Triplons in the excitonic Kitaev-Heisenberg model on the honeycomb lattice: condensation, interactions and topology

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Triplons in the excitonic

Kitaev-Heisenberg model on the

honeycomb lattice: condensation,

interactions and topology

Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart

zur Erlangung der W¨urde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Pavel S. Anisimov

aus Pereswet

Hauptberichter: Prof. Dr. Maria Daghofer Mitberichter: Prof. Dr. Hans Peter B¨uchler Vorsitzender: Prof. Dr. Martin Dressel

Tag der m¨undlichen Pr¨ufung: 26.02.2019

Universit¨at Stuttgart

Institut f¨ur Funktionelle Materie und Quantentechnologien 2019

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Inhaltsangabe

Diese Dissertation betrachtet Mott-Isolatoren vom Van-Vleck Typ, bei denen starke Spin-Bahn-Kopplung den nicht magnetischen J = 0 Zustand einer unvollständig besetzten d-Schale bevorzugt. Magnetismus tritt dann durch Singulett-Triplett Anregungen in Erscheinung (z.B. Verbindungen mit den Übergangsmetallen ( ÜM) Ru4+, Os4+oder Ir5+ Ionen). Wir konzentrieren uns auf Materialien, bei denen die Auf-spaltung der zwei tiefsten Energieniveaus mit der Superaustauschen-ergieskala ∼ 4t2_{/U vergleichbar ist, was die Bose-Einstein-Kondensation} von magnetischen Anregungen (Exzitonen) ermöglicht.

Wir betrachten ein Gitter, das aus Oktaedern mit gemeinsamen Kan-ten besteht (90◦UM-O- ¨¨ UM Bindungsgeometrie), wobei die ¨ Ubergangs-metalle in der Mitte der Oktaeder platziert und von sechs Sauerstoff-atomen umgeben sind. Wir leiten effektive mikroskopische Theorie für die t42g Elektronenkonfiguration her, wobei wir sowohl durch Sauer-stoff vermittelten Superaustausch, als auch direkten Austausch benach-barter d-Orbitale berücksichtigen. Im Rahmen eines Formalismus mit Bondoperatoren, führen wir eine Projektion auf den Niederenergieun-terraum durch, der aus einem J = 0 Singulett und einem J = 1 Triplett besteht. Daraus resultiert eine effektive Theorie von Triplonen (bosonische Quasiteilchen mit hard-core Zwangsbedingung). In diesem Modell konkurrieren zwei Beiträge, eine Heisenbergwechselwirkung von Vektor-Triplonen und ein richtungsabhängiger Kitaev Term, was an das J = 1/2 Kitaev-Heisenberg Modell auf dem Sechseckgitter erinnert. Wir untersuchen das Modell sowohl klassisch als auch quantenmecha-nisch, analysieren die entsprechenden magnetischen Phasen und

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vergle-Spin-Bahn-Kopplung) und eine neue Phase im Parameterbereich der Kitaev Spinfl¨ussigkeit, eine durch ’order-by-disorder’ erzeugte Triplon-fl¨ussigkeit.

Darüber hinaus finden wir topologisch nicht triviale Anregungen in magnetischen Feldern. Trotz großer Freiheiten im Parameterraum scheint es kaum möglich, topologisch triviale Phasen zu produzieren, solange das Magnetfeld eine zur Ebene orthogonale Komponente besitzt. Des Weiteren treten die Chernzahlen der Bänder in einer Vielzahl von Kom-binationen auf. Wir klassifizieren diese Vielfalt an ’topologische Phasen’ für ausgezeichnete Parameterbereiche der Theorie. Der hier betra-chtete Mechanismus für topologisch nicht triviale Bänder in Triplon-systemen scheint vielversprechender als der auf Dzyaloshinskii-Moriya-Wechselwirkung basierende Ansatz und könnte eine experimentelle Beobach-tung von Triplonbänder mit Chernzahlen bis zu C = 8 ermöglichen.

Das Modell sollte auf Materialien wie Li2RuO3 oder Ag₃LiRuO6 anwendbar sein, wobei der Transport von triplonischen Randzust¨ an-den durch die Messung der thermischen Hall-Leitf¨ahigkeit nachgewiesen werden k¨onnte.

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Abstract

This work deals with Van Vleck-type d4 _{Mott insulators, i.e.} transi-tion metal (TM) oxides with partially filled d shell, where strong spin-orbit coupling imposes a nonmagnetic ground state with total angular momentum J = 0. Magnetism is represented only by singlet-triplet excitations (e.g. compounds with Ru4+, Os4+ or Ir5+ ions). We focus on such materials, where the splitting between two lowest energy levels is comparable to the superexchange energy scale ∼ 4t2_{/U , giving rise} to the Bose-Einstein condensation of excitons and leading to a finite staggered magnetization.

We consider the edge-sharing octahedra lattice (90◦_TM-O-TM bond-ing geometry), where transition metal ions are placed into the octahe-dra centers and are surrounded by six oxygen atoms. We derive an effective microscopic theory for t4

2g electronic configuration, including both superexchange electron transport mediated by oxygens and direct overlap between two neighbouring d orbitals. In terms of the bond op-erator formalism we perform a projection onto the low-energy subspace spanned by a J = 0 singlet and a J = 1 triplet and thus get an effective theory expressed in triplons (hard-core boson quasiparticles). Within the resulting model on the honeycomb lattice, the Heisenberg vector-triplon interaction competes with the single-flavour bond-dependent Kitaev one, which is reminiscent of the J = 1/2 Kitaev-Heisenberg model. We study the model using both classical and quantum me-chanical techniques, obtain its magnetic phase diagram and compare it to the conventional Kitaev-Heisenberg model. Main differences can be represented by a spin-orbit coupling dominated paramagnetic sector

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In addition to the phase diagram, we find that in magnetic fields the model exhibits topologically nontrivial excitations. Despite the total parameter space being broad, it appears difficult to find topologically trivial triplon bands if the magnetic field has a finite out-of-plane com-ponent. Furthermore, Chern numbers acquired by the bands can take many values under parameter variation and arise in various combina-tions. They thus form a rich variety of ’topological phases’, which we classify for particularly important regimes of the theory. The introduced mechanism inducing nontrivial topology in triplon systems appears to be more promising than the Dzyaloshinskii-Moriya-interaction based approach, and suggests a possible experimental observation of triplon bands with Chern numbers up to C = 8.

The model should be appropriate to compounds like Li2RuO3 or Ag₃LiRuO6, where triplonic edge transport might then be evidenced by measuring the thermal Hall conductivity.

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1

Introduction

Transition metal oxides are known to incarnate strongly correlated systems, where two principal interactions compete – delocalizing hop-pings and the localizing Coulomb repulsion. If the latter dominates, metallic properties of the material vanish, it becomes a Mott insula-tor and exhibits intriguing properties [1–3]. In particular, in recent years Mott insulators with partially filled d shells got a lot of atten-tion both from theorists [4–21] and experimentalists [22–34]. Indeed, the first possible realization of the Kitaev-model [35] was proposed for d5 _{Mott insulators with a strong spin-orbit coupling [4]. Such} materi-als are considered promising candidates for an experimental probing of Majorana fermions and Z2gauge fields, quantum spin-liquids, and can be potentially useful for topological quantum computing [35, 36].

In [4] one considers a d5 _{system, where the spin-orbit coupling} lo-cally entangles spins and angular momenta. An effective Hamiltonian is given by a projection of a superexchange theory onto the low-energy subspace, spanned by an effective Kramers pseudospin doublet. Pseu-dospins inherit the orbital spacial dependencies and frustrations, thus resulting Hamiltonians possess a nontrivial structure and are lattice de-pendent. For 90◦ bonding geometry the isotropic part of the effective Hamiltonian exactly cancels and the Kitaev anisotropic interaction is defined by bond-directional character of orbitals.

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Spin liquid Quadrupolar Axion insulator Topological Mott insulator Mott

insulator Spin-orbit coupled_{Mott insulator}

Simple metal or band insulator Topological insulator or semimetal λ/t U/t Weyl semimetal

Figure 1.1.: (Taken from [52]) Figurative phase diagram for electronic materials expressed through Coulomb repulsion U and spin-orbit coupling λ. Red circle approximately indicates the region considered within the scope of this work.

Superexchange, considered together with the direct overlap of the d orbitals, provides a model applicable to iridium oxides A2IrO3, where Ir4+ ions form a honeycomb lattice. The resulting effective Hamilto-nian determines a J = 1/2 Kitaev-Heisenberg model [5]. A compelling feature of this theory is a striking interplay between isotropic Heisen-berg and bond-dependent Kitaev interactions. This model, with exten-sions, was widely used to describe materials like Na2IrO3, Li2IrO3 or α-RuCl3[37].

