Spin-Orbital Order and Condensation in 4D and 5D Transition Metal Oxides

Spin-Orbital Order and Condensation in 4d and 5d Transition Metal Oxides

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Christopher Svoboda, M.S., B.S. Graduate Program in Physics

The Ohio State University 2017

Dissertation Committee:

Professor Nandini Trivedi, Advisor

Professor Mohit Randeria

Professor Jay Gupta

Professor Linda Carpenter c Copyright by

Christopher Svoboda

2017 Abstract

Strong correlations and strong spin-orbit coupling are important areas of research in condensed matter physics with many open questions. Transition metal oxides provide a natural way to combine these strong correlations and strong spin-orbit coupling in electronic systems. Iridium compounds in the d5 conﬁguration (Ir4+) have received most of the focus in this area for the last decade, yet there remains much more unexplored territory with other electron counts. Here we explore the magnetism in several classes of 4d and 5d

Mott insulating transition metal oxides with d1, d2, and d4 electron counts. We ﬁrst cover

1 2 double perovskites A2BB’O6 where the B’ ion is in either the d and d conﬁguration and the other ions are nonmagnetic. We develop and solve magnetic models with both spin and orbital degrees of freedom within mean ﬁeld theory. The anisotropic orbital degrees of freedom play a crucial role in resolving some outstanding puzzles in these compounds including why ferromagnetism is common in d1 but not d2 and why the d1 ferromagnets have negative

Curie-Weiss temperatures. Then we cover a broad class of compounds in the d4 configuration. Despite the fact that Hund’s rules dictate the ground state should be nonmagnetic, we find that superexchange may induce magnetic moments and magnetic ordering through the condensation of triplon excitations. We find condensation occurs only at ~k = ~π which generates antiferromagnetic order in the models we consider, and strong Hund’s coupling does not induce ferromagnetism in the large spin-orbit coupling limit even though it induces ferromagnetic interactions in the absence of spin-orbit coupling. We then apply our results

4 to the d double perovskite Ba2YIrO6. Although experimental observations indicate the material possesses magnetic moments, we show that these magnetic moments are likely not

due to condensation induced by superexchange.

ii Acknowledgments

I am grateful to my graduate advisor, Professor Nandini Trivedi, for the support she provided throughout the entire course of my PhD. I would like to thank Professor Mohit

Randeria for his wisdom and perspective on several projects. I would also like to thank our experimental collaborators, Professor Patrick Woodward, Professor Jiaqiang Yan, and Professor Rolando Vald´esAguilar, for their insights and their commitment to our multidisciplinary endeavors. Finally I would like to thank my undergraduate advisor,

Professor Thomas Vojta, for the opportunities he provided during my undergraduate years.

I acknowledge the support of the Center for Emergent Materials, an NSF MRSEC, under Award Number DMR-1420451.

iii Vita

December, 2011 ...... B.S., Missouri University of Science and Technology, Rolla, MO May, 2015 ...... M.S., The Ohio State University, Colum- bus, OH

Publications

C. Svoboda, M. Randeria, N. Trivedi. “Orbital and spin order in spin-orbit coupled d1 and d2 double perovskites”. arXiv:1702.03199

C. Svoboda, M. Randeria, N. Trivedi. “Eﬀective magnetic interactions in spin-orbit coupled d4 Mott insulators”. Phys. Rev. B 95, 014409 (2017)

T. T. Mai, C. Svoboda, M. T. Warren, T.-H. Jang, J. Brangham, Y. H. Jeong, S-W. Cheong, R. Vald´esAguilar. “Terahertz Spin-Orbital Excitations in the paramagnetic state of multiferroic Sr2FeSi2O7”. Phys. Rev. B 94, 224416 (2016)

W. Tian, C. Svoboda, M. Ochi, M. Matsuda, H. B. Cao, J.-G. Cheng, B. C. Sales, D. G. Mandrus, R. Arita, N. Trivedi, J.-Q. Yan. “High antiferromagnetic transition temperature of a honeycomb compound SrRu2O6”. Phys. Rev. B 92, 100404(R) (2015)

L. Demkó,S. Bordács,T. Vojta, D. Nozadze, F. Hrahsheh, C. Svoboda, B. Dóra,H. Yamada, M. Kawasaki, Y. Tokura, I. Kézsmárki. “Disorder Promotes Ferromagnetism: Rounding of the Quantum Phase Transition in Sr1−xCaxRuO3”. Phys. Rev. Lett. 108, 185701 (2012)

C. Svoboda, D. Nozadze, F. Hrahsheh, and T. Vojta. “Disorder correlations at smeared phase transitions”. Europhys. Lett. 97, 20007 (2012)

iv Fields of Study

Major Field: Physics

v Table of Contents

Page Abstract...... ii Acknowledgments...... iii Vita...... iv List of Figures ...... viii

Chapters

1 Introduction ...... 1 1.1 Historical Motivation...... 1 1.2 General Overview...... 4 1.3 Summary of Results ...... 5

2 Transition Metal Oxides ...... 9 2.1 Materials and the Hubbard Model ...... 9 2.2 Crystal Field ...... 11 2.3 Coulomb Interactions ...... 13 2.4 Spin-Orbit Coupling...... 17 2.5 Band Limit versus Mott Limit ...... 18

3 d1 and d2 Double Perovskites ...... 22 3.1 Introduction...... 22 3.2 d1 Double Perovskites ...... 26 3.2.1 Model...... 27 3.2.2 Zero Temperature Mean Field Theory...... 29 3.2.3 Finite Temperature Mean Field Theory ...... 33 3.2.4 Simpliﬁed Model at Finite Temperature...... 38 3.3 d2 Double Perovskites ...... 40 3.3.1 Model...... 41 3.3.2 Zero Temperature Mean Field Theory...... 41 3.3.3 Finite Temperature Mean Field Theory ...... 44 3.4 Discussion...... 46

4 d4 Mott Insulators ...... 48 4.1 Introduction...... 48 4.2 Model...... 51

vi 4.3 Exact diagonalization ...... 54 4.4 Eﬀective Magnetic Hamiltonian...... 57 4.4.1 Norb = 3...... 59 4.4.2 Norb = 2...... 60 4.4.3 Norb = 1...... 62 4.5 Orbital Frustration...... 64 4.6 Triplon Condensation ...... 65 4.6.1 Overview of the Mechanism...... 66 4.6.2 Results ...... 68 4.6.3 Local Interactions versus Condensation ...... 70 4.7 Materials and Experiments ...... 73 4.8 Conclusions...... 75

5 Magnetic Condensation in Ba2YIrO6 ...... 76 5.1 Introduction...... 76 5.2 Absence of Condensation ...... 78 5.3 Application to Experiments...... 82

Bibliography ...... 85

Appendices

A Calculation Details for Transition Metal Oxides ...... 92 A.1 t2g Orbital Angular Momentum...... 92 A.2 Multi-Orbital Hubbard Interaction...... 94

B Calculation Details for d1 and d2 Double Perovskites ...... 97 B.1 d1 Superexchange...... 97 1 B.2 µeﬀ enhancement and To for d model ...... 98 B.3 Susceptibility in the Simpliﬁed Model ...... 100 B.4 Projection to j = 3/2...... 101

C Calculations Details for d4 Mott Insulators ...... 104 C.1 Eﬀective Hamiltonian ...... 104 C.2 Condensation Formalism...... 106 C.3 Condensation from Spin-Orbital Superexchange ...... 110

vii List of Figures

Figure Page

1.1 Conventional band insulators and simple metals are found when spin-orbit coupling and Coulomb interactions are small. When spin-orbit coupling is tuned to be large, the result is still either a band insulator or metal, however, the result may be topologically non-trivial. When Coulomb interactions are tuned to be large, the result is a Mott insulator. Both 4d and 5d transition metal oxides combine both strong spin-orbit coupling and strong correlations. Adapted from reference [1]...... 2 1.2 Honeycomb structure formed out of edge-sharing oxygen (purple) octahedra each enclosing a transition metal site (yellow). The Kitaev model is formed from three types of bond-dependent Ising interactions between sites (yellow) on the honeycomb lattice. The Ising interactions are Si,xSj,x along red bonds, Si,ySj,y along green bonds, and Si,zSj,z along blue bonds...... 3

2.1 d orbitals on the transition metal site (purple) are degenerate under spherical symmetry. When this symmetry is reduced to Oh due to the presence of neighboring ions (red), the d orbitals split into lower energy t2g and higher energy eg states. The energy difference, ∆CF, is the crystal field splitting. . 12 2.2 After crystal field splitting, spin-orbit coupling further splits the t2g orbitals with spin degeneracy into lower energy j = 3/2 states and higher energy j = 1/2 states...... 18 2.3 When t = 0 at half-filling, there is one electron per site in the ground state. Perturbing to second order in t U, charge fluctuation is allowed when the spins on nearby sites are antiparallel. The result of this perturbation may either result in the original spin configuration or a configuration with reversed spins...... 20

3.1 (a) Crystal lattice for double perovskite A2BB’O6. (b) The simpliﬁed tight- binding model takes hopping between xy orbitals (purple) on B’ sites within an xy plane. Similarly, zx orbitals are active in zx planes, and yz orbitals are active in yz planes...... 28

viii 3.2 (a) FCC lattice decoupled into four inequivalent sites shown by four different colors. (b) The orbital ordering pattern driven by both JSE and V constrains the direction of orbital angular momentum. The magnetization operator is shown as M = 2S L. (c) The zero temperature phase diagram shows − phases where the spin S and orbital L moments in each plane are collinear and the moments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing orbital repulsion V between sites reduces the minimum strength of Hund’s coupling required to induce FM. (d) Mean field values for the bottom sites (black, yellow) are shown as a function of temperature. The nyz orbital (red) has the largest occupancy followed by the xy orbital (blue). (e) With JSE = 0, we calculate the orbital ordering temperature To and effective Curie moment enhancement µeff for different values of V ...... 30 1 3.3 Typical susceptibility, χ = 3 (χxx + χyy + χzz), and inverse susceptibility are plotted against temperature. The susceptibility curves are shown both without the correction due to covalency, γ = 1, and with the correction, γ = 0.536. We have chosen JSE = 0 and left V finite to illustrate the consequence of high temperature orbital order on the susceptibility. By choosing JSE = 0, we show that although Tc = 0 while To = 0, the fitted Curie- 6 Weiss temperature appears to be negative. Note that a single Curie-Weiss fit cannot span the entire range below To...... 35 3.4 (a) The canted ferromagnetic solution to equation (3.6) is shown. (b) For J > 0 and K = 0, susceptibility along [110], [110], and [001] is shown for the antiferromagnet with φ = π/4. (c) For J > 0 and K > 0, the susceptibility − diverges at Tc. The canting angle satisfies π/4 < φ < 0. Note that the − Curie-Weiss law still holds at temperatures well above J and K, and the Curie-Weiss intercept is still negative...... 37 3.5 (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent the most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature phase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and FM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in each region. Increasing orbital repulsion V moves the phase boundary between AFM 4-sublattice phase and the AFM [110] phase down to favor the AFM 4-sublattice phase. The phase boundary between the AFM 4-sublattice phase and the FM [100] phase moves up in favor of the AFM 4-sublattice phase...... 40 3.6 Characteristic inverse susceptibility (blue/green) and orbital occupation (purple) curves are plotted against temperature for the three phases in Fig. 3.5: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility −1 −1 is averaged over all three directions, χ = 3(χxx + χyy + χzz) , and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured above each plot...... 44

ix 4.1 (a) The single site total angular momentum is zero in both the jj and LS coupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (4.3) pitting spin-orbit coupling λ against superexchange JSE where λ is the spin-orbit coupling energy scale and JSE is the superexchange energy scale with z being the coordination number. Starting with a van Vleck phase with no atomic moments at large λ we find a triplon condensate at k = ~π for all values of the Hund’s coupling JH /U. The intermediate regime where λ zJSE has not been explored. At large JSE we obtain effec- ≈ tive magnetic Hamiltonians that have isotropic Heisenberg spin interactions (antiferromagnetic for small JH /U and ferromagnetic for large JH /U) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and large JSE/λ regimes...... 52 4.2 (a) The Norb = 2 model is an approximation of oxygen mediated electron hopping between t2g orbitals in a simple cubic lattice. Both dxy and dyz orbitals participate in hopping along the y direction. (b) The Norb = 1 model is an approximation of direct hopping between t2g orbitals on the face of a face-centered cubic lattice. The dxy orbitals are most relevant for hopping in the xy plane...... 53 4.3 The Hamiltonian in (4.1) is solved for a two-site system. The local total angular momentum squared on one site, J 2 , is plotted for small and large h i i values of Hund’s coupling, JH /U = 0.1 and JH /U = 0.2, for the three types of hopping matrices used in the text. (a-b) Hopping using Norb = 3 produces sizable local moments. For small Hund’s coupling, the local moment gradually forms as t is turned on. For large Hund’s coupling, there is an abrupt formation of large local moments due to an energy level crossing. (c- d) Hopping using Norb = 2 produces qualitatively similar behavior to the Norb = 3 case. (e-f) Hopping using Norb = 1 produces negligible moments. 55 1 4.4 (a) The virtual process leaves the first site in a low spin, S = 2 , configuration and results in antiferromagnetic superexchange. (b) The virtual process 3 leaves the first site in a high spin, S = 2 , configuration and results in ferromagnetic superexchange...... 56 4.5 Energy eigenvalues of the two-site superexchange Hamiltonian (4.5) are plotted for (a) Norb = 3 using (4.6), (b) Norb = 2 using (4.8), and (c) Norb = 1 using (4.10). In addition to a spin-AF ground state, a spin-F ground state can be favored when Hund’s coupling is large in both the Norb = 3 and Norb = 2 models...... 59 4.6 Expectation values of different angular momentum correlators are plotted for the two-site effective Hamiltonian in (4.3) using the three different Norb 2 models with the parameterization λ = cos θ, t /U = sin θ, JH /U = 0.1 and λ/JH = 1. The Norb = 3 model features full rotational symmetry while the Norb = 2 and Norb = 1 models only have one axis of rotational symmetry to make the z correlators different than the x and y correlators. The effect of increasing JH /U is to push the crossing point from spin-AF to spin-F behavior further left in these plots...... 61

x 4.7 Orbital frustration is graphically illustrated for the Norb = 2 model. The orbitals shown on the vertices of the plaquette are the doubly occupied orbital on each site in a square lattice. Once the first bond, labeled as 1, is chosen to be of a particular type, either (a) AF or (b) F, the next bonds, labeled as 2, are immediately fixed by this choice. The result is that the last bond on the plaquette, labeled as 3, then takes a configuration which is neither the most energetically favorable AF bond nor the most energetically favorable F bond...... 63 4.8 The triplon condensation mechanism is graphically illustrated. When there exists a triplon excitation on a site, superexchange can move the excitation to neighboring sites. This effective hopping causes the triplon’s energy to disperse in k-space. When superexchange becomes large enough, condensation of triplon excitations occurs as the bottom of the triplon band becomes lower in energy than the original Ji = 0 level...... 67 4.9 Energy levels of the effective Hamiltonian HSE+HSOC appearing in equations (4.18) and (4.19) with the parameterization λ = cos θ and JSE = sin θ. The levels are labeled by their good quantum numbers. In the θ = 0 limit, the eigenstates of spin-orbit coupling are used, and, in the θ = π/2 limit, the eigenstates of the spin-orbital superexchange Hamiltonian are used. The interpretations of the states are discussed in the main text...... 72

5.1 The GGA band structure without spin-orbit coupling is shown. A tight- binding model with t2g Wannier orbitals is ﬁt to the three bands pictured...... 79 5.2 The triplet excitation spectrum is plotted using a typical value for the spin- orbit gap in 5d oxides of λ/2 = 200 meV...... 81

B.1 Examples of superexchange processes are shown graphically. Of the three t2g orbitals shown by three levels, the active orbitals along a particular bond direction are highlighted in green. (a) Ferromagnetic spin exchange occurs when only one site contains an electron on the active orbital. The virtual d2 state is in an S = 1 configuration. (b) Antiferromagnetic spin exchange occurs with the overlap of half filled orbitals. The virtual d2 state is in a total S = 0 configuration...... 98

C.1 The ﬂow of angular momentum is graphically shown where ingoing arrows are incoming angular momentum and outgoing arrows are outgoing angular momenta. Wigner-3j symbols and Clebsch-Gordan coeﬃcients are vertices with three legs while the scalar contraction of four Wigner-3j symbols (right) is a Wigner-6j symbol [2,3]. (a) Equation (C.17) is shown in graphical form † for the Tζ part of the equation. A J = 0 state is decomposed into its L = 1 l s and S = 1 components which are acted on by the (Li)m and (Si)σ operators. The resulting L = 1 and S = 1 are combined together to give a J = 1 state with quantum number ζ = m + σ. (b) The projection of equation (C.16) to (C.18) conserves angular momentum. Equation (C.19) will appear similarly except that 1m0 and 1m add to yield LM instead...... 108

xi Chapter 1 Introduction

1.1 Historical Motivation

Two sets of topics have come to dominate the study of electronic systems in condensed matter: strong correlations and strong spin-orbit coupling. Strong correlations in electronic systems refers to situations where interactions between particles drive the system away from the Fermi liquid regime. Phenomenology associated with strong correlations typically includes magnetism, high-temperature superconductivity, Mott physics, and more. Many of the most recent advancements have been in areas involving strong spin-orbit coupling. The associated phenomenology includes topological insulators, topological superconductors, and

Weyl and Dirac semimetals. While each of these subjects has drawn an enormous amount of interest individually, systems which combine these two aspects have come to the forefront of condensed matter research in the last decade. [1]

From a materials perspective, there are many routes to realizing electronic systems with strong correlations and strong spin-orbit coupling. Strong correlations typically occur in compounds containing atoms with valence d or f shells, ie. transition metals and rare earth elements. Strong spin-orbit coupling intrinsically occurs in materials with heavy nuclei, including materials with d and f shell atoms. Materials with heavy transition metal ions then naturally combine the two aspects.

The most well known class of transition metal compounds are the transition metal oxides which consistently support strongly correlated phenomena. While magnetism has been a staple of research in transition metal oxides, only recently have the eﬀects of strong spin-

1 Figure 1.1: Conventional band insulators and simple metals are found when spin-orbit coupling and Coulomb interactions are small. When spin-orbit coupling is tuned to be large, the result is still either a band insulator or metal, however, the result may be topologically non-trivial. When Coulomb interactions are tuned to be large, the result is a Mott insulator. Both 4d and 5d transition metal oxides combine both strong spin-orbit coupling and strong correlations. Adapted from reference [1].

orbit coupling drawn significant interest. The iridates (iridium compounds with oxygen; see [4] for a review) were perhaps the first class of transition metal oxides studied where the effects of spin-orbit coupling were fully appreciated due to some pivotal observations.

First was the experimental observation that Sr2IrO4 is a Mott insulator. By comparison,

Sr2RhO4 is metallic, and the bandwidth W was expected to increase moving from Rh to Ir. But reducing the bandwidth is key to achieve a Mott insulator as the Coulomb energy scale U needs to dominate. Density functional theory calculations could only achieve an insulating ground state if spin-orbit coupling was included [5]. This turned Sr2IrO4 from

5 effectively a three band system with five electrons (t2g configuration) to an effective half-

ﬁlled j = 1/2 band with a narrow bandwidth. Due to this eﬀect, Sr2IrO4 was dubbed a “spin-orbit assisted” Mott insulator.

The importance of spin-orbit coupled j = 1/2 bands on the magnetic properties of the iridates was quickly realized. In contrast to spin s = 1/2 systems, the j = 1/2 systems are an entangled mixture of spin and orbital characters. Spins are insensitive to the anisotropies of

2 Figure 1.2: Honeycomb structure formed out of edge-sharing oxygen (purple) octahedra each enclosing a transition metal site (yellow). The Kitaev model is formed from three types of bond-dependent Ising interactions between sites (yellow) on the honeycomb lattice. The Ising interactions are Si,xSj,x along red bonds, Si,ySj,y along green bonds, and Si,zSj,z along blue bonds.

crystal structures but orbital degrees of freedom are explicitly anisotropic. These spin-orbit coupled j = 1/2 Mott insulators then allow for highly anisotropic magnetic interactions which would otherwise not be possible in pure spin systems. While these anisotropic interactions do have measurable effects on the magentism in Sr2IrO4, they do not play a fundamentally important role. Advancements in the study of these highly anisotropic magnetic models actually came from iridate materials with honeycomb geometries, such as those found in Li2IrO3 and Na2IrO3. Nominally, honeycomb lattices are simple because they are bipartite and support conventional Néelorder from antiferromagnetic Heisenberg interactions. Yet the calculation for the effective magnetic interactions between nearest-neighbor sites with j = 1/2 moments showed there was no Heisenberg term due to a cancellation of superexchange paths in edge-sharing octahedra [6]. Instead, only bond-dependent Ising interactions result with three types of interactions: Si,xSj,x, Si,ySj,y, and Si,zSj,z. See Fig- ure 1.2. For example, along an x-bond, the interaction only involves the x components of

the j = 1/2 pseudo-spin. The importance of the resulting model is that it had been solved

3 exactly by Kitaev and shown to have a quantum spin liquid ground state [7]. This prompted the experimental search to realize the Kitaev model in transition metal oxides.

The j = 1/2 iridates have provided many opportunities for new types of magnetism due

5 to strong spin-orbit coupling, yet these materials in the t2g conﬁguration represent only a fraction of the possibilities. The central theme of this work is to explore the possibilities of

n other electron counts, t2g, in the context of transition metal oxides. Due to the anisotropic nature of orbital degrees of freedom, these strongly spin-orbit coupled systems will result in unexpected magnetic order in the other electron counts.

1.2 General Overview

In this work, we study two classes of strongly spin-orbit coupled transition metal oxides. However, there is a large amount of background material required to begin work on magnetism in these compounds. For this reason, the next chapter, Chapter2, provides background information required for the later chapters. This amounts to a standalone introduction to some of the general aspects of transition metal oxides. We will begin with a phenomenological description of the Hubbard model and how it applies to transition metal oxides. Unlike the one band (single orbital) Hubbard model, the presence of multiple orbitals per lattice site brings in new terms in the Hamiltonian and new possibilities for ordering. Next we discuss crystal ﬁeld eﬀects and how symmetry can be used to understand them. Then we review how short-ranged Coulomb interactions act in multi-orbital systems.

Spin-orbit coupling is then introduced as an important relativistic correction, especially in

4d and 5d systems. Finally, we compare how magnetism occurs in the band limit versus the Mott limit to set the stage for why our results in later chapters are non-trivial. Readers who are already familiar with these basic aspects of transition metal oxides may skip the chapter without loss of continuity.

Next we address the two main topics of research presented in this document: (1) d1

and d2 double perovskites and (2) d4 Mott insulators. Typically, each transition metal site in the crystal lattice contains a magnetic moment due to electron spin degrees of freedom,

4 electron orbital degrees of freedom, or both. Then nearby magnetic moments interact through spin-spin interactions which causes magnetic order at some ordering temperature

Tc. In the case of the j = 1/2 iridates on a honeycomb lattice, we just showed an example where this paradigm breaks down. Even though there are preexisting pseudo-spin 1/2 magnetic moments and the honeycomb lattice geometry supports ordering, the novel nature of superexchange interactions between j = 1/2 moments prevents magnetic order at all temperatures. In the classes of materials considered here, we will indeed ﬁnd magnetic order, but the paradigm for how to achieve magnetic order will be altered.

Here we give four key changes to this paradigm from our investigation of these (1) d1

and d2 double perovskites and (2) d4 Mott insulators. First, the d4 systems we consider

do not nominally possess preexisting magnetic moments, and we explore how magnetic

moments may be generated and also order order magnetically. Second, in both double

perovskites and d4 Mott insulators, the interactions between sites do not take the form

of spin-spin interactions. Instead, since orbital degrees of freedom are also involved, we

must consider spin-orbital superexchange, and this type of superexchange simultaneously

supports both ferromagnetic and antiferromagnetic interactions between sites. Third, the

d1 and d2 double perovskites we consider support both magnetic order at some temperature

Tc and orbital order at another temperature To. When spin-orbit coupling is involved, these two types of orders are tied together, and orbital interactions play just as important of a

role in determining the magnetic order as the magnetic interactions do. Fourth, although

strong Hund’s coupling is usually a viable way to achieving ferromagnetic interactions and

consequently ferromagnetic order, spin-orbit coupling and orbital geometry play a crucial

role in determining when ferromagnetism is allowed in spin-orbital systems.

1.3 Summary of Results

In Chapter3, we consider Mott insulators with either one or two electrons in the t2g (d) shell. While there are potentially many compounds to study, we focus on the strongly spin-

1 2 orbit coupled double perovskites A2BB’O6 with B’ magnetic ions in either d or d electronic

5 conﬁguration and non-magnetic B ions. The reason for looking at these compounds is that there are several experimental puzzles which have not yet been resolved by theory. These puzzles include the predominance of ferromagnetism in d1 versus antiferromagnetism in d2

systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials,

and the size of eﬀective magnetic moments. As we will show, the resolutions to these

puzzles lie in the orbital degrees of freedom. We develop and solve a microscopic model

with both spin and orbital degrees of freedom within the Mott insulating regime at ﬁnite

temperature using mean ﬁeld theory. The interplay between anisotropic orbital degrees of

freedom and spin-orbit coupling results in complex ground states in both d1 and d2 systems.

Although the models for these two electron counts are similar, their zero temperature phase

diagrams are quite diﬀerent. We show that the ordering of orbital degrees of freedom

in d1 systems results in coplanar canted ferromagnetic and 4-sublattice antiferromagnetic

structures. In d2 systems we ﬁnd additional collinear antiferromagnetic and ferromagnetic

phases not appearing in d1 systems. At ﬁnite temperatures, we ﬁnd that orbital ordering driven by both superexchange and Coulomb interactions may lead to both deviations from

Curie-Weiss law both at high temperature due to orbital ordering and at low temperature due to the anisotropy induced by orbital order.

Our results from Chapter3 are immediately applicable to experiments on 5 d1 and 5d2

double perovskites. Despite calculations by another group [8] which suggest ferromagnetism

should also be common in 5d2 double perovskites, the calculations presented here clearly

show that ferromagnetism is favored in 5d1 systems but not 5d2 systems. We go further to

provide a clear explanation for this general trend in terms of the constraints on magnetic

interactions imposed by orbital conﬁgurations. Others groups have previously canted fer-

romagnetic phases [9–11] in models for d1 spin-orbit coupled double perovskites. However,

we go further to show how orbital order resolves the paradox of having a material with a

diverging magnetic susceptibility (typically associated with a ferromagnet) yet a negative

Curie-Weiss temperature (typically associated with an antiferromagnet). Finally, we pro-

pose that the orbital order found in the canted ferromagnetic phase may also be present in

antiferromagnetically ordered compounds as well.

