Spin and Orbital Polarons in Strongly Correlated Electron Systems

Home , KANT (software)

Jagiellonian University Faculty of Physics, Astronomy & Applied Computer Science Marian Smoluchowski Institute of Physics

Krzysztof Bieniasz

A Ph.D. Thesis prepared under the supervision of prof. dr hab. Andrzej Michał Oleś

Kraków 4239

Oświadczenie

Ja, niżej podpisany, Krzysztof Bieniasz (nr indeksu: 323673;), doktorant Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiel- lońskiego, oświadczam, że przedłożona przeze mnie rozprawa doktorska pt. „Polarony spinowe i orbitalne w układach silnie skorelowanych elektronów” (tytuł w języku angielskim: “Spin and Orbital Polarons in Strongly Correlated Electron Systems”) jest oryginalna i przedstawia wyniki badań wykonanych przeze mnie osobiście, pod kierunkiem prof. dra hab. Andrzeja M. Olesia. Pracę napisałem samodzielnie. Oświadczam, że moja rozprawa doktorska została opracowana zgodnie z Ustawą o prawie autorskim i prawach pokrewnych z dnia 6 lutego 3;;6 r. (Dziennik Ustaw 3;;6 nr 46 poz. :5 wraz z późniejszymi zmianami). Jestem świadom, że niezgodność niniejszego oświadczenia z prawdą, ujaw- niona w dowolnym czasie, niezależnie od skutków prawnych wynikających z wyżej wymienionej ustawy, może spowodować unieważnienie stopnia naby- tego na podstawie tej rozprawy.

Kraków, 45 czerwca 4239 r...... Streszczenie

Właściwości układów silnie skorelowanych elektronów, a w szczególności skorelowanych izolatorów Motta, które są często spotykane wśród tlenków metali przejściowych, są istotnym zagadnieniem w ﬁzyce fazy skonden- sowanej. Jednym z głównych problemów w tej dziedzinie, zwłaszcza ze względu na jego znaczenie dla eksperymentów fotoemisyjnych, jest zachowanie się ładunku poruszającego się w izolatorze Motta, sprzęga- jącego się do spinowych i orbitalnych stopni swobody, tworząc polaron. Zagadnienie istnienia oraz dynamicznych właściwości powstającej w ten sposób kwazicząstki da się rozwiązać za pomocą obliczeń funkcji Greena z modeli efektywnych. Niniejsza rozprawa doktorska poświęcona jest teorii polaronów spinowych i orbitalnych, t.j. kwazicząstek powstają- cych przez oddziaływanie z magnetycznym lub orbitalnym porządkiem dalekozasięgowym. W ramach tych badań zaproponowano modele efek- tywne opisujące dwa strukturalnie podobne układy: rodzinę związków miedziowo-tlenowych (zwanych kupratami, opisywaną za pomocą modelu t-J oraz jego rozszerzeń) oraz perowskit miedziowo-ﬂuorowy (KCuF3, opisywany za pomocą modelu Kugela-Khomskiego). Rozwinięte zostały również metody analityczne i numeryczne, służące do obliczania jednoelek- tronowych funkcji Greena za pomocą rozwinięcia wokół stanu uporząd- kowanego, t.j. przybliżenie wariacyjne funkcji Greena oraz samozgodne przybliżenie Borna. Metody te zostały następnie użyte do rozwiązania trzech blisko spokrewnionych modeli układów polaronowych: dwuwymiarowego modelu spinowego reprezentującego kupraty, dwuwymiarowego modelu orbitalnego zainspirowanego płaszczyznami CuF2 w układzie KCuF3 oraz pełnego trójwymiarowego modelu spinowo-orbitalnego opi- sującego związek KCuF3, który nigdy wcześniej nie został rozwiązany. Poprzez porównanie wyników dla układów z jednym stopniem swobody wyciągnięte są wnioski na temat wpływu oddziaływań ze spinowymi i orbitalnymi stopniami swobody na właściwości polaronów, co stanowi podstawę dla rozważań dotyczących modelu spinowo-orbitalnego. Po- nadto przedstawione są interesujące zjawiska pojawiające się w modelu spinowo-orbitalnym, takie jak efekty wymiarowe w konkurencji między spinowymi i orbitalnymi stopniami swobody, zmiana charakteru orbitalnego na spinowy pod wpływem oddziaływania wymiennego czy znaczenie oddziaływania Hunda dla magnetycznego stanu podstawowego układu. Praca kończy się krótką dyskusją wciąż otwartych problemów oraz możli- wości dalszych badań w tej tematyce. Abstract

The properties of strongly correlated electron systems, in particular the correlated Mott insulators commonly encountered among transition metal oxides, are at the forefront of current research in condensed matter physics. One of the central problems in the field, particularly due to its relevance to photoemission experiments, is the behaviour of a charge injected into a Mott insulator which can couple to the ordered spin-orbital degrees of freedom to form a polaron. The questions of the existence and the dynamical properties of the ensuing quasiparticle state can be elucidated by means of Green’s function calculations from effective models of the system. In this thesis we explore the theory of spin and orbital polarons, i.e., quasiparticles resulting from the charge coupling to magnetic or orbital long range order. To this end, we develop effective models for two structurally similar systems: the copper-oxide series of high temperature superconductors (or cuprates, modeled using the t-J model and its extensions), and the copper-fluoride perovskite (KCuF3, modeled using the Kugel-Khomskii model). We then develop analytical and numerical methods for calculating single electron Green’s functions by means of expansion around an ordered ground state, namely the Green’s function variational approximation and the self-consistent Born approximation. Subsequently, we apply these methods to solve three related polaronic model systems: purely spin planar model based on cuprates, purely orbital planar model inspired by CuF2 planes of KCuF3, and the full three dimensional spin-orbital model for KCuF3 which has never been solved before. By comparing the results for the two cases with a single degree of freedom we demonstrate the differences between the spin and orbital interactions for the polaronic properties and draw general conclusions about the spin-orbital model. Further, we demonstrate a number of interesting effects encountered in the spin-orbital problem, such as dimensional interplay between orbitals and spins leading to polarons of predominantly orbital nature in the strong coupling regime; the orbital to spin polaron crossover under varying superexchange strength; or the importance of the Hund’s exchange in the settling of the magnetic ground state. We conclude by discussing open problems and proposing possible routes of continuation of the present work.

Acknowledgments

I would like to thank my advisor, Professor Andrzej M. Oleś, whose help and support was the fuel that propelled this work to its conclusion. His guidance and knowledge were invaluable motivators in the process of my self-development. I would also like to thank my colleagues and scientific collaborators, without whose expertise and cooperation it would not have been possible for me to complete this thesis. I am particularly indebted to Professor Mona Berciu from the University of British Columbia, the ongoing collaboration with whom produced most of the results that went into this thesis. I would also like to thank my co-advisor, Doctor Krzysztof Wohlfeld, for insightful discussions and words of encouragement. Usually what follows is a long list of relatives and friends whose support had an impact on the author’s work. I will not list them here, however, I would like to thank all the people without whose involvement this thesis would not have been completed in the timely manner that it was. Finally, I would like to acknowledge financial support from the Polish National Science Centre (ncn) under the “Etiuda” scholarship, project number 4237/38/t/st5/22725. I would further like to acknowledge support from ncn under project number 4234/26/a/st5/22553 as well as from the Pol- ish Ministry of Science and Higher Education under project number n n424 28;85;. Lastly, I would like to thank the Quantum Matter Institute, which has generously sponsored my first visit to the University of British Columbia in 4236.

Contents

Acknowledgments 9

Contents ;

Motivation 33

I Preliminaries 35

3 Introduction 37 3.3 Correlated Insulators ...... 37 3.4 Cuprates ...... 39 3.5 Orbital Ordered Systems and KCuF3 ...... 47

4 The Models 4; 4.3 The t-J Model ...... 4; 4.4 The Spin-Orbital Model ...... 55 4.5 The Planar Orbital Model ...... 5:

5 Methodology 63 5.3 The Green’s Function ...... 63 5.4 Self-Consistent Born Approximation ...... 67 5.5 Variational Method ...... 6;

II Results 77

6 Cuprates: Purely Spin Systems 79 6.3 Introductory Information ...... 79 6.4 Convergence of the Variational Results ...... 7; 6.5 Three Site Terms ...... 85 6.6 Conclusions ...... 88

7 KCuF3 Planes: Purely Orbital Systems 89 7.3 Convergence of the Ising Model ...... 89

; 32 Contents

7.4 The Role of Variational Constraints ...... 94 7.5 Signiﬁcance of Orbital Fluctuations ...... 96 7.6 Conclusions ...... 98

8 Spin-Orbital Polarons in KCuF3 9; 8.3 The Orbital-Flop State ...... :2 8.4 Self-Energy Analysis ...... :5 8.5 Weak Interaction Limit ...... :7 8.6 The Kugel-Khomskii State ...... :9 8.7 Many Magnon Expansion ...... :; 8.8 The Hund Exchange ...... ;2 8.9 Conclusions ...... ;6

9 Summary ;9

A Real Space Propagators 323

B SAGE 329

Bibliography 333

List of Publications 33; Motivation

I killed him for money and for a woman. I didn’t get the money and I didn’t get the woman.

Fred MacMurray as Walter Neﬀ in Billy Wilder’s Double Indemnity (3;66)

Transition metal oxides are a varied group of chemical compounds notable for their huge diversity of interesting and potentially useful physical properties, ranging from superconductivity in cuprate systems to colossal magnetoresist- ance in cubic manganites LaMnO3. They exhibit a full range of conduction properties, from insulators to metals, some of them even displaying a crossover under metal-insulator transitions, like for instance vanadates V2O3. In terms of magnetic properties they also show an unprecedented diversity, from ferro- magnetism to a number of different antiferromagnetic phases, and everything in between. A wide variety of exotic phases can also be observed, such as charge density waves, charge ordering, stripe phases, &c. The common element behind all those systems is the presence of the transition metal d valence electron shell, central to their physical properties. The d shell is known to be strongly localised, causing the electrons located there to experience very strong mutual Coulomb repulsion, which in a solid causes strong correlations between charges, and consequently between other degrees of freedom associated with them. Hence transitional metal oxides are quintessential strongly correlated systems, whose extraordinary properties arise from the emergent responses of an ensemble of interacting particles—as Phil Anderson quipped, more is different. Somewhat more recently, it has been noted that if the d shell experiences at least a partial degeneracy, the electrons will have the freedom to occupy orbitals of a certain type, usually reflecting the symmetry of the underlying lattice, leading to orbital ordering in the system. Thus the coupling between orbital degrees of freedom and lattice degrees of freedom is realised, leading to the Jahn-Teller effect, where orbital ordering is coupled with a structural transition of the system. In turn, the magnetic order and orbital order are in- trinsically linked to each other, subject to the so-called Goodenough-Kanamori rules. Thus, transition metal oxides are systems located at the intersection

33 34 Motivation

of charge, magnetic, orbital, and lattice degrees of freedom, all of which are mutually dependent and affecting each other. Moreover, the interplay of com- peting effective interactions often means that not only are such systems not solvable exactly, but even their approximate treatment poses great challenges to established theoretical frameworks, as they are often difficult to model both in the strong coupling and weak coupling regimes, while there even are no standard methods for dealing with the intermediate interaction region. It is for these reasons that the field of strongly correlated matter is as much about modelling real materials and explaining the exotic effects they exhibit, as it is about developing novel theoretical apparatus for addressing the pressing questions of the field. In this thesis we strive to cover, if only to a modest extent, both of these aims. On one hand, this work is motivated by correspondence to real materials, in particular spin ordered systems such as cuprates (whose importance and continued interest stems from high temperature superconductivity they exhibit), and orbital ordered systems, in this case the copper-fluoride perovskite KCuF3 (which is interesting for its clear interplay of orbital and spin degrees of freedom, as well as one-dimensional magnetism). On the other hand, we develop the powerful Green’s function variational approximation and adapt it to the slave boson formalism. We then compare it with other methods, firmly established in the field of polaronic physics, such as the self-consistent Born approximation or spectral moment calculation. In this way we are able to probe the dynamical properties of electrons coupling to spin and orbital polarized states in correlated systems, and obtain the so-called spectral function, which is directly related to spectroscopic experiments such as the angle resolved photoelectron spectroscopy (arpes), inverse photoelectron spectroscopy (ipes) or resonant X-ray scattering (rxs). Part I

Preliminaries

CHAPTER 3 Introduction

In Italy, for thirty years under the Borgias, they had warfare, terror, murder and bloodshed, but they produced Michelangelo, Leonardo da Vinci and the Renaissance. In Switzerland, they had brotherly love, they had ﬁve hundred years of democracy and peace—and what did that produce? The cuckoo clock.

Orson Welles as Harry Lime in Carol Reed’s The Third Man (3;6;)

We start our discussion with some basic background information about the systems under consideration. We will begin with presenting the Zaanen-Sawatzky- Allen scheme of classiﬁcation of correlated insulators, which is crucial for the understanding of the limits of applicability of various models of strongly correlated systems. We will then proceed to present the current state of knowledge concerning the two basic systems that are the motivation for our considerations, namely the cuprate series of high-Tc superconducting parent compounds, and the copper-ﬂuoride perovskite KCuF3, the quintessential spin-orbital compound.

3.3 Correlated Insulators

In the conventional sense of band theory, the term insulator refers to a system of multiple electronic bands, with electron ﬁlling such that the bands are either completely full or completely empty, and separated by a prohibitively wide gap, so that thermal excitations are an ineﬀective doping mechanism. However, it has been discovered that many transition metal oxides, in which band theory predicts very small or nonexistent gaps and therefore at least weak conductivity,

37 38 Introduction

Figure 3.3: Comparison of Mott-Hubbard and charge-transfer insulators. Ad- apted from [6]. (a) A broad metallic band is split into the upper and lower Hubbard bands under the inﬂuence of strong Coulomb repulsion U; (b) The dominating charge ﬂuctuations in Mott-Hubbard and charge transfer insulators; (c) If the ligand band is located above the lower Hubbard band, the gap is determined by the charge transfer energy ∆.

are in fact insulators with very big band gaps. This longstanding discrepancy has been elucidated in the works of Mott [3], who noted that 3d states near the Fermi energy are strongly localized, which means their Coulomb repulsion, denoted U, is large which suppresses charge fluctuations. These findings have been later formalised as a mathematical model by Hubbard [4, 5]. It is for this reason that such systems are called Mott-Hubbard Insulators (mhi). A simplified picture of the Mott-Hubbard mechanism is illustrated in figure 3.3(a). An otherwise broad metallic band is split into and upper and a lower Hubbard band under the influence of the Coulomb repulsion U. Without any doping the lower band is completely filled, and the upper one is empty. The size of the gap is U, which is the Coulomb energy penalty for adding an additional electron into the system. This framework was largely successful in understanding of correlated insulators and helped establish the field of strongly correlated systems. However, when applied to certain materials, such as NiO, it was found that this theory is insufficient, as the Coulomb repulsion U is actually much larger than the electronic band gap [7]. Furthermore, it was discovered that the size of the band gap correlates with the anion electronegativity rather than with the Coulomb repulsion on the cation site. These considerations led to the development of the Zaanen-Sawatzky-Allen (zsa) theory of correlated compounds [8]. This seminal study found that in transition metal oxides, apart from the Coulomb repulsion U, there is another fundamental energy scale, the so-called charge transfer energy ∆. While U is the energy of adding an electron to a transition metal 3d state, possibly removed from another 3d site, as shown on figure 3.3(b), the charge transfer energy ∆ is the cost of transferring an electron from a ligand (i.e., the anion neighbouring the transition metal) 2p 3.4. Cuprates 39

Figure 3.4: Crystal structures of the representative superconducting copper- oxide systems, adapted from [9]. (A) presents the single unit cell for the respective system, (B) is the common feature of the cuprates, the copper-oxide CuO2 planes, along with the crucial atomic orbitals overlaid on a single unit plaquette. state to a 3d orbital. An illustration of a charge transfer insulator (cti) is presented in ﬁgure 3.3(c). Again, Coulomb repulsion splits the 3d band into an upper and a lower Hubbard band (red). However, if the 2p band (blue) is situated above the lower Hubbard band, then the charge transfer energy is smaller then the Coulomb repulsion, ∆ < U, and so the low energy excitations, and thus the electronic gap, will be determined by the charge-transfer processes p6dn → p5dn+1. The zsa theory allowed for a comprehensive description of many classes of transition metal oxides by distinguishing the two classes of insulators, the Mott-Hubbard insulators when ∆ > U and the charge transfer insulators for ∆ < U.

3.4 Cuprates

Ever since the discovery of high temperature superconducting oxides (htsc) by Bednorz and Müller [:], an enormous amount of research has been devoted to the understanding of the physical properties of these systems. In the years following the discovery a number of other htsc systems has been found, with progressively higher critical temperatures Tc. Nevertheless, they all share 3: Introduction

9 2 2 Figure 3.5: Bonding between 3d and 2pxpy states in a tetragonal crystal ﬁeld. Numbers in parentheses indicate electron occupations in the undoped system. Adapted from [;].

common characteristics: all of them are insulating oxide materials; they are layered tetragonal systems, composed of 4d square planes of CuO2 interspersed with separating cations such as La3+, Sr2+,Y3+, or Ba2+. Some examples of the most important systems in this series have been illustrated in figure 3.4, along with a visualisation of a single CuO2 plane. It has been suggested early on that superconductivity is realized in carrier doped correlated insulators near a metal-insulator transition [32]. For this reason the properties of strongly correlated 4d systems under doping are of special interest here [33]. The undoped system is determined by copper Cu2+(d9) ions, and the crystal field splits the d state spectrum such that the highest lying is the x2 − y2 state, which is occupied by the single hole on the site, as illustrated in figure 3.5. For this reason it is believed that these systems are sufficiently described by the single band Hubbard model, or effectively the Haisenberg model for the strong coupling limit [34]. From the very beginning it was evident, both from experimental data obtained in spectral experiments, like xps, xas, or Auger spectroscopy [35, 36], as well as comparison with numerical calculations for cluster models [37, 38], that cuprates are in fact ct insulators with a gap ∆ = 2–4 eV and a big Cou- lomb repulsion U = 7–10 eV. This means that there is a qualitative difference between electron and hole doping of the system. If we dope the system with electrons, they will be situated on the copper site, creating a Cu(d10) state [39]. On the other hand, hole doping removes electrons from the highest occupied state in the system, i.e., from the O(2p) orbital, which has been confirmed by a number of Fermi energy studies of hole doped systems [3:–42], and in fact they locate primarily in the in-plane 2p states that connect the copper sites [43, 44]. For this reason, a complete description of a hole in the cuprate CuO2 plane involves a three band Hubbard-like Hamiltonian for the dx2−y2 and pσ orbitals, first discussed by Emery [45, 46] and nowadays often called the d-p model: 3.4. Cuprates 3;

Table 3.3: The d-p Hamiltonian parameter values, as calculated in two diﬀerent studies. All values given in eV.

Hybertsen [47] McMahan [48]

tdp 1.3 1.5 tp 0.65 0.6 ∆ 3.6 3.5 Ud 10.5 9.4 Up 4.0 4.7 V 1.5 0.8 0 tp 0.35 [49]

X † X † Hdp = −tdp (diσpjσ + H.c.) − tp (pjσpj0σ + H.c.) hi,jiσ hj,j0iσ 0 X † X p + tp (pjσpj00σ + H.c.) + ∆ njσ hj,j00iσ jσ X d d X p p X d p + Ud ni↑ni↓ + Up nj↑nj↓ + V ni nj , (3.3) i j hi,ji

d p where niσ, njσ is the particle number operator for d states and for p states, P respectively, and ni = σ niσ. The Hamiltonian involves both on-site (Ud for Cu(3d) and Up for O(2p)) and inter-site (V ) Coulomb repulsion, p level on-site 0 energy ∆, p-p hopping (tp for nearest neighbour, tp for the neighbour across the Cu site), as well as d-p hopping (tdp), which is the source of band hybridization. All these parameters can be estimated by fitting the tight binding models to lda calculations and comparing cluster calculations with spectroscopic results. The parameters, as calculated by two different studies [47, 48], are presented in table 3.3. Since the Coulomb repulsion is usually big compared to the kinetic parameters, in particular Ud tdp, it is natural to derive an effective, second-order perturbation Hamiltonian, by projecting out the higher energy states and only considering the lowest energy excitations [4:]. For half filling on d states this leads to an effective Heisenberg interaction between the Cu sites:

4 ! 4tdp 1 2 X H = + S · S , (3.4) H V 2 U U i j (∆ + ) d 2∆ + p hiji with the superexchange constant resulting from the fourth order perturbation expansion, involving the p sites as intermediary states. Based on the known values of the d-p model parameters, it can be easily calculated that J = 0.13 eV, in agreement with values obtained from cluster calculations [4;] as well as experimentally, based on Raman scattering [52, 53]. It was also shown that this value does not depend on the presence of the apical pz orbitals [54]. 42 Introduction

Away from half ﬁlling, when the system is doped with holes occupying the O(2p), the eﬀective Hamiltonian acquires several complicated terms, all involving second order p-p exchange interactions [4:, 55]. In their critical paper, Zhang and Rice proposed that a hole doped into the p state mixes with the spin localised on the Cu site, and forms a Kondo-like singlet [56]. A symmetric (+) or antisymmetric (−) combination of the p orbitals is:

1 h x x y y i ciσ = p ± p + p ± p , (3.5) 2 ri+x/2,σ ri−x/2,σ ri+y/2,σ ri−y/2,σ however this transformation is not canonical, because diﬀerent Cu sites share the neighbouring p orbitals, and so {ciσ, cjσ0 }= 6 0 when hi, ji. For this reason it is convenient to introduce the Wannier orbitals, in the usual way [57, 58], by Fourier transforming the p orbitals:

χ 1 X −ikrj χ pkσ = √ e pj,σ, (3.6) N j for χ = {x, y} indicating the p orbital symmetry. Next we (anti)symmetrize these operators in Fourier space: q ± 1 kx x ky y 2 kx 2 ky pkσ = cos 2 pkσ ± cos 2 pkσ , tk = cos 2 + cos 2 , (3.7) tk and transform them back to real space to obtain the Wannier orbitals:

± 1 X ikrl ± plσ = √ e pkσ. (3.8) N k

With these orbitals the previous ciσ orbitals can be represented as: X 1 X c = f p+ , f = eik(rl−ri)t . (3.9) iσ i,l l,σ i,l N k l 2 k

The fi,l coefficients decay rapidly [58], and so the spatial extension of the Wannier orbitals is rather small. Under these transformations the primary term in the effective second order Hamiltonian takes the form: 2 1 1 X H = 8t + f f 0 eff dp U − − V i,l i,l d ∆ 2 ∆ i,l,l0 ! 1 X +† + 1 X +† + × Si · plσ σσσ0 pl0σ0 − niplσ pl0σ , (3.:) 2 σσ0 4 σ where σ is the vector of Pauli operators. The other terms are much more − complicated since they involve also the antisymmetric operators piσ [34], and we will omit them here. One can now form the singlet and triplet states for a given Cu site and its surrounding O sites:

1 ±† † ±† † √ p d ± p d |0i, (3.;) 2 i↑ i↓ i↓ i↑ 3.4. Cuprates 43 with + phase for triplet and − phase for singlet pairing. It is easy to calculate the energy of these four states using the eﬀective Hamiltonian (3.:) for the special case i = l = l0, yielding:

+ + − − Es = −5.05 eV,Et = 1.12 eV,Es = 1.03 eV,Et = 0.75 eV. Clearly, the symmetric singlet state has a substantially lower energy then the other states, and thus this so-called Zhang-Rice singlet is very stable, forming a bound state of two holes on neighbouring Cu and O sites. It is also for this reason that the other terms of the effective Hamiltonian are relatively unimportant, since they involve high energy excitations. In band-structure terminology the symmetric singlet is the bonding state and the symmetric triplet state is antibonding, while the antisymmetric states are nonbonding. One could now consider the effects of the nondiagonal terms of the Hamilto- nian (3.:). These will in general cause the hopping of the p hole from site l0 to l, by an exchange process between a hole and a localised spin. However, because of the rapid decay of the fi,l coefficients, this will involve at most second neighbour hopping. This can be written as: X † Ht = tl,l0 (clσcl0,σ + H.c.), (3.32) l,l0 with the hopping constant 2 1 1 1 tl,l0 = tdp + δhl,l0i + 6f0,0fl,l0 + ..., (3.33) Ud − ∆ − 2V ∆ 4 where δhl,l0i is 1 for nearest neighbour sites, and 0 otherwise, and the dots represent many other secondary coefficients resulting from the omitted terms in the effective Hamiltonian. Using the d-p model parameters given in table 3.3, we can estimate the hopping integral for the nearest neighbour hopping t = 0.76 eV, however this value depends strongly on the model parameters and should be regarded more as a crude approximation. In this way, the effective Hamiltonian for the d-p model is finally mapped onto the t-J model [59, 5:], represented by the terms (3.4) and (3.32). Physically, this can be interpreted as the result of the d-p hybridization of the Cu and O bands and the projection onto the low energy state, which is the Zhang-Rice singlet of (3.;). It is worth noting here that although the Zhang-Rice formalism is widely employed, recent exact diagonalization studies as well as variational calculations indicate that to obtain a realistic description of a hole doped into a CuO2 plane it is necessary to include the full scope of p degrees of freedom. These model calculations found interesting effects, like three-spin polaron ground state in some regions of the Brillouin zone [5;, 62] as well as good agreement with arpes spectroscopic results even in the absence of transverse fluctuations [63]. Since calculating the hopping integral from eq. (3.33) is unreliable, it has been concluded that it would be more reasonable to determine it from numerical cluster calculations [64, 65] or by fitting the model to the Fermi 44 Introduction

surface, obtained either from arpes measurements [66] or from band calculations [67]. The currently widely accepted values of the t-J model parameters are t = 0.4 eV, J = 0.13 eV [68]. Moreover, it is often regarded that to obtain a realistic description of cuprates, especially in the superconducting regime, longer range hopping parameters (t0 for second neighbours or sometimes even t00 for third neighbours) need to be included. The model is then sometimes called the extended t-J model or the t-t0-J model. In this work however we neglect such terms for simplicity. Since the exchange constant in eq. (3.4) is positive, the Heisenberg Hamilto- nian for the undoped system will favour an antiferromagnetic (af) ground state. This has been indeed confirmed in cuprate systems using neutron scattering experiments, which found a Néel transition temperature TN = 195 K in lsco, although there is a long range af magnetic correlation in 4d as high as 300 K [69]. The low energy magnetic moment is strongly reduced (from 2+ 1.1 µB for a Cu ion to 0.5 µB in lsco) because of quantum fluctuations [6:]. However, this is not a spin density wave (sdw) transition, caused by the Fermi surface nesting in the presence of weak coupling, since in the temperature T > TN the system is an insulator, not a Fermi liquid. Therefore, the transition is from the localized spins already present above the Néel temperature [34]. At strong coupling, U → ∞, the Heisenberg model is equivalent to the Hubbard model. However, it has been showed with quantum Monte Carlo simulations [6;] that also at intermediate and weak coupling, even as small as U/t = 2, when double occupancies are allowed, the af ground state is maintained in the system, although the value of the local moment is reduced with decreasing coupling strength. In this regime the order can be well described as an sdw state. Similar conclusions have been drawn from Lanczos calculations [72]. Furthermore, numerical studies based on the d-p model have yielded an antiferromagnetic ground state as well [73, 74]. Let us now consider the motion of a charge doped into the CuO2 plane. Within the Hubbard model it has been found that in the presence of af order the bandwidth√ is substantially reduced and has a sharp square-root edge at ω = 2 z − 1t [75]. In the strong coupling limit, if we take the electron picture, the system is described by the standard t-J model; if the system is doped with a hole instead, the charge will delocalize over the four planar oxygen sites around a d site and form a Zhang-Rice singlet on it, so the problem can be mapped onto the t-J model again. The doped charge will remove a spin in the ordered af magnetic system, and by moving from site to site it will rearrange the nearby localized spins through which it passes. This will leave behind a trace of misaligned spins over the path of the moving hole, the so-called string, which has been illustrated in fig. 3.6. Since the Heisenberg Hamiltonian works to enforce an af arrangement over any given bond in the system, the energy of the system will grow linearly with the string length. In the absence of the transverse quantum fluctuations there is no way of relaxing the system, and so the moving hole is, to a large degree, confined by the potential of the string. 3.4. Cuprates 45

Figure 3.6: Motion of a hole in a classical antiferromagnet. The hole (black circle) moves along the path (dashed black line, starting from the site marked with a dashed circle) leaving behind a trail of misaligned spins (red arrows) in a classical Néel state. The excited bonds are marked by dashed red lines—note that the bonds lying on the hole’s path are not excited since the sites in question have opposite spins.

More precisely, there are certain paths, the so-called Trugman loops, that allow the hole to move coherently by removing its own string [76]. However, their role is limited because of their high perturbative order—the simplest Trugman loop is a 8th order process.∗ ∗See also fig. 5.6, and the Consider now the action of the t-J model, restricted to the Ising part, on discussion in sec. 5.5 |li of the significance of the state with a single hole added. If we denote by a state with a string Trugman loops for the of length l then, neglecting the possibility that the string might cut through methods used herein. † itself, on a square lattice there are three ways in which the string can be †Technically this means extended to l + 1, one in which it can be shortened to l − 1, and the Ising that the calculation energy is the cost of the string, proportional to its length. This leads to the is done on a four-fold Bethe lattice. following algebraic equation: √ 3 H|li = J( 2 + l)|li − 3t(|l − 1i + |l + 1i), (3.34) valid for l ≥ 2. Solving this problem, either numerically or by dimensional analysis, can be shown to yield the ground state energy of a single hole 2/3 −1/3 E0 ∝ (J/t) and an average string length ¯l ∝ (J/t) [77]. This character- istic behaviour, although derived under very specific assumptions, was found to be a very good approximation for models with strictly treated paths as well as with the quantum fluctuations included [78, 79]. It is noteworthy that the ground state of a hole in the t-J model seems to be given by the wavevector k = (π/2, π/2), although states with k = (π, 0) and k = (0, π) have an energy only slightly higher, which can even become lower in the extended model. This has been shown by a combination of results obtained with a number of different techniques, such as spin-wave [7:, 7;], variational [82], and numerical methods [83]. This result can be easily understood by noting that for the weak coupling Hubbard model the Fermi surface, i.e., the energy near which the hole is added when doped to the half-filled system, is defined by cos kx + cos ky = 0 which is satisfied by all the points mentioned above. 46 Introduction

It is useful to calculate the dispersion relation of a single charge doped to an af system. The properties of the interacting solution provide an insight into the degree of renormalization caused by the fermion coupling to the polarized background. This has been analyzed in the linear spin wave approximation using the Holstein-Primakoff transformation and 1/S expansion of spin waves [7:]. This approach has also proven fruitful in the dominant pole approximation, assuming the weight beyond the first pole of the Green’s function was incoherent, to obtain analytical results for a single hole [84]. Later studies of this problem applied the same principle in a numerical setting, the so-called self ‡This method is one of consistent Born approximation,‡ to obtain the spectral function in a wide range the techniques used of (k, ω) points [85–87]. These results are in surprising agreement with exact in this thesis. Its details are discussed diagonalization calculations [78, 88]. The general findings for a single hole are 0.79 at length in sec. 5.4. the scaling of the quasiparticle bandwidth W ∼ 1.5J [86], resulting from mass renormalization due to the confinement in the the string potential, and the near degeneracy between the Fermi surface momenta, mentioned previously. Interestingly, a good fit to numerical results at J/t = 0.4 is:

εk = −1.255 + 0.34 cos kx cos ky + 0.13(cos 2kx + cos 2ky), (3.35) indicating that the holes travel at doubled momenta, i.e., over sites belonging to a single sublattice and avoiding the other ones [89]. This dispersion relation also shows that the energy difference between the global minimum at k = (π/2, π/2) and the other Fermi momenta is tiny compared to the total bandwidth. Therefore, the pocket states located at the bottom of the band [8:] are very vulnerable both to temperatures and to doping rates. The Green’s function method also allows for the calculation of the dynamical properties of quasiparticles (qp), i.e., effective particles resulting from the hole’s dressing into the magnetic excitations of the polarized background. A quasiparticle is indicated by the lowest lying, coherent (i.e., not scattered by coupling to magnons with random momentum) peak on the photoemission spectrum, corresponding to a bound state of an electron coupled to a cloud of misaligned spins. At large J the spectral function has a relatively simple structure, with a huge qp spike at the bottom and a couple of smaller peaks in the higher energies. Extensive studies show that the dominant peak corresponds to a nearly localized hole with a large mass, and the higher peaks are associated with short string states of 3 and 4. If the exchange constant J is reduced, the peak height (the qp spectral weight, Z) is gradually reduced, and the weight is transferred to the higher energy states. This is easy to understand, as for smaller J the cost of generating magnons is reduced, and thus longer strings become more likely to occur. Finally, at J = 0, the spectral function is reduced to a pair of symmetric incoherent bands, separated by a pseudogap, however this solution should be treated with reserve, since for negligibly small J the hole wavefunction becomes delocalized and so the finite size effects start playing an important role. Different authors have tried fitting a power law Z ∝ J α, however the results remain inconclusive, with the 3.5. Orbital Ordered Systems and KCuF3 47 exponent varying from α = 1/2 obtained from exact diagonalization [8;, 92] to α = 2/3 extracted from Born approximation [86, 93]. On a final note, let us mention that upon doping additional hopping chan- nels are made possible, the so-called three-site terms. These processes are similar to the exchange processes, in that they involve a virtual excitation, but unlike in the exchange, the relaxation does not remove the hole created by the excitation, but rather the one doped from outside the system. This effectively introduces second and third neighbour hopping with the hopping integral ∝ −J. It has long been suggested that such processes are crucial for the description of low energy physics in cuprates [94]. More recently, the improvement of resolution of angle-resolved photoemission (arpes) allowed for precise measurements of the low energy excitations in htsc, revealing the so-called high-energy anomaly around the Γ = (0, 0) point of the band [95– 97]. Subsequent numerical studies have confirmed its existence [98, 99] and semi-analytical studies based on scba indicate, that this feature cannot be explained without employing the three site terms [9:].

3.5 Orbital Ordered Systems and KCuF3

Because of their specific symmetry and the Cu(d9) configuration cuprates are single d band systems. However, in general this does not have to be the case. Most transition metal oxides exhibit some degree of d state degeneracy as well as various electron configurations. Moreover, charge doping affects their electron configuration causing their behaviour to be very different from the undoped case [9;]. Since the orbital represents the shape of the electron cloud in space, in a solid this can lead to steric interference between the electrons occupying different atoms, affecting their energy levels and properties. This is known as crystal field splitting which, coupled with the d electron occupation number, leads to the emergence of different spin and orbital degrees of freedom [:2]. In a cubic crystal, which is the system of highest symmetry, the splitting separates the d states into two families, called the t2g orbitals (xy, 2 2 2 2 yz, and zx) and the eg orbitals (3z − r and x − y ). The d level degeneracy can be lifted by further lowering the symmetry of the system. Similarly to the case of cuprates, because of the strong 3d shell localization, the on-site Coulomb interaction is very strong, leading to the multiorbital Coulomb interaction:

X X 1 HU = U niα↑niα↓ + (Uαβ − 2 Jαβ)niαniβ iα i,α<β X † † X + Jαβ(diα↑diα↓diβ↓diβ↑ + H.c.) − 2 JαβSiα · Siβ, (3.36) i,α<β i,α<β where the Greek indices run over the d orbitals; the intraorbital Coulomb repulsion Uαβ and the exchange integral Jαβ are generally anisotropic and 48 Introduction

Table 3.4: On-site interorbital exchange elements Jαβ expressed in terms of the Racah parameters. Adapted from [:4], for more details see ref. [:3].

d orbital xy yz zx x2 − y2 3z2 − r2 xy 0 3B + C 3B + CC 4B + C yz 3B + C 0 3B + C 3B + CB + C zx 3B + C 3B + C 0 3B + CB + C x2 − y2 C 3B + C 3B + C 0 4B + C 3z2 − r2 4B + CB + CB + C 4B + C 0

dependent on the orbitals involved. Of course, due to the d orbital symmetries, the actual number of distinct values of these parameters is much smaller than it seems and it is common to express them using the so-called Racah parameters [:3]; their expressions have been tabulated in table 3.4. Furthermore, the intraorbital Coulomb integral is expressed as:

U = A + 4B + 3C, (3.37)

and, resulting from the invariance of the interactions in the orbital space, the interorbital repulsion is constrained to:

Uαβ = U − 2Jαβ. (3.38)

As before, the electrons in such a system can hop from site to site, possibly hybridizing different orbitals, governed by the kinetic integral t. In the strong interaction limit an effective exchange Hamiltonian can be derived, however this time, because of the interplay between the spin and orbital degrees of freedom, the exchange will be of a spin-orbital nature [:5–;2]. The exact form of this Hamiltonian depends strongly on the crystal structure and the electronic character of the system, and we will therefore not attempt to present here the full scope of diversity such models can exhibit. Some general guidelines for a variety of systems can be found in [:4] and for detailed derivations of con- crete models one should consult the relevant literature on manganites [;3–;5], vanadates [:8, ;6, ;7], copper fluorides [;8, ;9] and other systems [:7, ;:]. The effective model relevant for our case (a single charge in an eg orbital system, such as KCuF3) will be derived in sec. 4.4. The copper fluoride perovskite KCuF3 is a pseudocubic (actually tetragonal) system that has attracted attention since the 3;82s, after it was discovered that it exhibits a number of interesting phenomena relating to magnetic and orbital physics, such as orbital ordering, cooperative Jahn-Teller effect, and low-dimensional antiferromagnetism. Since then it has come to be viewed as a textbook example of spin-orbital physics. The crystal structure has been first studied with single crystal X-ray and neutron diffraction and a number of phases have been identified [;;]. Two basic polytype structures distinguished 3.5. Orbital Ordered Systems and KCuF3 49 by their orbital order exist, for obscure historical reasons called the a and d polytypes.§ The orbital order sets in at a very high temperature, essentially §The source of this convention seems to be persisting up to the structural phase transition at Ts ≈ 800 K at which the ref. [;;], where the au- Jahn-Teller effect leads to the distortions of the CuF6 octahedra. Below this thor denoted the poly- temperature the structure is tetragonal, with the in-plane Cu-Cu distance types as a and c, with d⊥ = 4.14 Å and the c direction distance shortened to dc = 3.93 Å [322, 323]. b and d being their re- The distance disproportion of dc/d⊥ ≈ 0.95 indicates that the system is very spective versions with close to cubic and such an approximation, at least for the sake of model stacking disorder. The notation for c and d calculations, is justified. was later reversed in an The polytype structures of KCuF3 differ by the orbital order, namely the erratum. a type exhibits alternating orbital order of the g type (g-ao), with the occupied orbitals alternating in all three directions, while the d structure exhibits alternating order of the c type (c-ao), with occupied orbitals alternating within the ab plane, and the order repeated in the c direction. In this thesis we will focus on the d phase, because of its quasi-4d nature of the orbital order, which can be viewed as an orbital counterpart of the magnetic structure of cuprates. More recently, advances in resonant X-ray scattering (rxs) techniques have made it possible to directly probe the orbital symmetry and correlations of orbitally ordered states. These reveal a strong coupling between spin and orbital degrees of freedom, with a change of orbital symmetry occurring just above the Néel temperature, paving the way for the magnetic transition [324, 325]. Sim- ilar conclusions have been reached with optical Raman spectra measurements, indicating a lattice symmetry reduction associated with a structural transition to an orthorhombic phase [326] and the stabilization of GdFeO3-type rotations of the CuF6 octahedra [327]. In the low temperature regime the magnetic moments of Cu also order, and the magnetic order is of a type (a-af), with spins ordering antiferromag- netically along the c direction and ferromagnetically in the ab plane [323, 328], in accordance with the Goodenough-Kanamori rules [329, 32:]. The Néel transition temperatures were determined to be Ta = 38 K for the a polytype and Td = 22 K for the d polytype; the magnetic moment is µ = 0.48 µB and it is oriented within the ab plane [323]. A number of neutron scattering studies have shown that magnetic structure of KCuF3 is composed of quasi-3d af chains which are weakly coupled along the in-plane bonds, with the ratio of exchange couplings J⊥/Jc ≈ 0.01 [32;]. The quantum spin fluctuations are strong and the system can be viewed as a textbook example of a 3d Heisenberg antiferromagnet. The magnetic excitations have been confirmed by neutron scattering to have spinon character [332]. The exact nature of the orbital order has been a point of debate for a number o years although many questions in this regard remain open. For convenience, let us parameterize a general combination of eg orbitals in terms of a rotations angle θ: θ 2 2 θ 2 2 |θi = cos 2 |3z − r i + sin 2 |x − y i, (3.39) 4: Introduction

(a) (b)

Figure 3.7: The orbital ordering for a unit plaquette in an ab plane of KCuF3, (a) is the state θ = π/3, (b) is for θ = π/2.

with the alternating orbitals given by the angles ±θ, i.e., in general the bases on different sublattices might not be mutually orthogonal. Early studies sug- 1 gested that the single hole might occupy orbitals corresponding to cos θ = 3 (θ ≈ 0.4π), which follows from the local Jahn-Teller lattice distortions [328]. On the other hand, based on the electronic interactions in the magnetic phase, the Kugel-Khomskii type orbitals |y2 − z2i/|z2 − x2i were suggested [:5], corresponding to θ = π/3. Over the years, experimental data [324, 325, 332] as well as various lda+u calculations have come to support this hypothesis [333– 335]. However, a recent analysis of Raman and X-ray scattering experiments probing the above-mentioned orthorhombic transition have found an oscillation between two nearly degenerate states very near the type |3z2 − r2i ± |x2 − y2i, corresponding to θ = π/2 ± φ, with a small detuning angle φ ≈ 0.012π [327]. This study, based on a model involving a direct orbital exchange process which is believed to explain the huge disparity between the orbital and magnetic ordering temperatures [336], suggests that the Kugel-Khomskii state might have a significantly higher energy. In any case, because of the Jahn-Teller distortion, both of the states are similar in the general characteristics of having a pronounced in-plane direc- tionality and a strong component along the c direction. The actual ground state lies somewhere in the vicinity of the region marked by θ = π/3 and θ = π/2. For illustration, in fig. 3.7 we have rendered the orbital order on a unit plaquette of the ab-plane for the two extreme values of θ. CHAPTER 4 The Models

—What will your ﬁrst step be? —The usual one. —I didn’t know there was a usual one. —Well, sure there is. It comes complete with diagrams on page 69 of “How to be a Detective in 32 Easy Lessons” correspondent school textbook. . . —You must’ve read another one on how to be a comedian.

Lauren Bacall as Vivian Rutledge & Humphrey Bogart as Philip Marlowe in Howard Hawks’ The Big Sleep (3;68)

In this chapter we will present the models which are the starting point for the Green’s function calculations in this thesis. First we present the t-J model and its derivation, which is the simplest eﬀective exchange model. The derivation of the spin-orbital model of KCuF3 follows in similar steps, only taking into account the multiplet structure of a partially ﬁlled d shell. The models are developed into the form necessary for the methodology used, i.e., the electronic operators are decoupled into spinless fermions and bosons describing excitations in the polarized background.