In contrast to d5 _{models, magnetism in d}4 _{Mott insulators of} Van-Vleck type (e.g. transition metal ions with t4_2gelectronic configuration: Re3+, Ru4+, Os4+, Ir5+) is driven by the Bose-Einstein condensation of excitonic bosonic quasiparticles (triplons) [20, 21, 38], which is one the examples of exciton condensation in condensed matter physics [39–51].

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The low-energy subspace of such systems is given by a nonmagnetic J = 0 singlet and a J = 1 triplet [20]. It turns out to be convenient to use a bond-operator formalism, originally developed for coupled dimers (on each spins form a singlet). The ground state singlet, triplet ex-citations and transitions between them are expressed in the language of core triplons. The prefix ’core’ refers to a so-called hard-core constraint, which postulates that not more than one (out of three) excitation can be found on a given site. Projecting a corresponding superexchange Hamiltonian onto this subspace yields an effective the-ory. The triplons have gapped dispersions, however, since the spin-orbit coupling energy scale is comparable to (or greater than) that of superexchange ∼ 4t2_{/U , the gap can close yielding a finite staggered} magnetisation [20]. Its length fluctuations represent a gapped ampli-tude Higgs mode, while the remaining two components become gapless Goldstone magnons [53].

Based on the superexchange Hamiltonians, this theory is strongly lattice dependent, thus two common bonding geometries are considered separately (see for example Fig. 3.3). The corner-sharing octahedra (180◦-bonding geometry) applies to perovskites ABO3 or A2BO4. The Higgs mode and its decay into two Goldstone modes at the antiferro-magnetic ordering vector was experimentally observed in Ca2RuO4[34]. However, the 90◦ case, which is applicable to a honeycomb lattice (e.g. Li2RuO3, Ag₃LiRuO6), is a completely different story. The bond-directional interaction character of triplons tends to organize a set of three single-flavoured one-dimensional zigzag chains. The linear spin-wave studies of a Hamiltonian quadratic in triplons do not provide any reasonable answer about the order, since in the momentum space con-densation is expected to happen along the edges of the First Brillouin zone.

Furthermore, topological effects in bosonic systems currently gain high interest, e.g. photons [54–60], phonons [61–63], magnons on

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py-rochlore [64–66], kagome [67–76], honeycomb [77–87] lattices and the most relevant for this work – topologically nontrivial bands in triplonic systems [88–90].

A prominent example of realization of the triplon Hall effect was pro-posed for a frustrated quantum magnet SrCu2(BO3)2 [88] described by a Shastry-Sutherland model [91]. This material has a gapped dimer-singlet ground state with triplet excitations. The authors included sym-metry allowed small Dzyaloshinskii-Moriya anisotropic interactions [92, 93] and a small magnetic field perpendicular to the SrCu2(BO3)2plane. They found appropriate fields, which provide topological transitions of the bands with resulting Chern numbers C = 0, ± 2. A further in-vestigation with an arbitrary field direction revealed other topological characters of the bands C = 0, ± 1 [89]. Neutron scattering data support topologically nontrivial bandstructures [90].

In contrast to electronic systems, triplon edge modes cannot be probed by doping. In the first case topological properties can be detected by integer quantum Hall measurements (the quantized transverse conduc-tivity is evidenced if the Fermi level lies in the gap and signifies the presence of edge states) but triplons are electrically neutral and are not affected by any voltage. Therefore, for detecting edge modes in bosonic systems another approach is used. It is called thermal Hall effect [94]. One can show that magnon wave packets in Chern bands exhibit ro-tational motion [95]. Applying a thermal gradient populates the edge states differently, so the magnon rotational motion becomes unbalanced and yields a transversal current. This scheme turns out to be valid for triplons as well [88, 89].

In this thesis we consider d4 _{Mott insulators of Van Vleck type with} the 90◦ geometry. Similar to [5] we include the direct overlap between d orbitals. Reproducing calculations from [20] we develop a superex-change theory, project the corresponding Hamiltonian onto the low-energy subspace and obtain an effective theory in the bond operator

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representation. We investigate magnetic properties of the system both classically and quantum mechanically and obtain magnetic phase dia-grams. Additionally, we apply external magnetic fields and find out, whether topologically nontrivial excitations occur in the scope of d4 Mott Insulators of Van-Vleck type.

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2

Theoretical basis

In this Chapter we briefly review main concepts from atomic physics and use them discussing the crystal field splitting, the Hubbard model, Mott insulators and the Kitaev model.

2.1 Classification of single atom

elec-tronic states

We start with a free hydrogen atom, i.e. one electron in a central po-tential. Since the system is rotationally invariant, we introduce spher-ical coordinates and the time-independent Schr¨odinger equation reads as HΨ(r, θ, φ) = EΨ(r, θ, φ), H = −~ 2 2m∇ 2₋e2 r. (2.1)

We notice two conserved quantities in the system, namely the square of the orbital angular momentum and its projection onto the quantiza-tion axis (we choose the z direcquantiza-tion), i.e.

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where Lz= −i~∂φ, L2= −~2∆θφ, (2.3) ∆ = 1 r∂ 2 rr + 1 r2∆θφ, ∆θφ= 1 sin θ∂θ sin θ∂θ + 1 sin2θ∂ 2 φ, [Li, Lj_{] = i~ε}ijkLk, [L2, Li] = 0.

The latter line means that we may find eigenvalues of only one L component and of its square at the same time. The eigenvectors and eigenvalues of these operators are

Lz= m~, Ψm= 1 √ 2πe imφ_, m ∈ Z, (2.4) and

L2Ylm_{= ~}2l(l + 1)Ylm, Ylm(θ, φ) = P_l(m)(cos θ)eimφ, (2.5) with l = 0, 1, 2, . . . and m = 0, ±1, ±2, . . . , ±l. Furthermore, Ylm are eigenfunctions of Lz _{with eigenvalues m~. One calls l the azimuthal or} orbital quantum number whereas m is called magnetic quantum number. Wave functions with different values of l are called orbitals and have a historical notation

l= 0 1 2 3 4

s p d f g

The value |Ylm(θ, φ)|2 _{defines the angular distribution of electron} density in the state with a given l and are axially-symmetric. Moreover, substituting r → −r or, in other words, (r, θ, φ) → (r, π − θ, φ + π) and using Yl,m(π − θ, φ + π) = (−1)lYlm(θ, φ), one concludes that all states with an even (odd) value of l have the positive (negative) parity.

Another important property of the system is the orbital degeneracy. Since appropriate values for m are m = 0, ±1, ±2, . . . , ±l, there are 2l+1

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2.1. Classification of single atom electronic states

levels characterized by the same l (hence (2l + 1)-fold degenerate). For example, in the case of a d-shell l = 2, which yields 2l + 1 = 5-fold degeneracy.

The angular momentum operator is an analogue of the classical ro-tational momentum of a particle moving along its trajectory in the central field. Still, electrodynamics postulates that any such particle with a charge e possesses a magnetic moment

µ = j

cΣ, (2.6)

where j = −e/T is a current, T = 2π/ω is a period and Σ = πr2_{n is an} infinitesimal oriented surface element. Using L = r × m ˙r one obtains

µ = −e

2mcL. (2.7)

The same holds for quantum mechanics as well: µz= − e~

2mcml, |µl| = µB p

l(l + 1), (2.8)

where µB= e~/2mc is the Bohr magneton.

At first glance, we already have enough tools to describe magnetic properties of a single atom within the quantum paradigm. However, there is another contribution into the magnetic moment, first pointed out by Stern and Gerlach in 1922.

It is known that every particle with a magnetic moment µ moving in a magnetic field H gets an energy shift ∆E = −µH. Without loss of generality we set the field H = (0, 0, H) and hence the corresponding energy reads as ∆E = −µzH = mlµBH, where ml = 0, ±1, . . . , ±l. Thus, one expects the energy to be split into an odd number of compo-nents, but various experiments demonstrate that an even number takes place as well. This contradiction was resolved by means of an intrinsic angular momentum, called spin, and associated with it intrinsic mag-netic momentum. An essential property of every quantum particle, it

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may provide a quantum particle classification: those with an integer spin are called bosons, while those with a half-integer are fermions, e.g. electrons with s = 1/2. By analogy with L2 one obtains

S2_{= ~}2s(s + 1) = 3 4~ 2_, _S z= ms~ = ± 1 2~ 2_, _(2.9)

where s is called the spin quantum number and ms= −s, −s+1, . . . , s− 1, s (for electrons ms= ±1/2) are new quantum numbers. Analogously to (2.8) the corresponding magnetic moment reads as

µ_S = −e

mcS (2.10)

and the spin-associated energy splitting becomes ∆E = −µSzH =

±µBH.

