare readers the details of this rather technical derivation if they so choose, we include a short summary (Section IV.6) of the form of the low energy model for TbTiO. Finally, in Section IV.7, we summarize some results that have been obtained from the effective Hamiltonian which have already appeared in the literature. Molavian et al. (2007); Molavian and Gingras (2009) All in all, we shall see that the DSIM couplings are renormalized by VCFEs and that effective anisotropic spinspin couplings appear in addition to the Ising interactions of the DSIM. In other words, the effective theory allows for fluctuations of the moments perpendicular to the local axes. We shall study the variation of the effective couplings in as is varied. This information will be useful in the interpretation of the semiclassical ground states of the effective model (Section V) and hence in assessing the effects of VCFEs on the physics of TbTiO.
Iii Classical part of
iii.1 Ising model for TbTiO
In this subsection, we consider the (lowest order) term in Eq. (7). The effective Hamiltonian derived from for TbTiO can be rendered in the form of a spin onehalf model by rewriting the model space operators in Eq. (7) in terms of Pauli matrices. This is possible because the model space, in our case, is a direct product of two dimensional Hilbert spaces spanned by the ground state crystal field doublet. The correspondence between Pauli matrices and operators on the crystal field ground state:
(9)  
(10)  
(11) 
together with the unit operator . Note, however, that despite the fact they do satisfy the commutation rules
where is the LeviCivita symbol, the do not swap sign under time reversal so they are not true angular momentum operators. For this reason, we shall call them pseudospins or effective spins. It is helpful for later sections to give their properties under , the time reversal transformation:
(12)  
(13)  
(14) 
because .
If we apply the projector to the full Hamiltonian to obtain , we find that the crystal field part becomes with . From now on, we omit this constant energy shift. To project the interaction part , we write the angular momentum components in the local coordinate system with local axes in the directions given in Table 1: . All operator components that refer to the local coordinate systems are labeled with a tilde. Also, when it is not important to distinguish different sublattices, we abbreviate with the site index . We add further numerical subscripts to to label different sites. With this notation, the projector acting on gives
where . Owing to , all matrix elements of the other angular momentum components vanish. So, the isotropic exchange becomes
(15) 
and the dipoledipole interaction becomes
(16) 
The renormalized, or effective, exchange and dipoledipole couplings are, respectively, and . is the celebrated DSIM. Bramwell and Gingras (2001); Gingras ; Bramwell et al. (2004); den Hertog and Gingras (2000); Melko et al. (2001); Fennell et al. (2004); Yavors’kii et al. (2008) It is a classical (local Ising) model because all the terms mutually commute as they solely consist of operators. This model exhibits two different ground states depending on the ratio of the exchange to the dipolar coupling; these are shown in the inset of Fig. 2. When , the ground state has ordering wavevector with the spins on a single tetrahedron in the state or the state the allin/allout phase. den Hertog and Gingras (2000) When , the ordering wavevector of the ground state is and each tetrahedron has spins satisfying the twoin/twoout ice rule; we refer to this state as the LRSI phase, Bramwell and Gingras (2001); Melko et al. (2001); Melko and Gingras (2004) with one of the domains shown in Fig. 1. Tra Above a nonzero critical temperature, the LRSI phase gives way Bramwell and Gingras (2001) to a spin ice state with no conventional longrange order (Fig. 2).
Formally speaking, the spin ice state is a collective paramagnetic state Villain (1979) a classical spin liquid of sorts. That the DSIM has proved to be a good model for spin ice materials is largely due to the substantial gap between the crystal field ground state doublet and first excited state which results in a roughly suppression of VCFEs. Spi (a) This model is not a good description for TbTiO. Indeed, if we consider the estimated couplings given in Section II.1, we find (recalling K and K as stated in Section II.1), which would put TbTiO in the allin/allout phase with a critical temperature into this phase from the paramagnetic phase at K (see vertical dashed line in the inset to Fig. 2). den Hertog and Gingras (2000) This is in contradiction with neutron scattering experiments which find no magnetic Bragg peaks in zero field. Spi (b) If we allow for inaccuracies in the estimate of , Mirebeau et al. (2007); Spi (b) such that a classical, dipolar spin ice state is implied by the coupling, we find various properties of spin ices that are not compatible with those of TbTiO. Some of these conflicting properties the diffuse neutron scattering pattern and differing spin anisotropies were discussed in the Introduction. Therefore, in the next section, we investigate what happens when is small enough that the lowest order fluctuation term in Eq. (7) becomes important.
iii.2 Exchange convention
In Eq. (5), we use the opposite sign convention for the exchange coupling to the one used in Refs. Melko et al., 2001, den Hertog and Gingras, 2000 and Melko and Gingras, 2004. Tra The convention in these works is to include a minus sign in front of the exchange coupling in contrast to our Eq. (5). In this article, in the global coordinate system, antiferromagnetic corresponds to .
A warning must be made regarding the convention within the local coordinate system. In rotating to the local system, geometrical factors appear in front of the couplings. For example, as shown in Section III, the local Ising exchange part of the coupling is where arises from the fact that the local axes are not collinear. In the following pages, we adopt the simplifying scheme of absorbing the geometrical factors into the couplings. In doing so, it will be useful to describe how to go from the sign of the local effective Ising coupling, , in front of to the type of order that is energetically favored by the coupling. Thus, when the local coupling is said to be ferromagnetic, is negative and the Ising components of the spins prefer to lie in an allin/allout configuration. When, instead, is positive, it is said to be antiferromagnetic and the local Ising components are frustrated, leading to a spin ice configuration on each tetrahedron.
Iv Lowest order quantum fluctuations
iv.1 General Considerations
In this section we present a derivation of the quantum terms in the effective Hamiltonian. We refer those readers interested only in the results of this technical derivation to Section IV.6. We begin by introducing a little more notation to describe the structure of the term which we shall refer to as . We write the interaction term in the form
(17) 
where, in the second line, we have absorbed the rotation matrices into the definition of . When the spins interact via nearest neighbor isotropic exchange and longranged dipoledipole interactions as in Eq. (5), we have for :
(18) 
with unit vectors for in the laboratory directions respectively (see Table 1). The prefactors of onehalf cure the double counting of pairs in Eq. (17).
The model space basis states are products of ground state doublet states and over all lattice sites while excited crystal field states are denoted for on each site . The state in Eq. (8) is a direct product of crystal field states on different sites with the condition that at least one of the states in lies outside the ground state crystal field doublet; in other words, at least one Tb ion must be virtually excited in a state with . With this notation in hand, we write the quantum term as
(19) 
There are a few observations that we can make from Eq. (19) that identify classes of nonvanishing terms. Suppose we choose magnetic sites on the pyrochlore lattice for in Eq.(19). Then, when we evaluate Eq.(19) for all other sites, we obtain unit operators for all sites with . This follows because the resolvent operator and projection operators are diagonal on each site. In the following, we do not write out all these unit operators explicitly. A second observation is that when we consider a term with magnetic sites () all different, we find that such a term vanishes. The reason for this is that the resolvent and angular momentum operators are sandwiched by projectors into the model space. That way, a virtual excitation induced, for example, by in the bilinear operator must be “deexcited” by an angular momentum operator in the other bilinear operator , (with , for example). If all are different there can be no virtual excitations and, because the resolvent operator is orthogonal to the model space states, such terms must vanish.
Having found those terms that must always vanish, we now divide all the potentially nonvanishing terms into three classes that we shall analyze in turn in the next three subsections.