6 In Chapter4, we discuss the next class of materials which are transition metal oxides with four electrons in the t2g shell. The rationale for studying these transition metal oxides

4 with the t2g electronic conﬁguration is that they are expected to be nonmagnetic atomic singlets. In the weakly interacting regime, spin-orbit coupling creates a situation with a fully

ﬁlled j = 3/2 band and an empty j = 1/2 band which leads to the absence of magnetism.

In the Mott insulating regime, the total L = 1 and S = 1 angular momenta anti-align

on every site to give a total J = 0 on every site and the absence of magnetic moments.

Starting with the full multi-orbital electronic Hamiltonian, we calculate the low energy eﬀective magnetic Hamiltonian which contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling JSE/λ, we obtain a phase transition from nonmagnetic atomic singlets to novel magnetic phases. The phase transition is always to an antiferromagnetic ground state which is surprising since large Hund’s coupling typically induces ferromagnetic interactions and ferromagnetic order. Spin-orbit coupling plays a non-trivial role in generating a triplon condensate of weakly interacting excitations at antiferromagnetic ordering vector ~k = ~π, regardless of whether the local spin interactions are ferromagnetic or antiferromagnetic. In the large JSE/λ regime, the localized spin and orbital moments produce anisotropic orbital interactions that are frustrated or constrained even in the absence of geometric frustration.

Our results from Chapter4 help resolve a conflict in the literature about whether ferromagnetic condensation will occur in strongly spin-orbit coupled d4 Mott insulators. In contrast to Ref. [12], we show that symmetry actually requires that ferromagnetic interactions turn into antiferromagnetic interactions as spin-orbit coupling increases. This highly unexpected result shows that rotationally invariant single-site effects can effectively negate the sign of magnetic interactions in spin-orbital systems, a feature which is impossible in spin-only systems. The immediate consequence for experiments is that most d4 Mott insu-

lators with strong spin-orbit coupling will not show ferromagnetism even if Hund’s coupling

is large.

In Chapter5, we apply the triplon condensation formalism from Chapter4 to the ma-

terial Ba2YIrO6. Using density functional theory, we calculate tight-binding parameters in

7 the eﬀective triplon Hamiltonian. The calculation shows that despite the experimental observation of magnetism in this compound, it is unlikely the result of superexchange induced condensation. Lastly, we comment on other possible sources of magnetism in Ba2YIrO6. Our results from this chapter complement a DMFT study by Ref. [13] and provides a simpler explanation for why 5d4 double perovskites with only magnetic B’ ions will not likely develop magnetism due to superexchange induced condensation.

8 Chapter 2 Transition Metal Oxides

2.1 Materials and the Hubbard Model

We begin by considering a simple starting Hamiltonian for non-relativistic electronic systems in ﬁrst quantized notation. It consists of the kinetic energy, an external potential, and electron-electron interaction.

2 X pi X 1 X H = + V (~ri) + U(~ri, ~rj) (2.1) 2me 2 i i i6=j

Here the sum over i is over all electrons in the system. In this starting example, the

e2 interaction term is simply the Coulomb interaction, U(~ri, ~rj) = , and the principle |~ri−~rj | source of the external potential comes from the positively charged nuclei of the ions. For simplicity, motion of the nuclei is not considered. At even a simple level like this, no closed form solution to this many-body problem is possible, and approximations are needed.

The first approximation is to restrict our calculations to only the electrons which are most relevant for determining the properties of solid-state materials systems. For an isolated atom in free space with many electrons, most of the electrons will be highly localized due to the strength of the Coulomb interaction from the nuclear charge, and they will not contribute to most observable properties of the atom. Only the highest energy electrons will play a role. Thus the first approximation is that only the valence electrons on each ion will be important, and we should seek a new effective model for just the valence electrons.

As the starting example contained the sum of two single-electron terms and an electron-

9 electron interaction, our new effective Hamiltonian should contain both a quadratic piece serving as the effective kinetic energy and a quartic piece serving as the new electron-electron interactions. Additionally, we know two things which will greatly assist in the simplification of the model. First, the potential V is periodic in space due to the crystalline nature of solids, and the new Hamiltonian must express this periodicity in some way. Second, the valence electrons in the d shells of these transition metal ions tend to be localized to their ions. We then consider a Hamiltonian where the relevant single-particle states that an electron may occupy are localized near transition metal ions. These single particle states † are expressed using fermion creation and annihilation operators. Let ci,α be a fermion creation operator for an electron localized to the α-th state on site i, and the annihilation operator ci,α is defined similarly. Then effective model will take the form below.

X X † X X † † H = tij,αβ c c + Uijkl,αβγδ c c c c (2.2) − i,α j,β i,α j,β k,γ l,δ ij αβ ijkl αβγδ

The quadratic term is the so called “tight-binding” term. (The minus sign in front is convention.) It acts as an eﬀective kinetic energy giving the electrons an energy dispersion in k-space. The quartic term is then the eﬀective Coulomb interaction between fermionic states.

There are two questions that immediately arise. The first question is obvious: how pre- cisely is this effective model obtained? Typically in solid-state physics, density functional theory (DFT) tools are used to approximate the t values. The effective single-electron

“Wannier” states states are computed within DFT, and then the matrix elements of t are determined by computing an integral involving the Wannier states, ψi,α V ψj,β .[14] h | | i While this method has been successfully used for decades to give good estimates of the t

values, obtaining accurate U values is still a current subject of research today. Unfortu-

nately, this limitation is one of several major impediments to microscopically determining

the properties of strongly correlated systems in solid-state physics.

This leads to the second question: what physical principles could be used to determine

the eﬀective model in lieu of microscopically exact calculations? Our ﬁrst assumption is

10 that the single-electron wavefunctions decay quickly in space, and only the overlaps between wavefunctions on atoms nearby will be important in determining t. The next assumption is that the long-ranged Coulomb interaction is screened so that the short-range part of the

Coulomb interaction is most relevant, although this assumption breaks down in the Mott limit. The simplest non-trivial model is where t is only nonzero between nearest neighbor sites and U is only relevant on-site. This is referred to as a Hubbard model, and the simplest

Hubbard model is given below.

X X † X H = t c c + h.c. + U ni,↑ni,↓ (2.3) − i,σ j,σ hiji σ=↑,↓ i

Here ij refers to all pairs nearest-neighbor pairs of sites, and the number operator is deﬁned h i † as ni,σ = ci,σci,σ. This simple Hubbard model is arguably the single most important model in all of condensed matter physics. Here each lattice site only possesses four types of states:

unoccupied 0 , spin-up , spin-down , or double occupancy . However, in addition | i |↑i |↓i |↑↓i to the spin degree of freedom, orbital degrees of freedom will need to be introduced in order

to correctly describe most transition metal oxides.

2.2 Crystal Field

In the limit of a single atom, the single-electron states on that atom are described

by atomic orbitals. Since the isolated atom is in a spherically symmetric environment,

we can label the single-electron states by orbital angular momentum. Equivalently, the

single-electron states can be labeled by the type of orbital corresponding to that angular

momentum. (ie. s corresponds to L = 0, p corresponds to L = 1, etc.) For the transition

metals, the valence electron shell consists of L = 2 or d orbitals so that there are 5 single-

electron orbital degrees of freedom per atom.

While it is clear that this works for an isolated atom, ions in a crystal are not in

spherically symmetric environments. In transition metal oxides, the transition metal ions

are surrounded by neighboring oxygen ions, and the presence of these ions can change

the relative energies of the d orbitals on the transition metal ion. The most common

11 Figure 2.1: d orbitals on the transition metal site (purple) are degenerate under spherical symmetry. When this symmetry is reduced to Oh due to the presence of neighboring ions (red), the d orbitals split into lower energy t2g and higher energy eg states. The energy diﬀerence, ∆CF, is the crystal ﬁeld splitting.

situation in oxides is where the transition metal ion is surrounded by six oxygens forming

an octahedral complex shown in Figure 2.1. Also shown in Figure 2.1 is the splitting of

d states due to this octahedral crystal ﬁeld. This occurs due to bonding of the transition

metal ion’s d orbitals with the neighboring oxygen p orbitals. Since two orbitals labeled as eg extend directly toward the neighboring oxygen sites, the resulting σ-bonding/antibonding is strong. However since the three orbitals labeled as t2g extend outward at 45 degrees from the neighboring p sites, their π-bonding is weak compared to that of the eg orbitals. This diﬀerence in bonding strength causes a diﬀerence in the energy shift of the two types of d

orbitals. The result is that the eﬀective antibonding eg orbitals move higher in energy than

the antibonding t2g orbitals. The diﬀerence in energy between states is octahedral crystal ﬁeld splitting of d-orbitals.

The eﬀect considered so far is essentially a linear correction in perturbation theory due

to bonding/antibonding. What are the eﬀects of higher order corrections? Or what about

other eﬀects which could split otherwise degenerate orbitals? It would be extremely diﬃcult

12 to track down every microscopic detail. Clearly a more economical approach is needed.

Group theory provides the framework to resolve these problems. In the present problem, we can identify that although the transition metal ion is not in a spherically symmetric environment, it is in an environment with octahedral symmetry. More speciﬁcally, the point group symmetry associated with transition metal site is Oh (full octahedral symmetry).

Both the eg orbitals and t2g orbitals are labeled as such because they correspond to the Eg

and T2g irreducible representations of the Oh point group. [15] We can then be sure that

as long as Oh symmetry is present, the eg and t2g levels will not further split. However, there are two notes about using group theory to determine the results. First,

it cannot determine whether the eg or t2g level is higher in energy. This can only be

accomplished through calculation. Second, the resulting eg and t2g wavefunctions from the crystal ﬁeld splitting do not necessarily correspond to atomic orbitals. The only requirement is that the resulting wavefunctions share a minimum set of symmetry properties with their atomic orbital counterparts.

We have just shown how group theory can be applied to understand why degenerate orbitals split when symmetry is reduced. It is easy to imagine that crystal ﬁeld splitting will alter other properties as well. For example, the eﬀective Hubbard model in equation (2.2) will now involve fewer degrees of freedom since some orbitals become energetically irrele- vant. More importantly, the tight-binding parameters tij,αβ and the Coulomb interaction

parameters Uijkl,αβγδ in the eﬀective Hubbard model may also change. In principle, there are well established methods to reliably estimate tight-binding parameters using electronic

structure calculations which makes the tight-binding term less challenging to deal with.

However, the eﬀective Coulomb interactions should be handled up-front.

2.3 Coulomb Interactions

A recurring theme in transition metal oxides is the importance of strong correlations

due to Coulomb interactions. In the chapters to come, we will frequently be considering

materials where the eg orbitals are higher enough in energy to project out, and the t2g

13 orbitals are the relevant orbital degrees of freedom. Here we derive the eﬀective on-site

Coulomb interaction for t2g orbitals for use in the later chapters. [16] In other words, we derive the Hubbard interaction for triply degenerate t2g orbitals, a multiorbital version of P the Hubbard interaction (ie. U i ni,↑ni,↓) appearing in the single orbital Hubbard model in equation (2.3). Coulomb interactions between diﬀerent sites will be treated in later chapters

as necessary.

In this derivation, let the indices (a, b, c) be an arbitrary permutation of the t2g orbital

labels (yz, zx, xy). The corresponding wavefunctions are denoted as φa, φb, and φc. Since the orbitals in a solid do not necessarily correspond to atomic orbitals, we can only assume

a minimal set of requirements about the t2g orbitals. The assumptions are then that these

∗ orbitals are real-valued (ie. φa = φa), and they transform as the T2g irreducible representa-

tion of the Oh point group. This immediately gives some important symmetry properties

for t2g orbitals. For example, let σa be the a-plane mirror symmetry (ie. σxy is a mirror

about the xy plane). Then σaφa = φa and σaφb = φb for a = b. − 6 Now we consider the matrix elements of the Coulomb interaction between t2g states. Let αβ refer to a two electron state where the ﬁrst electron is in state α and the second | i | i electron is in state β . Then a generic matrix element between the two electron states αβ | i | i and γδ is given below. | i Z αβ V γδ = dr1 dr2 φα(r1)φβ(r2)V (r1 r2)φγ(r1)φδ(r2) (2.4) h | | i −

2 Although it is easy to calculate matrix elements using V (r1 r2) = e / r1 r2 , the − | − | Coulomb interaction between electrons is screened by other electrons in the solid. This way, the eﬀective Coulomb interaction may diﬀer from the bare Coulomb interaction. While the exact form may be material dependent, we may still proceed using symmetry arguments.

The screened Coulomb interaction V must still be invariant under the symmetries of the

Hamiltonian. As an example, V (r1 r2) should at least be invariant under the mirror − plane operation σa. Given the symmetries that the interaction must obey, we will be able to determine which matrix elements are generally zero and which matrix elements are not.

Since there are only three types of orbitals, there can be at most three mutually distinct

14 indices appearing in the matrix elements αβ V γδ . This results in four types of cases. h | | i First, consider the case where all indices are identical. This is the direct Hubbard interaction

(where both electrons are on the same orbital but have opposite spins) and is deﬁned as

aa V aa U. Second, consider the case where three indices are the same and one index h | | i ≡ is different: aa V ab . This must be identically zero since it is odd under the mirror plane h | | i operation σb. More generally, the same reasoning holds for any permutation of the four indices (a, a, a, b). Third, consider the case where two of the indices are the same and the other two are different, aa V bc . Again this matrix element must be identically zero due h | | i to mirror symmetry, and all permutations of the four indices must also be zero. Fourth, consider situations where there are two pairs of indices which are the same. They are defined in the following way: ab V ab U 0, ab V ba J, and aa V bb J 0. However, since h | | i ≡ h | | i ≡ h | | i ≡ the wavefunctions φ(r) are real valued, it immediately follows that J = J 0. This leaves a total of three undetermined Coulomb parameters: U, U 0, and J.

Given the deﬁnitions of the Coulomb parameters, we may now write down the second- † quantized form of the on-site Coulomb interaction (ie. Hubbard interaction). Let φασ be a fermion creation operator for a state in orbital α and spin σ.

X X X X H = φ† φ† V γσ3,δσ4 φ φ (2.5) Hubbard ασ1 βσ2 ασ1,βσ2 δσ4 γσ3 ασ1 βσ2 γσ3 δσ4

The matrix elements between two-particle fermionic states are given by

γσ3,δσ4 V = (ασ1)(βσ2) V (γσ3)(δσ4) (2.6) ασ1,βσ2 { | | } using antisymmetric wavefunctions AB = √1 ( AB BA ). The result simpliﬁes to the | } 2 | i − | i following.

γσ3,δσ4 V = αβ V γδ δσ σ δσ σ αβ V δγ δσ σ δσ σ (2.7) ασ1,βσ2 h | | i 1 3 2 4 − h | | i 1 4 2 3 By explicitly writing out the Hubbard interaction in terms of the Coulomb parameters, the

15 following form results.

X 0 X 0 1 X X HHubbard = U na↑na↓ + U na↑nb↓ + (U J) naσnbσ − 2 a a6=b a6=b σ (2.8) X X J φ† φ† φ φ + J φ† φ† φ φ − a↑ b↓ b↑ a↓ a↑ a↓ b↓ b↑ a6=b a6=b

While this form is very general, it isn’t practical to leave three independent parameters characterizing merely one type of term in a model Hamiltonian. The most common ﬁx is to assume spherical symmetry as in the case of an isolated transition metal ion, which amounts to setting U 0 = U 2J.[16] Given that the result must be spherically symmetric, − the on-site Coulomb interaction can be rewritten in a form explicitly showing this. This

is accomplished using a total number operator N, total spin operator S, and total orbital

angular momentum operator in the t2g orbital subspace L = (Lx,Ly,Lz).

X N = naσ (2.9) a σ

1 X X † S = 2 φaστσσ0 φaσ0 (2.10) a σ σ0

X X † Lα = i εαβγφβσφγσ (2.11) β γ σ Here τ is a vector of Pauli matrices and ε is the Levi-Civita tensor. Note there is a commonly

used mapping between t2g orbital indices and Cartesian indices (xy z, yz x, and ↔ ↔ zx y). This is in fact due to the similarity between t2g orbital angular momentum and p ↔ orbital angular momentum. However, the t2g orbital angular momentum operators satisfy the negative of the usual commutation relations for angular momenta: L L = iL. (Note × − that ~ is factored out of all angular momentum operators in this text.) A derivation of the

t2g orbital angular momentum operators is provided in Appendix A.1. Given the N, S, and L operators, we may now write the spherically symmetric form of the on-site Coulomb

interaction. The derivation is given in Appendix A.2.

1 5 2 1 2 HHubbard = (U 3JH ) N(N 1) + JH N 2JH S JH L (2.12) − 2 − 2 − − 2 16 Since the symbol J appears in many contexts in magnetism, we have instead used the symbol J JH for Hund’s coupling. →

2.4 Spin-Orbit Coupling

So far we have built up our eﬀective Hamiltonian for t2g orbitals using only non- relativistic terms. This non-relativistic approach is a good approximation for materials with only light nuclei. Transition metal ions have heavy nuclei that generate strong electric

ﬁelds which lead to relativistic corrections to the Hamiltonian. In general, there are three types of corrections: kinetic, Darwin, and spin-orbit coupling. The kinetic term and Darwin terms can be absorbed into the tight-binding model we choose to work with. However, the spin-orbit coupling term acts in a non-trivial way. This is because it reduces the symmetry of our Hamiltonian by explicitly coupling the orbital and spin parts together which would have otherwise been independent.

2 e~ 1 ∂V HSO = S L (2.13) −m2c2 r ∂r ·

Here V is the electric potential the electrons experience due to the positively charged nu-

1 ∂V cleus. For atomic orbitals, the expectation value of r ∂r may be used, and the terms to the left of S L are lumped into an overall positive constant, λ, which characterizes the · strength of spin-orbit coupling for a set of degenerate orbitals. This single-particle eﬀect is most accurately expressed in the form below.

X X † HSO = λ c 0 0 Sσ0,σ Lm0,mc (2.14) m ,σ · m,σ m0,m σ0,σ

Since λ is positive, we would naively expect that states where the orbital angular mo-

mentum was anti-aligned with the spin would be the lowest in energy, and the states where

they were aligned would be the highest in energy. For t2g orbitals, this is exactly opposite to the correct result as shown in Figure 2.2. This is due to the fact that the commutation

relations for t2g angular momentum operators imply they are reversed with respect to the mathematical angular momentum operators for an L = 1 subspace. (See Appendix A.1.)

17 crystal spin-orbit field coupling

eg L S d j=1/2

t2g L S j=3/2

Figure 2.2: After crystal ﬁeld splitting, spin-orbit coupling further splits the t2g orbitals with spin degeneracy into lower energy j = 3/2 states and higher energy j = 1/2 states.

When dealing with strongly correlated systems, we must inherently deal with many-body wavefunctions. For example, when Hund’s coupling is strong, the degenerate multi-electron wavefunctions are labeled by total S and total L, not single particle S and L. We should ask how the single-particle spin-orbit coupling operator eﬀectively acts on these multi-electron states? Due to the fact that S and L are spherical tensors, the Wigner-Eckart theorem guarantees that the projection of HSO to the subspace of total S and total L must be composed of spherical tensors of the same rank. The only operators which satisfy this constraint are proportional to the spin-orbit coupling operators for total S and total L, ie.

S L. The only diﬀerence is that the energy scale λ now acquires a constant prefactor ∝ · which is equal to the reduced matrix element in the Wigner-Eckart theorem. This prefactor can be calculated for each relevant multiplet of total S and total L.

2.5 Band Limit versus Mott Limit

Up to this point, we have referenced many diﬀerent energy scales: t, ∆CF, U, JH , and λ. Although the relative magnitudes of all of these parameters are important, perhaps the most important ratio is between the tight-binding term, t, and the on-site Coulomb interaction,

U. For simplicity, we just consider the single orbital Hubbard model in equation (2.3) where there are exactly N electrons in the system of N sites (ie. half-ﬁlling). In the band limit, 18 U 0, the Hamiltonian is quadratic and exactly diagonalized by Fourier transform → X X H = E(k) nk,σ (2.15) k σ

where E(k) is the energy dispersion of the single-particle electron states in k-space. In the

Mott limit, t 0, the problem is already diagonal in the site occupation basis. → X H = U ni,↑ni,↓ (2.16) i The solutions in the two limits diﬀer substantially. In the band limit, the total energy is

just the sum of the energies of the individual electrons in the system, and the non-degenerate

ground state simply consists of ﬁlling the lowest N occupied energy levels. In the Mott limit,

the total energy explicitly depends on the correlations between electron locations, and the

ground states all have exactly one electron per site. Notice that this condition is fulﬁlled

without regard to spin on each site so that the t = 0 limit has an extensive degeneracy of

2N .

Due to the diﬀerence in degeneracy, the route to magnetism in these two limits diﬀers

substantially. In the band limit, susceptibility in the random phase approximation is given

by (0) 2 χ (q) χ(q) = (gµB) (2.17) 1 Uχ(0)(q) − (0) P f(p)−f(p+q) where χ (q) = p E(p+q)−E(q) is the magnetic magnetic susceptibility of the non- interacting system (with U = 0). The Fermi surface then becomes unstable to magnetism when the generalized Stoner criterion Uχ(0)(q) 1 is satisﬁed. [17] Unlike in the band limit, ≥ the presence of degeneracy in the Mott limit always allows for some type of magnetism to

occur for arbitrarily small t. However, when competing interactions, geometric frustration,

or low dimensionality is involved, the result is not necessarily classical magnetic order. In

the Mott limit, these eﬀects can lead to exotic ground states such as valence-bond crystals,

spin liquids, and multipolar order. [1]

Aside from degeneracy, there is another crucial diﬀerence in the way magnetism forms in

these two limits. In the band limit, the details of the Fermi surface determine which q vector

19 Figure 2.3: When t = 0 at half-filling, there is one electron per site in the ground state. Perturbing to second order in t U, charge fluctuation is allowed when the spins on nearby sites are antiparallel. The result of this perturbation may either result in the original spin configuration or a configuration with reversed spins.

ﬁrst results in an instability to the formation of magnetic ordering. For instance, when U

is large enough, an instability occurs at q = 0 resulting in ferromagnetism. For q = 0, an 6 instability will ﬁrst occur if the nesting condition, E(q + k) = E(k), is met on the Fermi − surface. So it would appear that the band limit allows for all types of classical magnetic

order as long as the Fermi surface is shaped appropriately. This is in stark contrast with

the Mott limit at half-filling. When t U, degeneracy in the Mott limit is broken due to perturbative charge fluctuations between nearby sites. These fluctuations are only possible

when the spins on these nearby sites are antiparallel due to the Pauli exclusion principle.

See Figure 2.3. Since the single orbital Hubbard model in equation (2.3) is invariant under

global spin rotations, total spin is conserved. Then the results of these charge ﬂuctuations

in perturbation theory must also be rotationally invariant. The eﬀective Hamiltonian from

these second order perturbation processes is called a superexchange Hamiltonian, and it

has the form given below. (2) 1 H = JSE Si Sj (2.18) SE · − 4 2 Here JSE = 4t /U is positive and dictates an antiferromagnetic interactions between spins. Notice that this result is quite general as it is insensitive to the lattice structure, or the value of t.

We might ask how ferromagnetism is possible in the Mott limit. The first remark is that 20 ferromagnetism can be achieved away from half-filing (ie. the total electron count is either less than or greater than N). In fact, there is a rigorous theorem, the Nagaoka theorem, which proves the ground state is ferromagnetic when exactly one electron is removed from the system at half-filling in the limit that U + . However, even if the possibility → ∞ of doping is excluded, ferromagnetism can still be achieved in the Mott limit if multiple orbitals are involved. This is one of the central themes covered here, and we will focus on microscopic routes to ferromagnetism next two chapters.

21 Chapter 3 d1 and d2 Double Perovskites

3.1 Introduction

Strong spin-orbit coupling in correlated materials has provided new platforms to study quantum spin liquids and correlated topological phases of matter [1,4]. Among the strongly spin-orbit coupled materials include 4d and 5d double perovskites of the form A2BB’O6. Here we study the special cases where only the B’ ions are magnetic with a particular focus on cases where the B’ ion is in the 5d1 or 5d2 electronic conﬁguration. Due to large distances between the magnetic B’ ions, these materials are usually Mott insulators and present a promising class of materials to explore the interplay of spin-orbit coupling and strong correlations.

Here we study magnetic spin-orbital models in the Mott regime applicable to this class of double perovskties. These models involve highly anisotropic interactions due to the orbital degrees of freedom. We ﬁnd that orbital order accompanies magnetic order and may occur at temperatures much higher than the magnetic ordering temperature. This orbital ordering results in deviations from Curie-Weiss behavior both near the orbital ordering temperature and near the magnetic ordering temperature. At low temperatures, we show how the magnetic phases are dictated by the orbital degrees of freedom resulting in unusual magnetic ordering patterns. Our results go beyond previous theoretical works on both 5d1 and 5d2 compounds and address several experimental puzzles.

To set the stage, we contrast these results with what would be observed in systems

22 with other electron counts. In the context of iridates, 5d5 systems have been extensively studied using pseudospin j = 1/2 models. In particular, double perovskites may oﬀer a route to realizing spin liquids and other interesting phenomena in three dimensions [18–20].

Moving to the next electron count, 5d4 systems are quite unique since spin-orbit coupling

dictates that local moments should be absent and magnetism is forbidden. However several

theoretical [12, 13, 21–23] and experimental [24–26] studies have examined the possibility of

obtaining local moments in these otherwise nonmagnetic systems. The next electron count,

3 5d , involves half-ﬁlled t2g shells nominally resulting in an eﬀective spin-3/2 model which is expected to be described as a classically frustrated spin system [27, 28]. The 5d2 and

5d1 electron counts stand out in that they combine aspects of the former electron counts.

First, they possess local angular momenta large enough to support orbital (quadrupolar)

order. Second, they possess unquenched orbital degrees of freedom that result in highly

anisotropic interactions between magnetic ions [29, 30]. Both of these aspects allow for the

orbital degrees of freedom to play a signiﬁcant role in determining the spin, orbital, and

spin-orbital ordering.

In the limit of large spin-orbit coupling, the spin S = 1/2 and orbital Leﬀ = 1 angular momenta add to a total angular momenta of j = 3/2. Within the jj-coupling scheme,

magnetic moments are identically zero due to cancellation of the spin and orbital moments,

M = 2S L = 0. On the other hand, d2 systems allow for a nonzero moment within the − √ 6 LS-coupling scheme of M = 2 µBJ for total J = 2. However both systems are experimentally observed to be magnetic. In this chapter, we visit several questions surrounding the magnetically ordered phases of the 5d1 and 5d2 materials and attempt to self-consistently resolve a number of puzzles described below.

d1 versus d2. Currently there are many experimental examples of ferromagnetic Mott insulating d1 systems for the present class of double perovskites, but there are few ex-

perimental examples (if any) of ferromagnetism in d2. Although we start with the same

electronic model for both d1 and d2 systems, the energetics of the ground states strongly

depend on the electron count. This is reﬂected in how the spin and orbital degrees of

freedom order and provides a qualitative understanding for why ferromagnetism has been

23 repeatedly observed in d1 systems while antiferromagnetic interactions tend to dominate in d2 systems.