Now we consider an atom and two non-interacting electrons with quantum numbers |l1, m1i and |l2, m2i. A two-electron state can be described by

ψ(1, 2) = ψl₁,m1(1)ψl2,m2(2) ≡ |l1, m1i|l2, m2i, (2.11)

where 1 and 2 represent sets of coordinates of the first and second particles correspondingly. In total there are (2l1+ 1)(2l2+ 1) such states. We introduce the total angular momentum L = l1 + l2, its projection onto the quantization axis Lz = l1,z + l2,z and find some important commutation rules:

[l2_i, l2_j] = [li,z, lj,z] = [li2, lj,z] = 0, i, j = 1, 2, (2.12) [L2, l_i2] = 0, [L2, li,z] 6= 0.

Instead of |l1, m1i|l2, m2i the two-electron system can be character-ized by another equivalent quantum number set |l1, l2, L, MLi, where

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2.1. Classification of single atom electronic states

ML can be easily defined from 





Lz|l1, l2, L, MLi = ~ML|l1, l2, L, MLi,

Lz|l1, m1i|l2, m2i = (l1,z+ l2,z)|l1, m1i|l2, m2i

(2.13)

⇒ Lz|l1, m1i|l2, m2i = ~(m1+ m2)|l1, m1i|l2, m2i ⇒ ML= m1+ m2.

One can see that Lmax= l1+ l2 and Lmin= |l1− l2|, hence Lmin< L < Lmaxwith integer steps. Additionally

l1+l2

X

L=|l1−l2|

(2L + 1) = (2l1+ 1)(2l2+ 1). (2.14)

In this manner one can introduce a so-called total angular momentum j = l + s. In the absence of interactions between l and s, the bases |n, l, ml, msi and |n, l, j, mji are equivalent.

Filling of the atomic levels with the lowest energy has to fulfil the Hund’s rules. The first rule postulates that for less-than-half-filled shells the electrons fill it with maximal possible spin, while for more-than-half-filled shells the total spin decreases (follows from the Pauli principle).

In order to discuss the second rule, we introduce another important ef-fect originating from the presence of the spin degree of freedom, namely the relativistic interaction between the spin and angular magnetic mo-ments. It is makes two spin directions (along the angular momentum and the opposite one) unequal and the difference is governed by the spin-orbit coupling. The spin-orbit coupling contribution for a single particle is given by HSOC= ξ(r)ls, ξ(r) = −e 2mc 1 r ∂V (r) ∂r , (2.15)

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Considering this operator as a perturbation for the main contribution of the Coulomb field yields the so-called spin-orbit coupling constant

ξnl= Ze 2 ~3 2m2_c2_a3 0 Z r−3Rnlr2dr = e 2 ~2 2m2_c2_a2 0 Z4 n3_{l(l + 1)(l +}1 2) , (2.16) where a0is the Bohr radius, Rnlrefers to the radial wave function and Z is the atomic number. We assume that l 6= 0 otherwise HSOC= 0.

The Eq. (2.16) shows that the value of the spin-orbit coupling con-stant strongly depends on the atomic number Z, i.e. for heavy atoms (ions) the Hamiltonian HSOC cannot be considered as a perturbation anymore. Consequently, the total wave function is not a product of the orbital wave function and the spin associated one.

A multi electron atomic wave function is characterized by the follow-ing set of quantum numbers: L - the total moment, M =P

imi, which may take 2L + 1 values from −L, . . . 0, . . . , L and the same holds for the spin. The set of states with the same L and S (but with different ML and MS) is called the atomic term symbol and is labelled S, P, D, F, . . . according to2S+1L as for L = 0, 1, 2, 3, . . . correspondingly.

The spin-orbit coupling splits the terms into components described by the total orbital momentum J = L + S with quantum numbers J = L+S, L+S −1, . . . , |L−S|. For not ’too heavy’ atoms (the middle part of the periodic table) spin-orbit coupling weaker than the electron-electron interaction. This case is called the LS or Russell-Saunders scheme, in which one implies an interaction between the corresponding total angular momentum L (the sum of those of all electrons) and total spin S. One can find the splitting energy values according to standard perturbation theory calculations, which in the LS- scheme read as

HSOC= λLS (2.17)

where the spin-orbit coupling constant of an ion λ is given by a combi-nation of radial integrals. This constant unlike ξ can be both negative

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2.2. Transition metal ions in solids

or positive and can be estimated by means of the following expressions:

J2= L2+ S2+ 2LS, where (2.18)

J2= J (J + 1), L2= L(L + 1), S2= S(S + 1).

Then the splitting energy difference is

∆EJ= 1

2λ J (J + 1) − L(L + 1) − S(S + 1), (2.19) EJ +1− EJ= λ(J + 1).

The last expression is called Land´e interval rule. This difference al-lows measurement of λ1_{. For heavy ions its value grows drastically} and in this case one follows the so called jj-scheme, which implies first summation of all the orbital and spin moments into total ones, and consequently the whole atom is described by a sum of all the total mo-ments. Within the scope of this work we consider the LS-interaction, as it is applicable for many transition metals.

Therefore, we may formulate the second Hund’s rule: for λ > 0 the lowest energy of less-than-half-filled shells corresponds to the smallest value of J and increases with it (regular multiplet), while for λ < 0 the lowest level is defined by the maximal J (inverted multiplet).

2.2 Transition metal ions in solids

Now we put a single transition metal ion in a solid. The ion does not possess a rotational symmetry anymore but obeys the local symmetry of the crystal [99, 100]. We consider an ion, surrounded by an O6 octa-hedron of oxygen atoms (ligands), which is typical for many transition metal compounds. The first effect to be observed is the splitting of the

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Figure 2.1.: (Taken from [99]) (a) Illustration of d-levels for different surrounding charge distributions. One can see that includ-ing the electrostatic attractive interactions does not affect the splitting but only shifts all d-levels down in energy. (b) eg (t2g) orbitals orientation with orbital lobes pointing to-wards (between) the ligands, increasing (decreasing) their energy relative to the original spherically symmetric case.

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2.2. Transition metal ions in solids

fivefold degenerate d-levels into two subsets: a lower threefold degen-erate t2g and an upper doublet eg (see Fig. 2.1(a)). This phenomenon is called crystal field splitting. In general, such splittings are classi-fied by the corresponding irreducible representations of the symmetry groups [101]2.

There are two processes behind this effect. First, the surrounding oxygens create the electric field and thus negatively charged ligands and d-orbitals repel. In Fig. 2.1(b) one can see that the d-levels (i.e. electron density distributions) have a specific spatial orientation and hence they do not interact with the ligand field equivalently, namely lobes of the eg (t2g) levels point towards (inbetween) the oxygens, which by means of the Coulomb repulsion increases (decreases) the energy of the orbitals. This contribution is called point-charge contribution [100].

The second process responsible for the splitting is a hybridization with the oxygen p-orbitals. That refers to a nonzero overlap of the ionic d and ligand p-orbitals. This covalency makes the d and p levels (lying much lower) mixed and repelled, and as a result shifts the corresponding d levels (see Section 2.4).

The resulting states can be expressed as (in the real basis) [100]

eg    |z2_{i ≡ |3z}2_{− r}2_{i = |l}z_{= 0i ∼} 1 2(3z 2_{− r}2_{) =} 1 2(2z 2_{− x}2_{− y}2_), |x2_{− y}2_{i =} _√1 2(|2i + | − 2i) ∼ √ 3 2 (x 2_{− y}2_), t2g        |xyi = −_√i 2(|2i − | − 2i) ∼ √ 3xy, |xzi = −_√1 2(|1i − | − 1i) ∼ √ 3xz, |yzi = _√i 2(|1i − | − 1i) ∼ √ 3yz.

It is often convenient to map the t2g triplet onto triplet states with

2_{Consequently the labels t}

2gand egare originating from group theory: t stands

for a triply degenerate representation of the Ohcubic point group, e for a doubly

degenerate one, while g refers to the fact that the wave function is even under the spatial inversion.

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an effective moment ˜l = 1. Then the spin-orbit coupling takes the form [100]

HSO(t2g) = ˜λ˜l · S, (2.20)

where the new constant ˜λ is proportional to the original one. But it should be taken with the opposite sign [102], which changes the second Hund’s rule for the t2g levels to the opposite.

In Chapter 3 we consider the t5

2g system (one t2g hole) with total degeneracy (2˜l + 1)(2S + 1) = 6. The spin-orbit coupling splits these levels into ˜J = ˜l + S = 3/2 quartet and ˜J = ˜l − S = 1/2 doublet, which lies lower and yields, in consonance with the so-called Kramers theorem, a Kramers degenerate ground state.

Kramers theorem. In the absence of a magnetic field (or a magnetic order in crystals) degeneracy of the energy levels of an atom (ion) with an odd number of electrons cannot be less than two. Thus, if the atomic (ionic) shells are filled with an odd number of electrons, the maximal possible multiplet splitting leads to a system with twofold degenerate energy levels or Kramers doublets.