4.3 The t-J Model

The t-J model is the generic single-band eﬀective exchange model. Its derivation begins with the Hubbard Hamiltonian:

X † X HH = −t (diσdjσ + H.c.) + U ni↑ni↓, (4.3) hijiσ i

4; 52 The Models

† where di /di is the electron creation/annihilation operator acting on site Ri, t is the intersite hopping integral, and U is the onsite Coulomb repulsion. The Hubbard model describes two electronic bands separated with a gap of size U. The properties of the system depend on the band filling. In particular, at half filling, i.e., with one electron at every lattice site, the lower Hubbard band is completely filled, and the upper band is empty, which means the system is a Mott insulator. To derive the t-J model we have to assume that the Coulomb repulsion U is very big compared to the hopping integral t, which at half filling means the electrons are frozen at their sites. Now consider virtual excitations resulting from the hopping of an electron from its site to the neighbouring one. Such a transfer is an excitation from the lower Hubbard band to the higher band, and costs the energy U; it is metastable and short-lived, and the electron has to relax back to the lower band by returning to the starting site. This process can be calculated in the second-order operator perturbation expansion to yield the t-J model [337, 338]:

X h † i Ht = −t (1 − niσ¯)diσdjσ(1 − njσ¯) + H.c. , (4.4a) hijiσ X h z z 1 α + − − + i HJ = 4J (Si Sj + 4 ) + 2 (Si Sj + Si Sj ) , (4.4b) hiji

t2 where J = U is the exchange constant. The electron operators in the kinetic term above are projected onto the space with singly occupied sites, thus formally ensuring the Hamiltonian applies to the lower Hubbard band. Henceforth we will omit this notation for simplicity, although it is always implicitly understood that the t-J Hamiltonian refers to the projected space. Equation (4.4b) is called the exchange term because it describes an exchange of an electron between two neighbouring sites. If after the excitation to the doubly occupied state the electron stays at the site, and the other electron goes back to the original site from which the excitation occurred, the spins of the neighbouring sites get exchanged, which is the source of the spin fluctuations in the model. The whole operator HJ is also called the Heisenberg Hamiltonian, while its sole first term is known as the Ising Hamiltonian (the t-J model with the exchange term restricted to the Ising Hamiltonian will be referred to as the t-J z model) and the second one describes the quantum spin fluctuations. For convenience, the Ising Hamiltonian is adjusted so that the mean field energy of the antiferromagnetic (af) ground state is 0. The α ∈ [0, 1] parameter is introduced so that it is possible to make a continuous interpolation from the Ising model to the Heisenberg model. If the system is doped with electrons (or holes), there are doubly occupied (empty) sites present in the system even in the ground state. This allows the charge to move by a second order process similar to the exchange process. An electron located at a site neighbouring with the doubly occupied site hops to 4.3. The t-J Model 53 another singly occupied site, leaving behind an empty site. This time however, it is an electron from the doped site that fills the empty site. Effectively this causes the electron to hop by two sites away from the doping site, which in an af state occurs within a single sublattice, allowing for a free electron propagation (not disturbing the ordered background). Such a process can be written as: X † † H3s = −J di+,σ0 diσ0 diσdi+δ,σ(1 − δδ), (4.5) iδσσ0 where the (1 − δδ) factor ensures that the final site is always different from the starting site, δ =6 . Assuming that the electron’s spin does not change in the process, i.e., σ = σ0, we can calculate the free propagation:

X † T3s = εkdkσdkσ, (4.6) kσ

2 where εk = −4J(4γk − 1) is the second order free electron dispersion, and P ikδ γk = δ e /4 is the structure factor for the case of a 4d plane. Finally, to transform the Hamiltonian to the form useful for the methods used herein, we need to decouple the standard fermionic operators diσ into spinless fermions fi and bosons bi, which will serve to represent the localised spin degree of freedom. The idea is to perform an expansion around the af ground state, which will therefore be represented as the the bosonic vacuum state |0i. A boson generated in this vacuum will represent a spin deviation from the ground state. It is convenient to ﬁrst perform a canonical transformation of the ordered state to a ferromagnetic (fm) state, which will eliminate the sublattice division of the system, at the cost of slightly complicating the Hamiltonian. The advantage of this approach is that in this way we can introduce a single boson representation for the entire system, thus avoiding the necessity of folding of the Brillouin zone [86]. The transformation is a standard rotation of the spin basis, parametrized by the lattice dependent rotation angle θi:

! θi θi ! ! |θii cos − sin |↑i = 2 2 , (4.7) |θ¯ i θi θi |↓i i sin 2 cos 2 which corresponds to the following transformation of the spin operators:

˜z ! ! z ! Si cos θi sin θi Si ˜x = x , (4.8) Si − sin θi cos θi Si where θi = Q · Ri [339], and Q = (π, π) is the so-called ordering wave vector for the case of a 4d af. Clearly if Ri points to the sublattice a, on which the spins are oriented upwards, the rotation angle θA = 0, whereas on sublattice 54 The Models

b, the angle θB = π. Therefore, this transformation rotates the spins on the b sublattice, so that now all the spins point upwards, like in a ferromagnetic state. Evaluating equations (4.8) for the af ordering vector leads to the following transformations of the Hamiltonian: † † z z ± ∓ djσ → djσ¯,Sj → −Sj ,Sj → −Sj , (4.9) which finally produces the fm representation of the t-J model: X h † i Ht = −t diσdjσ¯ + H.c. , (4.:a) hijiσ X h 1 z z α + + − − i HJ = 4J ( 4 − Si Sj ) − 2 (Si Sj + Si Sj ) , (4.:b) hiji while the three-site term (4.5) remains unaffected, since it acts within the Néel sublattices. Finally, we decouple the spin degree of freedom of the fermions by mapping the local fermion Hilbert space onto a product of a fermion occupation space and spin space: + di↑ = fi, di↓ = fiSi , (4.;) and represent the spins with bosons using the Holstein-Primakoff transformation: q √ + † Si = 2S − bi bibi ≈ 2Sbi, (4.32) q √ − † † † Si = bi 2S − bi bi ≈ 2Sbi , (4.33) z † Si = S − bi bi, (4.34) for a general spin of size S. However, it is vital to remember that these are restricted slave bosons, i.e., their maximal number on a site is not greater than 2S. The exact representation of the S± operators ensures that implicitly, however if the approximate, linear representation on the right-hand-side is used, such as we will later see is the case for the scba method, this rule is no longer ensured, and it might be necessary to enforce it by other means, if possible. Furthermore, another constraint is needed, namely the number of bosons and fermions at any given site is similarly restricted: † † fi fi + bi bi ≤ 2S. (4.35) Following these steps, the t-J Hamiltonian (4.:) can be expressed in the linear spin wave (lsw) formalism: X h † † i Ht = −t fi fi+δbi+δ + H.c. , (4.36a) iδ X h † † † † i HJ = J bi bi + bi+δbi+δ − α(bi bi+δ + bibi+δ) , (4.36b) iδ 4.4. The Spin-Orbital Model 55 and the three-site terms take the form:

X † † h † † † † i H3s = −J fi+fifi fi+δ 1 + bi bi + bi+δbi+ + bi+δbi + bibi+ (1 − δδ). iδ (4.37) Assuming there is only one electron doped into the system, we can simplify † it further by substituting fifi = 1. The ﬁrst term above is the free fermion term, which can be diagonalized exactly in Fourier space:

X † T3s = εkfkfk, (4.38) k which is the only free hopping term in the Hamiltonian, while the coupling to magnons is always higher order than in (4.36a).

4.4 The Spin-Orbital Model

The spin-orbital model of KCuF3 is developed along similar lines to the t-J model, however the crucial difference is that this system is multiband, i.e., has more than one active d state. More precisely, the Cu2+(d9) magnetic ion is located in the octahedral crystal field of the surrounding F– ions, which causes the 3d spectrum to split into the lower t2g states and the upper eg states. Given the site electron occupation, it is clear that the t2g states will be completely filled, while the site’s single hole will be located in one of the eg states, which are degenerate in the first approximation. Thus, the copper configuration can 3 1 be equivalently described as eg in terms of electron occupation, or eg in terms of hole occupation. The description of the charge dynamics starts with defining the kin- 9 etic Hamiltonian.√ We consider electron hopping between Cu(d ) sites along 2 2 |zγi = (3zγ − r )/ 6 type orbitals, where the direction zγ = x/y/z is paral- lel to the cubic directions γ = a/b/c,√ respectively. On the other hand, the 2 2 orthogonal orbitals |z¯γi = (xγ − yγ)/ 2 do not play any role in the kinetic processes, because the hopping is mediated by the ligand F(2p) orbitals, and the symmetry of the p-d bonds causes the hopping elements to cancel out. The kinetic Hamiltonian can thus be written as [:9]: H −t X X d† d . . . t = izγ jzγ + H c (4.39) γ hiji⊥γ

This formulation, while concise, is not very useful due to the basis being dependent on the bond direction. Therefore, the Hamiltonian has to be transformed to the standard basis {|zi, |z¯i}:

h X † 1 X † √ † √ i Ht = −t dizdjz + (diz ∓ 3diz¯)(djz ∓ 3djz¯) + H.c., (4.3:) hijikc 4 hiji⊥c 56 The Models

where the upper/lower sign corresponds to the directions a/b, respectively. The system is strongly correlated, with d electrons interacting with on-site and inter-site Coulomb repulsion, and with Hund interaction which drives the site towards maximal spin. This can be described with the multiorbital Hubbard model (3.36), which for the case of an exclusively eg orbital system simpliﬁes to:

X 5 X HU = U niα↑niα↓ + (U − 2 JH ) niαniβ iα i,α<β X † † X + JH (diα↑diα↓diβ↓diβ↑ + H.c.) − 2JH Siα · Siβ, (4.3;) i,α<β i,α<β

where U is the Coulomb repulsion and JH is the Hund exchange element. For the limit U t, and considering the d9d9 d8d10 virtual excitations relevant for the KCuF3 case, one can develop an eﬀective superexchange model similar to the t-J model. The spectrum of the d8 excitations, as determined 3 1 1 1 from (4.3;), consists of four spectroscopic terms A2, Eθ and E, and E1, with energies U − 3JH , U − JH (double), and U + JH , respectively. This leads to the following four superexchange terms:

γ X 3 1 γ γ H1 = −2Jr1 (Si · Sj + 4 )( 4 − τi τj ), (4.42a) hijikγ γ X 1 1 γ γ H2 = 2Jr2 (Si · Sj − 4 )( 4 − τi τj ), (4.42b) hijikγ γ X 1 1 γ 1 γ H3 = 2Jr3 (Si · Sj − 4 )( 2 − τi )( 2 − τj ), (4.42c) hijikγ γ X 1 1 γ 1 γ H4 = 2Jr4 (Si · Sj − 4 )( 2 − τi )( 2 − τj ), (4.42d) hijikγ

where the ri coeﬃcients serve to impose the Hund rule and follow from the 2 above-mentioned multiplet structure of the eg conﬁguration: 1 1 1 r = , r = r = , r = , (4.43) 1 1 − 3η 2 3 1 − η 4 1 + η γ where η = JH /U, and τi are direction dependent orbital operators which can be expressed in terms of pseudospin operators: √ a/b 1 z x c z τi = − 2 (Ti ∓ 3Ti ), τi = Ti , (4.44) under the standard convention

|z¯i ≡ |↑i, |zi ≡ |↓i. (4.45)

For simplicity, from here on we shall assume that the Hund exchange is negligibly small, JH = 0. This means that the multiplet structure collapses to 4.4. The Spin-Orbital Model 57

just one state of energy U and the coeﬃcients ri = 1. Equations (4.42) can then be simpliﬁed to just two terms:

γ X 1 γ γ 1 H1 = 4J (Si · Sj + 4 )(τi τj + 4 ), (4.46a) hijikγ γ X 1 γ γ H2 = 2J ( 4 − Si · Sj)(τi + τj ). (4.46b) hijikγ

Although the orbital ground state is known to be alternating, the actual occupied states do not necessarily have to be the basis (4.45). To ﬁnd the real occupied states we have to introduce an orbital crystal ﬁeld to the Hamiltonian, which will serve to suppress the orbital degeneration of the system:

X z Hz = −Ez Ti . (4.47) i This term simulates an axial pressure acting along the c direction, in the extreme limits Ez → ±∞ causing the system to order ferroorbitally (fo), with occupied states either |z¯i or |zi, respectively. Thus, tuning the orbital field allows one to drive the system from an ao to an fo order in a continuous manner. Note that the term (4.46b) has a somewhat similar form to the orbital field, particularly in that it is linear in pseudospin operators. It is therefore reasonable to expect that the classical ground state has alternating non-orthogonal orbitals, slightly tilted towards one of the ferroorbital states. To optimize the classical orbital ground state we start with parametrizing the orbital basis in terms of a rotation of the standard basis (4.45) in the same way as in (4.7). Nonetheless, it can be showed that the classical ground state does not depend on the rotation angle, i.e., the occupied states could be any pair of orthogonal orbitals, as long as the ground state is alternating. However, if an orbital field is present in the system, it breaks the rotational symmetry along the c axis, and so the ground√ state in the limit Ez → 0 is composed of the states |±i = (|z¯i ± |zi)/ 2, which corresponds to a rotation by an angle θ = π/2 [;8]. Therefore, to incorporate the orbital field, we will π parametrize the sublattice rotations with respect to this angle, θi = 2 + φi, iQR where φi = e i φ is the sublattice dependent detuning angle and Q = (π, π, 0) is the orbital ordering vector for the c-ao order. This approach means that the relative angle between occupied states at different sublattices equals π − 2φ and decreases, from π for ao to 0 for fo ordering. Now, transforming the superexchange Hamiltonian (4.46) according to the spin transformations given in (4.8) with the above rotation angle, we arrive at the exchange Hamiltonian dependent on the orbital field:

⊥ X 1 h z z x x H1 = J (Si · Sj + 4 ) 1 + (2 cos 2φ + 1)Ti Tj + (2 cos 2φ − 1)Ti Tj hiji⊥c √ x z z x x z z x i +2 sin 2φ(Ti Tj − Ti Tj ) ± 3(Ti Tj + Ti Tj ) , (4.48a) 58 The Models

k X 1 h 1 2 z z 2 x x H1 = 4J (Si · Sj + 4 ) 4 + sin φTi Tj + cos φTi Tj hijikc i iQRi 1 x z z x −e 2 sin 2φ(Ti Tj + Ti Tj ) , (4.48b)

√ ⊥ X 1 h z z H2 = J ( 4 − Si · Sj) (sin φ ∓ 3 cos φ)(Ti + Tj ) hiji⊥c √ x x x x i − cos φ(Ti + Tj ) ∓ 3 sin φ(Ti − Tj ) , (4.48c)

h i k X 1 Ez iQRi z z x x H2 = 2J ( 4 − Si · Sj − 4J ) e sin φ(Ti + Tj ) + cos φ(Ti + Tj ) , hijikc (4.48d)

where the last term incorporates the orbital field Hz. To find the optimal detuning angle φ, we evaluate the mean field energy for the classical ground state a-af/c-ao:

1 1 EMF = (J − 2 Ez) sin φ − 4 J(2 cos 2φ + 1), (4.49)

and then minimize it with respect to φ, which yields the relation between the orbital ﬁeld Ez and the detuning angle φ:

1 Ez = 4J(sin φ + 2 ). (4.4:)

Now if we are interested in the especially simple case of identical bases on both sublattices with alternating order (no effective orbital field), we need to set φ = 0, from which follows that Ez = 2J, i.e., the external orbital field Hz compensates the one originating in H2. Then the superexchange Hamiltonian takes the simplified form:

⊥ X h 1 z z 1 + − − + i H1 = J 4 + Si Sj + 2 (Si Sj + Si Sj ) hiji⊥c h √ i z z x x iQRi z x x z × 1 − 3Ti Tj + Ti Tj ± 3e (Ti Tj − Ti Tj ) , (4.4;a) k X h 1 z z 1 + + − − i h 1 x xi H1 = 4J 4 − Si Sj + 2 (Si Sj + Si Sj ) 4 + Ti Tj , (4.4;b) hijikc ⊥ X h 1 z z 1 + − − + i H2 = − J 4 − Si Sj − 2 (Si Sj + Si Sj ) hiji⊥c h √ i x x iQRi z z × (Ti + Tj ) ± 3e (Ti − Tj ) , (4.4;c) k X h 1 z z 1 + + − − i h x xi H2 = − 2J 4 − Si Sj − 2 (Si Sj + Si Sj ) Ti + Tj , (4.4;d) hijikc 4.4. The Spin-Orbital Model 59 already written in the |±i basis, which was the reference for the φ rotation. Moreover, the Hamiltonian has already been transformed to the fm/fo state, similarly to what has been done before for the t-J model, and Q is the ordering vector for the c-ao state. Lastly, we have to transform the kinetic Hamiltonian (4.3:) in the same way, so that both terms are expressed in identical representation:

⊥ t X h † † Ht = − (1 − 2 sin φ)diσ0djσ0 + (1 + 2 sin φ)diσ1djσ1 4 hiji⊥c,σ √ i iQRi † † +(2 cos φ ± 3e )(diσ0djσ1 + diσ1djσ0) + H.c., (4.52a)

k t X h † † Ht = − (1 + sin φ)diσ0djσ0 + (1 − sin φ)diσ1djσ1 2 hijikc,σ † † i − cos φ(diσ0djσ1 + diσ1djσ0) + H.c., (4.52b) where the 0/1 indices denote the orbital ground/excited states. Next we can introduce the bosons to represent the polaronic degrees of freedom: † † n m di,mn = fi ai bi , (4.53) † † where bi creates a spin excitation at site i, ai creates an orbital excitation, and m, n ∈ {0, 1} denote the local state of the polarized background, where 0 means the ground state is occupied, 1 that the state is excited. In other words, a charge can be added to the system only in the ground state; if there is an excitation already present, ﬁrst the system has to be relaxed to the ground state, and only afterwards can the charge be added. Decoupling the fermions from bosons and assuming, as before, orthogonal bases at φ = 0, we arrive at the polaronic form of the spin-orbital Hamiltonian:

t X † X † T = − fi+δfi = kfkfk, (4.54a) 4 i,δ⊥c k t X nh √ † √ † i H⊥ = − (2 ± 3eiQRi )a + (2 ∓ 3eiQRi )a + a a t 4 i i+δ i i+δ i,δ⊥c (4.54b) † † o † ×(1 + bi bi+δ) + bi bi+δ fi+δfi, k t X † † † Ht = − (ai − 1)(ai+δ − 1)(bi + bi+δ)fi+δfi, (4.54c) 2 i,δkc where 1 X ikδ k = −tγk, γk = e , (4.55) 4 δ⊥c is the free electron dispersion. Note that the free electron term T comes from hopping restricted to the ab plane, which is a result of the magnetic structure 5: The Models

Figure 4.3: Schematic representation of the orbital kinetic processes. Blue orbitals are in the ground state and red orbitals are excited, full orbitals denote a d9 configuration while a dashed contour is a doped state, without (pseudo)spin. The differences in hopping parameters are due to different orbital overlaps.

of the system, and therefore the momentum space is also restricted to the 4d subspace spanned by (kx, ky), even though the full model is 5d.

4.5 The Planar Orbital Model

Suppose now that we restrict the electron not to generate any magnetic excitations, only orbitonic ones. The Hamiltonian discussed in the previous section would still describe such a case, since this is only a subset of the possible kinetic processes. However, since the magnetic order is of a type, the electron would not be allowed to leave its initial ab plane. Such a toy model would be the orbitonic analogue of the t-J model, allowing for a direct comparison of magnetic and orbital excitations and their role in the propagation of a charge in a quasi-4d system. These constraints can be used to simplify the spin-orbital model by gauging away the magnetic degrees of freedom in the Hamiltonian. When this is done, the kinetic Hamiltonian is reduced to two terms:

t X † X † T = − fi+δfi = kfkfk, (4.56a) 4 i,δ⊥c k t h √ √ i ⊥ X iQRi † iQRi † † Ht = − (2 ± 3e )ai + (2 ∓ 3e )ai+δ + ai ai+δ fi+δfi, 4 i,δ⊥c (4.56b) whose physical meaning has been illustrated in ﬁg. 4.3. This illustration is also relevant for the full spin-orbital model from the previous section. At the same time, the superexchange Hamiltonian simpliﬁes to just one term [;4], with the ground state Ising energy adjusted to 0: J h √ i ⊥ X 1 z z x x iQRi z x x z H1 = 3( 4 − Ti Tj ) + Ti Tj ± 3e (Ti Tj − Ti Tj ) , (4.57) 2 hiji⊥c or in boson representation, up to second order:

⊥ J X h 3 † † 1 † † i H1 = 2 (ai ai + ai+δai+δ) + 4 (ai + ai)(ai+δ + ai+δ) . (4.58) 4 i,δ⊥c 4.5. The Planar Orbital Model 5;

It is perhaps worth mentioning that in this case, since there is only a single superexchange term corresponding to the energy U−3JH , it is of no consequence whether we neglect the Hund energy, as it is merely a simple renormalization of the exchange constant J. The same can be said about the average of the spin part of the Hamiltonian, which could be taken as an average over the full quantum state, not just the mean ﬁeld solution. However, here we are not primarily concerned with the exact values of the constants matching their real values, but rather with the qualitative physical eﬀects exhibited by the system.

CHAPTER 5 Methodology

Señor, I suspect that you were a very fine flyer and before that perhaps a promising shoe salesman, but you’re a gross amateur at intrigue. You cannot expect to catch a trout by shouting at it from the riverbank proclaiming that you’re a great fisherman. You need a hook with feathers on it.

Walter Slezak as Melchior Incza in Edward Dmytryk’s Cornered (3;67)

In this chapter we will cover in detail the methods used to calculate the single electron Green’s functions for the models. We will start with some general remarks about the Green’s function formalism, as applied to non-relativistic field theories relevant in condensed matter physics. We continue with a detailed description of the approximate, systematic methods of solving the Green’s function for a given model. First is the self-consistent Born approximation, which is a well-known, firmly established method used in polaronic models. Next we present the variational approximation, a more novel and flexible approach based on solving the equations of motion for the Green’s function. These methods are compared and their strong and weak sides are discussed.

5.3 The Green’s Function

Let us start with considering the time dependent Schrödinger equation:

(i∂t − H)|ψ(t)i = 0. (5.3) Usually it is not possible to ﬁnd an exact solution of this equation. However, it can be showed that given an initial condition |ψ(t0)i, a solution at any other

63 64 Methodology

point in time can be formally expressed as:

|ψ(t)i = U(t − t0)|ψ(t0)i, (5.4)

where −iH(t−t0) U(t − t0) = e (5.5) is called the time evolution operator, or the propagator. Then the Green’s function in the time domain is deﬁned as:

G(t − t0) = −iΘ(t − t0)U(t − t0), (5.6)

where Θ(t−t0), the Heaviside step function, acts as the time ordering operator, which ensures that t > t0, i.e., we are investigating the forward time evolution of the electron. Formally, this Green’s function is called the retarded function. Its backwards time evolution counterpart, when t < t0, called the advanced function, would describe the time evolution of a hole in a band. From (5.6) we see that this deﬁnition of the Green’s function matches that found in the study of ordinary diﬀerential equations:

(i∂t − H)G(t − t0) = δ(t − t0), (5.7)

and therefore the general solution of the Schrödinger equation can be found by convolution of the Green’s function with the initial condition ψ(t0). In solid state research we are usually more interested in the eigenenergies of the system rather than in its time evolution, since the solid’s properties are primarily determined by the equilibrium states with well deﬁned energies. To that end we need to calculate the Fourier transform of the Green’s function [33:]:

∞ ∞ Z Z G(ω) = dtG(t)eiωt = −i dteit(ω−H). (5.8) −∞ 0

For real values of ω − H this integral is divergent. Therefore, since the Hamilto- nian is a Hermitian operator, to ensure convergence we need to add a small broadening factor to the energy parameter ω → ω + iη. Physically this means that we are introducing a finite particle lifetime τ ∝ 1/η, where an infin- itesimally small broadening corresponds to an infinitely long lifetime. When the integral (5.8) is evaluated, the energy dependent Green’s function can be expressed in a compact form:

G(ω) = [ω + iη − H]−1, (5.9)

also known as the resolvent. Suppose now that we could ﬁnd the solution for the Hamiltonian, H|ψni = En|ψni, yielding a set of eigenenergies En and 5.3. The Green’s Function 65

the corresponding complete basis set |ψni. Then the Green’s function can be equivalently written as:

X |ψnihψn| G(ω) = , (5.:) n ω + iη − En known as the Lehmann representation [33;]. From this equation it is clear that the Green’s function has poles when the energy ω is equal to an eigenvalue of the Hamiltonian, ω = En. This gives us a simple way to ﬁnd the energy levels of the system without the need to explicitly solve the Schrödinger equation, provided we can calculate the Green’s function by other means. Until now we have been dealing with the operator form of the Green’s function, in particular the resolvent (5.9). In practice we will however be interested in its expectation value in the initial state, usually a free electron Bloch state:

† 1 X ikRi † |ki = fk|0i = √ e fi |0i. (5.;) N i Based on the Lehmann representation we can now write the expectation value for a free electron as: 2 |Zk| G(k, ω) = , (5.32) ω + iη − Ek to which we will henceforth refer as the Green’s function for short. Here Ek is the energy of a quasiparticle and Zk = hψn|ki is called the quasiparticle weight, describing the contribution of the free electron wavefunction to the full, interacting solution. The quasiparticle (qp) is understood as an eﬀect- ive particle arising in a correlated system when a physical particle (e.g., an electron) interacts with its surroundings, propagating coherently together with a cloud of excitations it generates in the system. For practical purposes it is useful to deﬁne the so called spectral function:

1 A(k, ω) = − =G(k, ω) (5.33) π |Z |2 η k . = 2 2 (5.34) π (ω − Ek) + η

The second form follows from the explicit expression for the imaginary part of (5.32), and it happens to be the Lorentzian peak function, which is one of the analytical representations of the Dirac delta. Therefore it follows that:

η→0 2 A(k, ω) −−−→|Zk| δ(ω − Ek). (5.35)

From its construction it is clear that the spectral function has a set of poles, whose location in the (k, ω) plane indicates the qp band. 66 Methodology

It is sometimes useful to evaluate the so-called spectral moments of the spectral function A(k, ω), which are similar to the moments of a statistical distribution: ∞ Z n Mn(k) = A(k, ω)ω dω. (5.36) −∞ On the other hand, the respective moment can be calculated analytically, based on the model in question, using the following formula [342]: † Mn(k) = h0| [[[fk,H],H],...], [..., [H, [H, fk]]] |0i, (5.37) | {z } | {z } n−p p where the value of 0 ≤ p ≤ n is arbitrary, i.e., it can be chosen in the most convenient way for the problem at hand. Obviously, M0(k) = 1 is just the spectral function normalization, which is consistent with the integral of the spectral function A(k, ω). In this way it is possible to quantify the agreement of the theoretical sum rules with those obtained with an approximate method, such as scba or the variational approximation. Note however, that although the agreement between a given method and the sum rules is desirable, its signiﬁcance can sometimes be hard to evaluate. Most approximate methods of solving the Green’s function are perturbative in nature, with the Hamiltonian H split into an exactly solvable part H0 corresponding to a free propagator G0, and the interaction V. The equation (5.9) can now be written in an equivalent form:

G(ω)(ω + iη − H0 − V) = I, (5.38) where I is the identity operator. Moving the interaction term to the right side and multiplying from the right with the free propagator G0(ω), we arrive at the Dyson equation:

G(ω) = G0(ω) + G(ω)VG0(ω), (5.39) and a similar expansion can be made with a reversed operator order. This equality is exact and true regardless of how the partition of the Hamiltonian is performed. Both of the approximate methods used in this thesis rely on the Dyson equality to perform a systematic expansion of the Green’s function in terms of the interaction V. Lastly, it is sometimes convenient to consider an auxiliary quantity called the self-energy Σ(k, ω), deﬁned with respect to the full interacting Green’s function by the equation [33;]: 1 G k, ω , ( ) = −1 (5.3:) G0 (k, ω) − Σ(k, ω) which describes all the interactions in the system, i.e., all the propagators making up the full Green’s function, excluding the free propagator G0(k, ω). 5.4. Self-Consistent Born Approximation 67

In terms of diagrammatic representation of the Green’s functions, the self- energy gathers all the irreducible diagrams of the expansion, i.e., ones that cannot be disconnected into disjoint parts by removing a single free propagator line. From the point of view of perturbation theory, this is equivalent to a sum over all the perturbation corrections to the ground state.