Entropy. In the case of Ba2NaOsO6, the entropy recovered through the magnetic transition is R ln 2 instead of R ln 4 expected for j = 3/2 systems [31]. This points toward the existence of a second phase transition at higher temperature where the remaining entropy is recovered. In our picture, this second transition comes from the orbital degrees of freedom and anisotropic interactions that accompany them. In particular, we show that the anisotropic interactions result in orbital order that stabilizes exotic magnetic order. The orbital ordering temperature scale is set both by superexchange interactions and by inter- site Coulomb repulsion. Orbital ordering may occur at temperatures much higher than the magnetic ordering temperatures if this ordering is primarily determined by orbital repulsion.

Susceptibility. The experimental observation of a negative Curie-Weiss temperature in the ferromagnet Ba2NaOsO6 seems to contradict simple mean-ﬁeld arguments [31]. Re- cently, non-Curie-Weiss behavior has been reported in two ferromagnetic d1 compounds

Ba2MgReO6 and Ba2ZnReO6 [32]. In our model, the onset of orbital order causes changes in magnetic susceptibility resulting in non-Curie-Weiss behavior. For situations where orbital order occurs at temperatures much higher than the magnetic ordering temperature, we show how a negative Curie-Weiss temperature could be extracted for the ferromagnetic phase using simple mean-ﬁeld calculations. We then use a simpliﬁed model to show how anisotropy induced by orbital order can result in a proper negative Curie-Weiss intercept despite diverging susceptibility at the transition temperature.

Magnetic Moment. Although the magnetic moment for a j = 3/2 system is nominally

zero, reduction of the orbital moment due to covalency allows for a ﬁnite moment. However,

if orbital order occurs, covalency with oxygen alone does not reproduce the experimentally

determined magnitudes of the magnetic moments in d1 systems. Further corrections are

necessary which may arise from dynamical Jahn-Teller eﬀects [33] and more generally with

mixing of the j = 3/2 and j = 1/2 states.

Lastly, we outline where our calculations stand with respect to other work. Density functional theory studies have so far revealed two important aspects of these compounds.

24 First, they have revealed the importance of oxygen covalency in suppressing the orbital moment so that a net moment results [33, 34]. Second, they have pointed out that spin-orbit coupling and hybridized orbitals play a major role in opening a gap within DFT+U [35–37].

Model Hamiltonian approaches have also shed some light on these materials by using spin- orbital Hamiltonians [11], projecting spin-orbital Hamiltonians to the lowest energy total angular momentum multiplet [8,9], lowest energy doublet [38], and other approaches [10].

In both electron counts, Chen et. al. [8,9] ﬁnd canted ferromagnetism accompanied by quadrupolar order occupies a majority of parameter space. Additionally they ﬁnd a novel non-collinear antiferromagnetic phase in d2 but not d1. Recently this phase was found in d1

as the most energetically favorable antiferromagnetic state [11]. Proposals for both valence

bond ground states [9, 11] and quantum spin liquids [9, 39] also exist.

Our ﬁndings are largely compatible with those of Romh´anyi et. al. [11], and we further

provide a clear interpretation of why these orbital ordering patterns occur, how they dictate

the magnetic ordering, and then extend our calculations to ﬁnite temperature. Like Chen et.

al., we ﬁnd that orbital ordering can occur at temperatures much higher than the magnetic

ordering temperature, however, we provide a clear interpretation of the negative Curie-

Weiss temperature in d1 ferromagnets. Furthermore, our spin-orbital approach includes

mixing between the j = 3/2 and j = 1/2 states induced by orbital order and intermediate

spin-orbit coupling energy scales. Our zero temperature phase diagrams diﬀer from those

of Chen et. al. [8,9], with striking diﬀerences in d2, which we discuss in detail in later

sections. The most signiﬁcant diﬀerence is in the energetics of antiferromagnetism versus

ferromagnetism in d2 systems which gives a qualitative explanation for the broadly observed

diﬀerences in ordering between 5d1 and 5d2 compounds. Finally, we do not consider valence

bond or spin liquid phases in this chapter although both may be applicable to d1 and d2

systems.

Many of our ﬁndings can be tested using multiple probes. At the orbital ordering

temperature, there will be a second order phase transition with a signature in heat capacity

as well as changes in the magnetic susceptibility which are relevant for both powder samples

and single crystals. NMR has recently found evidence of time-reversal invariant order above

25 the magnetic ordering temperature in Ba2NaOsO6 [40]. Resonant X-ray scattering may also provide crucial insights into this hidden order as it is sensitive to orbital occupancy. We show

that time-reversal invariant orbital order occurs in both ferromagnetic and antiferromagnetic

phases we ﬁnd, and we suggest that experimental probes which are sensitive to such order

should also be pointed at the antiferromagnetic compounds.

On the experimental side, many d1 and d2 compounds have already been investigated.

1 The 4d compound Ba2YMoO6 shows no long range magnetic order down to 2 K despite having a large Curie-Weiss temperature θ = 160 K and retaining cubic symmetry which − leads to the conclusion that the ground state consists of valence bonds [41–45]. Among

1 the 5d compounds are ferromagnetic Ba2NaOsO6 [31, 40, 46, 47], Ba2MgReO6 [32, 48],

and Ba2ZnReO6 [32] which is unusual since ferromagnetism in Mott insulators is uncom- mon. There are two additional twists to the story: ﬁrst, negative Curie-Weiss temperatures

have been observed in these ferromagnets, and, second, Ba2LiOsO6 is antiferromagnetic

2 despite sharing the same cubic structure as Ba2NaOsO6 [46]. The d compounds oﬀer a similar platform to search for unusual magnetism, however experimental studies seem to

suggest that antiferromagnetic interactions are dominant in d2 systems. Phase transitions

to antiferromagnetic order are reported in Ca3OsO6 [49], Ba2CaOsO6 [50], Ba2YReO6 [51],

and Sr2MgOsO6 [52, 53] while glass-like transitions are reported in Ca2MgOsO6 [52] and

Sr2YReO6 [54]. There are also several possible singlet ground states: La2LiReO6 [51],

SrLaMReO6 [55], and Sr2InReO6 [54].

3.2 d1 Double Perovskites

Here we develop a spin-orbital model for the d1 double perovskites with magnetic B’ ions

with spin-orbit coupling featuring both spin-orbital superexchange and inter-site Coulomb

repulsion between B’ ions. We then solve the model within mean ﬁeld theory at both zero

temperature and ﬁnite temperature. At zero temperature, we ﬁnd phases with orbital order

and show how this ordering restricts the magnetic order. At ﬁnite temperature, we examine

how orbital order modiﬁes magnetic susceptibility and the Curie-Weiss parameters.

26 3.2.1 Model

In the presence of cubic symmetry, the magnetic B’ ions form an FCC lattice and contain one electron in the outermost d shell. The ﬁve degenerate levels are split by the octahedral crystal ﬁeld into the higher energy eg orbitals and lower t2g orbitals so that the t2g shell contains one electron. The electronic structure for the t2g orbitals may be approximated by a nearest neighbor tight-binding model where only one of the three orbitals interacts along each direction. See Fig. 3.1.

X X X † HTB = t c c + h.c. (3.1) − i,α,σ j,α,σ α hiji∈α σ

Here the sum over α is over all yz, zx, and xy planes in the FCC lattice. As an example,

for B’ sites in an α = xy plane, the xy orbital on site i overlaps with the xy orbital on

site j. Each orbital has four neighboring orbitals of the same kind in its plane giving a

total of twelve relevant B’ neighbors per B’ site. In addition to the tight-binding term, the

unquenched t2g orbital angular momentum L = 1 results in a spin-orbit coupling on each P B’ ion [4] HSO = λ Li Si. Here the orbital L = 1 and spin S = 1/2 operators both − i · satisfy the usual commutation relations for angular momentum, L L = i~L. × P (i) The on-site multi-orbital Coulomb interaction is given by HU = H where HU i U → HHubbard from Section 2.3 in (2.12). Here U is the Coulomb repulsion and JH is Hund’s coupling [16]. Being in the Mott limit, we calculate the eﬀective spin-orbital superexchange

Hamiltonian within second order perturbation theory. The superexchange Hamiltonian is

given by the following

JSE X X 3 α α 2 HSE = r1( + Si Sj)(n n ) − 4 4 · i − j α hiji∈α (3.2) 1 α α 2 4 α α +( Si Sj) r2(n + n ) + (r3 r2)n n 4 − · i j 3 − i j 2 −1 −1 where JSE = 4t /U is the superexchange strength and r1 = (1 3η) , r2 = (1 η) , − − −1 and r3 = (1 + 2η) with η = JH /U [30]. Here the t2g orbital electron occupation numbers α P † are written as ni = σ ci,α,σci,α,σ. See Appendix B.1 for details. The top line of (3.2) contributes a ferromagnetic (FM) spin interaction which requires that one of the two orbitals

27 Figure 3.1: (a) Crystal lattice for double perovskite A2BB’O6. (b) The simpliﬁed tight- binding model takes hopping between xy orbitals (purple) on B’ sites within an xy plane. Similarly, zx orbitals are active in zx planes, and yz orbitals are active in yz planes.

participating in the interaction is occupied while the other is unoccupied. The bottom line of (3.2) contributes an antiferromagnetic (AFM) spin interaction which is maximized when both orbitals are occupied. The strength of Hund’s coupling, JH /U, determines the strength of the two interactions relative to each other. Additionally there is an eﬀective orbital

α α repulsion ni nj in the superexchange processes when ﬁnite Hund’s coupling is considered. Due to the large spatial extent of 5d orbitals from strong oxygen covalency, we include an additional term accounting for the direct Coulomb repulsion between orbitals on diﬀerent sites. From general symmetry constraints, we can express the repulsion within the xy plane for the t2g orbitals (yz, zx, xy).   V1 V2 V3   (xy) X X α β   H = ni n  V V V  (3.3) V j  2 1 3  hiji∈xy αβ   V3 V3 V4 αβ

To constrain the number of independent parameters, we take the matrix elements to be determined in the limit of electric quadrupolar interactions [9]. Now let (α, β, γ) be a cyclic permutation of the three t2g orbitals. The repulsion term for the entire lattice can then be

28 expressed in the following form.

X X h 9 α α 4 β γ β γ i HV = V n n (n n )(n n ) (3.4) 4 i j − 3 i − i j − j α hiji∈α

For example, within the xy plane, a pair of xy orbitals repel each other more than an xy

and yz orbital.

The total eﬀective magnetic interaction then reads H = HSO + HSE + HV. Of the three parameters, spin-orbit coupling has the largest energy scale λ 0.4 eV for the 5d oxides ≈ while superexchange and intersite Coulomb repulsion are taken to have energy scales on the

order of tens of meV. For 4d oxides, the spin-orbit energy scale is reduced to 0.1 0.2 eV so − that mixing between the j = 3/2 and j = 1/2 states is likely to occur. While our spin-orbit superexchange interaction is calculated in the LS-coupling scheme, recent evidence suggests that the true picture for the 5d oxides lies between the LS and jj limits [56].

α α We decouple HSE and HV into all possible on-site mean ﬁelds, i.e. Sin Sjn i j → α α α α α α Sin Sjn + Sin Sjn Sin Sjn . Since the FCC lattice is not bipartite, we i h j i h i i j − h i ih j i decouple into four inequivalent sites shown in Fig. 3.2(a) where each set of four inequiva-

lent neighbors forms a tetrahedron. Since the mean ﬁelds need not factor into the product

α α of spins and orbitals, Sin = Si n , there are a total of 15 mean ﬁelds per site com- h i i 6 h ih i i prised of three spin operators, three orbital operators, and products of the spin and orbital

operators. Applying the constraint that one electron resides on each site, there are 11 in-

dependent mean ﬁelds per site giving a total of 44 mean ﬁelds in the tetrahedron. We then

numerically solve for the lowest energy solutions of the mean ﬁeld equations.

3.2.2 Zero Temperature Mean Field Theory

In the limit where spin-orbit coupling λ is the dominant energy scale, the magnetically

ordered phases can be characterized by an arrangement of ordered j = 3/2 multipoles

[9]. However, when JSE and λ are comparable, a multipolar description within the j = 3/2 states breaks down and consequently both spin and orbital parts must be considered

independently. Furthermore, the orbital contributions come in the forms of both orbital

29 (a) (b) (c) 0.20 y

x Canted 0.18 FM

0.16 S L / U H J 0.14 AFM V/λ=0 (d) 0.8 (e) 0.20 0.06→ 4-sublattice Tc To 0.01 0.6 0.15 nyz 0.12 / λ

o 0.04 0.4 0.10 ← 0.02 n T xy B k S 0.2 0.05 ← 0.02=V/λ L 0.03 M 0.10 nzx 0.0 0.2 0.4 0.6 0.8 0.0 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.05 0.10 0.15 0.20 JSE /λ

kBT/λ μeff (μB)

Figure 3.2: (a) FCC lattice decoupled into four inequivalent sites shown by four different colors. (b) The orbital ordering pattern driven by both JSE and V constrains the direction of orbital angular momentum. The magnetization operator is shown as M = 2S L. (c) − The zero temperature phase diagram shows phases where the spin S and orbital L moments in each plane are collinear and the moments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing orbital repulsion V between sites reduces the minimum strength of Hund’s coupling required to induce FM. (d) Mean field values for the bottom sites (black, yellow) are shown as a function of temperature. The nyz orbital (red) has the largest occupancy followed by the xy orbital (blue). (e) With JSE = 0, we calculate the orbital ordering temperature To and effective Curie moment enhancement µeff for different values of V .

30 occupancy nα and orbital angular momentum L. Since nα, L, and S are coupled, there is

competition between order parameters which results in non-trivial ordering.

The zero temperature phase diagram is shown in Fig. 3.2(c) as a function of the strength

of Hund’s coupling η = JH /U and superexchange JSE/λ. Large values of η support a canted ferromagnetic (FM) structure while smaller values support an antiferromagnetic (AFM) structure. The spin-1/2 and orbital-1 angular momenta order parameters S and L are h i h i shown for each of the four inequivalent sites from Fig. 3.2(a). In both phases, one of the

three directions has no ordered angular momenta, e.g. Lz = Sz = 0, so that both h i h i magnetic structures are co-planar. Both phases feature some separation of the ordered

spin and orbital moments which increases as a function of JSE/λ. To understand why these magnetic structures emerge, we examine the orbital occupancy order parameters, nα, separately from the magnetic order parameters. In both the FM and AFM phases, there is an orbital ordering pattern pictured in Fig. 3.2(b). The two sites in the lower plane of Fig. 3.2(b) have the yz orbital (red) with the highest electron occupancy while the xy orbital (blue) receives the second highest and the zx orbital receives the lowest (green, not pictured). The two sites in the upper plane have identical ordering except the roles of the yz and zx orbitals are reversed. Qualitatively this orbital ordering pattern is favored by both the HV and HSE terms which pushes electrons onto orbitals that have small overlaps. This allows the electron on a green orbital to hop onto an unoccupied green orbital in the

plane directly above or below (and similarly for red orbitals). Since these mechanisms work

to suppress the overlap of half ﬁlled orbitals, FM interactions may become energetically

favorable. A derivation of the mean ﬁeld solution for HV is provided in Appendix B.2. Once orbital order sets in, the allowed magnetic phases are restricted by the direction

of orbital angular momentum. Full orbital polarization is time-reversal invariant and would

not allow orbital magnetic order. However Fig. 3.2(d) shows that each site has at least

two orbitals with non-negligible occupancy which allows for the development of an orbital

moment.

Thus the direction of the orbital moment is determined by the direction common to

the two planes of occupied orbitals with the overall sign of the direction (e.g. +x or x) − 31 left undetermined. Figure 3.2(c) shows that the orbital angular momenta between planes are close to 90 degrees apart for both FM and AFM phases. As spin and orbital angular momentum are coupled together, the spin moments will select which direction the orbital moments choose (i.e. +x or x). The decision to enter an FM or AFM state is then − determined by the spin interactions characterized both by the strength of η = JH /U and the magnitude of the orbital order parameter. If η is large, then FM spin interactions follow and result in both the spin and orbital degrees of freedom aligning within each xy plane producing a net canted FM structure. If η is small, then AFM spin interactions follow which result in the 4-sublattice AFM structure. We note that the Goodenough-Kanamori-

Anderson rules [57–59] are not enough to determine whether FM or AFM is favored since both magnetic structures have the same underlying orbital order. The interplay between spin-orbit coupling and orbital ordering plays a crucial role in tipping the energy scales in favor of one of the two magnetic structures.

There are two additional factors that determine if the FM or AFM state is selected.

The dominant effect is the degree of orbital polarization. When the strength of orbital repulsion V is increased, the tendency for orbitals to order becomes stronger. This disfavors the overlap of half filled orbitals causing AFM superexchange, and hence promotes FM superexchange. Figure 3.2(c) shows a dramatic shift toward FM when a small V interaction is included. The second effect comes from the separation of spin and orbital degrees of freedom. When JSE becomes comparable to λ, the spin moments can partially break away from the orbital moments tending more toward a regular spin FM state instead of a canted spin FM state. Since a spin AFM state does not benefit from this separation to the same extent, FM becomes increasingly energetically favorable.

Dimer phases have been proposed [9, 11] and oﬀer a way to explain the absence of

1 magnetic order in d materials. However when λ/JSE is large, these dimer phases only occur at very small values of η = JH /U [11]. Furthermore, orbital repulsion V acts to further suppress dimerization. Since our focus is on the magnetically ordered phases of these double perovskites, we will not pursue these possibilities in this chapter.

32 3.2.3 Finite Temperature Mean Field Theory

Orbital Order. We now examine the model at ﬁnite temperature. Figure 3.2(d) shows a characteristic order parameter versus temperature curve. At high temperatures all order parameters are trivial and each orbital occupancy takes a value of nyz = nzx = nxy = 1/3.

As temperature is lowered, the ﬁrst transition is to a time reversal invariant orbitally ordered state (see Fig. 3.2(b)) at temperature To whose scale is set both by V and JSE, including

when V = 0. At To, the entropy released is from orbital degeneracy with the spin entropy

remaining. Below the second transition at Tc whose energy scale is set only by JSE, time reversal symmetry is broken on each site with the development of magnetic order, and the

remaining entropy is released.

The fundamental question arises of how large the exchange interaction JSE and re-

pulsion V are in materials systems. Fits to experimental susceptibility show Ba2LiOsO6

and Ba2NaOsO6 have relatively small Curie-Weiss temperatures of θ = 40 K for AFM − Ba2LiOsO6 and θ = 32 K for FM Ba2NaOsO6 [46]. (Alternatively θ = 10 K for the FM − − 1 from a later study [31].) This indicates that JSE in cubic 5d double perovskites is weak.

However integrated heat capacity [31] of Ba2NaOsO6 shows an entropy release just short of

R ln 2 at Tc consistent with the splitting of a local Kramer’s doublet with no further anoma-

lies in heat capacity up to 300 K. This suggests To Tc so that V is the most relevant parameter for setting the scale of To. Magnetic Moment. Since the onset of orbital order necessarily alters the angular mo-

menta available to order and respond to an applied magnetic ﬁeld, we calculate how the

eﬀective Curie-Weiss constant depends on orbital ordering. Using JSE = 0, we calculate the temperature dependent susceptibility within mean ﬁeld theory as a function of tem-

perature for different values of V/λ. For each value of V/λ we calculate both the orbital p ordering temperature To and the effective Curie moment µeff = gµB j(j + 1) from a fit to low temperature inverse susceptibility. Figure 3.2(e) gives numerical results from our mean

ﬁeld theory that shows a linear relationship between To and µeﬀ . In the absence of orbital order, the projection of the magnetization operator to the j = 3/2 space is identically zero.

33 However once orbital order sets in, the j = 1/2 components of the wavefunction get mixed with the j = 3/2 components. The matrix elements that connect these two j spaces then acquire expectation values and allow the eﬀective Curie moment to become non-zero. An approximate derivation of this relation is provided in Appendix B.2, and the result is given by

µeﬀ 172V δnx µB/9λ (3.5) ≈ | | yz 1 where δnx = n is the deviation of yz occupancy from its high temperature value on h i − 3 the sites where nyz is the most occupied orbital. We note that the temperature dependence

of δnx makes the eﬀective moment dependent on temperature. In addition to the perturbative separation of L and S due to mixing of the j states, oxygen covalency has been shown to greatly reduce the orbital contribution to the moment

[33, 34]. Here the magnetization operator assumes the form M = 2S γL where γ = 0.536 − and results in an eﬀective Curie moment of 0.60µB compared to an experimental value of

0.67µB [46]. However the onset of quadrupolar order within the j = 3/2 states results in a

reduction of the nominal 0.60µB value. In general, the projection of a linear combination

of the nyz, nzx, and nxy operators to the j = 3/2 states is (up to a constant shift) a linear combination of the operators J 2 J 2 and J 2. By projecting to the lowest energy doublet x − y z induced by these operators, we may calculate the g factors for this pseudo-spin 1/2 space.

While the g factors are diﬀerent in the three cubic directions due to the anisotropic nature

of quadrupolar order, the sum of the squares is a constant, and the powder average is

2 1 2 2 2 g = 3 (gx + gy + gz ) = 3. Then splitting of the j = 3/2 states reduces the Curie moment p p p by a factor of (g 3/4)/( 15/4) = 3/5 which makes the calculated moment 0.47µB. Mixing between the j = 3/2 and j = 1/2 states brings the calculated moment closer to experimental values.

Susceptibility near To. There are more consequences of orbital ordering that are particularly important for the magnetic susceptibility of this spin-orbital system. The orbital order reduces the symmetry of the system and causes the susceptibility to become anisotropic.

Since the orbital ordering pattern tends to push angular momentum into the ordering planes,

34 without with covalency covalency

5 10

4 8

3 6 χ χ 2 4

1 2 T To o 0 0 0 0.04 0.08 0.12 0 0.04 0.08 0.12 k kBT/λ BT/λ 1 0.5 0.8 0.4 0.6 0.3 - 1 - 1 χ χ 0.4 0.2 0.2 0.1 T To o 0 0. 0 0.04 0.08 0.12 0 0.04 0.08 0.12 k kBT/λ BT/λ

1 Figure 3.3: Typical susceptibility, χ = 3 (χxx + χyy + χzz), and inverse susceptibility are plotted against temperature. The susceptibility curves are shown both without the correction due to covalency, γ = 1, and with the correction, γ = 0.536. We have chosen JSE = 0 and left V finite to illustrate the consequence of high temperature orbital order on the susceptibility. By choosing JSE = 0, we show that although Tc = 0 while To = 0, the fitted 6 Curie-Weiss temperature appears to be negative. Note that a single Curie-Weiss fit cannot span the entire range below To.

35 the susceptibility is enhanced in these two directions while reduced in the third direction.

Although anisotropic susceptibility is expected once cubic symmetry is broken, it is an easy test to determine at what temperature orbital order occurs. Additionally, when orbital order sets in at To, the eﬀective moment changes as the orbital degrees of freedom tend toward a (partially) quenched state which results in an eﬀective moment that changes with temperature. The non-Curie-Weiss behavior will be critical when interpreting the observed negative Curie-Weiss temperatures in 5d1 FM compounds.

To show this effect within our mean field theory, we now calculate the susceptibility both without and with the covalency correction γ. For clarity, we set JSE = 0 to isolate the contributions from orbital order from those of magnetic interactions. Figure 3.3 shows that below the orbital ordering temperature, the susceptibility deviates from the Curie-Weiss law. However the data below To can be fit over a large range to give a negative Curie-Weiss intercept despite the absence of magnetic interactions. In fact the region where the fit works the best is just below To where the orbital occupation is rapidly changing. To interpret this in a simple way, we will consider the case without covalency where the effective moment for the j = 3/2 states is identically zero. When orbital order occurs, there is mixing between the j = 3/2 and j = 1/2 states proportional to V δn /λ. Here δn h i refers to a change in orbital occupancy due to orbital order. Then below To, the effective magnetization operator for the lowest energy Kramer’s doublet increases in a way propor-

tional to δn due to the matrix elements between j = 3/2 and j = 1/2. The effective Curie h i moment goes as the square of magnetization and thus the enhancement is of order δn 2. h i 1/2 Since orbital order below To scales as δn To T within mean field theory, the effec- h i ∝ | − | tive Curie moment gains a contribution scaling as To T just below To. At temperatures | − | near To, the leading correction to susceptibility and inverse susceptibility is linear leading to the appearance of a negative Curie-Weiss intercept. We note, however, that this is not indicative of the physical magnetic interactions.

Despite using mean ﬁeld critical exponents, qualitatively we have understood how deviations from the Curie-Weiss law occur from changing orbital occupancy. Because we have used a simple model consisting of only λ and V with a-priori knowledge of the ideal

36 (a) Canted FM (b) J>0 K=0 (c) J>0 K>0

4 4 site 1 site 2 y y 3 3 ϕ / χ / χ

2 2 [110] ) )

B 2 B 2

( g μ ( g μ [110] ϕ S2 1 1 [001] S 0 0 x 1 x 0 1 2 3 4 0 1 2 3 4

kBT/J kBT/J (a) Canted FM (b) J>0 K=0 (c) J>0 K>0

4 4 site 1 site 2 y y 3 3 ϕ / χ / χ

2 2 [110] ) )

B 2 B 2

( g μ ( g μ [110] ϕ S2 1 1 [001] S 0 0 x 1 x 0 1 2 3 4 0 1 2 3 4

kBT/J kBT/J

Figure 3.4: (a) The canted ferromagnetic solution to equation (3.6) is shown. (b) For J > 0 and K = 0, susceptibility along [110], [110], and [001] is shown for the antiferromagnet with φ = π/4. (c) For J > 0 and K > 0, the susceptibility diverges at Tc. The canting angle − satisﬁes π/4 < φ < 0. Note that the Curie-Weiss law still holds at temperatures well − above J and K, and the Curie-Weiss intercept is still negative.

Curie-Weiss temperature of zero, we have been able to clearly interpret the non-Curie-Weiss

susceptibility. However the ﬁtting procedure must be performed with some caution since

both the ﬁt region and the chosen value of χ0 (temperature independent term) determine

the reported θCW and µeﬀ . In fact, experimental behavior may deviate even more strongly due to the quantitative details of how orbital occupancies change with temperature. In

particular, coupling between orbitals and phonons may be a crucial aspect here [33].