2.3 Hubbard model and Mott

insu-lators

The Hubbard model is the most prominent approach in studying of strongly-correlated electronic systems. In spite of its visual simplicity, it cannot be solved analytically in the general case. Following Ref. [103] we consider electrons in an ionic crystal with a periodic structure, where the lattice sites are static compared to the electrons. The one dimensional Hubbard model reads as

H = −tX

hiji

c†_iσcjσ+ UX i

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2.3. Hubbard model and Mott insulators

where c†_iσ, cjσ, niσare electrons creation, annihilation and number op-erators at site i with spin σ, hiji refers to nearest-neighbours pairs, t is the overlap integral, and U represents the Coulomb repulsion. Struc-turally it can be separated into two principal parts

H = H0+ U D, (2.22) H0= −t X hiji c†_i,σcj,σ, D = X i ni,σni,σ0.

We notice that these terms compete: H0tends to delocalize electrons, while D favours the localization. The Hamiltonian H0 is diagonal in the Bloch basis |k, σi = c†_k

N,σN. . . c

†

k1,σ1|vaci, where k = (k1, . . . , kN),

σ = (σ1, . . . , σN) and describes non-interacting electrons. The D-part is diagonal in the Wannier basis |x, σi = c†xN,σN. . . c

†

x1,σ1|vaci, with

x = (x1, . . . , xN) and counts the number of doubly occupied sites in the state |x, σi. These two operators do not commute and thus the Hubbard Hamiltonian (2.21) can neither be diagonalized in the Wannier nor in the Bloch basis.

If U = 0, one is left with non-interacting electrons, i.e. H = H0

H = −tX

hiji

c†_i,σcj,σ. (2.23)

By means of Fourier transformation ci,σ= √1 N X k eikxi_ck,σ, _(2.24) we get H =X k,σ

ε(k), with ε(k) = −2t(cos kx+cos ky+cos kz). (2.25)

A N -site lattice corresponds to N levels, which for N → ∞ gives a continuum spectrum. If there is only one electron per site, the band

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becomes half-filled and the crystal is metallic. These results do not depend on the lattice spacing, rather on the hopping constant t and on the number of the nearest-neighbours ζ yielding the bandwidth W = 2ζt (in 1D ζ = 2, in 2D ζ = 4, etc.).

The opposite case t → 0 is called the atomic limit and corresponds to a system with localized electrons. Considering the total Hamilto-nian (2.21) we see that due to the on-site repulsion, creation of an electron-hole pair costs energy U . The only gain in energy occurs in the kinetic term, when the electron and the hole start moving in the crystal, and is equal to W = 2ζt. On the other hand, the energy loss is driven by the Coulomb repulsion U . Hence, if U > W , the electrons are not itinerant and the system remains insulating. Thus, for a half-filled system with rather small values of t (compared to U ), the on-site repulsion makes the system insulating, despite the system seemingly being a conventional metal. Such insulators are called Mott insulators (or Mott-Hubbard insulators), which in contrast to the band insulators become insulating due to strong electron-electron interactions. More-over, the localized electrons in Mott insulators lead to the presence of localized magnetic moments and spin and orbital degrees of freedom remain active.

In the limit U t in second order perturbation theory in t/U one obtains the t-J Model, which describes interesting physics of high-Tc su-perconductivity in cuprates [104]. Its derivation can be performed either by means of Hubbard X operators [105–107] or by introducing projec-tors onto the subspace comprising no doubly-occupied states [103]. We follow the latter scheme and define

P0= N Y

i=1

(1 − ni,↓ni,↑), (2.26)

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2.3. Hubbard model and Mott insulators

The degenerate perturbation theory technique [103, 108] states that

H(2) = P0T P0− 1 U N X m=1 P0T PmT P0 m , (2.27) where Pm= (−1)m m! ∂m ∂αm N Y i=1 (1 − αni,↓ni,↑)

projects onto the subspace with m double occupancies. The resulting Hamiltonian reads as [103] Ht−J = N X i,j=1 i6=j

tijc†_i,σcj,σ(1 − ni) + N X i,j=1 i6=j 2|tij|2 U SiSj−ninj 4 (2.28) + 1 U N X i,j,k=1 i6=j6=k tijtjkc†_i,σσσ,σ0ck,σ0Sj− 1 2c † i,σck,σnj (1 − ni) with S = 1 2 X σ,σ0 c†_i,σσσ,σ0ci,σ0, (2.29)

where σ is the Pauli vector. Close to half-filling the three-site term becomes irrelevant. Introducing Jij ≡ 2|tij|2_{/U we rewrite the t-J} Hamiltonian Ht−J = N X i,j=1 i6=j

tijc†_i,σcj,σ(1 − ni) + N X i,j=1 i6=j Jij(SiSj−ninj 4 ). (2.30)

Setting ni= 1, i.e. the half-filling case, we notice that the first term vanishes and we are left with

HHeis = N X hiji JijSiSj−1 4 , (2.31)

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which is known as the Heisenberg model. It represents the exchange interactions between two neighbouring spins and its ground state is strongly dependent on the dimension of the system. In particular for D > 2 the ground state is the long-range antiferromagnetic (AF) order, whereas for D = 1 (D = 2) this scenario is not possible at T > 0 (T ≥ 0) [109].

One can keep the higher order corrections ∼ t4_/U3 _{and obtain a} biquadratic exchange and a four-spin-interaction terms, given by

H4−spin= Aijkl n

(SiSj)(SkSl) + (SiSl)(SjSk) + (SiSk)(SjSl) o

Hbiq= Bij(SiSj)2,

which regarded together with (2.31) give rise to nontrivial physics (e.g. [110, 111]).

2.4 Orbital degrees of freedom

Orbitals define electron density distributions, which connects the ma-terial structure with its magnetic properties. If orbital degeneracy is lifted, some states (and hence associated with them orbital quantum numbers) become more favourable, which makes it possible to observe a long-range orbital pattern called orbital ordering. In the dense sys-tems of magnetic ions the orbital degeneracy tends to be lifted and there are three conventional mechanisms for that – the dynamic Jahn-Teller effect, spin-orbital superexchange interaction and the spin-orbit coupling [3].

The first one is driven by the so-called Jahn-Teller theorem [112], which states: any non-linear molecular system in a degenerate elec-tronic state (except Kramers degeneracy) will be unstable and will un-dergo distortion to form a system of lower symmetry and lower energy

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2.4. Orbital degrees of freedom

thereby removing the degeneracy. Hence, systems with an even number of electrons have a singlet ground state.

The second scheme is represented by superexchange interactions. In contrast to the direct exchange, they imply a mediator, which could transfer electrons between two transition metal ions. This role in tran-sition metal oxides is played by oxygens, i.e. anions, located between two ions (see Fig.3.3). In Mott insulators a corresponding effective hop-ping is defined by the energy difference between the involved oxygen p and ionic d orbitals and reads as t = t2

pd/∆CT [100], where ∆CT is the so-called charge transfer gap.

As we mentioned above, orbitals are spacial distributions and their interactions are strongly dependent on their relative orientation, i.e. on the lattice geometry, which should be explicitly specified (e.g. in this thesis we consider two types of TM-O-TM bonding geometries, see Fig.3.3).

If superexchange dominates, one gets a group of Kugel-Khomskii-like theories [113–115]. These Hamiltonians can be obtained in the second order perturbation theory and in general they are of the form [3, 116]

H(γ)= −X n t2 εn X hijiγ Πij(S)O (γ) ij , (2.32)

where εn is the nth_{virtual process excitation energy and Πij(S) (O}(γ) ij ) is the spin (orbital) projection operator. Such Hamiltonians are mate-rial dependent and require a detailed analysis of the virtual processes occur, calculation of the corresponding excitation energies, etc. For example, the prominent Kugel-Khomskii model for KCuF3 [113] de-scribes a system, where Cu2+ has a d9 electronic configuration, while charge excitations are given by d9

id9j d10i d8j. The t2g levels are filled, while two-fold degenerate eg orbitals remain active and analogous to spins can be associated with τ = 1/2 pseudospin. The corresponding

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Hamiltonian (2.32) then reads as [116] H(γ)=X hijiγ n − t 2 U − 3JH SiSj+ 3 4 P_ij(γ)+ t 2 U − JH SiSj− 1 4 P_ij(γ) + t 2 U − JH + t 2 U + JH SiSj− 1 4 O(γ)_ij o+ Ez X i τ_ic, (2.33) with P_ij(γ) ≡1 2 + τ (γ) i 1 2 − τ (γ) j +1 2 − τ (γ) i 1 2 + τ (γ) j , (2.34) O(γ)_ij ≡ 21 2 + τ (γ) i 1 2+ τ (γ) j ,

where Ez is the crystal field splitting of eg orbitals.