5.4 Self-Consistent Born Approximation

We will now proceed to describe the self-consistent Born approximation [7:, 86, ;4, ;8], or scba for short, for the t-J model and the 4d orbital model. The full spin-orbital model is treatable only in the mean ﬁeld approximation, by separating the spin and orbital degrees of freedom. In scba the Hamiltonian is partitioned into the quadratic terms, which make up the free propagator, and the third order fermion-boson interaction vertex; higher order terms are neglected. The expansion is performed in Fourier space, where the quadratic terms are easily diagonalized. In the case of the t-J model, the only free electron Hamiltonian is T3s of (4.38), and the bosonic one is (4.36b), whose Fourier transform reads: X † 1 † † HJ = 2zJ [Aqbqbq + 2 Bq(bqb−q + bqb−q)], (5.3;) q where Aq = 1,Bq = αγq, (5.42) and γq is the 4d structure factor. This Hamiltonian is a quadratic form, but the double creation/annihilation terms (i.e., spin ﬂuctuations) indicate that the classical Néel state |Ni (or rather an fm state, after the spin rotation on the b sublattice) is in fact not the ground state. In order to diagonalize the Hamiltonian, we need to correct the ground state using the Bogoliubov transformation: ! ! ! βq uq −vq bq † = † , (5.43) β−q −vq uq b−q which for the case of bosons can be interpreted as a hiperbolic rotation 2 2 uq = cosh θq, vq = sinh θq, uq − vq = 1, (5.44) where the last equation is a consequence of unitarity. Under this transformation the Hamiltonian takes the form:

νq z }| { X † 2 2 HJ = 2zJ [βqβq (Aq(uq − vq) + 2Bquqvq) q † † 1 2 2 + (βqβ−q + βqβ−q)(Aquqvq + 2 Bq(uq + vq))], (5.45) | {z } 0 68 Methodology

where the conditions for diagonalization have been marked with the braces. It can now be easily deduced that the general solution to the Bogoliubov transformation is: q 2 2 νq = Aq − Bq, (5.46a) s s Aq 1 Aq 1 uq = + , vq = − sgn (Bq) − , (5.46b) 2νq 2 2νq 2

or explicitly for the t-J model: q 2 νq = 1 − (αγq) , (5.47a) s s 1 + νq 1 − νq uq = , vq = − sgn (γq) . (5.47b) 2νq 2νq

Now the exchange Hamiltonian can be written in its diagonal form:

X † HJ = ωqβqβq, (5.48) q

where ωq = 2zJνq is the 4d magnon dispersion, and the corresponding quantum ground state: ! X uq † † |0i = exp βqβ−q |Ni. (5.49) q vq Finally, the three body interaction (4.36a) needs to be transformed in the same manner, leading to:

zt X † Ht = −√ Mkqfkfk−qβq + H.c., (5.4:) N k,q

where Mkq = uqγk−q + vqγk (5.4;) is the fermion-boson vertex interaction function. The process is repeated nearly unaltered for the orbital model. The free fermionic Hamiltonian is (4.56a) and the exchange term (4.58) is Fourier transformed to yield:

z X † 1 † † HJ = J [Aqaqaq + 2 Bq(aqa−q + aqa−q)], (5.52) 4 q

with coeﬃcients 1 1 Aq = 3 + 2 γq,Bq = 2 γq. (5.53)

We introduce the Bogoliubov bosons αq in the same way as before, leading to the equation (5.45), whose general solution was already given in (5.46), 5.4. Self-Consistent Born Approximation 69 or explicitly after substituting the above coeﬃcients: q 1 νq = 3 1 + 3 γq, (5.54a) r s ! r s ! 1 νq 3 sgn(γq) νq 3 uq = + , vq = − . (5.54b) 2 3 νq 2 3 νq

The exchange Hamiltonian is now expressed as:

X † HJ = ωqαqαq, (5.55) q with the orbiton dispersion ωq = zJνq/4. Finally, the fermion-orbiton kinetic coupling (4.56b) takes a similar form as before:

⊥ zt X † Ht = − √ fkfk−q[Mkqαq + Nkqαq+Q] + H.c., (5.56) 4 N k,q however with an additional term coupling between ﬁrst and second pseudospin Brillouin zone, which results from the pseudospin non-conservation. The vertex functions are: √ Mkq = 2(uqγk−q + vqγk),Nkq = − 3(uqηk−q − vqηk), (5.57) where ηk = γk+πy , and πy = (0, π) accommodates the dependence on direction a/b. The scba Green’s function can be interpreted in terms of the second order symmetric Dyson expansion:

G(ω) = G0(ω) + G0(ω)VG(ω)VG0(ω), (5.58) which, for a free electron state |ki, is equivalent to a self consistent diagrammatic equation shown in ﬁg. 5.3. That is, the interacting Green’s function G(k, ω) is derived from the free function by adding a second order closed loop diagram, which is then renormalized on the intermediate electron state. In practical terms calculating the interacting diagram is done by iterating the self-energy:

2 2 (zt) X Mkq Σn+1(k, ω) = , (5.59) N q ω − ωq − Σn(k − q, ω − ωq) starting from Σ0(k, ω) = 0, i.e., the ﬁrst order self-energy can be expressed in terms of the interaction vertex and the free Green’s function, of which both are known. From its construction it is evident that the expansion is performed by adding closed boson loops around diagrams created in the previous steps. 6: Methodology

= + k k k k − q k

Figure 5.3: Self consistent equation for the Green’s function in scba. The double line is the interacting Green’s function G(k, ω) and a single line is the free function G0(k, ω); the wiggly line indicates a boson.

Therefore, none of the bosonic lines can ever cross, which is why scba is sometimes called the non-crossing approximation. Physically this means that the bosons can be removed from the system only in reversed order to how they were created—last created is always first to be removed. It is expected that the omitted crossing diagrams are of minor importance, especially for the case of the t-J model where spin conservation prohibits the leading crossing processes by symmetry. Some of the leading vertex corrections are displayed in fig. 5.4, namely the lowest (fourth order) crossing diagram, and the lowest (sixth order) nontrivial diagram resulting in coherent propagation of the fermion in the absence of quantum fluctuations. They are shown in real space representation, which allows for simple interpretation of the processes. Figure 5.4(a) illustrates the simplest fourth order crossing diagram, in which the fermion hops from site m to n and then back to the starting site m, generating two spin deviations at the sites in the process. This means that in the intermediate state, at site m there would be both a fermion and a boson present, which violates the constraint (4.35). The process is clearly unphysical, and thus needs to be excluded anyway, as does a whole class of similar higher order diagrams. However, not all crossing diagrams can be argued against in this way. There is in fact a whole class of crossing diagrams that are not only not forbidden by the occupation constraint, but can even lead to nontrivial physical effects. The simplest such example is illustrated in fig. 5.4(b); it is called a Trugman loop, after the author who first discussed its significance [76]. In this process the fermion takes a one-and-a-half loop around an elementary square plaquette, effectively moving diagonally across the square, from site m to p. In doing so, it generates three bosons, and subsequently removes them in the order of creation. In the end no bosons remain, so in a sixth order perturbation process the fermion has propagated coherently by two sites. In fact, any such looped movement in an alternating state leads to a coherent charge propagation by two sites. These are especially important for the Ising approximation, where there are no quantum fluctuations to remove excitations from the system, which reduces the importance of higher order terms. 5.5. Variational Method 6;

(a) (b)

m n m n m m n p r m n p

Figure 5.4: Leading crossing diagrams neglected in scba, in real space representation. Diagram (a) describes an intermediate state with a fermion and boson at site m, (b) describes a Trugman loop around a square plaquette, leading to coherent fermion propagation from site m to p.

5.5 Variational Method

The variational approximation is an exact, primarily analytical method of calculating Green’s functions, with a precisely controlled Hilbert space cutoff. It consists in generating a set of equations of motion (eom) by repeatedly applying the Dyson equation (5.39), and evaluating the effect of the interaction term [343–347]. What is exceptional about this method is the fact that the Hamiltonian is treated exactly, with no approximations introduced whatso- ever,∗ with the interaction treated in real space, unlike in scba. Instead, all the ∗In fact, for certain sys- approximations are controlled through the Hilbert space cutoff, which in turn is tems, such as a 3d ferromagnetic chain, the introduced by restricting the spread and size of the boson cloud. The rationale method allows to ob- is that the propagator decays with increasing distance, while the locality of tain an exact solution of the vertex interaction forces the fermion to return to the neighbourhood of the a single particle Green’s boson in order to interact. As a consequence, long distance fermion propagation function [348, 349]. is unlikely and at the same time risks leaving behind isolated bosons, which would invariably leave the system excited. Therefore, states with compact boson clouds are the ones determining the physical properties of the system. The Hamiltonian is partitioned into the free part H0, which in our case z includes any free fermion hopping T and the Ising term HJ , while the interaction V includes all the non-solvable terms, regardless of their vertex order—in principle all the terms can be treated exactly. In particular, quantum fluctuations do not have to be linearized, but can be evaluated according to the full Holstein-Primakoff representation. Furthermore, this means that different bosonic degrees of freedom do not have to be separated in the spin-orbital model, the way they would have to be in scba. In case where there is no free fermion hopping, such as in the t-J model without the three site terms, the free propagator is constant: 1 G (ω) = . (5.5:) 0 ω + iη 72 Methodology

We will now proceed to demonstrate how the expansion is performed, using the 4d orbital model as an example [34:, 34;]. The other models are treated analogically and the procedure should be clear enough for the reader to understand how it is applied there. We start by writing the expectation value for the Dyson expanded Green’s function:

G(k, ω) = hk|[1 + G(ω)V]G0(ω)|ki, (5.5;)

and since |ki is the eigenstate for G0(ω), this can be sipliﬁed to: 3 G(k, ω) = [1 + hk|G(ω)V|ki]G0(k, ω − 2 J), (5.62) where the shift in ω is the Ising energy of the ground state with an added fermion, which has a slightly higher energy due to the removal of the local spin at a single site. Next we need to evaluate V|ki in real space, using the deﬁnition in (5.;). The interaction will act directly on the position states under the sum, generating states with bosons (orbitons in this case) left behind by the moving fermion. Finally, the expansion yields: " √ # t 3t X X ¯ iπyδ 3 G(k, ω) = 1 − F1(k, ω, δ) − F1(k, ω, δ)e G0(k, ω − 2 J), 2 δ 4 δ (5.63) where we have introduced the generalized Green’s functions, which describe propagation between a free fermion state and a fermion-boson bound state:

1 X † † √ ikRi F1(k, ω, δ) = hk|G(ω) e fi+δai |0i, (5.64a) N i 1 X † † ¯ √ i(k+Q)Ri F1(k, ω, δ) = hk|G(ω) e fi+δai |0i, (5.64b) N i where the second function couples different orbital Brillouin zones. Clearly the polaronic states are coupled in real space, i.e., are always located on neighbouring sites, and the whole cloud is then transformed to Fourier space, sharing the wavevector k. This picture is quite different from the one in scba, where the wavevectors of the fermion and of the bosons are independent, as long as the total momentum in the interaction vertex is conserved. The functions in (5.64) are of course unknown and need to be expanded using the Dyson equation again. However, this time the free propagator will be acting on a state with a boson present. The location of the boson distinguishes a specific point in space, which the fermion is forbidden from entering, because of the constraint (4.35). This breaks the translational symmetry of the system, so that k is no longer a good quantum number. Instead, the propagators need to be evaluated in real space:

G˜0(n,i+δ,ω) z }| { 1 X † † † √ ikRi † F1(k, ω, δ) = hk|[1 + G(ω)V] e ai fn|0i h0|fnaiG0(ω)ai fi+δ|0i, N in (5.65) 5.5. Variational Method 73

(a) (b) 3 8 J

3 4 J

Figure 5.5: The Ising cost of a cloud of one orbiton and one fermion in diﬀerent 4d conﬁgurations. The energy cost of a single excited/broken bond is indicated next to it. In (a) the fermion far away from the boson gives 6 excited bonds 9 and 6 broken bonds, 2 J total cost; (b) if the fermion is neighbouring the boson, 15 one excited bond is replaced with a broken bond, 4 J total cost.

where G˜0(n, i + δ, ω) indicates a real space, non-interacting Green’s function, † corrected for the restrictions dictated by the presence of the boson ai . Let us consider how to incorporate those restrictions into the real space propagators, deﬁned as: π 1 Z G (n, m, ω) = h0|f G (ω)f † |0i = d2kG (k, ω)eik(Rn−Rm), (5.66) 0 n 0 m π2 0 0 which describes a propagation of a fermion from site m to site n in a 4d system. However, as the Fourier transform demonstrates, the real space Green’s functions do not depend on the speciﬁc locations of the sites in question, but rather on their relative distance, G0(Rn − Rm, ω), which is still translationally invariant. To include the spatial restrictions, we will again resort to the Dyson expansion. The restrictions that need to be included are twofold:

3. The fermion cannot ﬁnd itself at a site occupied by a boson, therefore the hopping elements for processes involving those sites have to be cancelled.

4. If the fermion ends up on a site neighbouring a boson, or a set of bosons, the Ising energy of such an arrangement will be smaller than in a situ- ation where the fermion is far away from the boson cloud (illustrated in ﬁg. 5.5). The reason is that all the inter-site bonds around a fermion are z broken (hTi i = 0 at the fermion site), which costs a smaller amount of energy than an excited bond between a boson and an empty site. The 3 energy diﬀerence for the 4d orbiton model is 4 J. These can formally be expressed as a Hamiltonian for a single site i, occupied by the boson:

t X † 3 X † Vi = (fi fi+ + H.c.) − 4 J fi+fi+. (5.67) 4 74 Methodology

Now we perform the Dyson expansion, with the above Hamiltonian acting as the interaction term:

G˜0(ω) = [1 + G˜0(ω)Vi]G0(ω), (5.68)

leading to the following set of equations:

˜ 9 X ˜ G0(n, m, ω) = G0(n, m, ω − 2 J) + G0(n, i + , ω) h t 9 3 9 i × 4 G0(i, m, ω − 2 J) − 4 JG0(i + , m, ω − 2 J) . (5.69)

This equation describes a general transfer between any two sites m and n. To solve it, we ﬁrst need to ﬁnd the propagators between sites neighbouring the bosons, from which the propagators for the general case can be extended. The propagators for the neighbours are mutually dependent and so have to be solved together, which can be represented as the following matrix equation:

h i−1 ˜ γδ γ δ t δ0 3 δ G0 = G0 I − 4 G0 + 4 JG0 . (5.6:)

In the general case there may be more bosons present in the system. In that instance, all the boson occupied sites are forbidden to the electron, and for every site neighbouring a boson there maybe one or more (up to four) bonds broken that would otherwise be excited. Then the procedure follows as before, only the interaction will have more terms. The general form of the corrected free propagator is:

0 X G˜0(n, m, ω) = G0(n, m, ω − rJ ) + G˜0(n, p, ω) hp,p˜i t 0 0 0 × 4 G0(˜p, m, ω − rJ ) − J G0(p, m, ω − rJ ) , (5.6;) where the summation goes over all such pairs of neighbouring sites for which p˜ is the boson occupied site and p is an adjacent empty site; r is the number of excited bonds plus 2 to account for the broken bonds, which cost half the 0 3 energy; J is the single bond energy cost, in this case 4 J. Finally, to complete the analytical description within the variational method, let us note that the non-interacting Green’s functions G0(n, m, ω) can be expressed exactly in terms of complex analytical continuations of the complete elliptical integrals of the first and second kind, K(κ), E(κ), with the elliptic modulus κ = t/ω [34:, 352]. The details of the derivation are rather tedious and of no importance here, therefore we relegate it to Appendix A. Now that we have corrected the free propagators, we can go back to the expansion of the single boson Green’s functions (5.65). After the propagation, the fermion can end up on any of the unoccupied sites n, however it only makes sense to consider sites that are near the bosonic cloud, as only these will contribute to the expansion, since the variational Hilbert space cutoff 5.5. Variational Method 75 is controlled by the spread of the cloud. For instance, if we decide on the minimal expansion, i.e., demand that all the bosons are located directly next to each other, then the electron can interact with the cloud only if it propagates directly next to it. If we would allow the cloud to have gaps between the bosons, then the electron can also interact if it ends up two sites away from the nearest boson. Here, for simplicity, we will assume the former. If we now † † evaluate Vai fn|0i, the expansion will yield: √ t X h iπy −ik F1(k, ω, δ) = − 2G(k, ω) + 3G¯(k, ω)e + F1(k, ω, −)e (5.72) 4 √ X iπyγ i + [2F2(k, ω, , γ) − 3F¯2(k, ω, , γ)e ] G˜0(i + , i + δ, ω), γ=6 √ t X h iπy −ik F¯1(k, ω, δ) = − 2G¯(k, ω) + 3G(k, ω)e − F¯1(k, ω, −)e (5.73) 4 √ X iπyγ i + [2F¯2(k, ω, , γ) − 3F2(k, ω, , γ)e ] G˜0(i + , i + δ, ω), γ=6 where the G/G¯ functions result from the boson annihilation, F1/F¯1 from the fermion-boson swap process, and F2/F¯2 from the creation of a second boson next to the first one. The G¯ function is defined similarly to G:

G¯(k, ω) = hk|G(ω)|k + Qi t h √ i X ¯ iπyδ 3 = − 2F1(k, ω, δ) + 3F1(k, ω, δ)e G0(k + Q, ω − 2 J), (5.74) 4 δ only it links between different regions of the orbital Brillouin zone. As before, the above expansion contains new unknown functions, for states with two bosons. These in principle should be expanded as the ones before, which would link them to the previous functions as well as new ones, containing three bosons, &c. This cascade of states, up to third order, is illustrated in fig. 5.6. In order to close the system of equations, this process has to be stopped at some point. For instance, to obtain a first order result, we can take F2 = 0, F¯2 = 0, thus obtaining a set of mutually-dependent functions, which can be solved as a regular linear system. Naturally, while the eom system is generated analytically it can rarely be solved exactly, even if in principle it is always possible. Instead, we represent the system as a sparse linear system of equations and solve it numerically to obtain the Green’s function on an arbitrarily dense grid on the (k, ω) plane. Up until now we have been only discussing the Ising limit, which has the advantage of being treatable exactly. While this limit is the most crucial for describing the physics of polaronic systems, there is no denying that quantum fluctuations are very important for the accurate description of the system, especially if quantitative agreement is sought. However, the difficulty with 76 Methodology

1 4 28

1 4 12 28 12

4 12 12

Figure 5.6: An example third order variational expansion. Only select representative processes are shown. The plaquette colours indicate states belonging to the same symmetry class (fourfold rotations of the boson cloud, coupled with fermion propagation around the cloud) and the number of states in the given symmetry class is indicated inside. In each case the expansion terminates when a branch hits a state encountered before. The middle path illustrates the Trugman process.

fluctuations is that they are non-local—they act on the polarized background regardless of the position of the doped charge. Their exact inclusion within the variational method is therefore not possible. Fortunately, it is fairly easy to include them approximately without any modification to the previously established framework. Namely, we include the fluctuations as yet another term in the interaction, but allow it to act only within the neighbourhood of the boson cloud. The states are then discriminated based on the same bosonic cloud configuration rules as before (for instance, by requiring the cloud to be connected, as in the earlier discussion). In this way we include only the fluctuations which can interact with the fermion, i.e., the ones that contribute to the polaronic physics of the system. Fluctuations that act farther away from the cloud play an increasingly less significant role, mostly contributing to the energy of the undoped ground state. It is expected that in this way we include the essential contributions to the quasiparticle state, without the need to treat the fluctuations in their full extent. However, the fluctuations add a substantial number of terms to the expansion, thus making the system noticeably more costly to solve, although in practice most of the cost comes from evaluating the real space Green’s functions. As a final note, it is clear that the variational expansion procedure can become complicated very fast. As the system of equations grows, the bookkeeping becomes increasingly tedious. In practical terms the expansion beyond a 4–5 boson approximation is close to impossible. It is for this reason that we have developed a program, called sage, to perform the expansion automatically with a computer. Some details of how this system works can be found in Appendix B. Part II

Results

CHAPTER 6 Cuprates: Purely Spin Systems

I distrust a close-mouthed man. He generally picks the wrong time to talk and says the wrong things. Talking’s something you can’t do judiciously unless you keep in practice. Now, sir, we’ll talk if you like. I’ll tell you right out—I’m a man who likes talking to a man who likes to talk.

Sydney Greenstreet as Kasper Gutman in John Huston’s The Maltese Falcon (3;63)

In this chapter we shall examine the spectral functions for various extensions of the t-J model, which is the primary model for a charge propagating in a 4d purely spin system. We will look at both the Ising and the Heisenberg limit of the exchange, as well as on the localized electron versus the three site hopping enabled model. We will investigate the convergence of the variational approximation and then compare the results with the scba. Discussion of the nature of the quasiparticle (qp) by way of its dispersion and spectral weight will follow from these results.