37 3.2.4 Simpliﬁed Model at Finite Temperature

Canting from Orbital Order. In our microscopic model, we found that orbital order induced deviations from Curie-Weiss behavior that resulted in ﬁtted negative Curie-Weiss constants despite diverging magnetic susceptibility at Tc. Although this explanation is self-consistent within the context of the model we study, it depends on the existence of a continuous phase transition for orbital order. However, experiments have not yet found a clear signature of high temperature orbital ordering. We then seek an explanation of the negative Curie-Weiss constant which is insensitive to the particular details of orbital ordering but still incorporates it in a phenomenological way. There are multiple reasons for this approach. Unlike models with only spin degrees of freedom, spin-orbital models are very sensitive to the tight-binding model used, but the results should not sensitively depend on the tight-binding parameters used. Our explanation should also not assume which mechanism lead to the orbital ordering nor require a continuous phase transition at

To. We then develop a minimal model to explain this phenomena. Our zero temperature results showed that the presence of orbital order tended to pin the orbital angular momentum along the axis common to the two most occupied orbitals.

This enters as an anisotropy that moments see, although an anisotropy which depends on the xy plane that each B’ site is in. In the canted FM phase, there are only two inequivalent sites, so our simpliﬁed model will consist of exactly two sites labeled as “site 1” and “site

2” each with eﬀective S = 3/2. The Hamiltonian for this model is then given below.

x 2 y 2 H12 = JS1 S2 K (S ) + (S ) (3.6) · − 1 2

The sign of J may be positive or negative while the anisotropy K is positive. The parameterization J = sin 2φ and K = cos 2φ where π/4 < φ < π/4 gives the classical ground − − state with S1 = (cos φ, sin φ, 0) and S2 = (sin φ, cos φ, 0). This state is shown in Fig. 3.4(a).

Susceptibility near Tc. For J > 0 and K = 0, the model is a simple AFM shown in

Fig. 3.4(b). When K > 0, susceptibility changes drastically at Tc shown in Fig. 3.4(c). The kink at the ordering temperature changes to a divergence which signals a transition

38 to a FM state instead of an AFM state. However, the divergence smoothly turns into a Curie-Weiss law as temperature is increased above the J and K energy scales. Only when the temperature approaches Tc does the deviation from Curie-Weiss behavior become important. Note that the Curie-Weiss temperature is still negative from ﬁtting the region well above these energy scales. Appendix B.3 gives an example of how this transition from

AFM Curie-Weiss behavior to a FM divergence in susceptibility may be calculated in closed form.

This simple model contains the essential explanation for why the 5d1 FM compounds have negative Curie-Weiss temperatures despite a diverging susceptibility. If orbital order occurs and creates a staggered anisotropy between xy planes, then canting immediately follows without further considering anisotropic Ising interactions or exotic octupolar interactions (see Appendix B.4). Positive Curie-Weiss temperatures can be associated with canting in the positive φ direction, and negative Curie-Weiss temperatures can be associated with canting in the negative φ direction. This suggests that Ba2NaOsO6 may be better identiﬁed as a canted AFM instead of a FM.

To complete the phase diagram for d1, we can create a simpliﬁed Hamiltonian that includes both the canted FM phase and the AFM 4-sublattice phase. This requires 4 inequivalent sites shown on the tetrahedron in Fig. 3.2(a).

H = J (S1 S2 + S1 S4 + S3 S2 + S3 S4) · · · · (3.7) 0 x 2 x 2 y 2 y 2 + J (S1 S3 + S2 S4) K[(S ) + (S ) + (S ) + (S ) ] · · − 1 3 2 4 Interactions between xy planes are characterized by J, and interactions within an xy plane are characterized by J 0. For K > 0, the ground states are then given either by a canted phase with two inequivalent sites or the AFM 4-sublattice structure. Again, this shows the magnetic structures are largely dictated by anisotropy from orbital ordering, and the microscopic form of the interactions are less important. Furthermore, to better match the microscopic model, a uniform anisotropy, K0 P (Sz)2, may also be introduced to ﬁne-tune − i i orbital occupancies and anisotropic susceptibilities.

Chen et. al. [9] claimed negative Curie-Weiss temperatures were achievable in their

39 (a) (b) 0.20 FM[100] (FM[110])

0.15 V/λ=0

AFM 4-sublattice / U 0.10

H (AFM[100]) J 0.01

0.02 0.05

0.03 y AFM[110] 0.00 x 0.0 0.2 0.4 0.6 0.8

JSE /λ

Figure 3.5: (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent the most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature phase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and FM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in each region. Increasing orbital repulsion V moves the phase boundary between AFM 4-sublattice phase and the AFM [110] phase down to favor the AFM 4-sublattice phase. The phase boundary between the AFM 4-sublattice phase and the FM [100] phase moves up in favor of the AFM 4-sublattice phase.

model for FM ground states, although this crucial result was not explicitly shown. Marjer- rison et. al. [32] have reproduced that model to generate FM with negative Curie-Weiss temperatures, and they ﬁnd near jump discontinuities in the magnetic susceptibility around

Tc. Such jump discontinuities are not seen in Ba2NaOsO6, Ba2MgReO6, or Ba2ZnReO6. We remark that a sharp feature in susceptibility occurs as K/J 0, but the sharp jump → is smoothed away from this limit. Additionally, Marjerrison et. al. [32] claim to see a pronounced feature in the heat capacity at temperatures just higher than Tc in both Ba2ZnReO6

and Ba2MgReO6. We note that this unusually pronounced feature in heat capacity is not

present in experimental data on Ba2NaOsO6.

40 3.3 d2 Double Perovskites

Here we modify the d1 spin-orbital model to accommodate two electrons. We then solve the model within mean ﬁeld theory at both zero temperature and ﬁnite temperature.

At zero temperature, we ﬁnd new orbital phases not found in our d1 phase diagram. For

completeness, we show susceptibilities and orbital occupancies at ﬁnite temperature.

3.3.1 Model

Our model for d2 is constructed from the same considerations used in d1 only changing the electron count. The tight-binding model HTB, the inter-site orbital repulsion HV,

2 and the on-site Coulomb interaction HU are valid for the d model without modiﬁcation. However spin-orbit coupling and superexchange will change since the total spin and orbital angular momentum on each site are now composed of two electrons. In the Mott limit,

Hund’s rules are enforced by HU resulting in a total spin S = 1 and total orbital angular momentum L = 1 on each lattice site. Within this space, the spin-orbit interaction takes

0 λ P the form H = Li Si. The superexchange Hamiltonian is given by the following SO − 2 i ·

0 JSE X X α α 2 H = r1(2 + Si Sj)(n n ) SE − 12 · i − j α hiji∈α (3.8) α α 2 3 5 α α +(1 Si Sj) (n + n ) + ( r3 )n n − · i j 2 − 2 i j where the deﬁnitions of JSE, r1, and r3 correspond to those used previously. As before, the top line in (3.8) gives a FM spin interaction when only one of the two interacting

orbitals is occupied while the second line gives an AFM spin interaction which is maximized

when two half ﬁlled orbitals overlap. The total eﬀective magnetic interaction then reads

0 0 0 0 H = HSO +HSE +HV. We decouple HSE and HV into all possible on-site mean ﬁelds using four inequivalent sites as before and then solve the mean ﬁeld equations numerically.

3.3.2 Zero Temperature Mean Field Theory

The zero temperature phase diagram is shown in Fig. 3.5(b) as a function of the strength

of Hund’s coupling η = JH /U and superexchange JSE/λ. The orbital ordering structures

41 corresponding to each magnetic phase are shown in Fig. 3.5(a). Even though the phases are labeled AFM and FM, the complex spin-orbital structures are described in detail below.

Similarly to our d1 treatment, we will again focus on how orbital order dictates magnetic order.

At small Hund’s coupling, the ground state is an AFM structure with the moment pointing either parallel or antiparallel to [110]. Within an xy plane, the moments point in the same direction, and the moments in neighboring xy planes point in opposite directions as shown in the the phase labeled “AFM 110” in Fig. 3.5(b). (Note that we use label [110] for the moment direction and not the structure factor.) To see why this phase occupies such a large region of phase space, we analyze the orbital structure that accompanies it, as shown in Fig. 3.5(a). On each site, one electron moves onto the yz orbital and the other onto the zx orbital. This corresponds to Lz = 0 on every site so that orbital angular momentum cannot point in the z direction. In this conﬁguration both occupied orbitals overlap with occupied orbitals on neighboring sites and unoccupied orbitals overlap with other unoccupied orbitals so that AFM superexchange is maximized. These orbitally controlled AFM interactions then take place between planes and not within planes resulting in AFM between planes and consequently FM alignment in each plane. Since this this orbital pattern is compatible with tetragonal distortion, as observed in Sr2MgOsO6 [53], we expect nominally cubic crystal structures to distort.

At intermediate Hund’s coupling and JSE/λ, we find the AFM 4-sublattice coplanar structure with the same accompanying orbital structure previously found in the d1 phase diagram. As before, the orbital angular momentum is closely aligned with the directions perpendicular to the occupied orbitals, and the spin and orbital moments tend to separate from each other with increasing superexchange. However there are important differences between this phase in the d2 and d1 cases. Although the underlying d1 and d2 orbital ordering structures possess the same symmetry, the electron count strongly influences the energetics. As in the d1 case, we find that this orbital structure supports a canted FM structure in d2 systems, but this solution to the mean field equations is significantly higher in energy than the other phases shown. In d1 systems, orbital order would create a situation

42 where occupied orbitals overlapped with unoccupied orbitals on other sites. While ordering of the yz and zx orbitals in Figure 3.5(a) is necessary for the canted FM structure, so is the elimination of electron occupancy from the xy orbitals so that AFM interactions do not take place within each plane. This is possible with one electron per site but not with two, and the AFM 4-sublattice magnetic ordering dominates over the canted FM magnetic order in d2 systems. It is also worth noting that in this region of the phase diagram, the next lowest energy phase is AFM [100] which can become a competitive ground state.

For large superexchange and Hund’s coupling, we find a FM phase with ordered moments parallel or antiparallel to [100]. This phase is best characterized as a “3-up, 1-down” collinear structure where three of the four moments order parallel to each other along the chosen direction and the fourth moment orders anti-parallel to the other three. It is worth noting that the second most energetically favorable phase in this region of the phase diagram is another “3-up, 1-down” structure where each moment is either approximately parallel or antiparallel to the [110] direction. The energy difference between the FM [100] and FM [110] phases is negligible and either phase is a suitable ground state. These two phases contain an underlying orbital structure which eliminates the overlap of half filled orbitals between three of the four sites so that FM interactions result between these three sites. However, the fourth site cannot be chosen in the same way as two orbitals must have AFM interactions with the other three sites, and consequently the fourth moment points antiparallel to the other three.

When inter-site orbital repulsion HV is included, the phase boundaries shift. The most dramatic effect is the recession of the boundary between AFM [110] and the AFM 4- sublattice structure. This becomes apparent by comparing the orbital configurations of the two phases as the AFM [110] structure maximizes the number of AFM bonds which are penalized by the orbital repulsion. Unlike in the d1 situation, we find that the inclusion of V does not enhance FM. Again, this is due to the constraints imposed by having two electrons per site. We also note that unlike the AFM [110] and AFM 4-sublattice structures, the FM/AFM [100] orbital structures feature more degenerate choices for orbital configurations. Of the four tetrahedral sites, three of them are able to minimize the repulsion and

43 (a) AFM 110 (b) AFM 4-sublattice (c) FM 100

0.30 1.0 1.0 1.0 nxy nyz 0.25 nyz,nzx 0.15 0.20 0.8 0.8 nyz 0.8 nxy 0.20 xy xy xy ( a.u. ) ( a.u. ) ( a.u. ) 0.6 , n 0.6 , n 0.15 0.6 , n 0.10 - 1 - 1 - 1 zx zx zx ) 0.15 nxy ) ) 0 0 0

0.4 , n nzx 0.4 , n 0.10 0.4 , n 0.10 yz yz nzx yz

n 0.05 n n ( χ - ( χ - ( χ - 0.05 0.05 0.2 0.2 0.2 Tc Tc To Tc To 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20

kBT/λ kBT/λ kBT/λ

Figure 3.6: Characteristic inverse susceptibility (blue/green) and orbital occupation (purple) curves are plotted against temperature for the three phases in Fig. 3.5: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility is averaged over all three directions, −1 −1 χ = 3(χxx + χyy + χzz) , and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured above each plot.

allow occupied orbitals to hop to unoccupied orbitals. Since the fourth site cannot minimize repulsion, its orbital occupancies are free and can be diﬀerent on every tetrahedron in the

FCC lattice. However, this degeneracy is then broken by magnetic ordering which selects only orbital conﬁgurations which are compatible with the magnetic order.

Although we have focused on spin-orbital magnetic order, it is necessary to remark that exotic singlet ground states are also possible. The Kramer’s theorem guarantees that trivial ionic singlets will not occur in d1 systems, and therefore the experimental observation of singlet behavior is an indication of a non-trivial ground state. Such considerations do not apply to d2, and experimental observations of singlet behavior may arise from trivial local magnetic singlets. Consequently this local non-magnetic singlet possibility must ﬁrst be ruled out when searching for exotic singlet behavior.

3.3.3 Finite Temperature Mean Field Theory

Here we consider the model at ﬁnite temperature. Figure 3.6 shows orbital occupations and inverse magnetic susceptibility as a function of temperature for the three ground state phases from the previous section. At high temperature, the orbitals have a uniform occu-

yz zx xy pancy of n = n = n = 2/3. There is a temperature To where time-reversal invariant

44 order sets in through the orbitals and a second temperature Tc where magnetic order sets in. In the case of the AFM [110] phase, Fig. 3.6(a) shows the two ordering temperatures coincide and that the electrons are pushed onto the nyz and nzx orbitals to maximize AFM superexchange. This is diﬀerent from the orbital ordering previously reported because this ordering maximizes orbital repulsion instead of minimizing it, so this orbital order is en- tirely driven by AFM superexchange. In this situation, the Curie-Weiss law with a negative

Curie-Weiss temperature occurs as expected.

The transition to an AFM 4-sublattice structure is shown in Fig. 3.6(b). Above To susceptibility follows the Curie-Weiss law with a negative Curie-Weiss constant. Below To the orbital occupancies change along with the inverse susceptibility to deviate from the high temperature behavior. Just below To, susceptibility may be fit to another Curie-Weiss law with another negative Curie-Weiss constant. Similarly to the d1 case, there is still deviation from the Curie-Weiss law in this regime, however, the deviations are smaller and so is the enhancement of the effective magnetic moment due to mixing of the J = 2 states with higher energy multiplets. But we note that when JSE = 0, we still find the appearance of a negative Curie-Weiss constant due to non-Curie-Weiss susceptibility as we did in the d1 model.

Finally, the transition to an FM [100] structure is shown in Fig. 3.6(c). Deviations from the Curie-Weiss law are seen below To, and the sign of the Curie-Weiss constant can switch from negative to positive depending which region ﬁtted. Unlike the other phases, magnetic order appears at Tc with a ﬁrst-order transition marked by the jumps in orbital occupancy and susceptibility. This arises from competition between having the most energetically favorable orbital structure at high temperature and the most energetically favorable magnetic structure at low temperature.

As in the d1 case, we compare values of the theoretical moments to those from experi- ment. Oxygen covalency will result in a Curie moment of µeﬀ = √6(1 γ/2)µB. Assuming − almost half of the moment resides on oxygen, the calculated moment is then µeﬀ 1.8µB. ≈ This is just short of the experimentally observed moments in Sr2MgOsO6 and Ca2MgOsO6

(1.87µB)[52], Ba2YReO6 (1.93µB)[51], and La2LiReO6 (1.97µB)[51].

45 3.4 Discussion

We have studied spin-orbital models for both d1 and d2 double perovskites where the

B’ ions are magnetic and have strong spin-orbit coupling. We found several non-trivial magnetically ordered phases characterized both by ordering of the spin/orbital angular momentum and ordering of the orbitals. We emphasize that examination of the spin and orbital degrees of freedom separately gives an enhanced qualitative understanding of the magnetism for this class of spin-orbit coupled double perovskites. Our results allow us to draw many conclusions that can be connected to particular materials and experiments.

First, the canted FM (canted AFM) phase may describe the known ferromagnets

Ba2NaOsO6, Ba2MgReO6, and Ba2ZnReO6. Experimental ﬁtting inverse susceptibility versus temperature data may strongly depend on the region ﬁtted due to deviations

from the Curie-Weiss law. Furthermore, high temperature susceptibility may lead to an

incorrect interpretation of the magnetic interactions in these materials. On the other hand,

our simpliﬁed model shows how canted states due to anisotropy from orbital order can

genuinely give negative Curie-Weiss temperatures while retaining a diverging susceptibility

at Tc. Second, while there is also strong evidence for orbital (quadrupolar) order in the fer-

romagnet Ba2NaOsO6, we propose the same order may be present in antiferromagnetic

Ba2LiOsO6. We found that the orbital structure which allows canted states can also support a unique antiferromagnetic state. This is due to the qualitative interpretation that

orbital order constrains the direction of orbital angular momentum which tends to favor

one of two types of coplanar structures.

Third, we predominantly found antiferromagnetism in d2 compounds since ferromag-

netism was not as favorable due to energetic constraints imposed by orbital occupancies.

We take this as an explanation for why so many d1 ferromagnets exist but few (if any) d2

ferromagnets exist. Furthermore, although the AFM [110] structure is likely to be found in

tetragonal crystals such as Sr2MgOsO6, the unusual 4-sublattice antiferromagnetic structure common to both d1 and d2 is likely to be found in cubic d2 materials due to the large

46 region of the phase diagram that it occupies.

Finally, in light of recent NMR work on Ba2NaOsO6 [40], we highlight an experimental puzzle. Above the magnetic ordering temperature of approximately 7 K, a broken local point group symmetry persists to higher temperatures between 10 K to 15 K. With the assumption that the signal for this broken local point group symmetry corresponds to an orbital (quadrupolar) order parameter, we would conclude that orbital order disappears above 15 K. In fact our microscopic model does allow for situations where To is close to

Tc. This could be used to show how antiferromagnetic interactions dominate above To, and only below To would the orbital occupancies change to favor ferromagnetism and a diverging magnetic susceptibility at Tc. The difficulty with these explanations is that it may be inconsistent with specific heat measurements [31, 32]. This leads us to favor a simplified interpretation of canted AFM over instead of low temperature orbital order.

We are then still left with the task of interpreting these seemingly contradictory results.

Using just NMR data, negative Curie-Weiss temperatures could be trivially explained using an orbital ordering energy scale near the magnetic ordering temperature. Using just speciﬁc heat data, this explanation instead suggests high temperature orbital order. This leaves several open questions. Does the broken local point group symmetry persist above 15 K?

At what temperature is the R ln 4 magnetic entropy recovered? Is there a relationship between the orbital order and the negative Curie-Weiss intercept?

47 Chapter 4 d4 Mott Insulators

4.1 Introduction

Between weakly correlated topological insulators and strongly correlated 3d transition metal oxides lie 5d compounds that combine both strong spin-orbit coupling and correlations on an equal footing [4]. In contrast to the well studied 5d5 materials, the effect of strong spin-orbit coupling in 4d4 and 5d4 systems have been sparsely studied due to expectations that these will naturally lead to non-magnetic insulating behavior. However there are several experimental counter examples to this notion. The first example is Ca2RuO4 which displays a moment of 1.3µB [60, 61]. Recently both double perovskite iridates [62–64] and honeycomb ruthenates [65] in the d4 configuration have been found to show magnetism. It has been argued that partial quenching of the orbital angular momentum from the presence of lattice distortions is the root cause. Very recent developments [13, 24–26] on Ba2YIrO6 have piqued interest on the origin of magnetism in this 5d4 system because the compound is negligibly distorted and still shows a Curie response.

In transition metal oxides with oxygen octahedra, the large crystal ﬁeld splitting puts

4 4 d ions into the t2g electronic conﬁguration. For materials with strong spin-orbit coupling, the j = 3/2 band is ﬁlled and the j = 1/2 band is empty leading to the conclusion that weakly correlated d4 materials are non-magnetic. However when Coulomb interactions are strong, a total spin S = 1 and orbital angular momentum L = 1 lead to a total angular momentum J = 0 on every d4 ion with no magnetism. Thus both jj coupling and LS

48 coupling schemes lead to the same conclusion that a single atom is in a J = 0 singlet state

and therefore trivially non-magnetic [8] as shown in Fig. 4.1(a).

We build on previous work by Khaliullin [21] that proposed an “exciton condensation”

mechanism, more accurately a condensation of J = 1 triplon excitations, to drive the onset

of antiferromagnetism in nominally non-magnetic d4 systems and our previous study [12]

showing that ferromagnetic superexchange interactions caused by strong Hund’s coupling

can precipitate ferromagnetic coupling. In this work we start with the atomic multi-orbital

Hamiltonian with intra- and inter-orbital Coulomb interactions and spin-orbital coupling

4 speciﬁcally for t2g systems. We next allow hopping between atoms and investigate all cases of orbital geometries– the idealized fully symmetric case when all orbitals participate in

hopping, as well as more realistic cases suitable for simple cubic and face-centered cubic

lattices. For each case, we derive the eﬀective spin-orbital superexchange Hamiltonian which

competes with spin-orbit coupling to produce strong deviations from the non-magnetic

atomic behavior. These results are obtained both using exact diagonalization on a two-site

problem and perturbation theory for the eﬀective magnetic interactions.

Tuning the superexchange interactions JSE relative to spin-orbit coupling λ, we ﬁrst see the formation of local moments followed by a Bose condensation of weakly interacting

J = 1 triplet excitations, or triplon condensation. Rather remarkably, regardless of the local

spin interactions favoring antiferromagnetic spin superexchange (spin-AF) at small Hund’s

coupling or ferromagnetic spin superexchange (spin-F) at large Hund’s coupling, the J = 1

triplons condense at the ~k = ~π point. This result that the rotationally invariant spin-orbit

coupling can eﬀectively ﬂip the sign of superexchange is unusual and unique to spin-orbital

coupled systems. In the opposite regime where JSE dominates, the orbital interactions are frustrated even in the absence of geometric frustration and can potentially lead to orbital

liquid phases. Even when λ = 0 and the local spin interactions are simple Heisenberg

FM or AFM, the frustrated orbital interactions generate frustration in the spin channel as

well, leading to the possibility of ground states with both orbital and spin entanglement on

lattices without geometric frustration. This is summarized schematically in Fig. 4.1(b).

The chapter is organized in the following way. In Section 4.2 we introduce the lattice

49 Hamiltonian used as the basis for the rest of the chapter which includes electron hopping, atomic spin-orbit coupling, and an eﬀective multi-orbital Coulomb interaction that captures

Hund’s rules. The orbital geometries for hopping used throughout the chapter include both a highly symmetric toy model to be used as a simpliﬁed diagnostic tool as well as two other more realistic cases found in perovskites.

In Section 4.3 we use exact diagonalization to study a two-site specialization of the problem introduced in Section 4.2. Isobe et. al. [66] has used a similar procedure to study transition metal systems with other electron counts. Local magnetic moments are absent when spin-orbit coupling is large, as expected in the atomic picture, but electron hopping introduces sizeable moments when t λ when two or three orbitals strongly ∼ overlap between sites. Although a single orbital overlap can also promote superexchange which competes with spin-orbit coupling, the number of superexchange pathways is limited and local moments do not form for any reasonable ratio of t/λ.

In Section 4.4 we derive an eﬀective magnetic Hamiltonian in terms of orbital angular momentum and spin operators using second order perturbation theory. We check that the spin-orbital superexchange Hamiltonian captures both spin-AF and spin-F interactions between spins depending on the value of Hund’s coupling, and the sum of spin-orbit coupling and the spin-orbital superexchange Hamiltonian reproduce the phases found in exact diagonalization of a two-site system.

In Section 4.5, we give a qualitative description of how bond-dependent spin-orbital superexchange results in orbital frustration. However, ﬁnding solutions to orbitally frustrated models can be challenging and is outside the scope of the present chapter [30, 67–69].

In Section 4.6 we ﬁrst review the “excitonic” condensation mechanism where the Bose condensation of van Vleck excitations gives magnetism to d4 systems with spin-orbit coupling. Although the condensation mechanism involves approximations to full spin-orbital models derived in the previous section, it gives valuable insight into the behavior at large spin-orbit coupling. Regardless of the nature of local interactions, only AF condensates are supported for the models studied, and we give the the critical superexchange required for

AF condensation for the three orbital geometries studied.

50 Section 4.7 discusses potential materials realizations and experiments beyond those men- tioned in the introduction.

4.2 Model

1 Our model Hamiltonian for t2g systems is composed of three parts: (i) kinetic part, (ii) Coulomb interaction, and (iii) spin-orbit coupling.

X (ij) X (i) X (i) H = Ht + Hint + Hso (4.1) hiji i i

The general form of the kinetic part

(ij) X X (ij) † Ht = tm0m cim0σcjmσ + h.c. (4.2) mm0 σ

(ij) 0 is given in terms of matrix elements tm0m between t2g orbitals m and m (with values yz, zx, and xy) on sites i and j. The index σ is for spin. We take the on-site Coulomb

interaction to be the t2g interaction Hamiltonian, Hint HHubbard, derived in Section 2.3 → and appearing in (2.12). The on-site intra-orbital Hubbard interaction is characterized U

and JH characterizes the strength of Hund’s coupling. We have chosen to use JH instead of J to avoid confusion with total angular momentum in the next two sections. The atomic

spin-orbit coupling has the form given in Section 2.4 and appearing in (2.14). (ij) We focus on three special cases of tm0m which diﬀer by the number of orbitals, Norb participating in hopping.

(ij) Norb = 3: First we consider the orbitally symmetric case where t 0 = tδm0m and all • m m orbitals participate in hopping. While this full rotational symmetry is not usually present

in material systems, the Norb = 3 case serves as a diagnostic tool where total angular momentum in the system is conserved and correlation functions have rotational symmetry.

(ij) Norb = 2: The next case, t 0 = tδm0m (1 δkm), uses two orbitals participating in • m m − hopping, Norb = 2, while one orbital is blocked. The blocked orbital k is determined by

1 This eﬀective model for t2g orbitals assumes the ligand orbitals have been eﬀectively integrated out, ie. the Mott-Hubbard limit.

51 (a) jj coupling LS coupling j = 1/2 S = L = 1

j = 3/2

J = 0 J = 0

(b)

Spin F → AF

/U van Vleck Triplon H PM J BEC Spin AF

λ zJ zJ λ ← SE SE → Figure 4.1: (a) The single site total angular momentum is zero in both the jj and LS coupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (4.3) pitting spin-orbit coupling λ against superexchange JSE where λ is the spin-orbit coupling energy scale and JSE is the superexchange energy scale with z being the coordination number. Starting with a van Vleck phase with no atomic moments at large λ we ﬁnd a triplon condensate at k = ~π for all values of the Hund’s coupling JH /U. The intermediate regime where λ zJSE has not been explored. At large JSE we obtain eﬀective magnetic ≈ Hamiltonians that have isotropic Heisenberg spin interactions (antiferromagnetic for small JH /U and ferromagnetic for large JH /U) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and large JSE/λ regimes.