The most relevant scenario for this work is the spin-orbit coupling being comparable to or larger than the superexchange energy scale. The spin and orbital degrees of freedom become locally entangled, which can be included within ionic multiplets of an effective pseudospin J [117, 118]3_.

Generally, the low-energy magnetic interactions can be described by effective Hamiltonians, obtained by projecting the corresponding su-perexchange Hamiltonians onto the low-energy subspace. This subspace is usually spanned by the lowest 2J + 1 multiplet (e.g. in Chapter 3 this multiplet is a twofold Kramers degenerate J = 1/2 doublet). However, via the spin-orbit coupling the new pseudospin degree of freedom inher-its the bond-directional structure and frustrations of the orbital interac-tions, which provides the resulting Hamiltonians a nontrivial structure. Moreover, any pseudospin ordering involves both S- and L-orderings.

The main principles, remarkable features and modern tendencies or effects governed by the orbital degrees of freedom can be found in re-views [3, 113, 116, 119, 120].

3_{In literature the term isospin is used as well. It should not be confused with}

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2.5. Kitaev model

2.5 Kitaev model

It is known that at low temperatures solids with localized electrons exhibit spin ordering, which allows them to minimize their energy. Still, there are states of matter, called spin liquids, where spins do not order down to T = 0 K but remain fluctuating [121]. If these fluctuations are of the thermal origin, the spin liquid is called classical spin liquid, whereas quantum spin liquids are highly entangled and are influenced by quantum fluctuations (see for example Review [122]). Quantum spin liquids imply no broken symmetries and hence are not predicted by the Landau-Ginzburg theory.

Quantum spin liquids are usually associated with valence bond solids [123, 124] or with the model of SU (2) spins with a bond-dependent interaction character proposed by Kitaev [35]. The Hamiltonian of the Kitaev model on the honeycomb lattice reads as4

HKitaev= −KxX hijia SixS x j − Ky X hijib S_iyS_jy− KzX hijic SizS z j, (2.35)

where γ = a, b, c refers to a given bond (see Fig. 2.2). The ground state of this model is exactly solvable in terms of Majorana fermions5 _[126] and Z2 gauge field degrees of freedom. It is represented either by a gapless or a gapped spin liquid (see Fig. 2.2(c)).

An experimental realization of this model could be of paramount im-portance, since the Kitaev model is argued to have possible applications in the field of topological quantum computing [35, 36]. Furthermore, apart from the quantum spin liquid states, with help of such materials one could experimentally probe Majorana fermions and the Z2 gauge

4_{Caused by the directional charater of orbitals, a model with pseudospin}

Ising-like anysotropic exchange was proposed in [113], which is known today as compass model [125].

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Figure 2.2.: (Reproduced from [127]) (a) A honeycomb lattice with bond-dependent spin interactions. Three inequivalent bond flavours are highlighted with colours. (b) The same lattice, reflecting Majorana fermions representation of spins (each Majorana is labelled with coloured circles). (c) Phase di-agram of the Kitaev model. If any Kx/y/z coupling domi-nates, a gapped spin-liquid is formed. The parameter space given by the red-shaded triangle corresponds to a gapless spin liquid, also known as Majorana metal.

fields driven physics. Luckily, there are possible candidates for realiza-tion of the Kitaev model [4]. We discuss it in more detail in the next Chapter.

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3

Mott insulators in presence of

the strong spin-orbit coupling

In this chapter we consider Mott insulators with the strong spin-orbit coupling, i.e. spin-orbit entangled J = 1/2 Mott insulators. We intro-duce the spin-1/2 Kitaev-Heisenberg model [5–19], discuss its principal properties and phase diagram. Supported by various experimental in-vestigations [25–33] this theory represents a minimal adequate model for describing honeycomb iridates or ruthenates, e.g. Na2IrO3, Li2IrO3 or α-RuCl3.

Then we focus on J = 1 Van-Vleck type ions and provide a short overview of the excitonic magnetism in such crystals [20, 21]. Finally, we derive an excitonic Kitaev-Heisenberg model.

3.1 Kitaev and Kitaev-Heisenberg

models in Mott insulators

High values of SOC constant λ are typical for late TM ions, like Ru4+ (λ ∼ 84meV), Os4+ (λ ∼ 186meV) or Ir4+ (λ ∼ 380meV) [97], which exceeds the t2gintersite interactions. Therefore, as we discussed above, we expect the total angular momentum to be formed locally. Here we

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d-orbitals octahedral crystal ﬁeld IrO6cage t2g eg ~ 3eV spin-orbit coupling j = 3/2 j = 1/2 ~ 400meV electronic correlations

U

j=1/2 Mott insulator

Figure 3.1.: (Taken from [127]). The general scheme illustrating forma-tion of a J = 1/2 Mott insulator, where spin and angular momentum are locally entangled into pseudospin doublets.

provide the original derivation of a superexchange theory for a single hole in a threefold degenerate t2g level (a low spin d5 configuration) in a strong octaheral field, performed in [4]. Furthermore, we demon-strate that in such systems at low energies one can obtain compass-like Hamiltonians, which can serve as a possible material realization of the Kitaev model [4].

We consider a single hole on the dxy, dxz and dyz orbitals and intro-duce the effective angular momentum ˜l = 1 (see Eq. (2.20)):

|lz= 0i ≡ |dxyi, |lz= ±1i = −√1

2(i|dxzi ± |dyzi). (3.1)

The total magnetic moment reads as M = 2s − l, where s refers to a hole spin operator. We include a tetragonal splitting ∆ > 0 of t2g orbitals, which elongates octahedrons along the z-axis. The single ion Hamiltonian on the site i reads as

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3.1. Kitaev and Kitaev-Heisenberg models in Mott insulators

Its ground state is given by a Kramers pseudospin doublet

|˜↑i = sin θ|0, ↑i − cos θ| + 1, ↓i, (3.3)

|˜↓i = sin θ|0, ↓i − cos θ| − 1, ↑i, with tan 2θ = 2 √

2λ λ − 2∆. The structure of the doublet leads to a nontrivial real space distri-bution (see Fig. 3.2) forming the bond geometry dependence of the exchange Hamiltonian.

isospin up spin up, lz=0 spin down, lz=1

+

=

Figure 3.2.: (Taken from [4]) Hole density cartoon in the pseudospin up state, which is a superposition of a spin up in |xyi orbital (i.e. |lz = 0, ↑i) and a spin down in (|yzi + i|xzi) with ∆ = 0.

There are two principal TM-O-TM bonding geometries (see Figs. 3.3 (a),(b)): a 180◦-bond (corner-sharing octahedra) and a 90◦ one (edge-shared geometry). In order to get the effective theory, one projects the corresponding exchange models onto the pseudospin subspace (3.3). For 180◦ we have [4]

H_ij180◦ = J1SiSj+ J2(Sirij)(rijSj), (3.4)

where S is the S = 1/2 pseudospin operator, rij is the unit vector along a given bond and J1= 4/9 and J2= 2η/9U if η = JH/U is small . This means that we obtain an isotropic Heisenberg-like Hamiltonian (2.31) with a negligibly small dipolar-like anisotropy interaction.

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In the 90◦- case the isotropic part of the Hamiltonian exactly vanishes and, as a result, the exchange interaction depends on the orientation. Thus, we are left with

H_ijγ,90◦= −J S_iγS_jγ (3.5)

where γ refers to a bond. This is precisely the compass-model Hamil-tonian [113] with only one certain spin (pseudospin) component active on the given bond. Figs. 3.3(c),(d) display two possible lattices for the ABO2 and A2BO3structures (with alkali A and a transition metal B). The second one coincides with the Kitaev model on the honeycomb lattice with the spin-liquid ground state.

interaction appears, however, due to the JH-multiplet

struc-ture of the excited levels. Most importantly, the very form of the exchange interaction depends on the spatial orienta-tion of a given bond. We label a bond ij laying in the plane perpendicular to the ð¼ x; y; zÞaxis by a ( )-bond. With this in mind, the Hamiltonian can be written as

H ð Þ

ij ¼ JSiSj; (3)

with J ¼4

3 2. Remarkably, thisHamiltonian is precisely a

quantumanalog of theso-called compassmodel. Thelatter, introduced originally for the orbital degrees of freedom in Jahn-Teller systems [5], has been the subject of numerous studies as a prototype model with protected ground state degeneracy of topological origin (see, e.g., Ref. [25]). However, to our knowledge, no magnetic realization of

py xy xy pz xz xz 180o (a) pz pz (b) xz yz yz xz o 90 Sy y 2 3 S S x x 1 2 S S Sz z_{1 3} _(b) xx zz yy (a) y z x 1 3 2

Figure 3.3.: (Taken from [4]) Two principal bonding geometries: (a) the 180◦corner-sharing octahedra and (b) the 90◦edge-sharing one. The large (small) dots represent the TM (oxygen) ions. (c) Triangular and (d) honeycomb unit cells realizable in ABO2 and A2BO3 compounds correspondingly. The big circles stand for the TM ions and small ones refer to oxy-gens.