6.3 Introductory Information

In the following discussion we will be plotting density maps of the spectral function over the (k, ω) plane. Note that in all the plots of the spectral functions presented hereafter (and in particular in later chapters) we will be using a nonlinear tanh-scale for the density scale. More precisely, instead of the spectral function A(k, ω) we will plot tanh[A(k, ω)], in order to bring out the

79 7: Cuprates: Purely Spin Systems

Figure 6.3: The color scale used throughout this work for the spectral density maps. The curve in green is tanh(x) and the one in blue is a linear function.

low amplitude incoherent part. The reason is that for very small (near zero) values of the argument x we have: x3 2x5 tanh(x) ≈ x − + + ..., (6.3) 3 15 which is to a large degree linear, so that the relative representation of the small values is mostly unaffected. However, for x > 1 the function becomes highly nonlinear, saturating at higher values, lim tanh(x) = 1, thus treating x→∞ the actual maxima uniformly, and normalizing them to 1. Therefore, the plots should be interpreted in such a way that the proper maxima are represented in pitch black, while the incoherent continuum is in colours ranging from white near zero through yellow and orange to brown. As a visual aid to the color scale, figure 6.3 presents the plot of tanh(x) along with the linear function for reference, plotted against the background of the color scale. Values beyond x = 2 are practically visually indistinguishable from one another. This scale will not be presented alongside the plots them- selves since the absolute values of the spectral function are irrelevant since their units are arbitrary—for instance, they depend on the broadening factor η, the choice of value of which is a matter of numerical convenience rather than any physical effect. It is therefore not important to know the absolute value of the maximum, but rather the area under the peak, which is a direct measure of the qp spectral weight Zk. For this, however, separate plots will be provided. As is usually the custom, we will not present the results for the whole Brillouin zone, but rather the vital cuts along the high symmetry directions on an “hourglass” path:

Y = (0, π) → M = (π, π) → Γ = (0, 0) → X = (π, 0) → Y. (6.4)

Note that there are additional symmetries leading to X ≡ Y , and for a bipartite af system Γ ≡ M. We will however not employ those symmetries, as it makes 6.4. Convergence of the Variational Results 7;

Figure 6.4: The spectral functions for the t-J z model obtained with the variational approximation and with scba at J = 0.1. The numbers in the upper-right corner indicate the size of the variational space.

for an easier comparison with other results later on. Additionally, let us deﬁne π π the special point S = ( 2 , 2 ), lying on the line intersection in the middle of the “hourglass”.

6.4 Convergence of the Variational Results

We will start with examining the convergence of the variational calculation, and further comparing the ground state qp with that found using scba. We begin with the simplest case of the t-J z model, comprising of the electron-magnon coupling (4.36a) and the Ising part of the exchange (4.4b); for scba eqs (5.4:) and (5.48) with α = 0 have to be used. Obviously, in this case there is no free hopping, the magnons are dispersionless and the Ising Hamiltonian merely provides the excitation energy associated with the creation of a magnon (or, more generally, the energy of a particular boson arrangement), and the kinetic Hamiltonian serves as the interaction between the electron and the magnons. 82 Cuprates: Purely Spin Systems

Figure 6.5: The qp ground state energy E(k) and spectral weight Z(k) in the t-J z model for J = 0.1.

Figure 6.4 presents the variational calculation for J = 0.1 expanded up to five magnons and compared to scba (last panel). The scba results show the familiar dispersionless ladder spectrum, associated with the electron confinement by the potential energy of the boson string. Each time the electron creates a new magnon in the system, the resulting qp energy is increased while its spectral weight (i.e., its bare electron contribution) decreases. On the other hand, the variational calculation shows a different number of states for each expansion level, which is increasing with the number of bosons in the system. This reflects the fact that the number of available qp states grows with the size of the boson cloud. Another important difference between the two methods is the tiny delocalization of the qp which can be observed in the variational result starting from the third expansion order. A close inspection of the shape of the dispersion (see also fig. 6.5) reveals the existence of minima at Γ and M points and maxima at the X ≡ Y point, with S being a saddle point. This is consistent with a −t cos kx cos ky dispersion relation, which is to be expected from the lowest order Trugman process—requiring at least three bosons in the cloud to appear, which leads to an effective h1, 1i hopping across the diagonal of a square plaquette. The next order contribution appears at the five boson level, and leads to hopping in directions h1, 1i and h2, 0i, although evidently this contribution is tiny enough to be neglected. To highlight the tiny differences between the qp ground states in the different expansion orders, as well as in scba, we have calculated the energy dispersion E(k) and the spectral weight Z(k) by a least squares fitting of a Lorentzian curve to the lowest energy peak of the spectral function. This has been presented in figure 6.5. Evidently, the difference between order 6 and 7 is negligibly small, and already at order 5 convergence can be considered adequate, as far as the dispersion relation is concerned. A somewhat bigger 6.4. Convergence of the Variational Results 83

Figure 6.6: The spectral functions for the t-J model obtained with the variational approximation and with scba for J = 0.1. The numbers in the upper-right corner indicate the size of the variational space. difference between the spectral weight curves reflects the fact that the boson cloud can still grow, however its contribution to the qp dynamics is negligible, since the particle is already largely localized. Next we turn our attention to the complete t-J model, including the transverse spin fluctuations. Figure 6.6 presents again the spectral functions for the two techniques under consideration. One can clearly see the qp delocalization appearing at the second expansion order, which is consistent with the spin fluctuation mechanism, i.e., the electron generating two magnons which are subsequently removed by fluctuations (or, as a matter of fact, the reverse process). Adding more magnons to the cloud seems to barely affect the shape of the qp, which is not unexpected, given that the fluctuations actively work to remove bosons from the cloud before it can grow substantially. Although the general features of the qp band seem to be well preserved in the variational calculation as compared to scba, there are significant differences. The most important one is the global minimum at the S point, which is a well known effect in cuprates, correctly reproduced by scba. Its absence in the variational solution points to the importance of longer range spin fluctuations for the correct description of the spin polaron and suggests that a more subtle 84 Cuprates: Purely Spin Systems

Figure 6.7: The qp ground state energy E(k) and spectral weight Z(k) in the t-J model for J = 0.1.

approach to spin ﬂuctuations might be needed in the case of the variational ∗The present calcula- approximation.∗ Indeed, by taking the Fourier transform of the Bogoliubov tions have been made Hamiltonian (5.48): with a compact boson cloud, i.e., all the X X † 1 X bosons need to form ω β†β = ω β β , ω = ω eiq(Rj −Ri), (6.5) q q q ij i j ij N q a connected cluster. q i,j q

we can convince ourselves that the diagonalization comes at the price of allowing arbitrarily long boson hoppings (unless ωq = ωc is constant, in which case ωij = ωcδij, i.e., the Ising limit where there are no fluctuations to consider), although only relatively short range hoppings are significant. Bear in mind, however, that these are Bogoliubov bosons, which are different from the bosons representing magnetic excitations in the classical Néel state. In the latter case the fluctuations are off-diagonal, two-boson processes which correct the Ising energy via an interaction term. The variational method includes the fluctuations only as corrections to the classical state, whereas scba employs the Bogoliubov ground state. This seemingly suggests that the deepening of the minimum at the S point is a nontrivial quantum effect which is not reproduced in the variational approximation. On the other hand, the variational method also produces a minimum at the S point, albeit a much more shallow one, resulting from the Trugman process. This, along with the fact that the addition of longer range hopping (i.e., treatment in the extended t-J model) can also deepen the minimum at S, makes it unclear whether the scba treatment of fluctuations is necessary to reproduce the experimental results. Moreover, the variational method has been independently verified with an exact diagonalization on a 54 site cluster [353]. The calculations agree to a striking degree, and the minimum at S is very shallow, in agreement with the Trugman mechanism. It is therefore not clear to what extent the 6.5. Three Site Terms 85 deepening of the minimum is an actual physical effect or simply a byproduct of the approximations employed in the scba method. Finally, we present the energy and spectral weight in figure 6.7 to see the details of the ground state. Although the convergence is again reached by the expansion order 6–7, further differences between the methods are clearly revealed. Both the energy and the spectral weight obtained within scba show a much bigger variation with k, although the general tendency is similar, with fairly big spectral weight in the vicinity of the energy minimum at S. However, the large drop in weight at the Γ and M points is in stark contrast to variational calculations. This can be partially explained by the mixing of the qp band with the incoherent region lying above it—since in the case of scba the incoherent region extends all the way from the ground state up, while in the variational method it is located higher, roughly in the ω ∈ [−1, 1] region, and thus is separated by a gap from the ground state. Since also the dispersion is larger in the scba, its high points start to enter the incoherent region and lose their spectral weight to the continuum, i.e., those states become short-lived due to scattering on highly energetic magnons around k = (0, 0) and k = (π, π).

6.5 Three Site Terms

Let us now examine the role of the three-site terms (4.37). This problem can be treated in two ways: either considering solely the effective hopping term (4.38), which is all that can be done within basic scba, or one can also include the spin-flip terms, i.e., processes where the hopping is accompanied by the creation/annihilation of a pair of magnons on the sites involved (here we also include the non-spin-flip term analogous to hopping, but working under the condition that there already are magnons present on the path of the electron). Figure 6.8 shows the spectral functions for the t-J z-3s and t-J-3s models, both of them without the spin-flip terms. Comparing the convergence of both series demonstrates that the three site terms are strongly suppressed by the immobile magnons in the t-J z-3s solution, so that the delocalization of the qp is relatively small, even though the bare electron dispersion εk is quite substantial, as demonstrated by the reference blue curve. This is evident even in the scba result where the qp dispersion is too small to allow mixing with the excited part of the spectrum. Therefore, we see that most of the dispersion in the t-J-3s solution is the result of the spin fluctuations, while the three site terms only enhance the effect slightly.† †Both the fluctuations In the case of the variational results, the combined effects lead to the fall and the 3s terms effectively contribute to the of the spectral weight at the Γ and M points, much like what was observed same kind of 4nd and in the regular t-J model. In fact, looking at the energy and spectral weight 5rd neighbour hopping, (fig. 6.9) one can see that the methods agree quite well on the ground state, which is due to their when compared to the regular t-J model. exchange interaction origin. 86 Cuprates: Purely Spin Systems

Figure 6.8: The spectral functions for the t-J z-3s model (up) and the t-J-3s model (down) obtained with the variational approximation and with scba for J = 0.1. The blue curve indicates the three site hopping dispersion. 6.5. Three Site Terms 87

Figure 6.9: The qp ground state energy E(k) and spectral weight Z(k) in the t-J-3s model for J = 0.1.

On the other hand, looking at the excited energy part of the spectrum, one can see a large discrepancy between the two results. Where the scba predicts a wide bulk of incoherent states with a fairly big dispersion, which mixes with the ground state to produce the “waterfall” feature, the variational result shows a ladder of very flat coherent states. To some degree this can be attributed to the still not completely converged variational result—while the ground state converges very fast, the excited spectrum requires a much bigger variational space to be obtained. At the same time, although scba is a self consistent method, such that continuing the calculation to higher order would not produce any different results from the present ones, the number of bosons in this expansion is typically very small, since the method usually converges after just 3–4 iterations. ‡ Since the excited spectrum is expected to involve ‡Usually the bigger the J an increasing number of bosons, both methods should be used cautiously in the more iterations are needed, which is hardly this regime. surprising. However, Lastly, let us look at the effects of the spin-flip terms. Figure 6.: presents here we are in the small the spectral functions for the full t-J-3s model with the spin-flips. For refer- J limit. ence also the previous scba result, which does not include the spin-flip terms, is repeated. We see that the general features are very similar to the ones found before, however the energy of the ground state is slightly higher. This small change pushes the upper part of the qp band into the incoherent region which produces an excited state separated from the ground state with a narrow pseudogap, while the spectral weight of the top of the qp band falls to zero. This produces a tiny “waterfall” feature through the mixing of the two states, although this is not related to the one found in scba, which should be clear from the sheer difference in their energy scales. The mechanism is nonetheless similar which suggests that the same kind of results could be obtained in the variational method in the right parameter regime. 88 Cuprates: Purely Spin Systems

Figure 6.:: The spectral functions for the t-J model obtained with the variational approximation and with scba for J = 0.1. The numbers in the upper-right corner indicate the size of the variational space.

6.6 Conclusions

In this chapter we have compared the variational approximation and the scba method using the extensively studied t-J model as a benchmark. We have demonstrated that the variational method produces satisfactory results, com- parable to the ones obtained in scba, although quantum fluctuations are treated only locally and are seemingly responsible for most of the difference between the two methods. However, it is hard to ascertain which method treats the fluctuations more effectively. Nonetheless, the relative importance of the fluctuations diminishes when the bare electron is mobile, which in our case is caused by the three site hopping, leading to a better agreement between our methods. This is because a mobile electron can follow the boson cloud, whereas a localized electron can be left stranded by fluctuations which can separate it from the rest of the boson cloud. It is therefore expected that the agreement between the two methods will in general be better for models with dispersionless bosons, as well as with mobile electrons. As we will see, these conditions are satisfied in the eg orbital models. CHAPTER 7

KCuF3 Planes: Purely Orbital Systems

I think I’m in a frame. . . I don’t know. All I can see is the frame. I’m going in there now to look at the picture.

Robert Mitchum as Jeﬀ Bailey in Jacques Tourneur’s Out of the Past (3;69)

In this chapter we will examine the spectral functions for the simpliﬁed, purely orbitonic planar model described in sec. 4.5. We then compare the variational results with scba to demonstrate that in this case the agreement between the methods is very good, which is the result of the peculiar properties of orbitons. Because of the relevance of this limit for the full spin-orbital model, which we will investigate in the next chapter, here we will also examine the large J limit of the problem, as well as the role of orbital ﬂuctuations. We will also demonstrate the consequences of the translational constraints, whose exact inclusion in the variational method is one of its advantages over scba.

7.3 Convergence of the Ising Model

We start by analyzing the convergence of the spectral function for the Ising version of the exchange Hamiltonian (4.57), at J = 0.1. Again, note that we will be plotting tanh [A(k, ω)], as described in the beginning of the previous chapter. Figure 7.3 presents the spectral function density maps obtained at diﬀerent levels of the variational expansion up to ﬁve bosons, together with the scba results in the last panel. The blue curve indicates the free electron

89 8: KCuF3 Planes: Purely Orbital Systems

Figure 7.3: The spectral functions for the Ising limit of the orbital model, obtained with the variational approximation and with scba at J = 0.1. The dashed blue curve indicates the free electron dispersion. The numbers in the upper-right corner indicate the size of the variational space.

3 dispersion Ek = k + 2 J, where k is the kinetic dispersion given by (4.56a) 3 and 2 J is the constant energy cost associated with the addition of the electron to the system. The spectrum consists of the ground state qp band in the low energy region, and a broad, low weight continuum above it, separated with a wide gap. The continuum itself shows the familiar ladder spectrum, with the number of states increasing with the expansion order. The variational calculation seemingly produces much better deﬁned excited states here, although the ladder can be clearly distinguished also in scba before the higher states dissolve into the incoherent continuum. Again, the slight diﬀerences in the continua between the methods are due to the not fully converged variational calculation. Let us turn our attention to the qp ground state. A clear evolution can be seen with the increasing expansion order. As the number of orbitons increases, the qp state lowers in energy and the dispersion becomes smaller, until around 6–7 bosons the result converges to a stable solution. This is all consistent with typical polaronic behaviour—the bigger the boson cloud the more stable the polaron is, but at the same time the heavier it becomes. Another important 7.3. Convergence of the Ising Model 8;

Figure 7.4: The qp ground state energy E(k) and spectral weight Z(k) in the Ising limit of the orbital model for J = 0.1.

feature is the transfer of the spectral weight at the M point from the qp state to the continuum, which leads to a sharp drop of spectral weight nearly to zero. Furthermore, the shape of the band changes dramatically. The free electron dispersion is exactly antisymmetric, with Γ = −M . However, under the interaction with the orbiton cloud the band becomes visibly asymmetric, with the shape at the M point becoming ﬂatter than at Γ, which indicates a heavier qp there than at the energy minimum. This is associated with the description in the extended Brillouin zone which is employed throughout this work, i.e., we are using the full structural zone, which is twice as large in every direction as the ao zone. In other words, the spectral function should be folded at the ao ordering vector Q = (π, π), which leads to the minimum at Γ being repeated at M, and vice versa. It is interesting though that those features were not present in the spin systems of the previous chapter. This can be traced to the fact that the t-J model has no free propagation connecting the two spin sublattices, a process forbidden by the spin conservation (or, equivalently, by the full SU(2) symmetry of spin 1/2). In our present case the orbital pseudospin of the particle does not have to be conserved, which means the electron is free to move around the system without generating any excitations—structurally the free electron does not feel the presence of the orbitally ordered state. As we saw in the previous paragraphs, our two methods seem to agree quite well, both with respect to the ground state as well as the general features of the ladder spectrum, such as the level separations. To plot the details of the energy dispersion and spectral weight of the qp state, we ﬁtted a Lorentzian peak to the lowest lying band of the spectral function. Figure 7.4 presents those results, demonstrating both the convergence of the variational method and the close comparison to scba. As can be seen, the methods disagree slightly around 92 KCuF3 Planes: Purely Orbital Systems

Figure 7.5: The spectral functions for the Ising limit of the orbital model, obtained with the variational approximation and with scba at J = 0.5. The dashed blue curve indicates the free electron dispersion. The numbers in the upper-right corner indicate the size of the variational space.

the M point, with the scba underestimating the qp bandwidth. However, the agreement around the energy minimum at Γ is near perfect. Lastly, a shallow minimum can be found in the variational solution around the S point due to Trugman processes, whereas the scba solution is ﬂat on the X–Y line. A diﬀerent way of looking at the convergence is through the cost of boson creation in the system. In our case the cost is given by the Ising energy and consequently for small J the boson cloud can grow substantially. Therefore, the electron becomes strongly localized by the boson cloud, which results in a heavy qp, which manifests itself as a big renormalization of the bandwidth. On the other hand, this means that the variational convergence will progress more slowly, as the polaron needs a big cloud to stabilize. However, the prob- ability that the electron will remove bosons in the exact reverse order of their creation (the basic assumption of scba) diminishes with the size of the cloud. Furthermore, the more bosons are present in the system the more sites are forbidden to the electron. Because of this it should be reasonably expected that in the small J limit scba could become very inaccurate. On the other hand, for large J one should expect a much faster convergence of the variational 7.3. Convergence of the Ising Model 93

Figure 7.6: The qp ground state energy E(k) and spectral weight Z(k) in the Ising limit of the orbital model for J = 0.5. method, a smaller renormalization of the qp bandwidth and a better agreement between the two methods, presumably due to more accurate behaviour of the scba. In ﬁgure 7.5 we demonstrate the convergence for a much bigger value J = 0.5. We see that the variational ground state converges almost immediately and at the 7-orbiton level even the excited spectrum seems to be settled. The ground states are very similar in both methods, and the band is much less renormalized than in the small J = 0.1 case, in agreement with the expectation outlined above. Figure 7.6 presents the ground state energy and spectral weight obtained from the spectral functions. We see that the results agree quite well with respect to the dispersion relation, although scba overestimates the bandwidth slightly, and there is also an energy shift which seems to place the scba energy in the vicinity of the 3-orbiton solution. This suggests that scba might be underestimating the number of bosons needed to obtain the right qp state. The discrepancy between the results is even greater when we consider the spectral weight, for which the scba result is completely wrong. Again, the spectral weight for the global minimum at Γ is very close to the variational 3-orbiton solution, indicating that the number of bosons in the system is too small. To reassess the convergence of our methods we have calculated the ﬁrst four spectral moments for the scba and the variational 7-orbiton solutions and compared them to the analytical moments calculated from eq. (5.37):

3 M1(k) = k + 2 J, (7.3) 3 2 7 2 M2(k) = (k + 2 J) + 4 t , (7.4) 3 3 57 2 189 2 M3(k) = (k + 2 J) + 16 t k + 16 t J, (7.5) 94 KCuF3 Planes: Purely Orbital Systems

Figure 7.7: Comparison of the ﬁrst four spectral moments for scba and the variational method with the analytical expressions, for J = 0.1 (up) and J = 0.5 (down).

3 4 7 2 21 2 7 2 3 2 M4(k) = (k + 2 J) + 4 t (k + 4 J) + 2 t (k + 2 J) 1 2 21 217 4 (7.6) + 8 t (k + 4 J)k + 32 t .

Figure 7.7 presents the ﬁrst four spectral moments compared with the analytical expressions for J = 0.1 and J = 0.5. We can see that the agreement with the variational solution is very good at small J, while the scba result is slightly oﬀ, especially for the even-valued moments. However, at large J the gap between the numerical and the analytical moments widens. Nonetheless, the general agreement seems quite good, with the variational method being slightly closer to the expected result. Note, however, that while agreement with sum rules is obviously desirable, nevertheless one has to be careful when assessing how meaningful such an agreement really is.

7.4 The Role of Variational Constraints

At this point it seems appropriate to use the example of the orbital model to illustrate the role of the variational constraints, i.e., the corrections introduced into the free propagators in equation (5.69), reflecting the fact that the electron and the orbiton cannot occupy the same site. We will also look at the importance of the cross-diagrams, which are not included in scba. In order to do this, in fig. 7.8 we plot the reference spectral function for the second order variational expansion (a), and three plots of the difference between the constraint-free results and the reference spectral function, panels (b)–(d). Note that here we use a different color scale than usual: shades of gray for the 7.4. The Role of Variational Constraints 95

Figure 7.8: Second order spectral functions for J = 0.1: (a) the full function, including cross-diagrams and constraints, (b) difference for the case without cross-diagrams, (c) difference for the case without constraints, (d) difference for the case without both effects. The dashed green line indicates the free 3 electron dispersion Ek = k + 2 J for reference. The colorbar refers only to panels (b)–(d). spectral function, and a red-white-blue scheme for the difference functions, with red indicating positive values and blue the negative ones (a colorbar is added for reference). As before, we plot the spectral functions using a nonlinear tanh-scale. Panel (b) shows the difference function for the case with the cross-diagrams removed, but with the translational constraints included. Somewhat unex- pectedly, we see that in this lowest order of expansion their role is almost negligible, with the maximal change in amplitude at around 7%. The qualitative change in the spectrum is also very small, with only a tiny transfer of spectral weight and small reduction of the distance between the lowest and highest state, indicated by pairs of red-blue areas which show that under the influence of the cross-terms the reference spectrum of (a) moves away from the blue region and towards the red region. It can be reasonably expected 96 KCuF3 Planes: Purely Orbital Systems

that the role of the cross-terms becomes slightly more important at higher orders of expansion, where their number relative to the non-crossing terms becomes much bigger. However, due to the increasingly unclear picture of what constitutes a cross-term in real space when the electron neighbours more than one boson, it is in general not an easy task to construct a variational expansion with a convincing scheme of exclusion of the cross-terms. On the other hand, the effects of the translational constraints are very strong. Panel (c) illustrates that neglecting to account for the these constraints causes the whole spectrum to shift upwards, especially at the Γ and M points, although the effect is much stronger at higher energies than for the ground state, with the shift at the highest band around 0.13t and in the ground state only 0.06t. In the incoherent part of the spectrum the effect is also very strong, although qualitatively quite complicated. Excluding the translational constraints seems to decrease the broadening of the features of the continuum, while at the same time tends to split them into additional subbands. This produces the effect that although the continuum becomes slightly more coherent, it appears even less so because the various resulting states blend together. In principle this picture is consistent with the results obtained within scba, which produces a broad incoherent continuum with a barely visible ladder of low amplitude bands. However, the effect cannot account for the whole difference between the methods, although a higher order expansion might produce a still better agreement, in particular because the more bosons there are in the system, the more sites are forbidden to the electron. Finally, panel (d) shows the combined effect of the translational constraints and the cross-terms. Since the two effects are independent it is no surprise that their combined effect is not much different from a simple sum of the effects treated separately. In particular, since in this case the cross-diagram terms have such a small effect, it is not unexpected that the results in panel (d) do not differ much from those in panel (c). A close inspection might reveal that some of the features are slightly more pronounced, especially in the ground state where the two effects combine positively in decreasing the band energy.