The first class of terms has two groups and of sites, with exactly one site in the first group in common with a site in the second group. In this case, the ion on the common site must be virtually excited and deexcited, and the other two ions remain in their ground doublets. This is because the resolvent operator demands that there be some virtual excitations and that the projectors require that any virtual excitation must be deexcited. So, only when two angular momentum operators (one in each operator of ) belong to a given site can that site be virtually excited.

This class of terms has identical pairs and regardless of label ordering, but with only one single ion ( or ) that is virtually excited.

Finally, we shall consider the case where and are identical pairs and where both ions are virtually excited.
The virtual excitations belonging to each of these three cases are illustrated in Fig. 5.
It will be convenient, while considering the possibilities enumerated above, to make use of the following explicit decomposition of the quantum term :
(20)  
We refer to as the exchangeexchange part, as the exchangedipole part and as the dipoledipole part.
iv.2 Case A
We will show that the situation in Case A described above leads to (i) effective Hamiltonian bilinear interactions between the local components of the spins and also to (ii) threebody interactions of the form where or , but not .
We write the bilinear operator on the lefthandside of Eq. (19) as and the other bilinear as with (, see Fig. 5(a)) with all angular momentum components referred to the local coordinate system. As we discussed above, the contribution of all the other sites gives identity operators for each site. Omitting these unit operators, we are left with
(21) 
with
(22) 
is the energy of an excited crystal field state on a single ion. Here, the angular momentum components are expressed in their respective local coordinate systems with local axes given in Table 1, the rotation matrices having been absorbed implicitly into . The integers run over and with the states lying within on their respective sites. We factor out the part for site : . Recalling the property, Eq. (4) and the mapping in Eq. (11), we obtain . We reach the same result for the sum over states and on site . Equation (21) then simplifies to
(23) 
in the local coordinate system where we have dropped the site labels from the state vectors enclosed by brackets. After summing over the excited states , and rendering the sum of operators in terms of Pauli matrices, we find an Ising interaction term and threebody operators with and since, if were to equal , the threebody term would violate time reversal invariance. The sum over virtual excited state and the subsequent rendering in terms of Pauli operators is discussed in some more detail in Appendix B
We have reduced the most general three ion terms in (case A) to interactions between pseudospins onehalf but we have not made any assumptions yet about the form of the interactions . In the following, we shall consider the four terms of Eq. (20) in turn within Case A. These terms determine the spatial range of the resulting effective interactions between the pseudospins and are obtained by distinguishing the exchange and dipolar parts of in Eq. (23).
Exchangeexchange part The exchangeexchange part (referring to the first term of Eq. (20)) which is nothing more than Eq. (23) for with is a shortrange, but not strictly nearest neighbor, effective interaction. If , and all lie on the same tetrahedron in the lattice, then the Ising interaction acts between nearest neighbors and, for given and , there are two choices for the position of the “mediating ion” as shown in Fig. 6. The thick lines in this figure join the and ions via two different choices for the ion