Further investigations were performed for A2IrO3(with A = Li, Na) in [5]. The background remains the same: the Ir4+ ions with a single hole residing on the threefold degenerate t2g levels with the effective angular momentum ˜l = 1. The strong spin-orbit coupling lifts this

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degeneracy and the ground state is the Kramers 1/2- doublet of pseu-dospins (which is further called ’spins’). In the layered iridates of the type A2IrO3, the IrO6 octahedra share edges but now two interaction channels between Ir are active: the direct orbital overlap between t2g levels and Ir-O-Ir superexchange. Projecting the corresponding multi-orbital Hubbard Hamiltonian onto the low-energy subspace yields

H_ij(γ) = −J1S_iγS_jγ+ J2SiSj. (3.6)

where J1 and J2 are governed by three different virtual processes [5]. We introduce the parametrization J1= 2α, J2= 1 − α, 0 ≤ α ≤ 1 and discuss properties of

H_ij(γ)= −2α S_iγS_jγ+ (1 − α)SiSj. (3.7)

The α = 0 regime describes the pure Heisenberg model with a N´eel ordered ground state. The opposite limit α = 1 reflects the Kitaev model, thus realizing a spin-liquid ground state. Besides, the model is exactly solvable for the intermediate value α = 1/2. We consider γ = z bond with the Hamiltonian (3.7)

H_ij(z) =1 2(S x iS x j + S y iS y j − S z iS z j). (3.8)

As the next step we divide the lattice into four sublattices and in-troduce a new rotated operator ˜S. It coincides with the original S in one of the sublattices (see Fig.3.5(b)) and in the remaining ones two appropriate components of ˜S differ from S by the sign (components should be chosen with respect to the given sublattice). That leads to the Hamiltonian

˜

H_ij(γ)= −2(2α − 1) ˜S_iγS˜γ_j − (1 − α)˜SjS˜j. (3.9) Setting α = 1/2 we are left with the ferromagnetically (FM) coupled Heisenberg Hamiltonian ˜H_ij(γ) = −1₂Sj˜ Sj. The inverse rotation yields a˜ Stripy ordering shown in Fig. 3.5(c).

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Figure 3.4.: A general overview of the A2IrO3 (with A = Na2+, Li2+) honeycomb structure from [15]. (b) In particular, a pic-ture of Na2IrO3 orbitals from [128]. For each kind of Ir-Ir bond only two appropriate hoppings between t2glevels are possible.

The numerical studies are represented by the Lanczos exact diago-nalization1 _{of a 24-site cluster (see Fig.3.5(b)), which accepts the} pre-viously discussed four-sublattice transformation. The results revealing the phase transitions are given in Fig.3.5.

Nevertheless, neutron scattering experiments in Na2IrO3 [31, 32] re-vealed that another candidate for the ground state of the Kitaev-Hei-senberg model is realized - the zigzag magnetic order. In fact, one can easily show that the α-parametrization introduced above leads to an oversight [12]. To demonstrate it, we consider the Hamiltonian (3.6) and introduce new parameters: energy scale A and J -K-tuning angle

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p y

the numerical result found below.

Now we discuss the phase behavior at1

2< < 1, i.e., in

between two exact solutions (stripy AF at ¼1

2and a

Kitaev spin liquid at ¼ 1). In terms of rotated spins, all

the couplings are of FM nature in this region [see Eq. (2)]. Thus, the FM order (read stripy AF of the original spins) is the only possible magnetic phase here to compete with the spin-liquid state. Since the latter is stable against a weak Heisenberg-type perturbation [7], a critical value

of for the spin order/disorder transition must be located

at some point less than 1. We give its naive estimate based 0.1

0.2 0.3 tions 〈˜SS 〉˜ 0.0 0.2 0.4 0.6 0.8 1.0

squared total spin

(b) 〈˜S2 tot〉 〈 S2 tot〉 F A l e e´ N stripy AF x x S S z z S S y y S S z ( ) y ( ) ( ) ) b ( ) a ( _x ( ) (c) stripy AF Heisenberg limit Kitaev limit spin liquid Neel AF α=0 α=1/2 α=1

Figure 3.5.: (Figure from [5]) (a) Three bond flavours on the honeycomb lattice labelled with the corresponding Kitaev anisotropic interaction. (b) The four-sublattice system supercell used in numerical simulations and in the transformation (3.9). (c) Schematic phase diagram depending on the exchange couplings parameter α. Exact diagonalization results of the 24-site cluster given in Fig.(b), using Lanczos algorithm (see Chapter 6).

ϕ as follows

H_ij(γ)= A(2 sin ϕ S_iγS_jγ+ cos ϕ SiSj), (3.10)

A =pJ2_{+ K}2_, _{J = A cos ϕ,} _{K = A sin ϕ.}

The sublattice transformation S → ˜S performed for the stripy phase leads to the Hamiltonian ˜H of the same structure but with new cou-plings

˜

K = K + J, J = −J˜ ⇔ tan ˜ϕ = − tan ϕ − 1. (3.11)

Exact diagonalization yields the phase diagram of the model, given in Fig. 3.6(a). The whole parameter space ϕ ∈ [0, 2π] can be divided into six different phases – two kinds of Kitaev spin-liquid and four ordered FM, AF, Stripy and Zigzag phases. Remarkably, we observe

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-18 -16 -14 -12 -10 -8 -6 0 π/2 π 3π/2 2π 0 100 200 300 400 500 EGS /A − d 2E GS / A d ϕ 2 ϕ l e e´ N y p ir t s d i u q il M F g a z g i z d i u q il l e e´ N

coupling constant I4 9t ð2pdþ UpJpH pdÞ is negative

t~ xz yz xz yz H2 H1 3 H H4 xy 3z −r22 xy xy ϕ t’ 3 4 1 4 2 3 yz xz xz yz 2 1 t FM Heisenberg AF Heisenberg FM Kitaev AF Kitaev Neel stripy ϕ liquid liquid zigzag FM

Figure 3.6.: (Taken from [12]) (a) The phase diagram of the Kitaev-Heisenberg model (3.10). Grey lines display the points con-nected by (3.11). (b) The second derivative −d2_EGS/dϕ2_of the ground state energy, whose peaks indicate phase transi-tions. (c) Four different exchange processes that contribute to K and J .

two different ϕ-intervals corresponding to the opposite Kitaev phases, i.e. FM and AF spin liquids are variously sensible to the J -perturbation. However, ab initio calculations show that the model (3.10) for the honeycomb iridates is not complete [15]. Introducing Ni, Si and Li, which are the total number, spin and angular momentum operators, the Coulomb interaction U and Hund’s coupling JH, we start with an atomic Hamiltonian of Kanamori form [129]

H_i(0)=U − 3JH 2 (Ni− 5) 2 − 2JHS_i2−JH 2 L 2 i, (3.12)

The perturbative kinetic terms are

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where operators d†_iα denote the creation operators for the t2g orbital α at site i and t1-t3 are functions of Slater-Koster parameters (tddσ, tddπ, tddδ, tpdπ) [130] and of the charge transfer gap ∆pd [100]. The effective Hamiltonian reads as [15] H_ij(γ)= J SiSj+ KS γ iSjγ + Γ(SiαS β j + S β iS α j), where J = 4 27 h6t1(t1+ 2t3) U − JH + 2(t1− t3)2 U − Jh + (2t1+ t3)2 U + 2JH i , (3.14) K = 8JH 9 h (t₁− t₃)2− 3t2₂ (U − 3JH)(U − JH) i , Γ = 16JH 9 h t2(t1− t3) (U − 3JH)(U − JH) i . By analogy with (3.10) we introduce

J = cos φ sin θ, K = sin φ sin θ, Γ = cos θ, (3.15) and want to obtain the corresponding phase diagram in the form of a filled circle. Its boundary must represent the previous Kitaev-Heisenberg diagram Fig. (3.6)(a), while the unknown inner part is regulated by θ-angle describing the influence of the cross-terms (Γ). The classical diagram Fig. 3.7(a) is obtained by means of Luttinger-Tisza approxi-mation [131] and a single-Q ansatz, i.e. considering spin configurations

Si= sin ηi ex_i cos(Qri) + ey_isin(Qri) + cos ηiez_i (3.16) where the canting angles ηi and local vectors e

γ

i are treated as indepen-dent variational parameters. Performing minimization with respect to these parameters and Q, one obtains the ground state energy for the given point (φ, θ). In Fig. 3.7(a) we find two new phases: 120◦ordering and a coplanar incommensurate spiral one.