7.5 Signiﬁcance of Orbital Fluctuations

Now we turn our attention to the full orbital model, including fluctuations, to determine their importance in the problem. Figure 7.9 presents the convergence of the variational method up to the 7-orbiton order, compared to the scba result, for the model with fluctuations. As before, we present the spectral function density maps using a nonlinear scale. As can easily be observed, the spectral functions are barely affected by the inclusion of fluctuations. In fact, the excited part of the spectrum changes so little it can barely be noticed. The biggest difference is in the ground state, although it is still very small. To get a better idea of the influence of 7.5. Significance of Orbital Fluctuations 97

Figure 7.9: The spectral functions for the orbital model with ﬂuctuations, obtained with the variational approximation and with scba at J = 0.1. The dashed blue curve indicates the free electron dispersion. The numbers in the upper-right corner indicate the size of the variational space.

fluctuations on the ground state, we again look at the qp band extracted from the spectral function by fitting a Lorentzian to the lowest energy peak. The results are shown in figure 7.:. What this plot demonstrates is that the changes observed in the variational calculation are negligible compared to the Ising limit of the model, and they mostly involve a tiny gain in energy at the Γ point and the slight deepening of the shallow minimum at S on the X–Y line (which in the Ising limit is the result of the Trugman processes). At the same time, scba closely resembles the variational result, providing an independent positive verification of the method, although here the effects of fluctuations are slightly more pronounced, with the qp dispersion visibly increased with respect to the Ising case. Interestingly, in contrast to our previous results for the t-J model, here the variational spectral function has a minimum at S (although only a local minimum, due to the free electron dispersion), while in scba we get a flat dispersion in the same part of the Brillouin zone. To understand the reason behind such a remarkable agreement between the two methods note that the fact of such tiny difference between the full model 98 KCuF3 Planes: Purely Orbital Systems

Figure 7.:: The qp ground state energy E(k) and spectral weight Z(k) in the orbital model with ﬂuctuations for J = 0.1.

and its Ising limit points to the irrelevance of fluctuations for the orbital waves. In other words, orbitons are to a great degree classical particles, and so there is not much difference between the resulting spectral functions. The reason for this is that the magnitude of the fluctuations barely equals 1/3 of the Ising term amplitude, and similarly the orbiton Bogoliubov dispersion (5.54) has a bandwidth even smaller than 1/3 of its maximal value. With these observations we can conclude that the orbiton fluctuations are of little importance in this problem, which gives reason to believe that also in the full 5d case the fluctuations can be neglected and an Ising limit of the model will be a good approximation of the full Hamiltonian. On the other hand, we see that the differences in the treatment of the fluctuations again result in slight discrepancies, which however in this case are minor.

7.6 Conclusions

In this chapter we have examined the purely orbitonic model, inspired by the copper-ﬂuoride planes found in KCuF3, which is the 4d limit of the full spin- orbital Hamiltonian. As such it provides us with insights into the full 5d model, which we will address in the next chapter. We have seen that the variational method converges very rapidly, so that the results are reliable already around the 6-orbiton mark, and that the renormalization of the qp band resulting from the interaction with orbitons is quite substantial. It splits the electron band into a ladder of states and causes a spectral weight transfer, depleting the spectral weight of the band maximum at the M point so that it almost disappears. Furthermore, by comparing the calculations with scba we have corroborated our results and demonstrated the reliability of the variational calculation for this kind of problems. 7.6. Conclusions 99

Next, we have established that at least part of the difference between the two methods can be explained by the simplifications employed in scba, which can be fairly easily avoided in the variational method. Finally, we have seen that the role of fluctuations for the eg orbitons is negligibly small, to the point where the results for the model with fluctuations are barely distinguishable from the Ising limit of the problem. Therefore, in principle the fluctuations can be neglected for orbital models, especially if our aim is not a quantitative calculation, but rather a demonstration of qualitative effects. Furthermore, we believe that it was conclusively established that the variational method is a perfectly reliable approach to the polaronic problems considered in this work. Therefore, we believe that fluctuations can be ignored in the calculations for the full 5d model of the next chapter, and that we can focus on the variational calculations of the spectral functions.

CHAPTER 8 Spin-Orbital Polarons in KCuF3

A man eats an apple. He gets a piece of the core stuck between his teeth, you know? He tries to work it out with some cellophane oﬀ a cigarette pack. What happens? The cellophane gets stuck in there too.

Burt Lancaster as Steve Thompson in Robert Siodmak’s Criss Cross (3;6;)

In this chapter we will examine the full 5d effective model of KCuF3, which was formulated in sec. 4.4. After having established the effectiveness of the variational method and the potential shortcomings of the scba, here we will only look at the variational results, since scba would need to employ unjustified approximations with respect to the polaronic interaction for a system with many bosonic flavours. Namely, the limitation of a three-operator interaction would exclude a plethora of higher order interactions coupling different bosonic Hilbert spaces, which are of vital importance to the full description of the system. We will investigate the convergence of the variational result in different partial variational spaces and compare them to the full expansion in order to demonstrate the roles played by magnons and orbitons in the system. By employing an analytical approach to obtain the self-energies at low order expansion we explain the differences between the properties of bosons and link them to the specific qualities of the ground state. When we consider the full spin-orbital model, the orbital-flop ground state of the purely orbitonic model of the previous chapter does not have to be the optimal one. In fact, it is still controversial which orbital states are the ones actually occupied in KCuF3. For this reason we will consider the two extreme

9; :2 Spin-Orbital Polarons in KCuF3

cases discussed in the introduction, quantified by the detuning angle φ, namely the orbital-flop state φ = 0 and the Kugel-Khomskii state φ = π/6 (defined by eq. (3.39) with θ = π/2 ± φ).

8.3 The Orbital-Flop State

As discussed in section 4.4, the purely orbitonic state composed of the orbitals |±i, corresponding to the detuning angle φ = 0, results from the orbital crystal field Hz suppressing the one already present in the superexchange Hamiltonian. This state, the orbital-flop state, can be therefore regarded as an orbital-field-free solution. Figure 8.3 presents the variational results for up to four bosons in the cloud. The four boson limit was used because of the impracticality of a higher order calculation. However, based on our results of the previous chapters, as well as the size of the variational space employed in these calculations, we expect that the four boson approximation is close enough to convergence for the higher expansion to be deemed unnecessary. The first five panels of the figure present the partial expansions in subspaces of different bosonic compositions (where (n, m) indicates that the maximal number of magnons allowed is n and the maximal number of orbitons is m) while the last one is the full 6-boson solution. This convention needs to be properly understood, in particular with respect to the size of the variational space. All the solutions presented allow at most 6 bosons in the polaronic cloud, but while the full solution makes no distinction of the bosonic flavours, the partial solutions additionally restrict the maximal number of magnons and orbitons separately, but such that their maximal total number does not exceed 6. Thus, the full solution includes all the states appearing in the partial solutions. Note that there can be some overlap between the partial variational spaces at lower order, for instance the states with 5 orbitons will appear both in the (0, 4) expansion as well as in the (1, 3) case. However, the full variational space is more than just a simple sum of the partial spaces, as one can readily verify by comparing the number of states in the different expansions. This is because there are certain states that formally belong to a particular magnon/orbiton composition but cannot be reached without first performing the expansion through states with a nonconforming composition. This is best explained with a simple example. Let us take the 3-orbiton expansion (0, 1), which includes all the states with up to one orbiton and no magnons. This includes all the states with the orbiton lying next to the electron within the ab plane (6 configurations), and in principle also the states where the orbiton is located above or below the electron in a different plane (4 configurations). However, as we know from the previous chapter, as well as from the discussion of the model presented in chapter 4, the electron moving out of plane would always generate magnons in the system. Therefore, 8.3. The Orbital-Flop State :3

Figure 8.3: The spectral functions for the Ising limit of the spin-orbital model, obtained with the variational approximation with up to 6 bosons at J = 0.1. The dashed blue curve indicates the free electron dispersion. The numbers in the upper-right corner indicate the size of the variational space. to reach one of the states with the out of plane orbitons, we have to first generate a magnon and then exchange it to an orbiton by acting again with the Hamiltonian (4.54c)∗, but this would violate the condition of a 3-orbiton ∗Which happens to real- expansion. This state cannot appear in the (0, 1) expansion, nor can it in the ize a process necessarily excluded in . (1, 0) case, but it can in the mixed (1, 1) expansion, which includes all the scba states in the previous two cases, as well as the ones which connect the two subspaces. As can be deduced from the number of states indicated in fig. 8.3, the number of such connecting states can be huge at higher expansion orders. Clearly, the spectra vary wildly between different variational subspaces, un- derlining the inherent distinctions between the orbiton and magnon properties. In all cases there is a broad continuum, roughly overlapping the free electron bandwidth. In the purely orbitonic (0, 4) case one can observe a ladder of qp states extending well below and above he incoherent continuum, in perfect agreement with the purely orbitonic 4d solution of the previous chapter, up to a constant shift of −J/2 resulting from the difference in the classical ground state energy. On the other hand, the purely magnonic case (4, 0) shows a pair of qp bands closely sandwiching the continuum, with a spectral weight :4 Spin-Orbital Polarons in KCuF3

Figure 8.4: The qp ground state energy E(k) and spectral weight Z(k) in the Ising limit of the spin-orbital model for J = 0.1.

transfer so large that the band practically disappears at certain regions of the Brillouin zone, giving the impression of qp pockets forming around the Γ and M points for the lowest and highest bands, respectively. The mixed solutions are interpolations between these two extremes, the orbitonic ladder slowly morphing with the continuum with increasing number of magnons, and the highest state gaining spectral weight at the expense of the ground state, until in the magnon rich case (3, 1) we get a narrow-spaced ladder of states nearly symmetric around the continuum, with a ground state shape starting to resemble the purely magnonic case. An interesting effect can be observed when the evolution of the ground state under magnon doping is examined closely. The purely orbitonic case describes a qp resulting from the electron coupling to orbitons within the plane in which it can also propagate freely. One would naïvely expect that the addition of a magnon, which requires the electron to hop to another layer, would localize the qp or make it incoherent by scattering on the localised magnon, because once the magnon is bound to the cloud it pins the electron which has to come back in order to remove it from the system. This would result in a much heavier polaron. However, one can see that the mixed qp is nearly as mobile as the purely orbitonic 4d polaron we saw before. In order to elucidate the details of the ground state, in figure 8.4 we present the qp energy and spectral weight calculated from the lowest energy peaks of the spectral functions. This plot demonstrates a number of interesting features of the partial results. First, we see that the lowest energy among the partial results is attained by the orbiton rich solutions (1, 3) and (2, 2), whose energy is practically identical, with a small additional mass renormalization from the magnons in (2, 2). Generally the shape of the band changes systematically with magnon doping, with the dispersion getting flatter with more magnons. 8.4. Self-Energy Analysis :5

The effect is especially important around the M point, where the dispersion practically becomes flat for the purely magnonic solution, indicating a localised qp. This reflects the fact that the magnons localise the electron by means of a string like confinement—the electron can still propagate freely in-plane, as long as its momentum is small. However, looking at the full solution we see that it most resembles the orbiton rich solutions, i.e., the properties of the system are dominated by orbitons, with magnons providing slight renormalization by means of the extension of the variational states which create new avenues for the electron-boson interactions. On the other hand, we see that energy of the full system is still slightly lower than that of any of the partial results, indicating that the additional states connecting the subspaces actively work to lower the polaron’s energy, but do not change its dynamics substantially. It is therefore reasonable to assume that the resulting polaron cloud will consist mostly of orbitons, however the addition of a small amount of magnons is also necessary for the system to reach its ground state. Finally, the spectral weight Z(k) plot further substantiates the above claims, with the full solution having a very similar spectral weight as all three of the orbital rich solutions, while the magnon rich ones experience a much bigger variation of spectral weight, again pointing to the restriction of the electron momentum due to localisation.

8.4 Self-Energy Analysis

To elucidate the characteristics of magnons and orbitons we perform a self- energy analysis of one boson models, which are simple enough to facilit- ate an exact analytical solution. Figure 8.5 presents the spectral function (left) and the real and imaginary part of the self-energy dependent function Ω(ω) = ω − Σ(ω)† (right) obtained by analytical solution of the corresponding †The plots actually eom system for (a) 3-magnon and (b) 3-orbiton problem. In these two cases the present the inverse re- k lation ω(Ω), to align self-energy happens to be -independent, which makes the analysis especially the Ω curve with the simple. Under the above notation, the Green’s function can be written as respective peaks of the −1 G(k, ω) = [Ω(ω + iη) − Ek] with the free electron dispersion Ek = k + J, spectral function. i.e., Ω serves as ω renormalized by the interaction with the boson. In this formalism a peak in the spectral function corresponds to two conditions being satisfied: <[Ω(ω + iη)] = Ek, =[Ω(ω + iη)] ≈ η, (8.3) where η is a small broadening factor (in all our calculations η = 0.05). In terms of the plot in fig. 8.5 this means that the real part of Ω has to cross the respective free electron energy (whose extreme values are marked by the black lines) at the given wavevector k, before the imaginary part of Ω grows large enough for the state to dissolve into the continuum due to the finite qp lifetime. We see that there is a fundamental difference between the Ω plots of the two cases :6 Spin-Orbital Polarons in KCuF3

Figure 8.5: The self-energy analysis of the 3-magnon (a) and 3-orbiton (b) analytical solutions of the spin-orbital model in the Ising limit at J = 0.1. Left: the spectral function, right: the inverse plot of Ω(ω), real part in blue, imaginary part in orange, dashed green line is the reference noninteracting system Ω = ω.

under consideration. The 3-magnon solution exhibits very small renormalization, so that in fact <[Ω(ω + iη)] ≈ ω, which means that the spectral function will behave similarly to the free electron case, with the total bandwidth slightly increased. However, =[Ω(ω + iη)] has a non-negligible value in the region of the electron band, so that the states there acquire a short lifetime. These two facts together produce the spectral function with pockets appearing where the qp band is stretched beyond the electron band limit, and the middle of the band is dissolved by the ﬁnite lifetime. On the other hand, the interaction with orbitons leads to strong renormalization, so that <[Ω(ω + iη)] crosses the highest electron energy (at the M point) before the =[Ω(ω + iη)] can grow substantially, leading to a fully developed qp band below the continuum. The same is repeated above the continuum, forming an excited qp band. To explain the diﬀerence between the results we examine the expressions for the self-energy for the two cases. The 3-magnon solution (1, 0) is given by

t 2 3 Σ(1,0)(ω) = 2( 2 ) G00(ω − 2 J), (8.4) 8.5. Weak Interaction Limit :7 while the 3-orbiton solution is

t 2 Σ(0,1)(ω) = 4( 2 ) [G00(ω − 4J) + 2G11(ω − 4J) + G20(ω − 4J)] √ 3t 2 + 4( 4 ) [G00(ω − 4J) − 2G11(ω − 4J) + G20(ω − 4J)], (8.5) where Gmn are real space Green’s functions for the in-plane propagation by the vector (m, n). The coefficient in front of the Green’s function in general can be expressed as zτ 2, where z is the number of directions in which the coupling to bosons occurs and τ is the respective coupling constant. The condition for the self-energy to cross the entire range of electron energies can now be restated as zτ 2 > t2. Clearly this condition cannot be met by the 3-magnon solution (∝ t2/2) while the 3-orbiton solution (effectively ∝ 7t2/4) greatly exceeds the requirement. The difference is therefore due to the interplay between the system dimensionality (in-plane vs. c direction hopping) and the specifics of the orbital physics, which does not conserve the orbital pseudospin, allowing for the free hopping in the alternating ordered state. On the other hand, there is also a secondary effect related to the magnitude of J, which causes a shift of the incoherent continuum to higher energies, facilitating the creation of the qp. Since orbitons are generally more costly, they will be significantly more affected by this effect than magnons. The above discussion also explains why magnons have such a small effect on the polaron dispersion. Because of the dimensional effects, the low order mixed boson diagrams are non-crossed, and thus the leading contributions to the self-energy are a simple sum of separate magnon and orbiton contributions. Clearly, the much smaller amplitude of the magnonic self-energy is going to make little difference when added to the dominating orbitonic self-energy.

8.5 Weak Interaction Limit

It is perhaps instructive to look at the results for a much larger value of the superexchange constant, which constitutes the weak interaction limit of the model. Figure 8.6 presents the partial and full results for up to four bosons at J = 0.5. Interestingly, there are significant differences as compared to the small J case. For instance, one can notice that in contrast to the previous results, here the orbiton rich spectra differ quite a lot from one another. In particular, the qp band noticeably changes under magnon doping, with the effective mass growing steadily and the spectral weight declining at the M point until the band almost disappears in that region of the Brillouin zone. Furthermore, although the general appearance of the full solution seems to resemble the orbiton rich solution (1, 3), the qp band most closely follows the magnon rich case (3, 1), rather than the orbiton rich ones as was the case in the small J limit. Interestingly, the purely magnonic solution (4, 0) is barely affected and looks very similar to the J = 0.1 case discussed before. :8 Spin-Orbital Polarons in KCuF3

Figure 8.6: The spectral functions for the Ising limit of the spin-orbital model, obtained with the variational approximation with up to 6 bosons at J = 0.5. The dashed blue curve indicates the free electron dispersion. The numbers in the upper-right corner indicate the size of the variational space.

As usual, in order to be able to take a closer look at the qp band we extract the energy and spectral weight by fitting a Lorentzian curve to the lowest peak of the spectrum. The result is presented in figure 8.7; the incomplete purely magnonic curve (4, 0) is due to the peak being to small to allow a reliable fit to the data. We immediately notice that the full solution closely resembles the magnon rich case (3, 1), although it has a slightly lower energy, which is to be expected since a bigger variational space always produces an energy gain. Therefore, the dynamical properties indicate that in the large J limit the polaron characteristics are dominated by magnons. The spectral weight Z(k) for the full solution further corroborates this claim, as it lies somewhere in between the (3, 1) and (2, 2) solutions. The observations made above can be easily explained if we recall that the cost of magnons is much smaller than the cost of orbitons. If the value of J is small, then the energy difference between the bosonic flavours can be neglected and the dimensional effects (the fact that there are more ways for the electron to couple to orbitons than to magnons) play the dominant role. However, when the value of J becomes large, creating a lot of costly orbitons is energetically 8.6. The Kugel-Khomskii State :9

Figure 8.7: The qp ground state energy E(k) and spectral weight Z(k) in the Ising limit of the spin-orbital model for J = 0.5. The (4, 0) curves are incomplete due to the peak amplitude being to low to ﬁnd a satisfying ﬁt.

unfavourable, so the cloud will contain substantially more magnons. Thus, J induces a crossover eﬀect between an orbital polaron and a spin polaron. On the other hand, larger J means that generating many bosons costs a lot of energy, and so the polaronic cloud will be smaller, mostly at the expense of orbitons. Therefore it is not so much the proportion of magnons to orbitons that changes with J as it is the upper bound on the number of dynamically functional orbitons in the system.

8.6 The Kugel-Khomskii State

Now we turn our attention to the other extreme on the orbital order spectrum, the Kugel-Khomskii (kk) state, defined by φ = π/6. Like with the |±i basis that was our focus before, so does this state have a simple interpretation in terms of orbital crystal field, namely it corresponds to no external crystal field, Hz = 0, so that the superexchange itself induces an orbital field driving the system away from the orbital-flop phase. Figure 8.8 presents the variational results with up to four bosons in the polaronic cloud, with the electron ground state taken to be the Kugel-Khomskii state. As evidenced by the nearly dispersionless spectra, in this case the bare electron is immobile, which is consistent with the free hopping (4.52a) proportional to 1 − 2 sin φ, which for φ = π/6 is equal to zero. This fact can be understood in simple geometric terms: since the kk state is composed of alternating y2 − z2 and z2 − x2 orbitals, free hopping would involve a transfer of electron across a lobe of one of the orbitals pointing towards the nodal point at the centre of the neighbouring orbital. Due to the orbital phase symmetry, :: Spin-Orbital Polarons in KCuF3

Figure 8.8: The spectral functions for the Ising limit of the spin-orbital model in the kk state, obtained using the variational approximation with up to 6 bosons at J = 0.1. The numbers in the upper-right corner indicate the size of the variational space.

the contributions from the transverse directions cancel out and the hopping is ‡In terms of superex- forbidden.‡ change the hopping A number of interesting features can be noticed in the spectra. First of all, is mediated by the F(2p) state connect- there is very little dispersion present in the system, and it is only present in ing the sites, whose the orbiton rich solutions. This obviously comes from the Trugman processes, symmetry further sub- which need at least three orbitons present in the plane with the electron. stantiates this claim. Interestingly, the addition of a magnon produces a big energy gain, at the same time suppressing the qp dispersion. This suggests that the Trugman processes are not the dominating factor in the qp properties—a huge number of processes which do not contribute to the coherent propagation of the electron nonetheless are energetically favourable to the creation of the qp. This observation is retained in the full solution, where a further energy gain can be observed, and the Trugman dispersion is barely visible. On the other hand, looking at the amplitude of the spectral function, for instance in the (0, 4) solution, we see that the ground state has very small spectral weight and there often are solutions higher in energy which have a much bigger weight—in particular the state located slightly above ω = 0 in the (0, 4) solution coincides with the 8.7. Many Magnon Expansion :; energy of a purely electronic (localized) state, suggesting a nearly bare electron solution there. Actually, a close comparison of the spectra for the kk and the orbital- flop states suggests a reasonable agreement between the state spacing and, to a degree, also the spectral weight outside of the continuum, which is easiest to observe on the X-Y line where there is no dispersion. The core of the difference between the two cases is in fact due to the specifics of the variational method, which approximates the full Green’s function in terms of real space propagators. In particular, for the case of a dispersionless electron, the real space function is given by (5.5:), which has a trivial imaginary part ∝ η. In other words, for localised particles the variational method produces a nearly coherent spectrum, or perhaps more graphically, the incoherence is confined to the region of the free electron bandwidth, which in this case is zero. Nowhere is this more evident than in the purely magnonic solution (4, 0), which for the orbital-flop phase is composed of small arcs pressed against the incoherent region in which we can see the hazy shadow of the free electron band, while for the kk state it consists of a number of well-defined, dispersionless levels.

8.7 Many Magnon Expansion

Due to the above-mentioned features of the purely magnonic solution, in this section we will look more closely at the intermediate orbital states, evenly distributed over φ ∈ [0, π/6], for a purely magnonic solution. Additionally, because of the 3d nature of magnons, it is feasible to perform the variational expansion up to a very large number of magnons in the system, in this case 322, which corresponds to the variational space spanned by 423 states. This number was chosen arbitrarily, i.e., it was not motivated by computer resources, however it seems to be high enough to guarantee convergence. Figure 8.9 presents the spectral functions for the six consecutive orbital ordered states φ = nπ/30, n = 0,..., 5. We see that with the growing detuning angle φ the bands gradually lose dispersion while the continuum in the middle develops into a fine ladder of states, corresponding to the standard string confinement picture. Interestingly, although for the orbiton-flop state φ = 0 the spectral function closely resembles the one for four magnons presented in sec. 8.3,§ the kk state seems to require a lot more magnons to reach a reason- §In fact it does not able convergence. This is a direct manifestation of the orbital and magnetic change substantially even for fewer magnons. order being strongly intertwined and suggests that in the kk state the classical 3d Néel order is not a good approximation of the real magnetic order. Another way of looking at this issue is through the magnon energy. The Ising cost of a 3d string containing n magnons can be calculated using eqs (4.48) and reads: J 2 2 En(φ) = [3 − 5 sin φ − n(1 + 4 sin φ)], (8.6) 2 ;2 Spin-Orbital Polarons in KCuF3

Figure 8.9: The spectral functions for the Ising limit of the spin-orbital model in the 322-magnon expansion for diﬀerent orbital ordered states φ = nπ/30, J = 0.1.

i.e., the string energy actually falls with the addition of new magnons, and at some point it can become negative, indicating that the creation of additional magnons is favourable. The critical number of bosons, at which the string energy En ≤ 0, is the biggest for the orbital-ﬂop state (E3 = 0) and becomes 9 smaller with increasing φ; for the kk state we have E2 = − 8 J. This produces the eﬀect of an inverted spectrum, with a lot of spectral weight in the high energy region at the top, and a highly magnonic ground state and is an indicator that our magnetic ground state is not in fact a ferromagnet in the ab plane.