52 (a) z dyz pz d x y yz

dxy px

dxy

(b) dxy

dxy

Figure 4.2: (a) The Norb = 2 model is an approximation of oxygen mediated electron hopping between t2g orbitals in a simple cubic lattice. Both dxy and dyz orbitals participate in hopping along the y direction. (b) The Norb = 1 model is an approximation of direct hopping between t2g orbitals on the face of a face-centered cubic lattice. The dxy orbitals are most relevant for hopping in the xy plane.

53 the direction of the line connecting sites i and j. This situation is commonly found in

simple cubic lattices where t comes from oxygen-mediated superexchange. See Fig. 4.2(a).

(ij) Norb = 1: The ﬁnal case, t 0 = tδm0mδkm, only has one orbital contributing, Norb = 1, • m m while two orbitals are blocked and approximates the hopping between nearest-neighbors

on a face-centered cubic lattice. The active orbital k is determined by which plane the

sites i and j share. See Fig. 4.2(b).

4.3 Exact diagonalization

Before analyzing the full lattice problem which will require approximations to be made, it is useful to examine exact results for a pair of interacting sites. We numerically diagonalize

(4.1) for a two-site site system, with site labels i and j, to extract the magnetic interactions in the Mott limit. We choose the blocked orbital k to be the xy orbital for the Norb = 2 and Norb = 1 models. Fig. 4.3 gives ground state values of the square of the local total angular momentum, J 2 , for the two-site specialization of (4.1). For all three types of h i i hopping matrices, small t compared to λ give negligible local moments since spin-orbit coupling keeps each site in a nonmagnetic Ji = 0 spin-orbital singlet. For larger values of t, local moments may form from the tendency of superexchange to cause spin and orbital ordering which is incompatible with local spin-orbital singlet behavior on each site. For both

Norb = 3 and Norb = 2, this eﬀect is pronounced and requires t/λ 2 at the two-site level. ≈ In a lattice, this critical ratio will be reduced due to presence of many neighboring sites contributing to superexchange, hence a smaller hopping t is able to destabilize the atomic singlet. For Norb = 1, the eﬀect is much less pronounced since the number of superexchange paths is limited.

When a single orbital is active, the results do not sensitively depend on JH /U, however, the presence of strong Hund’s coupling results in qualitatively diﬀerent behavior for the

Norb = 3 and Norb = 2 models. We expect that antiferromagnetic superexchange between spins (spin-AF) is responsible for moment formation and can qualitatively be understood in the following way. Each site has a local total spin Si = 1 and local orbital angular momen-

54 J 2 h i i 0 1 2 3 4

JH /U = 0.1 JH /U = 0.2 0.25 0.25 (a) (b) 0.20 0.20

0.15 0.15

0.10 0.10

0.05 0.05 Norb = 3 Norb = 3 0.00 0.00 0 1 2 3 4 5 0 1 2 3 4 5 0.25 0.25 (c) (d) 0.20 0.20

0.15 0.15

t / U 0.10 0.10

0.05 0.05 Norb = 2 Norb = 2 0.00 0.00 0 1 2 3 4 5 0 1 2 3 4 5 0.25 0.25 (e) (f) 0.20 0.20

0.15 0.15

0.10 0.10

0.05 0.05 Norb = 1 Norb = 1 0.00 0.00 0 1 2 3 4 5 0 1 2 3 4 5 t / λ

Figure 4.3: The Hamiltonian in (4.1) is solved for a two-site system. The local total angular momentum squared on one site, J 2 , is plotted for small and large values of Hund’s h i i coupling, JH /U = 0.1 and JH /U = 0.2, for the three types of hopping matrices used in the text. (a-b) Hopping using Norb = 3 produces sizable local moments. For small Hund’s coupling, the local moment gradually forms as t is turned on. For large Hund’s coupling, there is an abrupt formation of large local moments due to an energy level crossing. (c-d) Hopping using Norb = 2 produces qualitatively similar behavior to the Norb = 3 case. (e-f) Hopping using Norb = 1 produces negligible moments.

55 (a) (b)

1 Figure 4.4: (a) The virtual process leaves the first site in a low spin, S = 2 , configuration and results in antiferromagnetic superexchange. (b) The virtual process leaves the first site 3 in a high spin, S = 2 , configuration and results in ferromagnetic superexchange.

3 tum Li = 1 ( P conﬁguration) from each t2g orbital being at least singly occupied with one of the three orbitals doubly occupied. To maximize the number of superexchange paths,

the orbitals participating in antiferromagnetic superexchange should be singly occupied.

This means the doubly occupied orbitals try to match up between neighboring sites, see

Fig. 4.4(a). When each pair of singly occupied orbitals between sites is in a spin singlet, the

two-site system is a spin singlet, but the orbitals are in a ferro-orbital state. The combined

spin-AF and orbital-F interactions are responsible for moment formation in Figs. 4.3(a) and

4.3(c).

Large values of Hund’s coupling can produce a diﬀerent ground state via a level crossing

at the sharp boundaries in Figs. 4.3(b) and 4.3(d) which are not present in Figs. 4.3(a)

and 4.3(c). This behavior can be understood by examining the eﬀect of Hund’s coupling

on the d3d5 states during the virtual d4d4 d5d3 d4d4 process. The intermediate d3 → → 3 1 may have either a maximized spin state Si = 2 or a minimized spin state Si = 2 , and large 3 values of Hund’s coupling make the intermediate Si = 2 states energetically more favorable. Moving an electron oﬀ a doubly-occupied orbital leaves the ion in an energetically favorable

3 Si = 2 conﬁguration. For example, see Fig. 4.4(b). Since this requires electrons to move onto single-occupancy orbitals on the other site, the Goodenough-Kanamori-Anderson rules

[57–59] indicate the the spin interactions are ferromagnetic (spin-F) but the orbitals are in a singlet state (orbital-AF). Here too, as in the small Hund’s coupling case, on a lattice when an electron hops along diﬀerent directions, the doubly occupied orbitals become bond-

56 dependent and lead to anisotropic interactions.

4.4 Eﬀective Magnetic Hamiltonian

We now begin our analysis of the full lattice problem in (4.1) by calculating the eﬀective spin-orbital lattice model for each of the three Norb cases. Only the main results are presented here; the details of the calculation are presented in Appendix C.1.

To understand the superexchange mechanisms in the three diﬀerent Norb models and how they compete with spin-orbit coupling, we derive an eﬀective magnetic spin-orbital

3 2S+1 Hamiltonian [29, 70] within the local P space (spectroscopic notation LJ ) on each site. This eﬀective Hamiltonian is written as the sum of spin-orbit and superexchange

terms. X (i) X (ij) Heﬀ = HSOC + HSE (4.3) i hiji

(i) λ The ﬁrst order spin-orbit correction H = Li Si is qualitatively correct, but we give SOC 2 · the second order eﬀective spin-orbit interaction within the local 3P space to numerically

match the energies from exact diagonalization.

2 (i) λ 1 λ 7 λ 2 HSOC = 1 Li Si (Li Si) (4.4) 2 − 4 JH · − 40 JH ·

The spin-orbital superexchange Hamiltonian, HSE, is constructed using three diﬀerent virtual exchange processes each deﬁned by the energy values of intermediate multiplets [71–73].

In the present case, the d3 electron conﬁguration in the virtual process d4d4 d3d5 d4d4 → → is used to label these superexchange pathways [74, 75]. Each pathway yields a superexchange (ij) term which is the product of a spin interaction and a tm0m-dependent orbital interaction 4 2 2 3 ij. Since S, D, and P label the intermediate d conﬁgurations, we arrive at the three O corresponding superexchange terms.

(ij) t2 S H = (2 + Si Sj) ij SE − U−3JH · O t2 D (1 Si Sj) (4.5) − U − · Oij t2 P (1 Si Sj) ij − U+2JH − · O

57 The first pathway, S corresponding to the 4S state, has the lowest energy of all the Oij intermediate states since maximizing the spin of the d3 configuration minimizes the total energy. We see that maximizing the total spin favors spin-F behavior as in Fig. 4.4(b). The other two pathways, D and P corresponding to 2D and 2P , minimize the total spin with Oij Oij 3 Si = 1/2 in the d configuration and will favor spin-AF as in Fig. 4.4(a).

The spin-F and spin-AF behaviors may be veriﬁed by observing that both 2+Si Sj and · 1 Si Sj are non-negative. Then the energy due to each pathway may be minimized by − · simultaneously maximizing the spin part and the orbital part separately. Since each hopping matrix tm0m uniquely determines the resulting orbital interactions ij, we will compute O these orbital interactions explicitly for previous three choices of tm0m (Norb = 3, 2, 1). We

2 2 4 will ﬁnd that D and P pathways will together dominate over the S pathway when JH /U is small, however, this can change at larger values of JH /U. The spin-orbital models we calculate here are similar to those of d2 systems [74, 75]. This

4 2 follows from the fact that a (t2g) electronic system is the particle hole conjugate to a (t2g)

2 4 hole system. Formally every (t2g) spin-orbital model may be transformed into a (t2g) spin- orbital model so long as (a) the crystal ﬁeld splitting is large enough to prevent high spin conﬁgurations from becoming energetically relevant and (b) the fundamental parameters λ and tm0m are negated.

Before proceeding with the explicit calculations for ij, we note the intimate connection O between the type of spin state favored (ie. spin-AF or spin-F) and the orbital state pictured in Fig. 4.4 is now mathematically depicted in (4.5). While each pathway contributes a spin- orbital term which is the product of spin and an orbital term, the sum of the three pathways cannot generally be factored in this way. The consequence is that even without the spin- orbit interaction, the spins and orbitals are not independent on a site in the lattice [76] even though they are independent at the atomic level. This can have important consequences on the types of ordering allowed in lattices when the orbital part becomes frustrated due to orbital geometry even on geometrically unfrustrated lattices [69].

58 (a) Norb = 3 (b) Norb = 2 (c) Norb = 1 0.

) -2.

/U AF 2 -4. t

-6. AF AF

Energy ( -8. F F

-10. 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 JH /U

Figure 4.5: Energy eigenvalues of the two-site superexchange Hamiltonian (4.5) are plotted for (a) Norb = 3 using (4.6), (b) Norb = 2 using (4.8), and (c) Norb = 1 using (4.10). In addition to a spin-AF ground state, a spin-F ground state can be favored when Hund’s coupling is large in both the Norb = 3 and Norb = 2 models.

4.4.1 Norb = 3

For the orbitally symmetric model, tm0m = tδm0m, the total orbital angular momentum, L, is conserved. We obtain the eﬀective superexchange terms for this interaction term

below.

S 4 2 2 2 = Li Lj (Li Lj) (4.6a) Oij 3 − 3 · − 3 · D 4 1 1 2 = + Li Lj (Li Lj) (4.6b) Oij 3 3 · − 6 · P 1 2 = (Li Lj) (4.6c) Oij 2 ·

To understand these results, we exactly diagonalize the eﬀective superexchange Hamiltonian

in the context of a two-site system. It is useful to rewrite each pathway in terms of projection

59 operators, P, to a particular subspace of total angular momentum L = 0, 1, 2.

S 4 ij = P(L = 1) (4.7a) O 3 D 5 3 ij = P(L = 1) + P(L = 2) (4.7b) O 6 2 P 1 1 ij = 2P(L = 0) + P(L = 1) + P(L = 2) (4.7c) O 2 2

Fig. 4.5(a) shows the energy levels of this superexchange Hamiltonian for diﬀerent JH /U for a two-site problem. Owing to the fact that (4.6a) can be written as the projection to a total angular momentum L = 1 shared along a bond (up to a factor of 4/3), the ground states of the 4S pathway in (4.5) have total L = 1 and total S = 2 shared between the two sites. Spin-orbit coupling will split these states and make local interactions favor a total

4 J = 1 shared along a bond. For large values of JH /U, the S pathway will dominate and the non-zero angular momentum shared between sites gives an eﬀective Curie moment to

2 2 the two-site system. Small values of JH /U will be dominated by the D and P pathways which favor total L = 2 and S = 0 in opposition to 4S. The critical value of Hund’s coupling where the S = 2 quintet overtakes the S = 0 singlet, as seen in Fig. 4.5(a), can be computed

1 analytically as JH /U = (√505 17) 0.1. 54 − ≈

4.4.2 Norb = 2

The eﬀective superexchange interaction for the two orbital model, tm0m = tδm0m (1 δxy,m), can be expressed with operators acting on the two active orbitals. − 0 z Let (τi, τ ) be the 3+1 Pauli matrices for the L = 1 subspace corresponding to the two i i ± active orbitals. For convenience, we deﬁne the permutation operator on the two active

1 0 0 orbitals as Pij = (τi τj + τ τ ). Then the orbital part of the superexchange Hamiltonian 2 · i j can be expressed in the following way.

S 2 1 0 0 = Pij + (τ + τ ) (4.8a) Oij − 3 3 i j D 1 1 0 0 1 z z = 1 Pij (τ + τ ) + τ τ (4.8b) Oij − 6 − 6 i j 2 i j P 1 1 0 0 1 z z = 1 + Pij (τ + τ ) τ τ (4.8c) Oij 2 − 2 i j − 2 i j

60 (a) Norb = 3 1 1 1

0 0 0 J xJ x SxSx LxLx i j -1 i j -1 i j -1

0. 0.5 1. 0. 0.5 1. 0. 0.5 1. θ/(π/2)

(b) Norb = 2 1 1 1

x x x x x x Si Sj Li Lj J J 0 0 0 i j

z z z z Si Sj Li Lj z z -1 -1 -1 Ji Jj

0. 0.5 1. 0. 0.5 1. 0. 0.5 1. θ/(π/2)

SxSx LxLx J xJ x 0 i j 0 i j 0 i j

z z z z z z S S Li Lj J J -1 i j -1 -1 i j

0. 0.5 1. 0. 0.5 1. 0. 0.5 1. θ/(π/2)

Figure 4.6: Expectation values of different angular momentum correlators are plotted for the two-site effective Hamiltonian in (4.3) using the three different Norb models with the 2 parameterization λ = cos θ, t /U = sin θ, JH /U = 0.1 and λ/JH = 1. The Norb = 3 model features full rotational symmetry while the Norb = 2 and Norb = 1 models only have one axis of rotational symmetry to make the z correlators different than the x and y correlators. The effect of increasing JH /U is to push the crossing point from spin-AF to spin-F behavior further left in these plots.

61 4 When JH /U 0, we recover the d spin-orbital superexchange Hamiltonian used by Khal- → iullin [21]. This limit ignores the F spin interactions induced by Hund’s coupling [21]. The above equations for the orbital part combined with both spin-AF and spin-F spin components from (4.5) give the complete spin-orbital interactions for the 2-orbital model.

It is also worth noting that when the 2D and 2P intermediate states are taken to have the same coeﬃcients, rotational invariance within the active orbital subspace can be restored.

2 2 Since the D and P pathways have the same 1 Si Sj spin part, these two pathways may − · be easily combined

D P 1 2 0 0 + = 2 + Pij (τ + τ ) (4.9) Oij Oij 3 − 3 i j z z so that the τi τj Ising anisotropy has been eliminated. This allows us to draw a parallel between the Norb = 3 and Norb = 2 models. In the Norb = 3 model, the S = 0 state

(spin-AF) was a maximized L = 2 (orbital-F). In the Norb = 2 model, the S = 0 state

z z is L = L = 0 as seen in (4.9) since ij is to be maximized so that (4.5) is minimized. i j O Graphically this is shown in Fig. 4.4(a). This tendency for spin AF to be accompanied by aligned orbitals is common in spin-orbital models. Similarly, spin F tends to be accompanied by oﬀ-alignment of the orbitals as in Fig. 4.4(b).

D P Returning to the full Norb = 2 case where and are not combined, we diagonalize Oij Oij eﬀective superexchange Hamiltonian for a two-site system. For a critical value of Hund’s

1 coupling, JH /U = (√34 5) 0.09, the S = 2 quintet overtakes the S = 0 singlet as 9 − ≈ shown in Fig. 4.5(a).

4.4.3 Norb = 1

To complete the discussion, we calculate the eﬀective superexchange Hamiltonian for a single orbital hopping model tm0m = tδm0mδxy,m.

S = 1 (L2 + L2 ) 2 L2 L2 (4.10a) Oij 3 i,z j,z − 3 i,z j,z D = 1 (L2 + L2 ) 1 L2 L2 (4.10b) Oij 3 i,z j,z − 6 i,z j,z P = 1 L2 L2 (4.10c) Oij 2 i,z j,z

62 (a) AF (b) F 3 3 2 2 2 2 1 1

yz zx xy AF F unfavorable

(a) AF (b) F 3 3 2 2 2 2 1 1

yz zx xy AF F unfavorable

Figure 4.7: Orbital frustration is graphically illustrated for the Norb = 2 model. The orbitals shown on the vertices of the plaquette are the doubly occupied orbital on each site in a square lattice. Once the first bond, labeled as 1, is chosen to be of a particular type, either (a) AF or (b) F, the next bonds, labeled as 2, are immediately fixed by this choice. The result is that the last bond on the plaquette, labeled as 3, then takes a configuration which is neither the most energetically favorable AF bond nor the most energetically favorable F bond.

63 4 The S pathway is only active when one site is in a Li,z = 1 state and the other site is in ± a Li,z = 0 state reﬂecting that one hole needs to be shared between the sites to allow F. By combining the 2D and 2P pathways as before,

D + P = 1 (L2 + L2 ) + 1 L2 L2 (4.11) Oij Oij 3 i,z j,z 3 i,z j,z we see AF behavior is maximized when both sites are in the Li,z = 1 state reﬂecting that ± the xy orbitals participating in superexchange should be singly occupied, and the other two may be doubly occupied. After diagonalizing the full two-site Hamiltonian in Fig. 4.5(c), we

ﬁnd this case is qualitatively diﬀerent from the previous two cases in that F interactions are not supported for any reasonable value of JH /U. Fig. 4.4(b) shows the physical mechanism for F requires active orbitals on opposing sites to share a single hole. While spin-F states are supported by fewer superexchange paths compared to their spin-AF counterparts, there were many paths for spin-F states to reduce their energy in both the Norb = 3 and Norb = 2 models so that Hund’s coupling could still tip the balance in favor of spin-F. However, for

2 Norb = 1 model, the energy of a spin-F state can only be reduced by a single factor of t /U − per site, and favorable spin-F interactions require more than one orbital to be energetically favorable.

4.5 Orbital Frustration

While the orbitally symmetric Norb = 3 model features rotational symmetry, the Norb =

2 and Norb = 1 models do not due to their orbital geometries. The Norb = 1 model requires the single active orbital along a bond to be singly occupied so that AF spin superexchange interactions can minimize the energy. In d4 conﬁgurations, only two of the three orbitals can be singly occupied while one orbital must be doubly occupied. Since the doubly occupied orbital cannot participate in AF spin superexchange, one third of the bonds must be unsatisﬁed.

The Norb = 2 model extends this concept with the possibility for two diﬀerent low energy states depending on the value of Hund’s coupling. Fig. 4.4(a) shows that when

64 two orbitals are active, an AF spin interaction favors double occupancy on the inactive orbital. An AF bond in the Norb = 2 model then favors the orbitals perpendicular to the bond direction (xy doubly occupied along a z-direction bond) to be doubly occupied. Bond

1 in Fig. 4.7(a) is an example of such an AF bond (green). Choosing those two doubly occupied orbitals shown in the ﬁgure immediately restricts on other bonds emanating from these two sites. Since the doubly occupied orbitals are not perpendicular to the bonds labeled 2, a diﬀerent interaction must be favored along the bonds labeled 2. The next most energetically favorable interaction is the F bond (red) shown in Fig. 4.4(b). This places the double occupancies on the other two orbitals and requires the doubly occupied orbitals on each site to be opposite (ie. xz-yz along z-direction). However this leaves the

ﬁnal bond labeled 3 (blue) matching neither the criteria for the lowest energy AF or lowest

energy F bond and instead takes an energetically unfavorable AF conﬁguration. Similarly,

starting with an F bond in Fig. 4.7(b) as the most energetically favorable results in the same

conclusion. The orbital degrees of freedom then require one of the four bonds on a plaquette

to take a high energy conﬁguration in both scenarios. Regardless of the value of JH /U, the

Norb = 2 model again naturally yields frustration due to the orbital degrees of freedom even

on nominally unfrustrated lattices. When λ zJSE where z is the coordination number, these orbital eﬀects are very strong and, in the absence of large octahedral distortions, may

lead to orbital liquid states and perhaps highly entangled spin-orbital phases of matter due

to quantum ﬂuctuations.

4.6 Triplon Condensation

Here we study the case where spin-orbit coupling λ is signiﬁcantly larger than superex-

4 change JSE, particularly relevant to 5d materials. For JSE = 0 the ground state is the

product of Ji = 0 singlets and therefore non-magnetic. With increasing JSE/λ, a local moment starts to form continuously though long range magnetic order sets in at a ﬁnite

value of JSE/λ. This ordered region can be described as triplon condensation of weakly interacting J = 1 excitations that evolve to a strongly interacting regime. In this section,

65 we give a detailed introduction to the mechanism of triplon condensation and then apply the formalism to the three Norb models considered.

4.6.1 Overview of the Mechanism

With zero superexchange, the energy cost to make a Ji = 1 triplon excitation is λ/2. It was shown previously [21], using the bond operator formalism [77], that for spin-AF superexchange interactions that are substantially weaker than spin-orbit coupling, the superexchange interactions allow these triplet excitations to propagate from site to site and disperse in k-space to reduce the energy cost for the excitation around the π-point until condensation of these triplon excitations occur and order antiferromagnetically; (see Fig. 4.8).

One recent work has tested this mechanism with dynamical mean ﬁeld theory in the limit of inﬁnite dimensions [78]. Here we ask the question: Can spin-F interactions from the 4S pathway cause condensation for large JH /U? If so, then at which k-point does the condensation occur? We show that for spin-F interactions there is a condensate but surprisingly the condensate does not always occur at the expected k = 0 point.

When λ becomes much larger than the superexchange energy scale t2/U, the high energy

Ji = 2 states become energetically unfavorable and can be ignored. We project out Ji = 2 states from our spin-orbital superexchange Hamiltonians leaving just the Ji = 0 and Ji = 1 † parts. We utilize a set of operators Ti to describe the triplon excitations from Ji = 0 states † to Ji = 1 states. These operators are deﬁned by Ji = 1,Ji,z = m = T Ji = 0 . We | i i,m | i then project the superexchange Hamiltonian onto the space of triplon operators, keeping † † † only terms which are quadratic in the triplon operators (ie. Ti Tj, Ti Tj ) and throw away terms with three and four triplon operators which constitute eﬀective interactions between triplon excitations. See Appendix C.2 for calculation details. Since the projection of the

† i † magnetization operator is Mi = i√6(T T ) T T , the quadratic part describes − i − i − 2 i × i interactions between van Vleck excitations. This approximation cannot be justiﬁed deep in the condensed phase where interactions between excitations cannot be neglected, however, it can provide a good estimate of when the t2/U energy scale is large enough to support condensation and qualitatively what kind of magnetic ordering to expect.

66 J = 2 λ i

Ji = 1 Energy λ/2 T i+1 T i†+2

Ji = 0 πk π −

Figure 4.8: The triplon condensation mechanism is graphically illustrated. When there exists a triplon excitation on a site, superexchange can move the excitation to neighboring sites. This eﬀective hopping causes the triplon’s energy to disperse in k-space. When superexchange becomes large enough, condensation of triplon excitations occurs as the bottom of the triplon band becomes lower in energy than the original Ji = 0 level.

There is, however, a more subtle consequence of the quadratic approximation. Since each site may only accommodate at most one triplon, there is a hardcore Boson constraint on every site. Although neglecting this constraint and the triplon-triplon interactions is necessary to put the solutions in closed form, orbital frustration may be lost under these approximations. This is separate from the orbital anisotropy that will always remain present.

This is notable because anisotropic interactions usually cause frustration, yet here they may not due to the approximation that the triplons are non-interacting. Again, deep in the condensed phase (the unexplored region of Fig. 4.1(b)), the exact solutions may qualitatively diﬀer from the picture depicted here.

After projecting the superexchange Hamiltonian to this quadratic subspace and making † † † † † † † a transformation to cubic coordinates (ie. T ,T ,T Tx,Ty ,Tz ), the most general T T −1 0 1 → i j term can be decomposed into the three parts given below

(ij) † H = J T Tj + h.c. (4.12) iso i · (ij) † H = D T Tj + h.c. (4.13) skew · i × (ij) † H = T Γ Tj + h.c. (4.14) symm i · ·

67 † † and similarly for the Ti Tj terms. First consider the isotropic term in (4.12). When J † † † becomes large enough, all three ﬂavors of triplons (Tx, Ty , Tz ) condense simultaneously. Negative J causes condensation at the k point corresponding to a F condensate of van Vleck excitations while positive values cause condensation at the π point corresponding to AF van

Vleck excitations. Like the Heisenberg term, it comes from the orbitally symmetric component of the interactions like those considered in (4.6). The addition of further neighbor interactions can cause condensation at arbitrary q-vector. Next, the skew symmetric term in (4.13) results in a magnetic spiral condensate at q = π/2 points, and the addition of fur- ± ther neighbor interactions along with isotropic terms can make arbitrary q-spirals possible.

Like the Dzyaloshinskii-Moriya interaction, this term requires broken inversion symmetry.

Finally, the symmetric anisotropy in (4.14) picks one of the three ﬂavors of triplons as the favored condensate due to orbital anisotropy.

This qualitative picture can be extended to ﬁnite temperature since the condensation mechanism falls into the Bose-Einstein condensation universality class. Finite temperature condensation has been extensively studied [79], so we will instead focus the unique aspects of spin-orbital condensation from superexchange.

4.6.2 Results

We ﬁrst determine which orbital geometries allow for F condensation when the hopping matrix is diagonal. Then we decompose the hopping matrix into multipoles

(ij) X k k t = t0A0 (4.15) k where multipoles are deﬁned using Wigner-3j symbols   j k j 0 k j−m0   jm Aq jm = ( 1)   (4.16) h | | i − m0 q m −

68 k and t0 are the coeﬃcients of the decomposition. Then (4.2) is rewritten in the new form below.

(ij) X k X k † Ht = t0 A0 cimσcjmσ (4.17) mm k mσ This form is particularly convenient to calculate the resulting spin-orbital superexchange

k form for each of the t0 hopping matrices. Here we will only give the results, and details of the calculation are relegated to Appendices C.2 and C.3.

0 For Norb = 3, the hopping matrix tm0m is simply proportional to t0 in (4.15) and (4.17). 0 We ﬁnd that isotropic hopping, t0, only supports AF regardless of the value of Hund’s coupling. This result contradicts the claim of Meetei et. al. [12] that F condensation

1 results for orbitally symmetric hopping. If the hopping matrix is either skew-symmetric, t0, 2 4 or symmetrically anisotropic, t0, the overall sign of the S pathways is the opposite to that of the 2D and 2P pathways, and large Hund’s coupling can favor a F condensate. The main diﬀerence between the isotropic term and the anisotropic terms is that both anisotropic terms feature matrix elements of diﬀerent signs while the isotropic term does not. Both the

Norb = 2 and Norb = 1 models have hopping matrices described as linear combinations of

0 2 t0 and t0. However, only AF condensates result in these cases since their decompositions are closer to the isotropic hopping matrix instead of the anisotropic hopping matrix. The ferromagetic condensation of triplon excitations is possible, but, due to the isotropic term only favoring AF, special orbital geometries are required for F condensation.