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The Lanczos exact diagonalization provides a quantum phase dia-gram of the model, given in Fig. 3.7, whose rim reflects the same physics as Fig. 3.6(a) for the given value of φ. All the phases, except the in-commensurate one, can be similarly established by means of the static spin structure factor

SQ= 1 N

X

ij

eiQ(ri−rj)SiSj. _(3.17)

Classically defined as incommensurate, the IS - labelled phase cannot be correctly treated by exact diagonalization, since the cluster size is too small. As a result, in Fig. 3.7 one does not observe sharp peaks, hence distinct phase transitions around the spiral phase IS are not clearly defined.

3.2 Van-Vleck ions

There is a class of transition metal ions called Van Vleck ions or Van Vleck paramagnets. In these systems the ground state does not possess a magnetic moment but their paramagnetic susceptibility exceeds the diamagnetic one. This group is represented by rare-earth ions with an even number of electrons on the partially filled 4f orbital: Pr3+, Eu3+, Tb3+, Ho3+, Tm3+ or 4d and 5d transition metals with t4_2g electron configurations, such as Re3+, Ru4+, Os4+ or Ir5+. The crystal field splits their multiplets2S+1LJ so that the lowest energy level is a singlet or a non-magnetic doublet [132].

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3.2. Van-Vleck ions

only have the bond-dependent Γ term. Here, two ne states are introduced: 120° order and an incommensura

spiral. The 120° order with wave vector ◌◌⃗Q ¼ K appea

near the (antiferromagnetic) Kitaev limit at (π=2, π=2 This is a coplanar spiral, with the spins lying in th plane perpendicular to [111]. The spins are at relativ

angles 0, 2π=3 on the same sublattice [as shown

Fig. 2(f)], with the relative angle between sublattic unconstrained. An additional degenerate point appears

ðǺ; θÞ¼ ð3π=4; cos− 1_ð1=pffi₃ffiffi_{ÞÞwhere J ¼ − K ¼ − Γ, wi}

the 120°, FM, and zigzag phasesmeeting at a single poi [40]. The second large region of zigzag phase appearin when Γ ≫ jJj, jKj has the spins predominantly oriente along the [1¯1 ¯1], [ ¯11¯1], and [ ¯1 ¯11] directions, tilting awa slightly as one explores the phase. The IS phase remai

coplanar despite the ◌◌⃗Q vector varying throughout th

phase. The magnitude of the IS wave vector lies in th

range 1:2 j ◌◌⃗Qj 1:8 as shown in Fig. 2(g).

Exact diagonalization.— To gain an understanding the featuresof the classical resultsthat carry over to the fu quantum mechanical model, we have performed exa diagonalization. We consider a 24-site cluster that h been used previously to study the HK model [13,18,19 providing a reasonable description of the phases found the classical level as well as the Kitaev spin liquids. In th HK limit, the existence of a local spin rotation [13,41] th

mapsJ→ − J and K → K þ 2J givesfour well-understoo

magnetic limits in addition to the two exactly solvab Kitaev points. These are the FM, AFM, and their zigza and stripy images under the mapping. This transformatio is no longer useful as Γ is included [42], but the phas surrounding these points can still be identified with eac respective limit. While the IS phase is unlikely to be we represented on such a small cluster, the remaining phas such as the 120° phase are compatible with the clust geometry. We note that the transformation used to rela

Γ 0 to Γ 0 no longer applies in the quantum case, an

so both regions must be analyzed separately. To identify the phase boundaries we have computed th

FIG. 2 (color online). (a) Combined Luttinger-Tiszaand single-Q analysis. Solid colors correspond to exact classical ground states from Luttinger-Tisza while the region indicated by the white dashed line is the single-Q results. [(b)— (f)] Ground-state spin configurations in each phase. (g) Magnitude of the ordering

wave vector ◌◌⃗Q in the IS phase.

only have the bond-dependent Γ term. Here, states are introduced: 120° order and an incom spiral. The 120° order with wave vector ◌◌⃗Q ¼ near the (antiferromagnetic) Kitaev limit at (π This is a coplanar spiral, with the spins lyi plane perpendicular to [111]. The spins are angles 0, 2π=3 on the same sublattice [as Fig. 2(f)], with the relative angle between s unconstrained. An additional degenerate point ðǺ; θÞ¼ ð3π=4; cos− 1_ð1=pffi₃ffiffi_{ÞÞwhere J ¼ − K ¼}

the 120°, FM, and zigzag phasesmeeting at a si [40]. The second large region of zigzag phase when Γ ≫ jJj, jKj has the spins predominantl along the [1¯1 ¯1], [ ¯11¯1], and [ ¯1 ¯11] directions, ti slightly as one explores the phase. The IS phas coplanar despite the ◌◌⃗Q vector varying throu phase. The magnitude of the IS wave vector range 1:2 j ◌◌⃗Qj 1:8 as shown in Fig. 2(g). Exact diagonalization.— To gain an underst the featuresof the classical resultsthat carry over quantum mechanical model, we have perform diagonalization. We consider a 24-site cluste been used previously to study the HK model [ providing a reasonable description of the phase the classical level as well as the Kitaev spin liqu HK limit, the existence of a local spin rotation [1 mapsJ→ − J and K → K þ 2J givesfour well-u magnetic limits in addition to the two exactly Kitaev points. These are the FM, AFM, and th and stripy images under the mapping. This trans is no longer useful as Γ is included [42], but t surrounding these points can still be identified respective limit. While the IS phase is unlikely represented on such a small cluster, the remain

FIG. 2 (color online). (a) Combined Luttinger-Tiszaand

single-PRL 112, 077204 (2014) PH Y S I C A L R E V I E W L E T T E R S 21 FEBRweek

Figure 3.7.: (Figure from [15]) (a) Classical phase diagram based on both Luttinger-Tisza (solid lines) and single-Q analysis (surrounded by white dashed lines). (b)-(f) depict clas-sical ground state configurations of the system, whereas (g) shows the Q-magnitude in the incommensurate spiral phase. (h) Quantum phase diagram based on the exact di-agonalization of 24-site cluster for Γ > 0 and (i) Γ < 0. Colours represent SQ contour map magnitudes while white solid lines display its ”equipotential” contours.

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Figure 3.8.: (a) S = 1 and L = 1 coupled by LS-scheme yield J = 0 non-magnetic singlet and separated by λ J = 1 triplet (see Eq. (2.19)). We omit J = 2 levels, since λ is large. (b) The singlet ground state and three triplon flavours correspond-ing to the excited J = 1 levels. (c) Hole configurations |Ai, |Bi and |Ci, realized on the t2g subshell.

3.3 Singlet-triplet model for the case

of 180

◦

bonding geometry

In order to develop a microscopic theory of magnetism in these ma-terials, we derive a superexchange Hamiltonian [20]. First we study the corner-shared 180◦ bonding geometry (see Fig. 3.2), which applies to perovskites of the type ABO3 or A2BO4 (e.g. Ca2RuO4). Three t2g orbitals dyz, dxzand dxywe label with a, b and c correspondingly. Then hoppings along the c-bond are described by the Hamiltonian

H(c)= tX hiji

(a†_iaj+ b†_ibj+ h.c.). (3.18)

Following [20] we introduce operators of the t4

2g shell orbital configu-rations

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3.3. Singlet-triplet model for the case of 180◦ bonding geometry

and an effective orbital momentum operator (similar to ˜l = 1) [114] Lx= −i(B†C − C†B), Ly= −i(A†C − C†A), Lz= −i(A†B − B†A).