8.8 The Hund Exchange

The behaviour highlighted in the previous section stems from the simpliﬁc- ations implemented in our model, in particular the negligence of the Hund exchange, introduced with the parameter η = JH /U in eq. (4.42). It can be showed that for η = 0 the actual ground state is antiferromagnetic in all three cubic directions. In this section we will explore this idea within the context of our formalism. To this end we ﬁrst have to determine a reasonable 8.8. The Hund Exchange ;3

Figure 8.:: The spectral functions for the Ising limit of the spin-orbital model in the orbital-flop state, obtained using the variational approximation with up to 6 bosons at J = 0.1 and η = 0.12. The numbers in the upper-right corner indicate the size of the variational space. choice of the value of the Hund parameter η. Based on estimates found in the literature, derived from lda+u calculations of the microscopic parameters of the model, the value η = 0.12 is believed to correctly reflect the partly screened interactions within CuF6 octahedra [:4], which is what we will be using for the numerical results in this section. After repeating the calculations with the Hund exchange included we note that in the strong coupling regime J = 0.1 the spectra, shown in figure 8.:, are surprisingly similar to the ones presented earlier. This is probably due to the fact that, as noted before, in this limit the qp is dominated by the orbitonic effects, while the Hund interaction primarily affects the magnetic interaction. Its main effect on the orbital interaction is a fairly simple renormalization of the superexchange constant J (increase by more than 82%, depending on the orbital ordering involved), so the spectra simply resemble the ones from before with a slightly bigger value of J. However, in the weak coupling case J = 0.5, the increase of the energy scale tips the balance to the advantage of magnons, and so the magnetic effects will be more pronounced. Thus, it will be more instructive to discuss these results instead. ;4 Spin-Orbital Polarons in KCuF3

Figure 8.;: The spectral functions for the Ising limit of the spin-orbital model in the orbital-ﬂop state, obtained using the variational approximation with up to 6 bosons at J = 0.5 and η = 0.12. The numbers in the upper-right corner indicate the size of the variational space.

Figure 8.; presents the variational calculation for J = 0.5 and η = 0.12. The first thing that we need to notice is the increased free electron energy, marked by the dashed blue curve. The energy shift for a single electron added to the system with a nonzero Hund exchange is: 1 − 2η − η2 Ee(η) = J , (8.7) (1 − η2)(1 − 3η) which for the assumed value of η equates to 1.94J, nearly twice as much as for η = 0. Clearly the effects of the Hund exchange are quite significant. Along with the increase in energy for orbitons, discussed before, it is no surprise that the purely orbitonic solution has a much higher energy and bigger dispersion than before—the convergence is even more rapid than for η = 0 and so the qp corresponds to a state with very few orbitons. Overall, we see a similar behaviour with increasing magnon number, the band gradually morphing into the purely magnonic case. However, this time the solution without orbitons is qualitatively quite different, with a clearly visible ground state, separated with a gap from the continuum, and the excited state at M incorporated into 8.8. The Hund Exchange ;5

Figure 8.32: The spectral functions for the Ising limit of the spin-orbital model in the 322-magnon expansion for different orbital ordered states φ = nπ/30, J = 0.5, η = 0.12. the continuum. This is exactly due to the fact that this time the assumed a-af magnetic order is indeed the ground state of the system. Furthermore, the magnon rich solution (3, 1) now has a clearly developed ladder of states above the ground state, where in the Hund-free case we only had a single level mixing with the ground state, immediately followed by the continuum. This is again associated with the reversed energy spectrum of magnons pushing the continuum to higher energies. The full variational space expansion is again quite different from the Hund-free case, although the general tendency is very similar, i.e., the full solution closely follows the (3, 1) case, with a small gain in energy resulting from a bigger variational space. Thus, we see that most of the difference between the Hund-free and Hund-enabled solutions is the result of the character of magnons and the fact that only now are we performing an expansion around the real magnetic ground state. Having said that though, since for the Ising Hamiltonian the assumed magnetic state is always an eigenstate, the previous calculation was also consistent, only not performed with respect to the ground state. Lastly, it is interesting to see the effect of the orbital order on the many magnon solution, as we did before, but for finite Hund interaction. Figure 8.32 ;6 Spin-Orbital Polarons in KCuF3

presents the purely magnonic solutions for the variational expansion up to 322 magnons, with parameters J = 0.5 and η = 0.12. We can notice that in this case the behaviour is completely diﬀerent from what we observed in the Hund-free case. This time we have a well-developed qp ground state separated with a gap from the ladder-like continuum above and this state persists with the growing detuning angle φ. However, around φ = π/10 a qualitative change in the spectrum occurs. The ground state connects to the incoherent region and the ladder of states develops into a proper continuum lacking any structure. Then, at even bigger detuning angle the spectrum reverses and the state with the highest spectral weight becomes the high energy one. This again indicates that the assumed magnetic state is no longer a good approximation of the ground state. We can ﬁnd the magnon string energy by calculating the in-plane Ising cost of the creation of a single magnon using the Hund-enabled version of eqs (4.48). If we now examine this energy as a function of φ and η:

" J 3(1 + η) + (η − 5) sin2 φ En(φ, η) = 2 (1 + η)(1 − 3η) # 1 − 10η − 7η2 + 4(1 − 2η + η2) sin2 φ −n , (8.8) (1 − η2)(1 − 3η)

we see that, apart from the constant energy which in our regime of parameters is always positive, the term dependent on the string length n is always negative for η = 0, but it can change sign for some ﬁnite value of η. If φ = 0 this critical value is well below our value of η = 0.12. However, if φ is nonzero it can dominate the energy and change the sign back to negative—in our case this happens for φ just slightly greater than π/10. Thus we see that, indeed, the Hund interaction is crucial to the system actually attaining the ground state that is believed to be the one found in nature, although for the value η = 0.12 some orbital orderings, in particular the kk state, still preclude the assumed magnetic order. Therefore, at least within the superexchange model, the kk state cannot be the orbital ground state of the system. Possibly other eﬀects, such as coupling to the lattice through the Jahn-Teller interaction, would have to be invoked in order to allow this kind of state to become the ground state. The above also demonstrates that the orbital and magnetic orders are highly intertwined and cannot be treated separately in the spin-orbital models of strongly correlated systems.

8.9 Conclusions

We have performed variational calculations for up to four bosons for the two primary orbital orderings in KCuF3, the orbital-flop state φ = 0 and the Kugel-Khomskii state φ = π/6. We have shown that an electron injected into the orbital-flop phase produces a polaronic state by coupling to the orbital and 8.9. Conclusions ;7 magnetic order, and its properties are dominated by the interactions with the orbitons due to dimensional effects, i.e., the 4d nature of the coupling to the orbital excitations vs. the 3d character of the magnetic coupling. However, in the weak coupling regime, when the exchange constant is large, the magnetic excitations start to play an increasingly important role due to their lower energy cost. Thus, J controls a crossover effect between an orbital polaron and a spin polaron. Secondly, in the kk state we get a nearly dispersionless spectrum, apart from a small dispersion generated by the Trugman effect. However, the qp state separation is to a large extent similar to the one found in the spin-flop phase, with the major difference being the lack of the incoherent region, which is a consequence of the bare electron localization. On the other hand, we have seen that in both of these solutions the cost of the magnons can become negative at high enough expansion order, indicating that the magnetic ground state is in fact different from the one assumed, which is due to the negligence of the Hund interaction. For this reason, we have also explored the role of this effect and demonstrated that a finite Hund interaction has a huge albeit trivial effect on the orbiton energy. Nonetheless, it allows for the attainment of the proper magnetic ground state, although it keeps the magnon energy small. However, Hund interaction is not enough to reach the magnetic ground state in the vicinity of the Kugel-Khomskii state, indicating that other effects would have to be at play if the system is to actually occupy this particular orbital phase. Overall, we see that the qp dispersion measured in a spectroscopic experiment such as arpes could provide insight into the nature of the orbital order in KCuF3. However, the models used herein are highly idealized and they can only be used to demonstrate certain effects. In order to produce results that could be related to experiments, more general models including the coupling to lattice through Jahn-Teller effect, as well as charge transfer terms, would have to be employed. Finally, we saw that there is a radical difference between a purely magnonic qp and a one involving orbital excitations. An experiment designed to couple electrons only to magnetic excitations should in principle be able to observe a spectrum both completely different from the optimal, orbiton rich solution, as well as one that is highly susceptible to the orbital order in the system.

CHAPTER 9 Summary

I never make up my mind about anything at all until it’s over and done with.

Orson Welles as Michael O’Hara in his production of Tha Lady from Shanghai (3;69)

In this work we have set out to accomplish three main goals:

• Generalize the Green’s function variational approximation to polaronic models involving slave bosons which represent elementary excitation in a polarized system.

• Corroborate results obtained within this formalism, by solving two well- known problems which can be formulated in terms of slave bosons, namely the t-J model and an eﬀective eg orbital planar model of KCuF3, and comparing them against a well established numerical method, namely the scba.

• Employ the variational method to solve an interacting single-electron problem in a spin-orbital model, which until now proved too challenging for other methods, in particular the scba.

Over the course of six chapters we have addressed these issues and provided conclusive answers to the problems outlined above. In chapter 3 we have presented a short review of the relevant information concerning the physical systems under consideration, namely the cuprates and the copper-ﬂuoride perovskite KCuF3. Next, in chapter 4 we have introduced the models used throughout this work to represent the essential physics of those systems: the t-J model (including the three site hopping) for cuprates and the

;9 ;: Summary

effective spin-orbital model for KCuF3. We then further transformed these models to obtain their polaronic form, useful in the context of the methods used herein. Finally, in chapter 5 we have introduced the concept of the single-electron Green’s function and discussed its most important properties. We then proceeded to discuss in detail our two basic research methods, the self-consistent Born approximation and the variational approximation, which is a primarily analytical tool for calculating Green’s functions on an infinite lattice. We compared the two methods and concluded that the variational method should in principle be more accurate than scba, owing to the fact that it employs almost no approximations except for the single criterion controlling the geometry of the boson cloud. Its only drawback is the treatment of the quantum fluctuations which are included only locally around the electron site, while the scba handles them exactly, although only within the linear waves formalism which neglects any terms other than the quadratic ones. For these reasons it was determined that it is not clear which method should be deemed more trustworthy in general, however the variational method clearly has advantages which make it better suited for problems involving many types of bosons, and in particular when the fluctuations are of minor importance. Next, in chapter 6 we proceeded to present the comparison between the two methods on the basis of the t-J model. We have seen that the convergence of the ground state in the variational method is quite rapid, a stable state being reached around 6–7 magnons, and a good agreement with scba can be observed. However, the excited part of the spectrum differs quite a lot between the two methods, for which there are multiple reasons, not the least important of which being the basic difference in methodology. However, there are signs that a far higher order of expansion would be needed to reach a reliable convergence in the high energy region. In total, four different limits of the t-J model have been considered: the Ising limit versus the Heisenberg limit of the exchange interaction, as well as the localized electron versus the three site hopping enabled model. We have demonstrated, through both methods, that the quantum fluctuations have by far the greatest effect on the qp dynamics, although the details of the band dispersion depend on the method used, pointing to the importance of proper inclusion of fluctuations in the solution. On this issue we have demonstrated that in 4d magnetic systems, where fluctuations are strong, inclusion of effective longer range magnon hopping might be necessary. Finally, we have seen that some more subtle effects involving mixing of the ground state with excitations in the spectral function (in particular the “waterfall” feature) are not well reproduced in the variational calculation, at least as far as they can be believed to be correctly predicted by scba. In order to gain some first insight into the eg orbital physics without diving straight into the full spin-orbital problem, in chapter 7 we have examined the simplified, purely orbitonic 4d model, which results from the magnon-free limit of the full superexchange model of KCuF3. Due to the assumed magnetic ground state, the free propagation of the electron is only allowed within the ab Summary ;; planes where the orbital order is alternating. Because of this all the crucial orbital physics is contained within this 4d model. At the same time, since this model involves only a single type of bosons, it is also feasible to solve it using scba to compare the results for the case of orbitons. We have seen that in this case the agreement between the methods is exceptional, even in the higher energy region of the spectral function, in both cases showing a ladder of states gradually dissolving into the continuum, with the ground state dispersion strongly suppressed with respect to the free electron dispersion, and with a substantial decay of the spectral weight at the M point. We have also demonstrated that at least part of the difference between the the two methods can be ascribed to the electron-boson translational constraints, which are always neglected in scba, suggesting that at least in the case of orbitons the variational method is more reliable. Furthermore, we have examined the role of the orbital fluctuations and saw that their impact on the orbital excitations is negligibly small, suggesting that in practice orbitons are classical particles. These findings therefore unequivocally suggest that the variational method is very well suited for the calculations involving orbital models. Finally, in chapter 8 we have employed the variational method in order to solve the full spin-orbital model of KCuF3. In this case we had to deal with a 5d system and the model under consideration was much more complicated than in the previous chapters, involving the electron propagating freely in the ab plane but also coupling both to magnons and orbitons through directional kinetic interaction terms, which include multiple boson interactions and even processes changing the bosonic flavour, while the superexchange Hamiltonian mixes the orbital and spin degrees of freedom. We have analysed the fourth order expansion in terms of partial results, imposing limits on the number of bosons within individual flavours, as well as the full expansion without distinguishing the flavours (which includes a large number of states not ob- tainable within the individually restricted variational subspaces). In this way we have demonstrated that the results obtained for a realistic magnitude of the superexchange interaction J = 0.1 are dominated by the orbital physics, i.e., the full solution most resembles the partial results for the orbiton rich subspaces, although clearly the presence of magnons plays an important role in the system reaching its ground state. As we have demonstrated through the self-energy analysis, this behaviour is mostly due to the dimensional effects associated with the in-plane coupling to orbitons versus a 3d coupling to magnons. On the other hand, for a much bigger value of J = 0.5 the character of the full solution evolves towards the magnon rich solutions, due to the rising difference in cost of creation between orbitons and magnons. Thus, it can be said that the magnitude of the superexchange constant J controls a crossover effect between an orbiton polaron and a spin polaron. Finally, we have examined the role of the nature of the orbital ground state by repeating the calculation for a Kugel-Khomskii orbital ground state. In this case, due to orbital symmetry, free propagation is forbidden and therefore the resulting qp is mostly disper- 322 Summary

sionless, save for the Trugman effect. However, comparing the regions of the Brillouin zone which are the least affected by the change in the orbital state, namely the X–Y line, we can conclude that the general distribution of spectral weight is very similar to the earlier discussed orbital-flop state. We have also examined the effect of the many magnon expansion, facilitated by their 3d character which precludes the fast growth of the variational space. Here we have seen a clear evolution controlled by the mixing of the orbital state, with very little effect in the orbital-flop state but growing with the mixing angle φ. This behaviour is a direct indication that the magnetic and orbital state of the system are closely intertwined. On the other hand, the inverted energy scale found in the many magnon expansion indicates that the assumed magnetic ground state is not a good approximation of the actual magnetic state within our present model. We have concluded that this is due to the negligence of the Hund interaction in our first model. Therefore, we have also explored a model with a finite Hund interaction and concluded that, indeed, this produces the right magnetic ground state, although only near the orbital-flop state φ = 0, while in the vicinity of the kk state the ground state is still wrong. We have put forward a speculation that in order for the kk state to be a valid option for the orbital ground state, additional effects would have to be considered, such as the Jahn-Teller effect. On a final note, we would like to remark that the work of a scientist, like ∗To paraphrase the that of the police, is never done.∗ Thus, it is always advisable to be aware of narrator from Alfred the shortcomings as well as the possible ways of improving one’s work. In the Werker’s highly under- rated 3;69 film noir case of the research presented herein the most obvious modes of improvement He Walked by Night. would involve the inclusion of the quantum fluctuations, in particular the spin fluctuations which we have seen to potentially have a big effect on the result. Furthermore, the addition of the three-site terms for the spin-orbital model might prove to be quite significant, due in part to opening the possibility of coherent charge propagation in all three spatial directions, which would necessitate the analysis of the 5d Brillouin zone and heavily complicate the numerical calculations due to the need for 5d real space propagators. Another possibility is the explicit treatment of the fluorine p states, as well as the extension of the superexchange model to include the charge-transfer interaction. Finally, the ultimate challenge for a theory is the confrontation with experiment, which would entail the performing of a photoemission experiment and relating the theoretical calculations to it by finding a satisfying set of model parameter values that would reproduce the experimental results. To recapitulate, we have successfully extended the variational method to spin and orbital polaronic models and independently verified it using the scba. Furthermore, we have used it to address a notoriously hard problem of a mixed spin-orbital polaron, which has never been solved before, the natural realization of which would occur in the copper-fluoride perovskite KCuF3 system. APPENDIX A Real Space Propagators

Here we will derive the analytical representation of the real space Green’s functions in terms of the elliptic integrals. We will loosely follow in the steps outlined by Morita [352, 354]. We start by rewriting the Schrödinger equation as a discreet difference equation for the Green’s function on a square lattice: t ωG(m, n) − [G(m + 1, n) + G(m − 1, n) + G(m, n + 1) + G(m, n − 1)] = 4 = δm0δn0, (A.3) where we introduced a simplified notation of the Green’s function as dependent only on the propagation vector (m, n). The solution of this equation is: π 1 Z cos mx cos ny G(m, n) = dxdy . (A.4) π2 ω − t (cos x + cos y) 0 2 On the other hand, the problem can be expressed in a rotated coordinate system x0 = (x + y)/2, y0 = (x − y)/2, (A.5a) m0 = m + n, n0 = m − n, (A.5b) in which the difference equation has a similar form as before t ωG0(m0, n0)− [G0(m0+1, n0)+G0(m0−1, n0)+G0(m0, n0+1)+G0(m0, n0−1)] = 4 = δm00δn00, (A.6) but its solution takes the form: π 1 Z cos m0x0 cos n0y0 G0(m0, n0) = dx0dy0 , (A.7) π2 ω − t cos x0 cos y0 0

323 324 Real Space Propagators

and the relation between the two solutions can be easily found:

G0(2m0, 2n0) = G(m0 + n0, m0 − n0), (A.8a) G(m, n) = G0(m + n, m − n). (A.8b)

Using these relations, and after performing the integration over y, we can express the diagonal Green’s function as

π Z 0 1 cos 2mx 1 G(m, m) = G (2m, 0) = dx√ = F2m(κ), (A.9) π ω2 − t2 cos2 x πω 0 where in the last equality we have introduced an auxiliary function, deﬁned in the following way: π Z cos nx Fn(κ) = dx√ , (A.:) 1 − κ2 cos2 x 0 where κ = t/ω is called the elliptic modulus. It is easy to write the ﬁrst two of the above functions in terms of elliptic integrals: 2 G(0, 0) = K(κ), (A.;) πω " # 2 2 − κ2 2 G(1, 1) = K(κ) − E(κ) , (A.32) πω κ2 κ2

where

π/2 π/2 Z 1 Z p K(κ) = dx√ ,E(κ) = dx 1 − κ2 sin2 x, (A.33) 1 − κ2 sin2 x 0 0 are the complete elliptic integrals of the ﬁrst and second kind, respectively. In order to ﬁnd the higher diagonal functions, we need to derive the recurrence relations for Fn(κ). To do that, we note that the integral

π Z q 2 2 In(κ) = dx cos nx (1 − κ cos x), (A.34) 0 can be written in two diﬀerent ways. First, by simple trigonometry, we can rewrite it as π Z cos nx(1 − κ2 cos x) In(κ) = dx p (1 − κ2 cos2 x) 0 κ2 2 − κ2 κ2 = − Fn (κ) + Fn(κ) − Fn− (κ). (A.35) 4 +2 2 4 2 Real Space Propagators 325

On the other hand, performing an integration by parts yields:

π 1 Z κ2 cos x sin x In(κ) = − dx sin nxp n (1 − κ2 cos2 x) 0 κ2 = − [Fn− (κ) − Fn (κ)]. (A.36) 4n 2 +2 Equating (A.35) and (A.36) leads us to the following recurrence relation:

2n 2 − κ2 n − 1 Fn (κ) = Fn(κ) − Fn− (κ), (A.37) +2 n + 1 κ2 n + 1 2 or, after applying Equation (A.9), the recurrence relation for the diagonal Green’s functions: 4m 2 − κ2 2m − 1 G(m + 1, m + 1) = G(m, m) − G(m − 1, m − 1). (A.38) 2m + 1 κ2 2m + 1 Since we have already demonstrated that the ﬁrst two diagonal Green’s functions can be calculated from elliptic integrals, using this relation we can now express any other diagonal function in the same way. Next, we can use the diﬀerence equation (A.3) to extend the recurrence relations to the whole lattice. Note that the lattice Green’s functions are symmetric with respect to the main and diagonal axes of the lattice:

G(m, n) = G(±m, ±n),G(m, n) = G(n, m), (A.39) which means we only need to derive the relations in a single one-eighth triangle of the lattice. We start with the sub-diagonal: 1 1 G(1, 0) = G(0, 0) − , (A.3:) κ t 2 G(m + 1, m) = G(m, m) − G(m, m − 1). (A.3;) κ Next we can proceed to the lower sub-diagonals, m > n > 0: 4 G(m + 1, n) = G(m, n) − G(m − 1, n) − G(m, n + 1) − G(m, n − 1), (A.42) κ and ﬁnally derive the relations on the main axes: 4 G(m + 1, 0) = G(m, 0) − G(m − 1, 0) − 2G(m, 1). (A.43) κ In this way we can express any single lattice Green’s function in terms of the three functions in the tip at the origin, or equivalently, solely in terms of the two elliptic integrals (A.33). 326 Real Space Propagators

Lastly, we need to consider the three site terms (4.38) of the t-J model. This second order kinetic process, when written in the notation used in this Appendix, takes the form:

π 1 Z cos mx cos ny G (m, n) = dxdy , (A.44) 3s π2 ω − ε(x, y) 0

where the electron dispersion ε(x, y) has to be transformed so that we can take advantage of our previous results: 1 1 = G (x, y) = (A.45) ω − ε(x, y) 3s ω − 4J[(cos x + cos y)2 − 1] 1 ≡ G¯ x, y , 3s( ) = 2 t 2 (A.46) ω¯ − [ 2 (cos x + cos y)] √ √ where ω¯ = ω − 4J and t = 4 J. Therefore, we can now decompose this function into fractions: " # 1 1 1 G¯ (x, y) = + , (A.47) 3s ω t t 2¯ ω¯ − 2 (cos x + cos y) ω¯ + 2 (cos x + cos y)

which can now be used in Equation (A.44) to yield:

1 G (m, n) = G¯(m, n)δm n, N , (A.48) 3s ω¯ + 2

where G¯ indicates that this function depends on ω¯ and t as defined earlier. The δ function is nonzero only if the transfer vector (m, n) connects sites on the same af sublattice, which reflects the fact that the three site hopping is a second order kinetic process. Finally, although the methods described in this appendix were the basis for our calculations, it should be noted that there are certain issues with this approach. Namely, it can be showed that using the recurrence relations to extrapolate the Green’s functions starting from the origin leads to the breakdown of the method for distances greater than a couple of lattice constants. The reason is that the general solution of the differential equation is a combination of two conjugate solutions, with opposite asymptotic behaviour at great distances. Obviously, the physical solution has to be convergent at infinity, reflecting the decreasing likelihood of the propagation with distance, which is ensured by choosing the Green’s function for n = 0 as the purely convergent solution, thus implicitly setting the combination coefficient for the divergent function to zero. However, because of numerical rounding errors introduced when integrating the first Green’s function, the amplitude of the divergent solution is never strictly equal to 0 and for greater distances its contribution starts to dominate the result [355]. For this reason a more robust and general scheme was recently Real Space Propagators 327 developed, based on a generalisation of the continued fractions method. In this approach the recurrence relations are generated similarly as before, however this time they are used to extrapolate the shorter distance propagators starting from the boundary condition at infinity limn→∞ G(n, n) = 0, which ensures that only the convergent solution is used in the expansion. This method has the additional benefit of not requiring any integration, instead being purely algebraic [356].