The lack of F condensation for most common orbital geometries has an immediate consequence on the phase diagram for d4 materials. In the limit of large spin-orbit coupling for any value of JH /U, there is a PM to AF transition with increasing superexchange. However the limit of small λ allows for both F and AF interactions. Then there must be an additional phase transition between the AF condensate phase and a spin-orbital F phase at

2 intermediate values of (t /U)/λ when JH /U is large in the unexplored region of Fig. 4.1(b). In the case of AF condensation, we give the critical condensation value (t2/U)/λ for

each of the three models.

Norb = 3: Since this model possesses rotational invariance, each triplon ﬂavor condenses • 69 simultaneously. The eﬀective singlet-triplet gap from on-site interactions is ∆ = λ/2 − z 2 3 (t /U) where z is the coordination number and the inter-site interactions give aδ = 5/3 2 and bδ = 4/3 in (C.22). Condensation occurs at the (π, π, π) point at t /U = λ/40. −

Norb = 2: This model was studied by Khaliullin [21] where the orbital anisotropy was • averaged away so that condensation occurred at (π, π, π) at t2/U = λ/20 for simple cubic

lattices. We can conclude that triplon condensation is then likely to be active in both 4d

and 5d transition metal oxides.

Norb = 1: In this case, condensation occurs along a degenerate set of points: at (kx = • 2π/a, ky = 0, kz) where kz can be arbitrary for the z-boson and the other 3 degenerate

lines related by C4 symmetry. Here the 4 lines are parallel to the kz axis for the z

boson, and the x and y bosons condense along lines being parallel to the kx and ky axes

respectively. We ﬁnd that aδ = 1/6 for directions perpendicular to a bond and aδ = 2/3

in the direction of a bond. The values of bδ are just the negatives of the aδ values. On a face-centered cubic lattice, we ﬁnd a critical value of t2/U = (3/32)λ. With this value,

condensation is likely to occur in 4d compounds, but large values of spin-orbit coupling

in 5d compounds will likely prevent condensation from occurring.

4.6.3 Local Interactions versus Condensation

It is surprising that although local F interactions were found for large values of JH /U,

F condensation did not appear even in the isotropic Norb = 3 model which was free of orbital frustration. Even more surprising was that isotropic hopping is the cause of this unexpected result despite our calculations in Figs. 4.3(b). This discrepancy can be resolved by examining the two site problem more carefully, and our goal is to tie the two site and lattice condensation results together. To do this, we focus on the key problem presented: the lack of F condensation in a spin-orbital superexchange Hamiltonian which is explicitly

F by construction.

First we will rewrite the ferromagnetic part of the superexchange Hamiltonian for the

70 Norb = 3 model appearing in equations (4.5) and (4.6a)

JSE 2 HSE = (2 + Si Sj) 2 Li Lj (Li Lj) (4.18) − 2 · − · − · which is pitted against the lowest order spin-orbit correction.

λ HSOC = (Li Si + Lj Sj) (4.19) 2 · ·

Fig. 4.9 shows the energy spectra for a two site system parameterized with λ = cos θ and

JSE = sin θ. On the right hand side at θ = π/2, the lowest energy levels are the total S = 2 and L = 1 states. A small amount of spin-orbit coupling splits the states into total

J = 1, 2, 3 states as already stated in Section 4.4.1. On the left hand side at θ = 0, the

two lowest energy levels are as follows: a non-degenerate state with both sites in the non-

magnetic J = 0 singlet state and a six-fold degenerate ﬁrst excited level where one of the two

sites contains an excitation. These levels correspond to the vacuum and the triplon band

of excitations in the condensation picture. Introducing a small amount of superexchange

splits the ﬁrst excited states into symmetric and anti-symmetric states. The lower energy

states are the anti-symmetric ones which correspond to condensation at the π-point from a

triplon condensation Hamiltonian T † T + h.c. with a positive hopping coeﬃcient. i · j Here lies the source of the discrepancy between two site results and the lattice conden-

sation result. There are two sides to the lowest energy J = 1 (S = 2, L = 1) level: the

regime where λ and JSE are comparable and the regime where λ JSE (θ 0). Section ≈ 4.4.1 showed that the regime where the two interactions were comparable produced F spin

interactions. However these J = 1 states in the λ JSE regime correspond to triplon AF. Even though the two regimes are smoothly connected, the nonmagnetic Ji = 0 states are lower in energy than the triplon excitations. Then two site exact diagonalization covers

up this aspect of AF while the lattice limit allows the anti-symmetric level (triplon band)

decrease enough in energy to reveal the AF nature of the θ 0 part of the lowest energy ≈ J = 1 states.

To summarize, while from the exact diagonalization, it seemed like the J = 1 line in

Fig. 4.9 should have been FM for all θ. However, in the region where it was not FM and

71 J = 2 S = 1 -1.0 L = 1 J = 1

Ji/j = 1 J = 0 -1.5 symm

anti- symm -2.0 Ji = 0

Energy Jj = 0

2.5 - J = 3 J = 2 S = 2 -3.0 J = 1 L = 1 0.0 0.2 0.4 0.6 0.8 1.0 θ / (π/2)

Figure 4.9: Energy levels of the eﬀective Hamiltonian HSE + HSOC appearing in equations (4.18) and (4.19) with the parameterization λ = cos θ and JSE = sin θ. The levels are labeled by their good quantum numbers. In the θ = 0 limit, the eigenstates of spin-orbit coupling are used, and, in the θ = π/2 limit, the eigenstates of the spin-orbital superexchange Hamiltonian are used. The interpretations of the states are discussed in the main text.

72 was actually the anti-symmetric AFM triplon condensate (at the two-site level), a diﬀerent

AFM energy level (Ji = Jj = 0) was the one exact diagonalization was measuring, not the J = 1 AFM condensate line which was the next-lowest energy level. In the full lattice (not 2 site), the anti-symmetric level is the condensate (AFM) and drops below the (Ji = Jj = 0) energy level. This arises because on the right side for θ greater than the crossing, J = 1 comes from S = 2 and L = 2 (FM) while on the left side J = 1 comes from an antisymmetric splitting of Ji = 1 and Jj = 0 with Ji = 0 and Jj = 1 (AFM). Lastly Fig. 4.9 highlights a diﬀerence between local spin AF and triplon AF. The exact diagonalization results in Figs. 4.3 show that the two site model features a smooth transition between Ji = 0 states and AF behavior. The same behavior is found in Fig. 4.9 by following the Ji = 0 line to the J = 0 line which is the transition from local non-magnetic singlets to a total J = 0 singlet between sites. (For reference, the AF-F level crossing discussed in the ﬁrst two sections would be shown here by this AF J = 0 (S = 1, L = 1) level being

overtaken by the F J = 1 (S = 2, L = 1) level when the other two pathways in (4.5) are

included.) The AF from triplon condensation corresponds to the anti-symmetric level, and

these two types of AF are therefore diﬀerent. In fact, from the quantum numbers shown in

Fig. 4.9, the two types of AF belong to diﬀerent irreducible representations at the two site

level. It is then possible that there is an additional phase boundary in Fig. 4.1(b) which

separates AF triplon BEC and spin AF despite the fact that they are both AF phases.

4.7 Materials and Experiments

The d5 iridates (Ir4+) with half ﬁlled j = 1/2 bands have attracted a signiﬁcant amount

of attention due to the interplay between strong correlations and spin-orbit coupling. Al-

though experimental studies have recently been focused on these materials, many strongly

correlated oxides with moderate to strong spin-orbit coupling have d4 conﬁgurations. A well

studied example [60, 61] is Ca2RuO4 which necessarily violates Hund’s rules which require

4 non-magnetic Ji = 0 Ru sites, however there are many less well studied d materials. There

are many double perovskites of the form A2BB’O6 where both A and B have completely

73 4 ﬁlled valence shells and the B’ site is in the d conﬁguration including La2ZnRuO6 [80],

La2MgRuO6 [81], Sr2YIrO6 [62], Ba2YIrO6 [24–26], and a large array of compounds with

the form Sr2BIrO6 [63]. However, from our Norb = 1 results in the previous sections, 5d double perovskites are unlikely candidates for triplon condensation due to the small su-

perexchange energy scales when compared to spin-orbit coupling. Additionally, it has been

suggested that the observed magnetism in 5d double perovskites is due to disorder and/or

impurities [25]. However, both 4d compounds and compounds with more than one active

4 orbital (Norb = 2 or 3) should be good candidates. Honeycomb d oxides Li2RuO3 and

4 Na2RuO3 have been found to order antiferromagnetically [65]. Other d oxides include

post-perovskite NaIrO3 [82] and pyrochlore Y2Os2O7 [83]. Many probes can be used to deduce the existence of novel magnetism in d4 materials. In

particular, both magnetic susceptibility and x-ray absorption spectroscopy (XAS) have the

advantage that magnetic ordering is not required to infer the existence of moments. The

ﬁrst test for novel magnetism in d4 systems comes at the level of magnetic susceptibility

measurements. Curie Weiss ﬁts provide a measure of the eﬀective magnetic moment, and

measuring a non-zero eﬀective moment (of order 1 µB) is a direct indication that the ground

state is not the product of non-magnetic Ji = 0 singlets. Determining whether the ground state is not the product of non-magnetic singlets can also be probed by XAS [84]. In the

non-magnetic Ji = 0 singlet state, L2 absorption edge intensity is zero while the L3 edge is non-zero which leads to a diverging branching ratio

I 2 + r B.R. = L3 = (4.20) IL 1 r 2 − where r = ( Li) Si / nh . However, the Ji = 1 and Ji = 2 states lead to ﬁnite L2 edges h − · i h i with magnitudes comparable of that of the L3 edge. Thus measuring a branching ratio of order 1 is direct evidence against the non-magnetic singlet ground state.

74 4.8 Conclusions

We have studied how superexchange opposes the effect of spin-orbit coupling in d4 systems and induces local moments and interactions between them. If Hund’s coupling is large, the local interactions favor ferromagnetism instead of the expected antiferromagnetism. We also found that at least two orbitals need to be involved for this local ferromagnetic behavior to be energetically favorable. The condensation mechanism allows AF to generally be favorable in both 4d4 and 5d4 compounds. However because isotropic orbital interactions favor antiferromagnetic condensation regardless of how large Hund’s coupling is, ferromagnetic condensation is unlikely in materials systems. We would like to highlight again that the ability of rotationally invariant atomic spin-orbit coupling to flip the sign of the effective exchange constant on a superexchange interaction is a unique feature of spin-orbital systems and has no analog in pure spin systems.

The eﬀective magnetic Hamiltonians derived here for transition metal oxides with d4 occupancy can be directly used for the particle-hole symmetric d2 occupancy as well after changing the sign of the hopping and spin-orbit couplings. These Hamiltonians lay the foundation for spin-orbit coupled Hamiltonians in the t2g sector. Going forward, diﬀerent analytical and numerical methods can now be applied to obtain detailed phase diagrams.

75 Chapter 5 Magnetic Condensation in Ba2YIrO6

5.1 Introduction

The anomalous magnetic moments observed in Y-Ir double perovskites has been the subject of many theoretical and experimental investigations. Ir5+ ions are in the d4 (ie.

4 t2g) configuration and are expected to be non-magnetic in both the weakly and strongly correlated limits. In the weakly correlated picture, Ir t2g shells are split into a fully filled j = 3/2 shell and an empty j = 1/2 shell due to strong spin-orbit coupling present in 5d transition metals. Spin-orbit coupling then opens up a bandgap between the j = 3/2 and j = 1/2 bands leading to a non-magnetic insulating ground state. In the strongly correlated picture, the first two Hund’s rules require each Ir5+ site to be in a total S = 1 and total

L = 1. The strong spin-orbit coupling then couples the spin and orbital degrees of freedom to a rotationally invariant J = 0 state on every ion. Again the conclusion is that the ground state is non-magnetic.

Why then do Y-Ir double perovskites show Curie susceptibilities? There are two promi- nent explanations for the existence of magnetism in these 5d4 compounds. The ﬁrst mechanism is quenching of the orbital degrees of freedom through octahedral distortions. Uniaxial distortions are given as δ (L nˆ)2 where δ is the energy scale of the interaction and nˆ is · the direction of the distortion. This splits the t2g orbitals into orbital singlet and orbital doublet levels where the lower energy level is determined by the sign of δ. When the orbital

76 doublet level is lower in energy, those orbitals are doubly occupied while the orbital singlet is unoccupied. Again the conclusion is a non-magnetic ground state. However if the orbital singlet level is lower, it is occupied by two electrons while the other two electrons occupy the orbital doublet level. This half ﬁlled orbital doublet then has a total spin S = 1 leading

to a magnetic ground state.

Cao et. al. [62] proposed this distortion mechanism was the cause of the 0.9 µB/Ir moments in their Sr2YIrO6 samples. Additionally, they found unusual features in the magnetization and speciﬁc heat curves which appeared to signal a phase transition associated with the magnetism near temperatures of 1 K. However several later studies contradicted these claims. First, Ranjbar et. al. [24] and Phelan et. al [26] studied the series of compounds

Ba2−xSrxYIrO6 where Ba was substituted for Sr. While large compressive octahedral distortions are present in Sr2YIrO6, the Ba2YIrO6 compound is undistorted and should not gain moments from a distortion mechanism. Similar moments were found in both Sr2YIrO6

and Ba2YIrO6 leading to the conclusion that distortions are not responsible for moment

formation. A third study by Dey et. al. [25] looked at Ba2YIrO6 in detail and conﬁrmed the abscence of magnetic ordering and anomalies in the speciﬁc heat down to 0.48 K despite the presence of magnetic moments.

The second theoretical mechanism to explain the existence of magnetic moments is

“excitonic” condensation ﬁrst proposed by Khaliullin [21] in the context of ruthenates but still generally applicable to strongly correlated 4d4 and 5d4 systems. Each Ir5+ ion is nominally in a local non-magnetic J = 0 state with an energy gap of λ/2 to the next lowest energy state of J = 1 where λ is the spin-orbit coupling strength appearing as (λ/2)L S. · For very small superexchange interactions, the system remains in its unperturbed product state of non-magnetic J = 0 singlets. However, larger superexchange interactions can result in a second-order phase transition to a magnetic ground state. Superexchange eﬀectively allows a J = 1 excitation to move between sites giving it a k-space dispersion ω(k). The

bandwidth of this dispersion is directly proportional to the superexchange coupling constant

t2/U where t is the hopping and U is the on-site Coulomb repulsion. As the bandwidth of

ω(k) increases, the energy gap between non-magnetic J = 0 product state and excited J = 1

77 states is reduced until the gap closes at a critical value of (t2/U)/λ where the condensation

of these magnetic excitations occurs.

Despite the observation of Curie moments in both Sr2YIrO6 and Ba2YIrO6, density functional theory calculations have not explained the origin of magnetism. Initially, Bhowal

et. al. [64] performed GGA+SOC+U calculations within the plane wave basis to ﬁnd anti-

ferromagnetic (AFM) ground states in both Sr2YIrO6 and Ba2YIrO6 despite the absence of

distortions in Ba2YIrO6. However, a later study by Pajskr et. al. [13] rebutted this claim, and further employed dynamical mean ﬁeld theory (DMFT) to again obtain a non-magnetic

ground state. Furthermore, this study estimated the gap between the J = 0 singlets and

the J = 1 triplets to be more than 250 meV which marginalizes the prospect of condensa-

tion. We will argue on simpler grounds that this ﬁnding is qualitatively correct, and the

condensation of magnetism due to intrinsic eﬀects does not occur in Ba2YIrO6.

5.2 Absence of Condensation

We will revisit the DFT-DMFT theory results [13] which conclude that the ground state

of Ba2YIrO6 is non-magnetic. To gain insight into why magnetic condensation is absent, we must connect those results with the condensation mechanism which pits superexchange

with an energy scale of t2/U against the singlet-triplet energy gap of λ/2 from spin orbit

coupling. The critical ratio of superexchange to spin-orbit coupling required to produce

magnetic condensation in a single perovskite [21] is given approximately by 10t2/U λ/2. ≥ In this section, we will repeat this analysis for double perovskites and ﬁnd the criteria for

closing the singlet-triplet gap.

We ﬁrst obtain a tight-binding model relevant for Y-Ir double perovskites. The large

crystal ﬁeld splitting induced by 5d oxygen octahedral complexes separates the eg orbitals

in energy so that only the t2g orbitals are relevant. Then we can write our tight-binding

model as the sum of electron hopping between t2g orbitals in the yz, xz, and xy planes

as H = Hyz + Hxz + Hxy. For the xy plane, for example, we can restrict the form of the

78 Ba2YIrO6_soc atom 3D-t2g size 0.20

0.0 EF

Energy (eV) -1.0 W L Λ Γ ∆ X Z W K

Figure 5.1: The GGA band structure without spin-orbit coupling is shown. A tight-binding model with t2g Wannier orbitals is ﬁt to the three bands pictured.

tight-binding model by symmetry

X X X † Hxy = tαβ ciασcjβσ + h.c. (5.1) hiji ∈ xy αβ σ∈{↑,↓}

  t11 t12 0     tαβ =  t t 0  (5.2)  12 11    0 0 t33 αβ

where α and β index the t2g orbitals yz, xz, and xy in this order and ij ranges over nearest h i neighbors within an xy plane of the lattice of Ir ions. Next-nearest neighbor hopping is

ignored since the resulting superexchange constants will be negligible. We then perform a

DFT calculation [85] to obtain these tight-binding parameters without spin-orbit coupling

to separate the energy scales for superexchange and spin-orbit coupling. Fig. 5.1 shows

the band structure of the three (t2g) bands at the Fermi energy. A tight-binding ﬁt with

maximally localized Wannier orbitals [86] yields the following parameters: t11 = +23 meV,

t12 = 19 meV, and t33 = 131 meV. ± − We now calculate the triplet excitation Hamiltonian where triplet excitations from

the non-magnetic J = 0 to excited J = 1 states are described by Ji = 1,Ji,z = m = | i † T Ji = 0 . After performing a unitary transformation into cubic coordinates Tx = i,m | i

79 1 1 √ (T1 T−1), Ty = √ (T1 + T−1), and Tz = iT0, the quadratic part of the eﬀective i 2 − 2 triplet Hamiltonian has the following form

X 1 h i H0 = T † A T + h.c. + T † B T † + h.c. + T † C T + (i j) xy U i · · j i · · j i · · i ↔ hiji∈xy (5.3) † † † † where Ti = (Ti,x,Ti,y,Ti,z) and the matrices A, B, and C are given below.   2 1 2 2 1 2 1 t + t33t11 + t + t t11t12 t12t33 0  11 2 3 12 6 33 − 6   1 2 1 2 2 1 2  A =  t11t12 t12t33 t + t33t11 + t + t 0   − 6 11 2 3 12 6 33   2 2 1 2 2  0 0 3 t11 + 3 t33t11 + 3 t33 (5.4)

  5 2 1 5 2 1 2 1 t t33t11 t t t12t33 t11t12 0  − 6 11 − 3 − 6 12 − 6 33 3 −    B =  1 t t t t 5 t2 1 t t 5 t2 1 t2 0   3 12 33 11 12 6 11 3 33 11 6 12 6 33   − − − − −  0 0 2 t2 2 t2 − 3 11 − 3 33 (5.5)

 5 2 1 1 2 1 2 1 1  − 18 t11 − 9 t33t11 + 6 t12 + 18 t33 − 3 t11t12 − 3 t33t12 0   C =  1 1 5 2 1 1 2 1 2   − 3 t11t12 − 3 t33t12 − 18 t11 − 9 t33t11 + 6 t12 + 18 t33 0    2 2 4 1 2 0 0 9 t11 − 9 t33t11 − 9 t33 (5.6)

The triplet Hamiltonian for the other two planes can be obtained from cyclic permutations

0 on Hxy. The total Hamiltonian for the triplet excitations is given by the sum over all three planes and the singlet-triplet gap of λ/2.

λ X H0 = H0 + H0 + H0 + T † T (5.7) yz zx xy 2 i · i i

To determine when condensation occurs, we diagonalize the eﬀective Hamiltonian H0 and obtain the energy dispersion ω(k) for triplet excitations. Before performing the rigorous calculation with all parameters included, it is useful to obtain a simple estimate for the energy scales required to produce magnetic condensation. To obtain this estimate, ignore the superexchange dependent contributions to the singlet-triplet gap (ie. set C = 0) and

80 230

220

210

200 Energy ( meV )

190

180 W L Γ X W K

Figure 5.2: The triplet excitation spectrum is plotted using a typical value for the spin-orbit gap in 5d oxides of λ/2 = 200 meV.

only include the largest tight-binding parameter t33 (ie. set t11 = t12 = 0). In this simpliﬁed scenario, the A and B matrices are diagonal with A = B, and a closed form solution − can easily be obtained. The condition for closing of the singlet-triplet gap is given by the

following. 16 t2 λ 33 (5.8) 3 U ≥ 2 This value is approximately half of the estimate for single perovskites [21] (10t2/U λ/2) ≥ with the diﬀerence due to fewer superexchange paths available in face-centered cubic geometry compared to simple cubic geometry. With a U value taken to be 2 eV, the left-hand side is estimated to be 45 meV compared to a typical spin-orbit gap for 5d materials λ/2 200 ≈ meV. From this estimate, the condensation mechanism should be inactive in Ba2YIrO6. (Note that equation (5.8) only gives the criteria for a gap closing. The diﬀerence between the two sides of equation (5.8) does not give the value of the remaining gap.)

The triplet dispersion for the full model with all tight-binding parameters is shown in Fig. 5.2 using typical values for spin-orbit coupling and Coulomb repulsion given by

λ/2 = 200 meV and U = 2 eV. The lowest energy triplet excitation occurs along X-W at

81 about 20 meV below the spin-orbit gap of λ/2. We then see that the energy scale of the triplet dispersion is negligible compared to the spin-orbit coupling energy gap. Consequently magnetic condensation does not occur by a large margin.

Although we have so far neglected next-nearest neighbor interactions, we now show they only give a negligible correction to the estimate in (5.8). Without going through further formalism, we can estimate the effect of including both nearest neighbor (NN) and next-nearest neighbor (NNN) interactions. Since the NNN Ir ions are in an octahedral configuration around each Ir ion, the NNN case is exactly that of the single perovskite first obtained by Khaliullin. Since the dispersions of the NN and NNN interactions add linearly for each triplet excitation type, ω(k) = ωNN(k) + ωNNN(k), we can put quickly estimate the result by adding the two energy scales together

16 t2 t2 λ 33 + 10 NNN (5.9) 3 U U ≥ 2 where the relevant NNN tight-binding parameter obtained from our DFT calculation is tNNN = 18 meV. This contributes an extra 2 meV on the left hand side of (5.9) which is − a negligible correction.

5.3 Application to Experiments

Although the routes to intrinsic magnetism seem unlikely, extrinsic effects remain plau- sible. Dey et. al. [25] suggested the magnetism in Ba2YIrO6 was due to paramagnetic impurities. In a later study [87], they identified a Schottky anomaly in the specific heat of

Sr2YIrO6 pointing to the conclusion that paramagnetic impurities are responsible for the observed Curie susceptibilities without the onset of long range magnetic order. Synthe-

sis of double perovskites generally presents challenges due to disorder and impurities. We

speculate that antisite disorder or oﬀ-stoichiometry is responsible.

Since the absence of magnetism in d4 Mott insulators can be understood at the atomic

level, it would seem that antisite disorder would not alter the picture. However the argument

for generating magnetism from superexchange involves interactions between sites. While

82 the eﬀects of interactions between B’ sites is unlikely to generate a magnetic state, the situation is diﬀerent when considering neighboring B and B’ sites. First, the tight-binding values (t) become much larger due to the increased overlap of orbitals between sites. Hence

2 the superexchange interaction strength, JSE t /U, becomes signiﬁcantly larger. This ∼ changes the relative strength of JSE/λ which determines if the condensation of magnetic excitations becomes energetically favorable. Second, the orbital geometry now consists of

corner sharing octahedra with two active orbitals along a bond between d4 sites instead of

one type of orbital being involved. The diﬀerence between having one orbital mediate the

interactions between sites versus having two orbitals mediate the interactions between sites

is substantial. With twice as many orbitals, there are four times as many superexchange

processes involved, and hence magnetic interactions become much stronger. With both of

these contributions, the picture of local non-magnetic singlets will likely break down when

Ir5+ ions cluster together. In this situation, the bulk Curie moment from bulk susceptibility

scales with the number of B sites containing Ir5+ ions.

In the limit that these misplaced ions are isolated antisite defects (with no clustering

of misplaced ions), the total susceptibility is then proportional to the number of antisite

defects. 2 2 j(j + 1)g µB χ fmisplaced (5.10) ∝ 3kB(T θ) × − 5+ Here fmisplaced is the fraction of Ir ions appearing on B sites. As a rough estimate, assume that for each misplaced ion, the misplaced ion and its six surrounding B’ neighbors become magnetic j = 1 states with a corresponding g factor of one-half. Then the eﬀective Curie

2 7 2 moment from bulk susceptibility is given as µ = µ fmisplaced. In this scenario, even expt 2 B × when just 1 percent of the sites were disordered, the measured bulk µeﬀ would be 0.18µB. Although antisite defects are diﬃcult to avoid and will likely have some contribution to the magnetism, we have so far assumed that these defects preserve the stoichiometry of

Ba2YIrO6. However the more realistic scenario involves a change in the charge count. For example, an abundance if Ir atoms would result in an average dn ﬁlling where n is between

4 and 5. Each j = 1/2 electron carries a g-factor of 2 so that the Curie moment is that of a

83 5 pure spin, µeff 1.73µB. As before, the effect of 1 percent of these sites containing d ions ≈ would result in a measured bulk µeff of 0.17µB. While further experimental measurements are needed to verify these ideas, we can conclude that the effects of disorder and/or off- stoichiometry can easily be the root cause of the observed magnetic moments in Ba2YIrO6

4 and perhaps other d double perovskites such as Sr2YIrO6.

84 Bibliography

[1] William Witczak-Krempa, Gang Chen, Yong Baek Kim, and Leon Balents. Correlated quantum phenomena in the strong spin-orbit regime. Annual Review of Condensed Matter Physics, 5(1):57–82, 2014.

[2] A. P. Yutsis, I. B. Levinson, and V. V. Vanagas. Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Scientiﬁc Translations, 1962.

[3] D.M. Brink and G.R. Satchler. Angular Momentum. Oxford library of the physical sciences. Clarendon Press, 1962.

[4] Jeﬀrey G. Rau, Eric Kin-Ho Lee, and Hae-Young Kee. Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials. Annual Review of Condensed Matter Physics, 7(1):195–221, 2016.