(3.20) In the second-order perturbation theory one obtains the resulting spin-orbital Hamiltonian (for details see Appendix A)

H = t 2 U X γ X hijiγ h SiSj+ 1O (γ) ij + (L γ i) 2_{+ (L}γ j) 2i_, _(3.21)

where for the c-bond

O(c)_ij = (Lx_iLx_j)2+ (Ly_iLy_j)2+ Lx_iLy_iLy_jLx_j + L_iyLx_iLx_jLy_j,

and operators O(a)_ij and O_ij(b)can be obtained by means of cyclic permu-tations of Lx, Lyand Lz(compare the structure of (3.21) to Eq. (2.32)). The low-energy subspace is spanned by the ground state J = 0 singlet and the excited J = 1 triplet separated by λ (see Fig. 3.8). In the |MS, MLi basis these states are

|si = √1 3 |1, −1i − |0, 0i + | − 1, 1i, |t1i = 1 √ 2 |1, 0i − |0, 1i, (3.22) |t−1i = − 1 √ 2 | − 1, 0i − |0, −1i, |t0i = 1 √ 2 |1, −1i − | − 1, 1i. It is often convenient to use the Cartesian components |txi = 1 i√2(|t1i − |t−1i), |tyi = 1 √ 2(|t1i + |t−1i), |tzi = |it0i, (3.23) and to introduce bond operators [133]

s†|vaci = |si, t†_i|vaci = |tii, [s, s†] = 1, (3.24) [ti, t†_j] = δij, [s, ti] = [s, t†_i] = 0, s†s + t†_iti= 1,

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Since only one state can be found on a given site, an essential con-straint ns+ nt = 1 with nt = nx+ ny+ nz ≤ 1 has to be enforced. We also introduce ’transition’ hard-core triplon operators T = s†t and their cross product J = −iT†× T. Using commutation relations for s and t (3.24) one finds

nTi = nti, [Ti, T

†

j] = (1 − 2nTi)δij. (3.25)

The T-operators can be expressed in terms of two real fields u and v2 _{as follows [135],[136]:}

T = u + iv, u2+ v2≤ 1, u · v = 0. (3.26)

In order to get the effective theory we project the Hamiltonian (3.21) onto the given subspace, i.e. H(S, L) → H(T, J ), and calculate all ma-trix elements of its entries (see Appendix B). For example, the operators of spin, angular momentum and magnetic moment (M = 2S − L) read as S = −i r 2 3 T − T †₊ i 2J, (3.27) L = +i r 2 3 T − T †₊ i 2J, (3.28) M = −i√6 T − T† + i 2J. (3.29)

Finally, collecting all projections we end up with the Hamiltonian Heff= λX i ni+X γ H(γ), H(γ)= t 2 U X hiji (h2+ h3+ h4)(γ)_ij ,

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where (for the c-bond) h(c)₂ = 11 3 vi· vj− vizvjz+ 1 3(uiuj− uizujz), (3.30) h(c)₃ = √1 24(vi· Jj+ vizJjz+ uixQjx− uiyQjy) + (i ↔ j), h(c)₄ = 3 4d † ijdij+ 1 2Ji· Jj+ 1 4(JizJjz+ J 2 izJ 2 jz) − 5 36ninj n =X γ T_γ†Tγ, Qx= −(T_y†Tz+ T_z†Ty), etc.

We quantify the interaction by κ = 4t2/U . The density of out-of-condensate particles near the critical point is small, hence we neglect J and Qγ. Moreover, for the given geometry the bond-dependent terms are irrelevant [2], and hence we substitute viγvjγ with vivj/3. Thus, we are left with the quadratic Hamiltonian

H2= λ X i ni+ 2κ 9 X hiji n T†_iTj− 7 16TiTj+ h.c. o . (3.31)

One assumes that the hard-core constraint is fulfilled on average and neglects magnon interactions. Following the singlet-triplet exten-sion of the linear spin-wave theory [137, 138] one performs the Bogoli-ubov transformation for T-bosons (for details see Appendix D.1), which yields the excitation dispersions

ωγ(k) = λ r 1 + κ κcrφk, with φk = 1 2 X γ cos(kγ), (3.32) where κcr ∼ 2

5λ refers to some critical value of κ. Thus, if κ > κcr, the gap λp1 − (κ/κcr) closes and we observe a magnetic phase tran-sition driven by condensation of the dipolar part of one of the T-components [20, 43–45, 139]. This yields a finite staggered magneti-zation Mγ = 2 p 6ρ(1 − ρ), with ρ =1 2 1 −1 τ , τ = κ κcr > 1. (3.33)

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The M -length (or the Higgs-amplitude mode [53]) fluctuates with a gapped dispersion (for details see Appendix D.2)

ωγ(k) ' λ p

τ2_{+ φ}

k, (3.34)

while two other components represent gapless Goldstone modes with ωα/β(k) ' λτ + 1

2 p

1 + φk, γ 6= α 6= β 6= γ. (3.35) In [20] one applies the above results to Ca2RuO4 [22], a d4 Mott insulator with λ ' 75 meV [23] and with the staggered moment M ' 1.3µB[24]. First, this provides reasonable values for (4t2/U ) ' 53 meV, t ∼ 0.2 meV and U ∼ 3 − 4 eV. Second, the estimated product LS ' −0.2 [20] does not strongly deviate from the observed one −0.28± 0.07 [23]. Furthermore, one obtains the uniform magnetic susceptibil-ity [20] χ = 12µ 2 BNA λ(1 + τ ) ' 2.3 × 10 −3 _emu/mol, _(3.36)

while its experimental value above the N´eel temperature is ∼ 2.5 × 10−3 emu/mol [22, 24].

Motivated by these predictions, one considered an extension of this model including a tetragonal distortion essential for perovskites, which specifies one component of T [21]. Introducing the splitting between xy and xz/yz orbitals as δ = (Exz/yz− Exy)/2, we obtain the effective Hamiltonian Heff= δX i nz_i −1 3ni + λX i ni+ J X hijia H_ij(a)+ J X hijib H_ij(b), (3.37) with H_ij(a)= T†_iTj− 1 3T † i,xTj,x− 5 6TiTj+ 1 6Ti,xTj,x+ h.c., (3.38) ni= nx_i + ny_i + nz_i,

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where a and b denote two bonds of a square lattice, J = t2_{/U and H}(b) ij can be obtained by substituting Ti,x→ Ti,y. Here we do not consider the bond-dependent terms as the average value vivj/3.

ALIREZA AKBARI AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 90, 035137 (2

(a) ∆ Λ s Tx y Tz (b)

III. GROUND-STATE PROPERTIES

Depending on the relative strength of the exchange and SOC λ parameters, the ground state of effective Ha tonian (1) can be either paramagnetic or antiferromagne There are two different magnetic phases, with out-of-p M c and in-plane M ab orientations of the stagge moments. The M orientation is decided by the competi between the exchange J and the crystal-f eld δ coupli We calculate below classical energies of magnetically orde states and obtain from them a phase diagram and orde moment values. 0. 0.1 0.2 0.3 0.4 0.5 0. 0.2 0.4 Λ Ρ b 0.2Λ 2 Ρ1 0. 0.1 0.2 0.3 0.4 0.5 0. 0.6 1.2 1.8 2.4 Λ M c 0.2Λ M2 M1 0. 0.1 0.2 0.3 0.4 0.5 0. 0.2 0.4 ∆ Λ Ρ d _0.2Λ Ρ2 Ρ1 0. 0.1 0.2 0.3 0.4 0.5 0. 0.6 1.2 1.8 2.4 ∆ Λ M e _0.2Λ M2 M1

Figure 3.9.: (Taken from [21]): (a) A cartoon of energy levels highlight-ing the relative orientation of the ground-state shighlight-inglet and the excited triplet state split by the crystal field δ. (b) Magnetic phase diagram displaying both magnetic phases as a function of J and δ. (c) 3D plot of the magnetisation M , where grid distortions correspond to the phase transi-tions shown in (b). (d), (e) the condensate density and the magnetic moment for δ = 0.2λ vs. J . (f),(g) J = 0.2λ vs. δ.

One observes two different ordered phases depending on the J/δ ratio: an out-of-plane M k c and an in-plane M k ab one. In the first case the condensate density reads as [21]

ρ1=1 2 1 − β1 κ1 , where κ1= 22 3 J and β1= λ + 2 3δ, (3.39) which yields a critical value Jcr= (3λ+2δ)/22. The in-plane orientation

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is described by ρ2= 1 2 1 −β2 κ2 , where κ1=19 3 J and β1= λ − 1 3δ, (3.40) with Jcr= (3λ − δ)/19. The staggered magnetic moment is given by

Mi=p6(1 − ηi), (3.41)

where ηi ≡ β2

i/κ2i, i = 1, 2. These magnetic moments and corre-sponding condensate densities can be found in Fig. 3.9. In Ca2RuO4 the crystal field δ is relatively small, hence the in-plane order is more preferable.

In the relevant in-plane phase the amplitude x-mode dispersion reads as ωx(k) = q κ2 2+ β 2 2φk, (3.42)

while taking into account the chemical energy shift for the transverse y/z components we obtain

ωy(k) = ˜κ2p1 + φk, (3.43) ωz(k) = ( ˜κ2+ δ) r 1 + 22 19 ˜ κ2φk ˜ κ2+ δ, (3.44)

where ˜κ2= (κ1+ β1)/2. The corresponding spin excitation intensities are [21] Imχγ_k(ω) = |Fγ(k)| ωγ(k) δ(ω − ωγ(k)), γ = x, y, z, (3.45) Fx(k) = 3˜κ2 19(−1 + 19η2)φk, Fy(k) = 54˜κ2 19 φk, Fz(k) = 60˜κ2 19 φk. In Fig. 3.10 one can directly compare these intensities at the ordering wave vector M -point (π, π).

Triplons in the excitonic Kitaev-Heisenberg model on the honeycomb lattice: condensation, interactions and topology