APPENDIX B SAGE

This Appendix is dedicated to sage, which is the computer system developed as part of this thesis to perform automatic expansion of the equations of motion for the Green’s function. Its name derives from the recursive acronym “Sage is an Analytical Green’s function Evaluator”. It is a Python script which implements a class to represent states and another one for equations, as well as a number of functions which operate on those states.

>>> from sage import * >>> s=State(2) >>> s [0 0] x [((0, 0), [’e’])]

The listing above demonstrates an interactive session in a Python shell. After importing all the deﬁnitions from the sage package to the main namespace, we create a 4d state and then print its text representation. As can be seen, by default the state is initialized to a single electron doped system. Next we can use the methods built into the State class to perform elementary operations, such as excite the system with and orbiton o and then relax it back into the ground state.

>>> s.excite((1,0),"o") [0 0] x [((0, 0), [’e’]), ((1, 0), [’o’])] >>> s.relax((1,0),"o") [0 0] x [((0, 0), [’e’])]

In each case the method returns the class instance, i.e., the modiﬁed state. The class also handles mathematical operations by overloading standard operators, in which case it either returns an equation or a constant.

329 32: SAGE

>>> t=s+s >>> t {[0 0] x [((0, 0), [’e’])]: 2} >>> s*s 1 >>> t*t 4

As evident, the Equation class resembles more a linear combination of states than an actual equation, which was a more convenient design choice. The State class also provides a number of other methods which we will not present here. The only other relevant ones are the methods for handling the phases arising from shifting the particles back to the origin.

>>> s.relax((0,0),"e").excite((1,0),"e") [0 0] x [((1, 0), [’e’])] >>> s.reduce() {[0 0] x [((0, 0), [’e’])]: exp(I*dot(k, (-1, 0)))}

First we move the electron to the neighbouring site and then we use the reduce method to shift it back to the origin. However, reduce also adds a phase factor dependent on the vector k, which is a symbolic function, deﬁned using the sympy package (automatically imported by sage to the namespace sp).

>>> s.kadd((sp.pi,sp.pi)) [pi pi] x [((0, 0), [’e’])] >>> s.relax((0,0),"e").excite((1,0),"e") [pi pi] x [((1, 0), [’e’])] >>> s.reduce() {[pi pi] x [((0, 0), [’e’])]: -exp(I*dot(k, (-1, 0)))}

Furthermore, there is the kadd method, which adds the ordering vector (mod- ulo 2π) to the state—in this case we have added the af ordering vector Q = (π, π). Now, moving the electron and reducing it to the origin as before produces a similar phase factor, but with the ordering vector added and simpliﬁed to an additional factor of −1. This whole machinery is then used to implement functions performing elementary physical processes on the states. For instance, the function ElHop performs the processes illustrated in the previous two listings.

def ElHop(state,site): point = state.locate("e")[0] s = state.copy() s.relax(point,"e") s.excite(site,"e") return s.reduce() SAGE 32;

With these functions we can now implement the relevant free propagator and the interaction for a given Hamiltonian. A dedicated function, eom, performs the expansion up to the required order, by recursively applying the propagator and the interaction to a given starting state (usually a free electron), and then to every new state arising in the expansion. It keeps track of which states have already been expanded and eliminates states which do not conform to a set of rules governing the shape of the boson cloud, continuing the expansion until the set of equations is closed. It then returns two functions which, for a given set of argument values {k, ω, t, J}, produce the eom matrix and the matrix of free coeﬃcients, which can be solved numerically to yield the values of all the Green’s functions; it also returns a list of all the states involved in the expansion.

>>> A,B,L = eom(s,2,propAB,intactAB,("k","omega","t","J")) >>> len(L) 34 >>> spsolve(A((3.0,1.7),0.3+0.05j,1,0.1), B((3.0,1.7),0.3+0.05j,1,0.1))[0] (-0.14786811089063845-0.25598089461711049j)

In the listing above, we have performed an expansion up to second order for the purely orbital model (propAB and intactAB are the propagator and interaction functions for that model). We then demonstrate that the expansion involves 56 states, and solve the eom system for k = (3.0, 1.7) and ω = 0.3 + i0.05 (which includes the broadening factor), yielding the value of the ﬁrst Green’s function, G(k, ω).

Bibliography

[3] N. Mott, Proc. Phys. Soc. A 84, 638 (3;6;).

[4] J. Hubbard, Proc. Phys. Soc. A 499, 459 (3;86).

[5] J. Hubbard, Proc. Phys. Soc. A 4:3, 623 (3;86).

[6] E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky, eds., Quantum Materials: Experiments and Theory, Lecture Notes of the Autumn School on Correlated Electrons, Vol. 8 (Forschungszentrum Jülich GmbH, Jülich, 4238) Chap. The Explicit Role of Anion States in High-Valence Metal Oxides.

[7] G. A. Sawatzky and J. W. Allen, Phys. Rev. Lett. 75, 455; (3;:6).

[8] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 77, 63: (3;:7).

[9] N. Barišića, M. K. Chana, Y. Lie, G. Yua, X. Zhaoa, M. Dresselb, A. Smontarac, and M. Grevena, PNAS 332, 34457 (4235).

[:] J. G. Bednorz and K. A. Müller, Zeitschrift für Physik B 86, 3:; (3;:8).

[;] P. Fulde, Electron correlations in molecules and solids, Springer series in solid-state sciences (Springer-Verlag, 3;;3).

[32] P. Anderson, Science 457, 33;8 (3;:9).

[33] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 9:, 39 (4228).

[34] M. Ogata and H. Fukuyama, Rep. Prog. Phys. 93, 258723 (422:).

[35] A. Fujimori, E. Takayama-Muromachi, Y. Uchida, and B. Okai, Phys. Rev. B 57, ::36 (3;:9).

[36] A. Bianconi, A. Congiu Castellano, M. De Santis, P. Rudolf, P. Lagarde, A. Flank, and A. Marcelli, Solid State Commun. 85, 322; (3;:9).

[37] A. Fujimori, Phys. Rev. B 5;, 9;5 (3;:;).

333 334 Bibliography

[38] H. Eskes, L. H. Tjeng, and G. A. Sawatzky, Phys. Rev. B 63, 4:: (3;;2).

[39] V. Emery, Nature 559, 528 (3;:;).

[3:] N. Nücker, J. Fink, J. C. Fuggle, P. J. Durham, and W. M. Temmerman, Phys. Rev. B 59, 737: (3;::).

[3;] P. Kuiper, G. Kruizinga, J. Ghijsen, M. Grioni, P. J. W. Weijs, F. M. F. de Groot, G. A. Sawatzky, H. Verweij, L. F. Feiner, and H. Petersen, Phys. Rev. B 5:, 86:5 (3;::).

[42] C. T. Chen, F. Sette, Y. Ma, M. S. Hybertsen, E. B. Stechel, W. M. C. Foulkes, M. Schulter, S.-W. Cheong, A. S. Cooper, L. W. Rupp, B. Bat- logg, Y. L. Soo, Z. H. Ming, A. Krol, and Y. H. Kao, Phys. Rev. Lett. 88, 326 (3;;3).

[43] N. Nücker, H. Romberg, X. X. Xi, J. Fink, B. Gegenheimer, and Z. X. Zhao, Phys. Rev. B 5;, 883; (3;:;).

[44] Y. Seino, K. Okada, and A. Kotani, J. Phys. Soc. Japan 7;, 35:6 (3;;2).

[45] V. J. Emery, Phys. Rev. Lett. 7:, 49;6 (3;:9).

[46] J. E. Hirsch, Phys. Rev. Lett. 7;, 44: (3;:9).

[47] M. S. Hybertsen, M. Schlüter, and N. E. Christensen, Phys. Rev. B 5;, ;24: (3;:;).

[48] A. K. McMahan, J. F. Annett, and R. M. Martin, Phys. Rev. B 64, 848: (3;;2).

[49] H. Matsukawa and H. Fukuyama, J. Phys. Soc. Japan 7;, 3945 (3;;2).

[4:] V. J. Emery and G. Reiter, Phys. Rev. B 5:, 6769 (3;::).

[4;] Y. Mizuno, T. Tohyama, and S. Maekawa, Phys. Rev. B 7:,R36935 (3;;:).

[52] K. B. Lyons, P. A. Fleury, L. F. Schneemeyer, and J. V. Waszczak, Phys. Rev. Lett. 82, 954 (3;::).

[53] K. B. Lyons, P. A. Fleury, J. P. Remeika, A. S. Cooper, and T. J. Negran, Phys. Rev. B 59, 4575 (3;::).

[54] Y. Tokura, S. Koshihara, T. Arima, H. Takagi, S. Ishibashi, T. Ido, and S. Uchida, Phys. Rev. B 63, 33879 (3;;2).

[55] J. Zaanen and A. M. Oleś, Phys. Rev. B 59, ;645 (3;::). Bibliography 335

[56] F. C. Zhang and T. M. Rice, Phys. Rev. B 59, 597; (3;::).

[57] A. Ramšak and P. Prelovšek, Phys. Rev. B 62, 445; (3;:;).

[58] R. Raimondi, J. H. Jeﬀerson, and L. F. Feiner, Phys. Rev. B 75, :996 (3;;8).

[59] H. Fukuyama, H. Matsukawa, and Y. Hasegawa, J. Phys. Soc. Japan 7:, 586 (3;:;).

[5:] H. Matsukawa and H. Fukuyama, J. Phys. Soc. Japan 7:, 4:67 (3;:;).

[5;] B. Lau, M. Berciu, and G. A. Sawatzky, Phys. Rev. Lett. 328, 258623 (4233).

[62] B. Lau, M. Berciu, and G. A. Sawatzky, Phys. Rev. B :6, 387324 (4233).

[63] H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, Nature Phys. 32, ;73 (4236).

[64] H. Eskes, G. Sawatzky, and L. Feiner, Physica C 382, 646 (3;:;).

[65] M. S. Hybertsen, E. B. Stechel, M. Schluter, and D. R. Jennison, Phys. Rev. B 63, 3328: (3;;2).

[66] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 97, 695 (4225).

[67] Q. Si, Y. Zha, K. Levin, and J. P. Lu, Phys. Rev. B 69, ;277 (3;;5).

[68] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Phys. Rev. Lett. :9, 269225 (4223).

[69] G. Shirane, Y. Endoh, R. J. Birgeneau, M. A. Kastner, Y. Hidaka, M. Oda, M. Suzuki, and T. Murakami, Phys. Rev. Lett. 7;, 3835 (3;:9).

[6:] D. Vaknin, S. K. Sinha, D. E. Moncton, D. C. Johnston, J. M. Newsam, C. R. Saﬁnya, and H. E. King, Phys. Rev. Lett. 7:, 4:24 (3;:9).

[6;] J. E. Hirsch and S. Tang, Phys. Rev. Lett. 84, 7;3 (3;:;).

[72] G. Fano, F. Ortolani, and A. Parola, Phys. Rev. B 68, 326: (3;;4).

[73] G. Dopf, A. Muramatsu, and W. Hanke, Phys. Rev. B 63, ;486 (3;;2).

[74] R. T. Scalettar, D. J. Scalapino, R. L. Sugar, and S. R. White, Phys. Rev. B 66, 992 (3;;3).

[75] W. F. Brinkman and T. M. Rice, Phys. Rev. B 4, 3546 (3;92). 336 Bibliography

[76] S. A. Trugman, Phys. Rev. B 59, 37;9 (3;::).

[77] B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 82, 962 (3;::).

[78] E. Dagotto, R. Joynt, A. Moreo, S. Bacci, and E. Gagliano, Phys. Rev. B 63, ;26; (3;;2).

[79] E. Dagotto, Rev. Mod. Phys. 88, 985 (3;;6).

[7:] S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 82, 49;5 (3;::).

[7;] B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 83, 689 (3;::).

[82] S. Sachdev, Phys. Rev. B 5;, 34454 (3;:;).

[83] E. Dagotto, A. Moreo, and T. Barnes, Phys. Rev. B 62, 8943 (3;:;).

[84] C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 5;, 8::2 (3;:;).

[85] F. Marsiglio, A. E. Ruckenstein, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 65, 32::4 (3;;3).

[86] G. Martínez and P. Horsch, Phys. Rev. B 66, 539 (3;;3).

[87] Z. Liu and E. Manousakis, Phys. Rev. B 66, 4636 (3;;3).

[88] D. Poilblanc, T. Ziman, H. J. Schulz, and E. Dagotto, Phys. Rev. B 69, 36489 (3;;5).

[89] E. Dagotto, A. Nazarenko, and M. Boninsegni, Phys. Rev. Lett. 95, 94: (3;;6).

[8:] J. R. Schrieﬀer, X.-G. Wen, and S.-C. Zhang, Phys. Rev. Lett. 82, ;66 (3;::).

[8;] E. Dagotto and J. R. Schrieﬀer, Phys. Rev. B 65, :927 (3;;3).

[92] D. Poilblanc, H. J. Schulz, and T. Ziman, Phys. Rev. B 69, 548: (3;;5).

[93] Z. Liu and E. Manousakis, Phys. Rev. B 67, 4647 (3;;4).

[94] J. Bała, A. M. Oleś, and J. Zaanen, Phys. Rev. B 74, 67;9 (3;;7).

[95] K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. A. Nekrasov, T. Pruschke, and D. Vollhardt, Nat. Phys. 5, 38: (4229).

[96] F. Ronning, K. M. Shen, N. P. Armitage, A. Damascelli, D. H. Lu, Z.-X. Shen, L. L. Miller, and C. Kim, Phys. Rev. B 93, 2;673: (4227). Bibliography 337

[97] J. Graf, G.-H. Gweon, K. McElroy, S. Y. Zhou, C. Jozwiak, E. Rotenberg, A. Bill, T. Sasagawa, H. Eisaki, S. Uchida, H. Takagi, D.-H. Lee, and A. Lanzara, Phys. Rev. Lett. ;:, 289226 (4229).

[98] A. Macridin, M. Jarrell, T. Maier, and D. J. Scalapino, Phys. Rev. Lett. ;;, 459223 (4229).

[99] M. M. Zemljič, P. Prelovšek, and T. Tohyama, Phys. Rev. Lett. 322, 258624 (422:).

[9:] Y. Wang, K. Wohlfeld, B. Moritz, C. J. Jia, M. van Veenendaal, K. Wu, C.-C. Chen, and T. P. Devereaux, Phys. Rev. B ;4, 29733; (4237).

[9;] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 92, 325; (3;;:).

[:2] Y. Tokura and N. Nagaosa, Science 4::, 684 (4222).

[:3] J. S. Griﬃth, The Theory of Transition Metal Ions (Cambridge University Press, Cambridge, 422;).

[:4] A. M. Oleś, G. Khaliullin, P. Horsch, and L. F. Feiner, Phys. Rev. B 94, 436653 (4227).

[:5] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 47, 453 (3;:4).

[:6] L. F. Feiner, A. M. Oleś, and J. Zaanen, Phys. Rev. Lett. 9:, 49;; (3;;9).

[:7] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. :7, 5;72 (4222).

[:8] P. Horsch, A. M. Oleś, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. 322, 389427 (422:).

[:9] L. F. Feiner and A. M. Oleś, Phys. Rev. B 93, 366644 (4227).

[::] A. M. Oleś, P. Horsch, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. ;8, 369427 (4228).

[:;] J. Bała, G. A. Sawatzky, A. M. Oleś, and A. Macridin, Phys. Rev. Lett. :9, 289426 (4223).

[;2] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, and J. van den Brink, Phys. Rev. Lett. 329, 369423 (4233).

[;3] L. F. Feiner and A. M. Oleś, Phys. Rev. B 7;, 54;7 (3;;;).

[;4] J. van den Brink, P. Horsch, and A. M. Oleś, Phys. Rev. Lett. :7, 7396 (4222). 338 Bibliography

[;5] M. Daghofer, A. M. Oleś, and W. von der Linden, Phys. Rev. B 92, 3:6652 (4226).

[;6] G. Khaliullin, P. Horsch, and A. M. Oleś, Phys. Rev. Lett. :8, 5:9; (4223).

[;7] K. Wohlfeld, A. M. Oleś, and P. Horsch, Phys. Rev. B 9;, 446655 (422;).

[;8] J. van den Brink, P. Horsch, F. Mack, and A. M. Oleś, Phys. Rev. B 7;, 89;7 (3;;;).

[;9] A. M. Oleś, L. Felix Feiner, and J. Zaanen, Phys. Rev. B 83, 8479 (4222).

[;:] G. Khaliullin, Phys. Rev. B 86, 434627 (4223).

[;;] A. Okazaki, J. Phys. Soc. Japan 48, :92 (3;8;); A. Okazaki, J. Phys. Soc. Japan 49, 73:B(3;8;).

[322] A. Okazaki and Y. Suemune, J. Phys. Soc. Japan 38, 398 (3;83).

[323] M. T. Hutchings, E. J. Samuelsen, G. Shirane, and K. Hirakawa, Phys. Rev. 3::, ;3; (3;8;).

[324] L. Paolasini, R. Caciuﬀo, A. Sollier, P. Ghigna, and M. Altarelli, Phys. Rev. Lett. ::, 328625 (4224).

[325] R. Caciuﬀo, L. Paolasini, A. Sollier, P. Ghigna, E. Pavarini, J. van den Brink, and M. Altarelli, Phys. Rev. B 87, 396647 (4224).

[326] J. Deisenhofer, I. Leonov, M. V. Eremin, C. Kant, P. Ghigna, F. Mayr, V. V. Iglamov, V. I. Anisimov, and D. van der Marel, Phys. Rev. Lett. 323, 379628 (422:).

[327] J. C. T. Lee, S. Yuan, S. Lal, Y. I. Joe, Y. Gan, S. Smadici, K. Finkelstein, Y. Feng, A. Rusydi, P. M. Goldbart, S. L. Cooper, and P. Abbamonte, Nature Phys. :, 85 (4234).

[328] S. Kadota, I. Yamada, S. Yoneyama, and K. Hirakawa, J. Phys. Soc. Japan 45, 973 (3;89).

[329] J. Kanamori, J. Phys. Chem. Solids 32, :9 (3;7;).

[32:] J. B. Goodenough, Magnetism and the Chemical Bond (Wiley, New York, 3;85).

[32;] S. K. Satija, J. D. Axe, G. Shirane, H. Yoshizawa, and K. Hirakawa, Phys. Rev. B 43, 4223 (3;:2). Bibliography 339

[332] B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler, Nature Mat. 6, 54; (4227).

[333] N. Binggeli and M. Altarelli, Phys. Rev. B 92, 2:7339 (4226).

[334] E. Pavarini, E. Koch, and A. I. Lichtenstein, Phys. Rev. Lett. 323, 488627 (422:).

[335] I. Leonov, D. Korotin, N. Binggeli, V. I. Anisimov, and D. Vollhardt, Phys. Rev. B :3, 29732; (4232).

[336] M. V. Mostovoy and D. I. Khomskii, Phys. Rev. Lett. ;4, 389423 (4226).

[337] K. A. Chao, J. Spałek, and A. M. Oleś, J. Phys. C 32,L493 (3;99).

[338] K. A. Chao, J. Spałek, and A. M. Oleś, Phys. Rev. B 3:, 5675 (3;9:).

[339] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 9;, 366638 (422;).

[33:] G. D. Mahan, Many-Particle Physics (Springer, 4222).

[33;] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover Books, 4225).

[342] W. Nolting, Z. Phys. B 477, 47 (3;94).

[343] M. Berciu, Phys. Rev. Lett. ;9, 258624 (4228).

[344] G. L. Goodvin, M. Berciu, and G. A. Sawatzky, Phys. Rev. B 96, 467326 (4228).

[345] M. Berciu and G. L. Goodvin, Phys. Rev. B 98, 38732; (4229).

[346] M. Berciu and H. Fehske, Phys. Rev. B :4, 2:7338 (4232).

[347] M. Berciu and H. Fehske, Phys. Rev. B :6, 387326 (4233).

[348] K. Bieniasz and A. M. Oleś, Phys. Rev. B ::, 337354 (4235).

[349] K. Bieniasz and A. Oleś, Acta Phys. Pol. A 348,A(4236).

[34:] K. Bieniasz, M. Berciu, M. Daghofer, and A. M. Oleś, Phys. Rev. B ;6, 2:7339 (4238).

[34;] K. Bieniasz, M. Berciu, and A. Oleś, Acta Phys. Pol. A 352, 87; (4238).

[352] T. Morita and T. Horiguchi, J. Phys. Soc. Japan 52, ;79 (3;93).

[353] H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, J. Phys. Condens. Matter 4:, 327825 (4238). 33: Bibliography

[354] T. Morita, J. Math. Phys. 34, 3966 (3;93).

[355] M. Berciu, J. Phys. A 64, 5;7429 (422;).

[356] M. Berciu and A. M. Cook, EPL ;4, 62225 (4232). List of Publications

[3] K. Bieniasz and A. M. Oleś, “Exact spectral function for hole-magnon coupling in a ferromagnetic CuO3-like chain,” Phys. Rev. B ::, 337354 (4235).

[4] K. Bieniasz and A. Oleś, “Quantum versus Classical Polarons in a Ferro- magnetic CuO3-Like Chain,” Acta Phys. Pol. A 348, A–:2–A–:6 (4236). [5] K. Bieniasz and A. Oleś, “Polaron States in a CuO Chain,” Acta Phys. Pol. A 349, 48;–493 (4237).

[6] K. Bieniasz, M. Berciu, M. Daghofer, and A. M. Oleś, “Green’s function variational approach to orbital polarons in KCuF3,” Phys. Rev. B ;6, 2:7339 (4238).

[7] K. Bieniasz, M. Berciu, and A. Oleś, “The Green Function Variational Approximation: Signiﬁcance of Physical Constraints,” Acta Phys. Pol. A 352, 87;–885 (4238).

[8] K. Bieniasz, M. Berciu, and A. M. Oleś, “Orbiton-magnon interplay in the spin-orbital polarons of KCuF3 and LaMnO3,” Phys. Rev. B ;7, in press (4239), arXiv:3928.28293 .

33;