[5] B. J. Kim, Hosub Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, Jaejun Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg. Novel Jeﬀ = 1/2 mott state induced by relativistic spin-orbit coupling in Sr2IrO4. Phys. Rev. Lett., 101:076402, Aug 2008.

[6] G. Jackeli and G. Khaliullin. Mott insulators in the strong spin-orbit coupling limit: From heisenberg to a quantum compass and kitaev models. Phys. Rev. Lett., 102:017205, Jan 2009.

[7] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2 – 111, 2006. January Special Issue.

[8] Gang Chen and Leon Balents. Spin-orbit coupling in d2 ordered double perovskites. Phys. Rev. B, 84:094420, Sep 2011.

[9] Gang Chen, Rodrigo Pereira, and Leon Balents. Exotic phases induced by strong spin-orbit coupling in ordered double perovskites. Phys. Rev. B, 82:174440, Nov 2010.

1 [10] Hiroaki Ishizuka and Leon Balents. Magnetism in S = 2 double perovskites with strong spin-orbit interactions. Phys. Rev. B, 90:184422, Nov 2014.

[11] Judit Romh´anyi, Leon Balents, and George Jackeli. Spin-orbit dimers and noncollinear phases in d1 cubic double perovskites. Phys. Rev. Lett., 118:217202, May 2017.

85 [12] O. Nganba Meetei, William S. Cole, Mohit Randeria, and Nandini Trivedi. Novel magnetic state in d4 mott insulators. Phys. Rev. B, 91:054412, Feb 2015.

[13] K. Pajskr, P. Nov´ak,V. Pokorn´y,J. Kolorenˇc, R. Arita, and J. Kuneˇs.On the possibility of excitonic magnetism in Ir double perovskites. Phys. Rev. B, 93:035129, Jan 2016.

[14] N.W. Ashcroft and N.D. Mermin. Solid State Physics. Saunders College, Philadelphia, 1976.

[15] S. Sugano, Y. Tanabe, and H. Kamimura. Multiplets of transition-metal ions in crystals. Pure and applied physics. Academic Press, 1970.

[16] Antoine Georges, Luca de’ Medici, and Jernej Mravlje. Strong correlations from hunds coupling. Annual Review of Condensed Matter Physics, 4(1):137–178, 2013.

[17] P. Fazekas. Lecture Notes on Electron Correlation and Magnetism. Series in modern condensed matter physics. World Scientiﬁc, 1999.

[18] A. M. Cook, S. Matern, C. Hickey, A. A. Aczel, and A. Paramekanti. Spin-orbit coupled jeﬀ = 1/2 iridium moments on the geometrically frustrated fcc lattice. Phys. Rev. B, 92:020417, Jul 2015.

[19] A. A. Aczel, A. M. Cook, T. J. Williams, S. Calder, A. D. Christianson, G.-X. Cao, D. Mandrus, Yong-Baek Kim, and A. Paramekanti. Highly anisotropic exchange inter- 1 actions of jeﬀ = 2 iridium moments on the fcc lattice in La2BIrO6 (B=Mg, Zn). Phys. Rev. B, 93:214426, Jun 2016.

[20] O. Nganba Meetei, Onur Erten, Mohit Randeria, Nandini Trivedi, and Patrick Wood- ward. Theory of high Tc ferrimagnetism in a multiorbital mott insulator. Phys. Rev. Lett., 110:087203, Feb 2013.

[21] Giniyat Khaliullin. Excitonic magnetism in van vleck type d4 mott insulators. Phys. Rev. Lett., 111:197201, Nov 2013.

[22] Alireza Akbari and Giniyat Khaliullin. Magnetic excitations in a spin-orbit-coupled d4 mott insulator on the square lattice. Phys. Rev. B, 90:035137, Jul 2014.

[23] Christopher Svoboda, Mohit Randeria, and Nandini Trivedi. Eﬀective magnetic interactions in spin-orbit coupled d4 mott insulators. Phys. Rev. B, 95:014409, Jan 2017.

[24] Ben Ranjbar, Emily Reynolds, Paula Kayser, Brendan J. Kennedy, James R. Hester, and Justin A. Kimpton. Structural and magnetic properties of the iridium double perovskites Ba2xSrx YIrO6. Inorganic Chemistry, 54(21):10468–10476, 2015.

[25] T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass, C. G. F. Blum, F. Steckel, D. Gruner, T. Ritschel, A. U. B. Wolter, J. Geck, C. Hess, K. Koepernik, J. van den Brink, S. Wurmehl, and B. B¨uchner. Ba2YIrO6: A cubic double perovskite material with Ir5+ ions. Phys. Rev. B, 93:014434, Jan 2016.

86 [26] Brendan F. Phelan, Elizabeth M. Seibel, Daniel Badoe Jr., Weiwei Xie, and R.J. Cava. Inﬂuence of structural distortions on the Ir magnetism in Ba2xSrx YIrO6 double perovskites. Solid State Communications, 236:37 – 40, 2016.

[27] E. Kermarrec, C. A. Marjerrison, C. M. Thompson, D. D. Maharaj, K. Levin, S. Kroeker, G. E. Granroth, R. Flacau, Z. Yamani, J. E. Greedan, and B. D. Gaulin. Frustrated fcc antiferromagnet Ba2YOsO6: Structural characterization, magnetic properties, and neutron scattering studies. Phys. Rev. B, 91:075133, Feb 2015.

[28] A. E. Taylor, R. Morrow, R. S. Fishman, S. Calder, A. I. Kolesnikov, M. D. Lumsden, P. M. Woodward, and A. D. Christianson. Spin-orbit coupling controlled ground state in Sr2ScOsO6. Phys. Rev. B, 93:220408, Jun 2016.

[29] Kliment I Kugel’ and D I Khomski. The jahn-teller eﬀect and magnetism: transition metal compounds. Soviet Physics Uspekhi, 25(4):231, 1982.

[30] Giniyat Khaliullin. Orbital order and ﬂuctuations in mott insulators. Progress of Theoretical Physics Supplement, 160:155–202, 2005.

[31] A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta, Z. Schlesinger, W. A. Harrison, J. M. Kim, and I. R. Fisher. Ferromagnetism in the mott insulator Ba2NaOsO6. Phys. Rev. Lett., 99:016404, Jul 2007.

[32] Casey A. Marjerrison, Corey M. Thompson, Gabrielle Sala, Dalini D. Maharaj, Ed- win Kermarrec, Yipeng Cai, Alannah M. Hallas, Murray N. Wilson, Timothy J. S. Munsie, Garrett E. Granroth, Roxana Flacau, John E. Greedan, Bruce D. Gaulin, and Graeme M. Luke. Cubic Re6+ (5d1) double perovskites, Ba2MgReO6, Ba2ZnReO6, and Ba2Y2/3ReO6: Magnetism, heat capacity, µSR, and neutron scattering studies and comparison with theory. Inorganic Chemistry, 55(20):10701–10713, 2016.

[33] Lei Xu, Nikolay A. Bogdanov, Andrew Princep, Peter Fulde, Jeroen van den Brink, and Liviu Hozoi. Covalency and vibronic couplings make a nonmagnetic j = 3/2 ion magnetic. npj Quantum Materials, 1:16029, December 2016.

[34] Kyo-Hoon Ahn, Karel Pajskr, Kwan-Woo Lee, and Jan Kuneˇs. Calculated g-factors of 5d double perovskites Ba2NaOsO6 and Ba2YOsO6. Phys. Rev. B, 95:064416, Feb 2017.

[35] H. J. Xiang and M.-H. Whangbo. Cooperative eﬀect of electron correlation and spin- orbit coupling on the electronic and magnetic properties of Ba2NaOsO6. Phys. Rev. B, 75:052407, Feb 2007.

[36] K.-W. Lee and W. E. Pickett. Orbital-quenchinginduced magnetism in Ba2NaOsO6. EPL (Europhysics Letters), 80(3):37008, 2007.

[37] Shruba Gangopadhyay and Warren E. Pickett. Spin-orbit coupling, strong correla- 3 tion, and insulator-metal transitions: The jeﬀ = 2 ferromagnetic dirac-mott insulator Ba2NaOsO6. Phys. Rev. B, 91:045133, Jan 2015.

87 [38] Tyler Dodds, Ting-Pong Choy, and Yong Baek Kim. Interplay between lattice distortion and spin-orbit coupling in double perovskites. Phys. Rev. B, 84:104439, Sep 2011.

[39] W. M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira. Chiral spin-orbital liquids with nodal lines. Phys. Rev. Lett., 117:017204, Jul 2016.

[40] L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee, I. R. Fisher, and V. F. Mitrovi´c. Magnetism and local symmetry breaking in a Mott insulator with strong spin orbit interactions. Nat. Commun., 8:14407, Feb 2017.

[41] M. A. de Vries, A. C. Mclaughlin, and J.-W. G. Bos. Valence bond glass on an fcc lattice in the double perovskite Ba2YMoO6. Phys. Rev. Lett., 104:177202, Apr 2010.

[42] Tomoko Aharen, John E. Greedan, Craig A. Bridges, Adam A. Aczel, Jose Ro- driguez, Greg MacDougall, Graeme M. Luke, Takashi Imai, Vladimir K. Michaelis, Scott Kroeker, Haidong Zhou, Chris R. Wiebe, and Lachlan M. D. Cranswick. Mag- 1 netic properties of the geometrically frustrated S = 2 antiferromagnets, La2LiMoO6 and Ba2YMoO6, with the B-site ordered double perovskite structure: Evidence for a collective spin-singlet ground state. Phys. Rev. B, 81:224409, Jun 2010.

[43] J. P. Carlo, J. P. Clancy, T. Aharen, Z. Yamani, J. P. C. Ruﬀ, J. J. Wagman, G. J. Van Gastel, H. M. L. Noad, G. E. Granroth, J. E. Greedan, H. A. Dabkowska, and B. D. Gaulin. Triplet and in-gap magnetic states in the ground state of the quantum frustrated fcc antiferromagnet Ba2YMoO6. Phys. Rev. B, 84:100404, Sep 2011.

[44] M. A. de Vries, J. O. Piatek, M. Misek, J. S. Lord, H. M. Rnnow, and J.-W. G. Bos. Low-temperature spin dynamics of a valence bond glass in Ba2YMoO6. New Journal of Physics, 15(4):043024, 2013.

[45] Zhe Qu, Youming Zou, Shile Zhang, Langsheng Ling, Lei Zhang, and Yuheng Zhang. Spin-phonon coupling probed by infrared transmission spectroscopy in the double perovskite ba2ymoo6. Journal of Applied Physics, 113(17):17E137, 2013.

[46] Katharine E Stitzer, Mark D Smith, and Hans-Conrad zur Loye. Crystal growth of Ba2MOsO6 (M = Li, Na) from reactive hydroxide ﬂuxes. Solid State Sciences, 4(3):311 – 316, 2002.

[47] Andrew J. Steele, Peter J. Baker, Tom Lancaster, Francis L. Pratt, Isabel Franke, Saman Ghannadzadeh, Paul A. Goddard, William Hayes, D. Prabhakaran, and Stephen J. Blundell. Low-moment magnetism in the double perovskites Ba2MOsO6 (M=Li,Na). Phys. Rev. B, 84:144416, Oct 2011.

[48] Kirill G. Bramnik, Helmut Ehrenberg, Jakob K. Dehn, and Hartmut Fuess. Prepa- ration, crystal structure, and magnetic properties of double perovskites M2MgReO6 (M=Ca, Sr, Ba). Solid State Sciences, 5(1):235 – 241, 2003. Dedicated to Sten Ander- sson for his scientiﬁc contribution to Solid State and Structural Chemistry.

[49] Hai Luke Feng, Youguo Shi, Yanfeng Guo, Jun Li, Akira Sato, Ying Sun, Xia Wang, Shan Yu, Clastin I. Sathish, and Kazunari Yamaura. High-pressure crystal growth and

88 electromagnetic properties of 5d double-perovskite Ca3OsO6. Journal of Solid State Chemistry, 201:186 – 190, 2013.

[50] C M Thompson, J P Carlo, R Flacau, T Aharen, I A Leahy, J R Pollichemi, T J S Munsie, T Medina, G M Luke, J Munevar, S Cheung, T Goko, Y J Uemura, and 2 J E Greedan. Long-range magnetic order in the 5d double perovskite Ba2CaOsO6: comparison with spin-disordered Ba2YReO6. Journal of Physics: Condensed Matter, 26(30):306003, 2014.

[51] Tomoko Aharen, John E. Greedan, Craig A. Bridges, Adam A. Aczel, Jose Ro- driguez, Greg MacDougall, Graeme M. Luke, Vladimir K. Michaelis, Scott Kroeker, Chris R. Wiebe, Haidong Zhou, and Lachlan M. D. Cranswick. Structure and magnetic properties of the S = 1 geometrically frustrated double perovskites La2LiReO6 and Ba2YReO6. Phys. Rev. B, 81:064436, Feb 2010.

[52] Yahua Yuan, Hai L. Feng, Madhav Prasad Ghimire, Yoshitaka Matsushita, Yoshihiro Tsujimoto, Jianfeng He, Masahiko Tanaka, Yoshio Katsuya, and Kazunari Yamaura. High-pressure synthesis, crystal structures, and magnetic properties of 5d double- perovskite oxides Ca2MgOsO6 and Sr2MgOsO6. Inorganic Chemistry, 54(7):3422–3431, 2015. PMID: 25751088.

[53] Ryan Morrow, Alice E. Taylor, D. J. Singh, Jie Xiong, Steven Rodan, A. U. B. Wolter, Sabine Wurmehl, Bernd B¨uchner, M. B. Stone, A. I. Kolesnikov, Adam A. Aczel, A. D. Christianson, and Patrick M. Woodward. Spin-orbit coupling control of anisotropy, 2 ground state and frustration in 5d Sr2MgOsO6. Scientiﬁc Reports, 6:32462, aug 2016.

[54] A. A. Aczel, Z. Zhao, S. Calder, D. T. Adroja, P. J. Baker, and J.-Q. Yan. Structural 2 and magnetic properties of the 5d double perovskites Sr2BReO6 (B=Y, In). Phys. Rev. B, 93:214407, Jun 2016.

[55] Corey M. Thompson, Lisheng Chi, John R. Hayes, Alannah M. Hallas, Murray N. Wilson, Timothy J. S. Munsie, Ian P. Swainson, Andrew P. Grosvenor, Graeme M. Luke, and John E. Greedan. Synthesis, structure, and magnetic properties of novel B-site ordered double perovskites, SrLaMReO6 (M = Mg, Mn, Co and Ni). Dalton Trans., 44:10806–10816, 2015.

[56] Bo Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson, J. Greedan, G. Cao, B. C. Jeon, T. W. Noh, M. H. Upton, D. Casa, T. Gog, A. Paramekanti, and Young-June Kim. Determination of hund’s coupling in 5d oxides using resonant inelastic x-ray scattering. Phys. Rev. B, 95:235114, Jun 2017.

[57] J.B. Goodenough. Magnetism and the Chemical Bond. Interscience Publishers, 1963.

[58] J. Kanamori. Superexchange interaction and symmetry properties of electron orbitals. J. Phys. Chem. Solids, 10(2):87–98, 1959.

[59] P. W. Anderson. New approach to the theory of superexchange interactions. Phys. Rev., 115:2–13, Jul 1959.

89 [60] Satoru Nakatsuji, Shin ichi Ikeda, and Yoshiteru Maeno. Ca2RuO4: New mott insulators of layered ruthenate. Journal of the Physical Society of Japan, 66(7):1868–1871, 1997. [61] M. Braden, G. André,S. Nakatsuji, and Y. Maeno. Crystal and magnetic structure of Ca2RuO4: Magnetoelastic coupling and the metal-insulator transition. Phys. Rev. B, 58:847–861, Jul 1998. [62] G. Cao, T. F. Qi, L. Li, J. Terzic, S. J. Yuan, L. E. DeLong, G. Murthy, and R. K. 5+ 4 Kaul. Novel magnetism of Ir (5d ) ions in the double perovskite Sr2YIrO6. Phys. Rev. Lett., 112:056402, Feb 2014. [63] M. A. Laguna-Marco, P. Kayser, J. A. Alonso, M. J. Mart´ınez-Lope, M. van Veenen- daal, Y. Choi, and D. Haskel. Electronic structure, local magnetism, and spin-orbit effects of Ir(IV)-, Ir(V)-, and Ir(VI)-based compounds. Phys. Rev. B, 91:214433, Jun 2015. [64] Sayantika Bhowal, Santu Baidya, Indra Dasgupta, and Tanusri Saha-Dasgupta. Break- down of J = 0 nonmagnetic state in d4 iridate double perovskites: A first-principles study. Phys. Rev. B, 92:121113, Sep 2015. [65] J. C. Wang, J. Terzic, T. F. Qi, Feng Ye, S. J. Yuan, S. Aswartham, S. V. Streltsov, D. I. Khomskii, R. K. Kaul, and G. Cao. Lattice-tuned magnetism of Ru4+ (4d4) ions in single crystals of the layered honeycomb ruthenates Li2RuO3 and Na2RuO3. Phys. Rev. B, 90:161110, Oct 2014. [66] Hiroki Isobe and Naoto Nagaosa. Generalized hund’s rule for two-atom systems. Phys. Rev. B, 90:115122, Sep 2014. [67] G. Khaliullin and S. Maekawa. Orbital liquid in three-dimensional mott insulator: LaTiO3. Phys. Rev. Lett., 85:3950–3953, Oct 2000. [68] D I Khomskii and M V Mostovoy. Orbital ordering and frustrations. Journal of Physics A: Mathematical and General, 36(35):9197, 2003. [69] Andrzej M Ole´s. Fingerprints of spinorbital entanglement in transition metal oxides. Journal of Physics: Condensed Matter, 24(31):313201, 2012. [70] Daniel I. Khomskii. Transition Metal Compounds. Cambridge University Press, Cam- bridge, 10 2014. [71] S. Ishihara, J. Inoue, and S. Maekawa. Electronic structure and effective hamiltonian in perovskite Mn oxides. Physica C: Superconductivity, 263(14):130 – 133, 1996. [72] S. Ishihara, J. Inoue, and S. Maekawa. Effective hamiltonian in manganites: Study of the orbital and spin structures. Phys. Rev. B, 55:8280–8286, Apr 1997. [73] Louis Felix Feiner and Andrzej M. Ole´s. Electronic origin of magnetic and orbital ordering in insulating LaMnO3. Phys. Rev. B, 59:3295–3298, Feb 1999. [74] Giniyat Khaliullin, Peter Horsch, and Andrzej M. Ole´s. Spin order due to orbital fluctuations: Cubic vanadates. Phys. Rev. Lett., 86:3879–3882, Apr 2001. 90 [75] S. Di Matteo, N. B. Perkins, and C. R. Natoli. Spin-1 effective hamiltonian with three degenerate orbitals: An application to the case of V2O3. Phys. Rev. B, 65:054413, Jan 2002.

[76] Yan Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang. Spin-orbital entanglement and quantum phase transitions in a spin-orbital chain with SU(2) SU(2) symmetry. Phys. × Rev. B, 75:195113, May 2007.

[77] Subir Sachdev and R. N. Bhatt. Bond-operator representation of quantum spins: Mean- ﬁeld theory of frustrated quantum heisenberg antiferromagnets. Phys. Rev. B, 41:9323– 9329, May 1990.

[78] T. Sato, T. Shirakawa, and S. Yunoki. Spin-Orbital Entangled Excitonic Insulators in 4 (t2g) Correlated Electron Systems. ArXiv e-prints, Mar 2016. [79] Vivien Zapf, Marcelo Jaime, and C. D. Batista. Bose-Einstein condensation in quantum magnets. Rev. Mod. Phys., 86:563–614, May 2014.

[80] Peter D. Battle, James R. Frost, and Sang-Hyun Kim. Magnetic properties of La2CuTi1–x Rux O6 and La2ZnRuO6. J. Mater. Chem., 5:1003–1006, 1995. [81] R. I. Dass, J.-Q. Yan, and J. B. Goodenough. Ruthenium double perovskites: Transport and magnetic properties. Phys. Rev. B, 69:094416, Mar 2004.

[82] M. Bremholm, S.E. Dutton, P.W. Stephens, and R.J. Cava. NaIrO3-a pentavalent post-perovskite. Journal of Solid State Chemistry, 184(3):601 – 607, 2011.

[83] Z. Y. Zhao, S. Calder, A. A. Aczel, M. A. McGuire, B. C. Sales, D. G. Mandrus, G. Chen, N. Trivedi, H. D. Zhou, and J.-Q. Yan. Fragile singlet ground-state magnetism in the pyrochlore osmates R2Os2O7 (R= Y and Ho). Phys. Rev. B, 93:134426, Apr 2016.

[84] M. A. Laguna-Marco, D. Haskel, N. Souza-Neto, J. C. Lang, V. V. Krishnamurthy, S. Chikara, G. Cao, and M. van Veenendaal. Orbital magnetism and spin-orbit eﬀects in the electronic structure of BaIrO3. Phys. Rev. Lett., 105:216407, Nov 2010. [85] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz. WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties. Karlheinz Schwarz, Techn. Universit¨atWien, Austria, 2001.

[86] Arash A. Mostoﬁ, Jonathan R. Yates, Giovanni Pizzi, Young-Su Lee, Ivo Souza, David Vanderbilt, and Nicola Marzari. An updated version of wannier90: A tool for obtaining maximally-localised wannier functions. Computer Physics Communications, 185(8):2309 – 2310, 2014.

[87] L. T. Corredor, G. Aslan-Cansever, M. Sturza, Kaustuv Manna, A. Maljuk, S. Gass, T. Dey, A. U. B. Wolter, Olga Kataeva, A. Zimmermann, M. Geyer, C. G. F. Blum, S. Wurmehl, and B. B¨uchner. Iridium double perovskite Sr2YIrO6: A combined structural and speciﬁc heat study. Phys. Rev. B, 95:064418, Feb 2017.

[88] Giulio Racah. Theory of complex spectra. iii. Phys. Rev., 63:367–382, May 1943.

91 Appendix A Calculation Details for Transition Metal Oxides

A.1 t2g Orbital Angular Momentum

For a spherically symmetric environment, all degenerate energy levels can be labeled by angular momentum L. For d orbitals with L = 2, the part of the wavefunction is given by spherical harmonics Y m for 2 m +2. The transformation from spherical harmonics 2 − ≤ ≤ m Y2 to d orbitals is given below.

1 −2 +2 d 2 2 = √ Y + Y x −y 2 2 2 1 −1 +1 dzx = √ Y Y 2 2 − 2 0 dz2 = Y2 (A.1) d = √i Y −1 + Y +1 yz 2 2 2 i −2 +2 dxy = √ Y Y 2 2 − 2 The overall phases are chosen so the resulting wavefunctions are real-valued. We now can calculate the matrix representations of orbital angular momentum in this new basis. Matrix representations of the angular momentum operators are given below. (The factors of ~ are omitted.)

1 Lx = 2 (L+ + L−) (A.2)

1 Ly = (L+ L−) (A.3) 2i −

92 0 0 p l , m L± l, m = l(l + 1) m(m 1)δl0,lδm0,m±1 (A.4) h | | i − ±

0 0 l , m Lz l, m = m δl0,lδm0,m (A.5) h | | i

From unitary transformation, we obtain the matrix representations of the Lx, Ly,

and Lz operators in the L = 2 subspace. The basis order used below is given by dyz dzx dxy dz2 dx2−y2 so that the ﬁrst three entries are t2g orbitals and the last two entries are eg orbitals.   0 0 0 i√3 i  − −       0 0 +i 0 0      Lx ~  0 i 0 0 0  (A.6) →  −     +i√3 0 0 0 0      +i 0 0 0 0

  0 0 i 0 0  −     √   0 0 0 +i 3 i   −    Ly ~  +i 0 0 0 0  (A.7) →      0 i√3 0 0 0   −    0 +i 0 0 0   0 +i 0 0 0        i 0 0 0 0   −    Lz ~  0 0 0 0 +2i  (A.8) →      0 0 0 0 0      0 0 2i 0 0 − The eﬀective orbital angular momentum operators for t2g orbitals are given by projection: ˜ L = Pt2g LPt2g . These operators satisfy the reverse of the usual commutation relations, L˜ L˜ = iL˜. In the main text, we use the L symbol in place of L˜. × −

93 A.2 Multi-Orbital Hubbard Interaction

Same spins Diﬀerent Spins Exchanged Spins

Same orbitals V a↑,a↑ = 0 V a↑,a↓ = U V a↓,a↑ redundant a↑,a↑ a↑,a↓ a↑,a↓ →

Diﬀerent orbitals V a↑,b↑ = U 0 J V a↑,b↓ = U 0 V a↓,b↑ = J a↑,b↑ − a↑,b↓ a↑,b↓ −

Pair hop V b↑,b↑ = 0 V b↑,b↓ = J V b↓,b↑ redundant a↑,a↑ a↑,a↓ a↑,a↓ →

Starting Point: Beginning with the multi-orbital Hubbard interaction from the main text, we will rewrite this in terms of the N, S, and L operators.

X 0 X 0 1 X X HHubbard = U na↑na↓ + U na↑nb↓ + (U J) naσnbσ − 2 a a6=b a6=b σ (A.9) X X J φ† φ† φ φ + J φ† φ† φ φ − a↑ b↓ b↑ a↓ a↑ a↓ b↓ b↑ a6=b a6=b

Grouping Terms: We will group terms with occupancy operators n according the the following scheme.

1 X (S) 1 X X (A) 1 X X QU naσnaσ¯ Q 0 naσnbσ Q 0 naσnbσ¯ (A.10) ≡ 2 U ≡ 2 U ≡ 2 a σ a6=b σ a6=b σ

1 (S) (A) N(N 1) = Q 0 + Q 0 + QU (A.11) 2 − U U We will also group the other terms into the spin exchange operator X and the pair hop

operator P .

† † † † X(b, c) = φb↑φc↓φc↑φb↓ + h.c.P (b, c) = φb↑φb↓φc↓φc↑ + h.c. (A.12)

1 X 1 X X = 2 X(b, c) P = 2 P (b, c) (A.13) b6=c b6=c

94 Total S2: Next we write the square of the total spin in terms of these operators.

2 2 1 X X † † Sx + Sy = 2 φaσφaσ¯φbσ¯φbσ (A.14) a b σ

1 1 X X † † = 2 N + 2 φaσφbσ¯φbσφaσ¯ (A.15) a b σ

1 = N + X QU (A.16) 2 −

2 1 X X 1 S = naσnbσ naσnbσ¯ (A.17) z 4 − 4 a b σ

1 2 (A) = N Q 0 QU (A.18) 4 − U −

2 1 1 2 (A) S = X + N + N Q 0 2QU (A.19) 2 4 − U −

Total L2: First we show one component of orbital angular momentum squared.

2 X † † † † (La) = φ φ φ φ φ 0 φ 0 φ 0 φ 0 (A.20) − bσ cσ − cσ bσ bσ cσ − cσ bσ σσ0

Case σ = σ0:

X † † † † = φbσφcσφcσφbσ + φcσφbσφbσφcσ (A.21) ··· σ X = (nbσhcσ + ncσhbσ) (A.22) σ X 2 = (nbσ ncσ) (A.23) σ −

Caseσ ¯ = σ0:

X † † † † † † † † = φbσφcσφbσ¯φcσ¯ φcσφbσφcσ¯φbσ¯ + φbσφcσφcσ¯φbσ¯ + φcσφbσφbσ¯φcσ¯ (A.24) ··· σ − − X † † † † † † † † = φbσφbσ¯φcσ¯φcσ φcσφcσ¯φbσ¯φbσ φbσφcσ¯φcσφbσ¯ φcσφbσ¯φbσφcσ¯ (A.25) σ − − − − = 2 ( X(b, c) + P (b, c) ) (A.26) −

95 Then combine the two cases.

2 X 2 (La) = 2X(b, c) 2P (b, c) + (nbσ ncσ) (A.27) − − σ −

2 2 2 2 The total is then just L = Lx + Ly + Lz obtained by substituting for a in the previous expression.

2 (S) L = 2(N Q 0 X P ) (A.28) − U − − Rewriting the General Form: We rewrite equation (A.9) using the useful relations of the previous section.

0 (S) 0 (A) 0 HHubbard = UQU + (U J) Q 0 + U Q 0 JX + J P (A.29) − U U −

(S) (A) Notice equations (A.11), (A.19), and (A.28) form a linear system in QU , QU 0 , and QU 0 .

1 (S) (A) N(N 1) = QU + Q 0 + Q 0 (A.30) 2 − U U 2 1 3 (A) S X N(N 1) N = 2QU Q 0 (A.31) − − 4 − − 4 − − U 1 2 (S) L N + X + P = Q 0 (A.32) 2 − − U which is easily solved below.

1 7 2 1 2 QU = N(N 1) + N S L P (A.33) − 4 − 4 − − 2 − (S) 1 2 Q 0 = N L X P (A.34) U − 2 − − (A) 3 11 2 2 Q 0 = N(N 1) N + S + L + X + 2P (A.35) U 4 − − 4

Then the general form has been simpliﬁed to the following.

1 0 7 7 0 0 2 HHubbard = 4 (3U U)N(N 1) + ( 4 U 4 U J)N + (U U)S − − − − − (A.36) + 1 (U 0 U + J)L2 + (U 0 U + 2J)P 2 − − Spherical Symmetry: The coeﬃcient on P in equation (A.36) should be identically zero for spherically symmetric ions. Requiring this coeﬃcient to be zero, we obtain

1 5 2 1 2 HHubbard = (U 3J) N(N 1) + JN 2JS JL (A.37) − 2 − 2 − − 2

96 Appendix B Calculation Details for d1 and d2 Double Perovskites

B.1 d1 Superexchange

Here we derive the spin-orbital superexchange Hamiltonian for d1 systems in equation

(3.2) using the approximate tight-binding model in equation (3.1) and the on-site Coulomb interaction in equation (2.12). With P being the projection to the unperturbed ground states, the eﬀective Hamiltonian is expressed below.

1 P HSE =PHTB − HTB P (B.1) E0 HU − We calculate the second-order superexchange Hamiltonian within the xy plane as an example. X † 1 P † HSE = P ci,xy,σ0 cj,xy,σ0 − cj,xy,σci,xy,σ P + (i j) (B.2) E0 HU ↔ σ0σ − Let mi, σi; mj, σj describe an unperturbed ground state for sites i and j. Since the xy | i orbital is the m = 0 state, we can readily formulate the matrix elements of HSE in terms of

Clebsch-Gordan coeﬃcients j1m1, j2m2 j3m3 . h | i

m1, σ1; m2, σ2 HSE m3, σ3; m4, σ4 = h | | i 2 X t 2 2 2 1 1m1, 10 Lm1 δm2,0 + 1m2, 10 Lm2 δm1,0 HU(1, ) HU(L, S) × |h | i| |h | i| (L,S) 2 − 1 1 1 1 σ1, σ2 S(σ1 + σ2) σ3, σ4 S(σ3 + σ4) δσ +σ ,σ +σ δm ,m δm ,m (B.3) × h 2 2 | ih 2 2 | i × 1 2 3 4 1 3 2 4

97 Figure B.1: Examples of superexchange processes are shown graphically. Of the three t2g orbitals shown by three levels, the active orbitals along a particular bond direction are highlighted in green. (a) Ferromagnetic spin exchange occurs when only one site contains an electron on the active orbital. The virtual d2 state is in an S = 1 configuration. (b) Antiferromagnetic spin exchange occurs with the overlap of half filled orbitals. The virtual d2 state is in a total S = 0 configuration.

Here the virtual d2 states take on (L, S) values of (1, 1), (2, 0), and (0, 0). See Figure B.1.

The factor of 2 accounts for antisymmetrization of the d2 intermediate state. Rewriting these matrix elements in operator form yields the d1 superexchange Hamiltonian given in equation (3.2).

1 B.2 µeﬀ enhancement and To for d model

Here we calculate the effective moment and orbital ordering temperature due to the inter-site orbital repulsion in equation (3.4). To obtain the orbital ordering temperature To and the effective moment µeff as a function of V/λ, we will solve the mean field equations for HV + HSO analytically. The relevant mean field parameters for the four sites from Fig. 3.2(b) are given below

xy xy xy xy 1 n = n = n = n = + δnz (B.4) h 1 i h 2 i h 3 i h 4 i 3

yz yz zx zx 1 n = n = n = n = + δnx (B.5) h 1 i h 2 i h 3 i h 4 i 3 P α with the condition α ni = 1 determining the other four parameters. We obtain the single site mean ﬁeld Hamiltonian for V .

0 86 43 yz 43 53 xy H = V ( δnx + δnz)n + ( δnx + δnz)n (B.6) V − 3 3 3 3 98 0 0 Since above Tc, the high mean ﬁeld Hamiltonian HMF = HV +HSO is time reversal invariant, we rotate into the basis of total angular momentum j which factors into two 3 3 blocks × of doublets. The upper block may be chosen to be of the form below   43V (2δn +δn ) 3λ √x z 7V√ δnz  2 − 3 6 − 2   43V (2δn +δn ) 43V (2δn +δn )   √x z 7V δnz √x z  (B.7)  − 3 6 2 6 3   43V (2δn +δn )  7V√ δnz √x z 7V δnz − 2 6 3 − 2 where the basis j, mj is given by 1/2, +1/2 , 3/2, 3/2 , 3/2, +1/2 in this order. Using | i | i | − i | i θ = arctan 43√3 (2δnx + δnz) /63δnz, we diagonalize the Hamiltonian in the j = 3/2 block   3λ x y  2     x ∆ 0  (B.8)    −  y 0 ∆ where p 2 1849δnx(δnx + δnz) + 793δn ∆ = V z (B.9) 3√3

θ θ 43√3(2δnx + δnz) cos + 63δnz sin x = V 2 2 (B.10) − 9√2 and y is given by x with sin θ cos θ and cos θ sin θ applied. The lowest j = 3/2 → → − doublet with energy ∆ is mixed with the j = 1/2 doublet with amplitude 2x/3λ. − − We now obtain the eﬀective moment. We project the magnetization operator M =

2S L onto this lowest doublet. Since nominally g = 0 for the j = 3/2 states, the ﬁrst non- − zero correction to the wavefunction comes from mixing of the j = 3/2 and j = 1/2 states.

From the projection, we obtain the g factors for this doublet in all three directions (ie.

µB Mx = gx 2 σx, etc) and compute the average g factor obtained in a powder susceptibility 2 1 2 2 2 measurement g = 3 gx + gy + gz to obtain the powder average eﬀective moment for the

doublet. For the parameter regime we are interested in, δnz has a negligible contribution

to g, and the g factor is given approximately by g = 344V δnx /9√3λ so that the moment | | is µeﬀ = 172V δnx µB/9λ. | | Now we obtain the mean ﬁeld orbital ordering temperature To which occurs when the

99 j = 3/2 states split. In the limit that δnz is negligible, we self consistently solve for the

yz 1 expectation value of the operator the projections of the operator δnx n within the → − 3 2 2 subspace of energies ∆ and ∆ (ie. j = 3/2, jz = 3/2 and j = 3/2, jz = +1/2 ). × − | − i | i The projection of the δnx operator to this subspace is   √1 1  − 2 3 − 6  δnx   (B.11) → 1 √1 − 6 2 3

so that the mean ﬁeld equations for δnx read

1 δnx = tanh β∆ (B.12) 2√3

43V where ∆ √ δnx. Then we ﬁnd kBTo = 43V/18 which is consistent with Chen et. al. [9]. ≈ 3 3 However, in contrast to [9], our analysis shows that this orbital order is compatible with both the FM and AFM phases and does not disappear below Tc for the AFM phase. We can relate the ratios of these results as seen in Fig. 3.2(e) by

k To/λ 1 B = . (B.13) µeﬀ /µB 8 δnx

Using the zeroth order approximation for δnx as 1/2√3, this ratio becomes 0.43 which is close to that shown in Figure 3.2(e).

B.3 Susceptibility in the Simpliﬁed Model

For the phenomenological model, we analytically calculate the magnetic susceptibility above T > Tc. For simplicity, we consider susceptibility along the [100] direction. For zero anisotropy, K = 0, the susceptibility per magnetic site is given by the Curie-Weiss law.

(5/4)(gµ )2 χ = B (B.14) T + (5/4)J

100 The temperature T is given in units of energy, ie. T kBT . When a positive anisotropy → is included, K > 0, the form is modiﬁed as follows.

χ 2K 2 = 2(9JK 6JT 7KT ) + 2(J(K + 6T ) 7KT ) cosh (gµB) − − − T 2K 2JK 15JT + 4KT + 6T 2 sinh − − T (B.15) 2K 2 J 2(9K 6T ) 8KT 2 + 2 J 2(K + 6T ) 8KT 2 cosh − − − T 2K +J 2(15T 2K) sinh − T

The Curie-Weiss temperature takes the following modiﬁcation: θCW = (5/4)J (1/5)K. − 2 (gµB) T + θCW 2(125J 28T ) = − K2 + O(K3) (B.16) χ (5/4) − 125(4T 5J)T − Note that the K2 correction is at high temperature and does not accurately give the new

Tc at low temperature.

B.4 Projection to j = 3/2

For reference, we give the fully anisotropic Hamiltonian projected to the j = 3/2 states.

To project the HSE and HV to the low energy j = 3/2 states, we ﬁrst derive the projections of single site operators to the j = 3/2 states. The ﬁrst three spin-orbital projections are

given below.

yz 1 2 α Sxn Jx T (B.17) → 15 − 15 x

zx 2 1 α 1 β Sxn Jx + Tx Tx (B.18) → 15 15 − 3√15

xy 2 1 α 1 β Sxn Jx + Tx + Tx (B.19) → 15 15 3√15 The T symbols refer to octupole operators which can be written as the products of angular momentum J operators.

α 3 1 2 2 T = J (JxJ + J Jx) (B.20) x x − 2 y z

101 √ β 15 2 2 T = (JxJ J Jx) (B.21) x 6 y − z The overbar means symmetrization of the operators under the bar, ie. AB2 = ABB +

BAB + BBA. The remaining six spin-orbital projections are given by cyclic permutations of the (x, y, z) indices. The projections of the orbital operators are given below.

yz/zx 1 1 1 n + Oz2 Ox2−y2 (B.22) → 3 9 ∓ 3√3

xy 1 2 n O 2 (B.23) → 3 − 9 z The quadrupole operators, O, are also given as products of J operators.

2 1 2 2 O 2 = J (J + J ) (B.24) z z − 2 x y

√3 2 2 O 2 2 = (J J ) (B.25) x −y 2 x − y The result for projecting the interactions between neighboring B0 sites in an xy plane,

102 (xy) (xy) (xy) Hij = HV + HSE , are given below.

(xy) Pj=3/2Hij Pj=3/2 = r1 r2 2r3 2V JSE + + O 2 + O 2 − 9 81 81 − 3 i,z j,z 2r1 2r2 4r3 4V 64 + JSE + O 2 O 2 VO 2 2 O 2 2 27 − 243 − 243 9 i,z j,z − 81 i,x −y j,x −y 2r1 2r2 4r3 α β α β + JSE + + Ti,xT Ti,yT + i j 45√15 135√15 135√15 j,x − j,y ↔ 2r1 2r2 4r3 α α α α 8r1 8r2 16r3 α α + JSE + + T T + T T + JSE + + T T 225 675 675 i,x j,x i,y j,y 225 675 675 i,z j,z 2r1 2r2 4r3 β β β β + JSE + + T T + T T 135 405 405 i,x j,x i,y j,y r1 19r2 8r3 β β + JSE + + Ji,xT Ji,yT + i j −45√15 135√15 135√15 j,x − j,y ↔ r1 19r2 8r3 α α + JSE + + Ji,xT + Ji,yT + i j −225 675 675 j,x j,y ↔ 2r1 34r2 8r3 α + JSE Ji,zT + i j 75 − 675 − 675 j,z ↔ 4r1 68r2 16r3 8r1 32r2 4r3 + JSE + + (Ji,xJj,x + Ji,yJj,y) + JSE + + Ji,zJj,z − 75 675 675 −225 675 675 (B.26)

Our simpliﬁed model in equation (3.6) essentially discards the highly anisotropic nature of the magnetic dipolar and octupolar interactions.

103 Appendix C Calculations Details for d4 Mott Insulators

C.1 Eﬀective Hamiltonian

We calculate the eﬀective spin-orbit coupling in the 3P subspace. The linear correction for spin-orbit coupling is given by

3 3 P Hso P H = h || || i L S (C.1) SOC,(1) 3P L S 3P · h || · || i 3 3 3 3 where P Hso P and P L S P are the reduced matrix elements of the two oper- h || || i h || · || i 3 3 3 3 ators respectively. Evaluating this ratio, we obtain P Hso P / P L S P = λ/2. h || || i h || · || i Now we calculate second order energy corrections for the 3P levels due to spin-orbit

coupling. Since Hso conserves total angular momentum, only energy levels of the same

3 1 3 total angular momentum J are coupled together. Then P2 couples to D2 and P0 couples

1 3 to S0. The P1 level remains unshifted.

3 The second order correction for P2 requires us to calculate the matrix elements

1 3 3 D2 Hso P2 for the P2 energy shift, and it suﬃces to calculate the matrix element h | | i given below

1 3 † † † D2,Jz = +2 Hso P2,Jz = +2 = vac c c λc (lmm0 sσσ0 ) c 0 0 c c vac h | | i h | 1↑ 1↓ mσ · m σ 0↑ 1↑ | i (C.2)

1 + − which is only nonzero for l10 s↓↑ = l s = 1/√2. Then with an energy denominator of · 2 10 ↓↑

104 1 3 E( D) E( P ) = 2JH , we have a second order energy shift of: − √ 2 (2) 3 λ/ 2 ∆E P2 = (C.3) − 2JH

1 3 Similarly we can calculate the matrix element S0 Hso P0 = λ√2, and, with an energy h | | i 1 3 denominator of E( S) E( P ) = 5JH , we obtain: − √ 2 (2) 3 λ 2 ∆E P0 = (C.4) − 5JH

This is represented in operator form below.

2 λ 1 1 7 2 HSOC,(2) = L S (L S) (C.5) JH 20 − 8 · − 40 ·

The total effective spin-orbit interaction used in (4.4) is just HSOC,(1) + HSOC,(2). The effective superexchange Hamiltonian is broken into three parts based on the energy value of the intermediate d3 configuration in the process d4d4 d3d5 d4d4. Energies of → → the relevant states for computing this Hamiltonian are given below.

4 3 4 3 E[d ( P ), d ( P )] = 12U 26JH (C.6) −

3 4 5 E[d ( S), d ] = 13U 29JH − 3 2 5 E[d ( D), d ] = 13U 26JH (C.7) − 3 2 5 E[d ( P ), d ] = 13U 24JH − The three energy diﬀerences, E(d3, d5) E(d4, d4), are given below. −

3 4 5 4 3 4 3 E[d ( S), d ] E[d ( P ), d ( P )] = U 3JH − − E[d3(2D), d5] E[d4(3P ), d4(3P )] = U (C.8) − 3 2 5 4 3 4 3 E[d ( P ), d ] E[d ( P ), d ( P )] = U + 2JH −

Using these energy diﬀerences, HSE is computed using the following scheme.

1 HSE = PHt 4 3 4 3 HtP (C.9) − Hint E[d ( P ), d ( P )] −

105 Since the denominator must take on one of the three energy diﬀerences in (C.8), we can separate (C.9) into the three pathways.

t2 t2 t2 HSE = HSE,(4S) HSE,(2D) HSE,(2P ) (C.10) − U−3JH − U − U+2JH

Spin symmetric interactions constrain each pathway to the form

(ij) ξ H = (αξ + βξSi Sj) (C.11) SE,(ξ) · Oij

(ij) where α and β are real coeﬃcients and ij is an orbital interaction determined by t 0 . O m m

C.2 Condensation Formalism

Here we derive the general triplon condensation Hamiltonian from a spin-conserving superexchange Hamiltonian. Since each superexchange pathway can be written as the product of orbital interactions and spin interactions as in (4.5), the eﬀective pathways can be decomposed into the product of orbital and spin multipole operators on each site.

(ij) X ll0 l l0 X ss0 s s0 H = xmm0 (Li)m(Lj)m0 yσσ0 (Si)σ(Sj)σ0 (C.12) ll0mm0 ss0σσ0

Here Li and Si are the multipole operators for the orbital and spin parts of site i, and

ll0 ss0 x 0 and y 0 are coeﬃcients of the decomposition with l 0, 1, 2 and l m l, etc. mm σσ ∈ { } − ≤ ≤ We rewrite the superexchange Hamiltonian using total orbital operators and total spin O operators . S X 0 0 X 0 0 H(ij) = αll L ll L βss S ss S (C.13) M OM Σ SΣ ll0LM ss0SΣ

ll0L X 0 0 l l0 = lm, l m LM (Li) (Lj) 0 (C.14) OM | m m mm0

ss0S X 0 0 s s0 = sσ, s σ SΣ (Si) (Sj) 0 (C.15) SΣ | σ σ σσ0 0 0 The symbol lm, l0m0 LM is a Clebsch-Gordon coeﬃcient, and αll L and βss S are the h | i M Σ new coeﬃcients of the decomposition. Since S = Σ = 0 for superexchange interactions

106 preserving spin symmetry as in (4.5), we have the following superexchange interaction.

X 0 0 X H(ij) = αll L ll L βss0 ss0 (C.16) M OM 0 S0 ll0LM s

We now project out the high energy Ji = 2 states from the Hamiltonian and only

leave the Ji = 0 and 1 components. Since we are only interested in the quadratic part of † † † the result which couples sites together (ie. Ti Tj, Ti Tj ) and captures the condensation of † triplon excitations, we project directly to this subspace and ignore terms like Ti Ti since they only amount to energy shifts. This projection is accomplished by ﬁrst projecting the spin-

orbital operators on each site to the space of triplon creation and annihilation operators.

The Wigner-Eckart theorem requires this projection of the product of multipole operators

l s † (Li)m(Si)σ is proportional to Ti,ζ and Ti,−ζ where ζ = m + σ,       l s 1 l s  1 l s  h l+1−ζ † s i (Li) (Si)   ( 1) T + ( 1) T (C.17) m σ →   × − i,ζ − i,−ζ ζ m σ  1 1 1  − and the factor in braces is a Wigner-6j symbol. The ﬂow of angular momentum due to the † Tζ operator from this projection is represented graphically in Fig. C.1(a). Now we combine the terms from sites i and j with the constraint from (C.16). It will be more convenient to project each type of term appearing in (C.16) individually, so we deﬁne

0 the operator H0 = ll L ss0. Note that H0 is not generally Hermitian, but, for every LM OM S0 term, there is a complementing L( M) term so that H(ij) in (C.16) is Hermitian. Then − the projection of H0 takes the two forms below.

0 X † 0 L H † = g1 Ti,m0 √2L + 1 1m AM 1m Tj,m (C.18) Ti Tj h | | i m0m

0 X † 0 † H † † = g2 Ti,m0 1m , 1m LM Tj,m (C.19) Ti Tj | m0m The 1m0 AL 1m coeﬃcients are the multipole matrix elements previously deﬁned. The h | M | i

107 (a) lm 1 lm 1 1 l s 1ζ 0 = 11 1 1 1 1ζ sσ sσ

(b) LM s LM l0 l 1 1

1m0 1m = l0 L l s s 1m0 1m 0

Figure C.1: The ﬂow of angular momentum is graphically shown where ingoing arrows are incoming angular momentum and outgoing arrows are outgoing angular momenta. Wigner- 3j symbols and Clebsch-Gordan coeﬃcients are vertices with three legs while the scalar contraction of four Wigner-3j symbols (right) is a Wigner-6j symbol [2,3]. (a) Equation † (C.17) is shown in graphical form for the Tζ part of the equation. A J = 0 state is l decomposed into its L = 1 and S = 1 components which are acted on by the (Li)m and s (Si)σ operators. The resulting L = 1 and S = 1 are combined together to give a J = 1 state with quantum number ζ = m + σ. (b) The projection of equation (C.16) to (C.18) conserves angular momentum. Equation (C.19) will appear similarly except that 1m0 and 1m add to yield LM instead.

108 l0 −s coeﬃcients g1 = ( 1) g and g2 = ( 1) g are given below. − −       0 ( 1)l+L+1  1 L 1   1 l s   1 l s  g = − (C.20) √2s + 1 0  l s l   1 1 1   1 1 1 

It is important to note that the angular momentum contained in H0 (and also H(ij))

is conserved under projection. Furthermore, after making the transformation to cubic

coordinates, we immediately have our result: the L = 0 component gives the coeﬃcient

J in (4.12), the L = 1 components give D in (4.13), and the L = 2 components give the

irreducible components of k in (4.14). This is the advantage of writing the orbital and spin parts using their total angular momenta.

While we have restricted the calculation to spin-symmetric superexchange interactions, the concept applies more generally. If the total spin operators are non-trivial, further combine the total orbital and total spin operators into a total spin-orbital operator. The total angular momentum contained in the total spin-orbital operator will be that which appears in the condensation Hamiltonian.

Now we determine the critical value of (t2/U)/λ where condensation occurs. To sim- plify, we consider the condensation of a single ﬂavor of triplon. Then the condensation

Hamiltonian has the following form

2 X t X aδ bδ H = ∆ T †T + T † T + T T + h.c. (C.21) i i U 2 i+δ i 2 i+δ i i i,δ where aδ and bδ are constants which depend on the bond direction δ. Here we have assumed that the singlet-triplet gap is given by ∆ = λ/2 + ν(t2/U) where a correction due to superexchange has been included. Then the criteria for triplon condensation is given by

λ/2 t2/U = (C.22) −min(φa φb) + ν q ± q

a X φ = aδ cos(q δ) (C.23) q · δ where “min” refers to the most negative value of the argument and corresponds to the lowest

109 part of the triplon energy band in Fig. 4.8(a). The extra ν term allows for the center of the triplon band to shift in energy with superexchange. Since t2/U is positive by deﬁnition,

min(φa φb) + ν must be negative for condensation to be possible. q ± q

C.3 Condensation from Spin-Orbital Superexchange

We apply our formalism from Section C.2 to diagonal hopping so that tm m = 0 if m1 = 1 2 6 m2 to quantitatively obtain the key result of Section 4.6.2. This includes the three special cases (Norb = 3, 2, 1) from before, but generally applies to systems with corner sharing and face sharing octahedra. The previous section showed expressing the Hamiltonian in

the form of (C.16) had an immediate connection to the condensation Hamiltonian. In

ll0L ss0 this section, we explicitly calculate the coeﬃcients αM and β0 in (C.16) from hopping matrices to determine when the 4S pathway allows a F condensate. For the scope of this section, we will make the additional simpliﬁcation to throw away single site orbital

2 2 anisotropy in the superexchange Hamiltonian (ie. terms like Li,z+Lj,z). Using this condition and the restriction that the Hamiltonian must preserve time reversal symmetry, we are left

llL ss0 to calculate α0 and β0 in (C.16) for arbitrary l, L, and s for diagonal hopping matrices. These conditions allow us to correctly guess the coeﬃcients simply using conservation of

angular momentum without resorting to more involved formalisms.

In second order perturbation theory, there are two applications of the Ht operator. The

k ﬁrst application contributes an angular momentum k with amplitude t0 while the second 0 † k0 application contributes k with amplitude (t )0 . Together this angular momentum is shared l l between (Li)m and (Lj)m0 and can be recast in terms of a total angular momentum using (C.14). The resulting orbital coeﬃcient αllL will be proportional to k0, k00 L0 . 0 h | i We denote the intermediate multiplet for the site which gives up an electron during the

4 2 2 ﬁrst virtual hop ( S, D, P ) as having orbital angular momentum ξ1 and spin ξ2 so that

2ξ2+1 llL ss0 the multiplet is expressed as (ξ1). Then the product of the coeﬃcients α0 β0 is given below.

llL ss0 llL ss0 2 1 3 2 α β = 12α ˜ β˜ p (1, 1); p( , 1) p (ξ2, ξ1) (C.24) 0 0 0 0 h 2 | i

110 Here p2(S0,L0); p(S00,L00) p3(S000,L000) are the coeﬃcients of fractional parentage [88] for h | i a p shell and account for the total angular momenta being composed of identical particles llL ˜ss0 whileα ˜0 and β0 are quantities which only depend on the recoupling of orbital and spin angular momentum previously described.

0 llL k +ξ1+1 2 0 α˜ = 3( 1) (2ξ1 + 1)(2l + 1) k0, k 0 L0 0 − h | i      1 k 1          † k k0  1 l 1     1 l 1  (t )0(t)0 l L l (C.25) ×        1 ξ1 1     1 1 1   1 k0 1 

The new symbol in braces is a Wigner-9j symbol. The 9j symbol and the Clebsh-Gordan coefficient together contain the information of of how the angular momenta in the hopping matrix t adds to give the orbital angular momenta in the resulting orbital part llL of the O0 superexchange Hamiltonian. The two 6j symbols along with the coefficients of fractional parentage contain the information of how angular moment is transferred between sites through the transfer of identical particles. Lastly, our spin part is given similarly except that the restriction S = Σ = 0 allows some simplification.     1 1 1 1  s  1  s  ˜ss0 ξ2+3/2 2 2 2 2 2 β0 = 3( 1) (2ξ2 + 1)(2s + 1) (C.26) −   √2s + 1  1   1 ξ2 1   1 2 1 

With the coeﬃcients αβ determined, we can now use (C.18) to determine when the ferro-

4 3 magnetic mechanism mediated by the S channel (ξ1 = 0, ξ2 = 2 ) is active.

111