Spin and Orbital Polarons in Strongly Correlated Electron Systems

(1)

Jagiellonian University

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

Spin and Orbital Polarons

in Strongly Correlated Electron Systems

Krzysztof Bieniasz

A Ph.D. Thesis

prepared under the supervision of prof. dr hab. Andrzej Michał Oleś

(2)

(3)

Oświadczenie

Ja, niżej podpisany, Krzysztof Bieniasz (nr indeksu: 1014519), 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 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 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.

(4)

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: dwuwymia-rowego modelu spinowego reprezentującego kupraty, dwuwymiadwuwymia-rowego modelu orbitalnego zainspirowanego płaszczyznami CuF2 w układzie

KCuF3oraz 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 orbital-nego na spinowy pod wpływem oddziaływania wymienorbital-nego 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.

(5)

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 temperat-ure 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 or-bital planar model inspired by CuF2planes 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 inter-esting 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.

(6)

(7)

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 2015/16/t/st3/00503. I would further like to acknowledge support from ncn under project number 2012/04/a/st3/00331 as well as from the Pol-ish Ministry of Science and Higher Education under project number n n202 069639. Lastly, I would like to thank the Quantum Matter Institute, which has generously sponsored my first visit to the University of British Columbia in 2014.

(8)

(9)

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 (1944) 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 trans-ition 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 quintessen-tial 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 11

(12)

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 tem-perature 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 elec-trons 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).

(13)

Part I

Preliminaries

(14)

(15)

CHAPTER

1

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 (1949) We start our discussion with some basic background information about the sys-tems 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.

1.1 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, 15

(16)

Figure 1.1: Comparison of Mott-Hubbard and charge-transfer insulators. Ad-apted from [4]. (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 [1], 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 [2, 3]. 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 fig-ure 1.1(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 insu-lators 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 [5]. 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 develop-ment of the Zaanen-Sawatzky-Allen (zsa) theory of correlated compounds [6]. 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 1.1(b), the charge transfer energy ∆ is the cost of transferring an electron from a ligand (i.e., the anion neighbouring the transition metal) 2p

(17)

1.2. Cuprates 17

Figure 1.2: Crystal structures of the representative superconducting copper-oxide systems, adapted from [7]. (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 1.1(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 _{→ p}5_dn+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.

1.2 Cuprates

Ever since the discovery of high temperature superconducting oxides (htsc) by Bednorz and Müller [8], 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

(18)

Figure 1.3: Bonding between 3d9 _{and 2p}2

xp2y states in a tetragonal crystal field. Numbers in parentheses indicate electron occupations in the undoped system. Adapted from [9].

common characteristics: all of them are insulating oxide materials; they are layered tetragonal systems, composed of 2d square planes of CuO2 interspersed

with separating cations such as La3+_{, Sr}2+_{, Y}3+_{, or Ba}2+_{. Some examples of}

the most important systems in this series have been illustrated in figure 1.2, 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 [10]. For this reason the properties of strongly correlated 2d systems under doping are of special interest here [11]. 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_{− y}2 _{state, which is occupied by the single hole on the}

site, as illustrated in figure 1.3. 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 [12].

From the very beginning it was evident, both from experimental data obtained in spectral experiments, like xps, xas, or Auger spectroscopy [13, 14], as well as comparison with numerical calculations for cluster models [15, 16], 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 [17].}

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 [18–20], and in fact they locate primarily in the in-plane 2p states that connect the copper sites [21, 22].

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 [23, 24] and nowadays often called the d-p model:

(19)

1.2. Cuprates 19

Table 1.1: The d-p Hamiltonian parameter values, as calculated in two different studies. All values given in eV.

Hybertsen [25] McMahan [26] 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 t0_p 0.35 [27] H_dp = −tdp X hi,jiσ (d† iσpjσ+ H.c.) − tp X hj,j0_iσ (p† jσpj0_σ+ H.c.) + t0 p X hj,j00_iσ (p† jσpj00_σ+ H.c.) + ∆ X jσ np_jσ + UdX i nd_i↑nd_i↓+ UpX j np_j↑np_j↓+ V X hi,ji nd_inp_j, (1.1) where nd_{iσ, n}p

jσ is the particle number operator for d states and for p states, respectively, and ni =P

σniσ. 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 energy ∆, p-p hopping (tpfor nearest neighbour, t0

p 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 [25, 26], are presented in table 1.1.

Since the Coulomb repulsion is usually big compared to the kinetic para-meters, 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 [28]. For half filling on d states this leads to an effective Heisenberg interaction between the Cu sites:

H_H= 4t 4 dp (∆ + V )2 _U1_d +_{2∆ + Up}2 ! X hiji Si· Sj, (1.2) 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 [29] as well as experimentally, based on Raman scattering [30, 31]. It was also shown that this value does not depend on the presence of the apical pz orbitals [32].

(20)

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 [28, 33]. 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 [34].

A symmetric (+) or antisymmetric (−) combination of the p orbitals is: ciσ = 1 2 h px_r_i_+x/2,σ± px_r i−x/2,σ+ p y ri+y/2,σ± p y ri−y/2,σ i , (1.3)

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 [35, 36], by Fourier transforming the p orbitals:

pχ_kσ= √1 N X j e−ikrj_pχ j,σ, (1.4)

for χ = {x, y} indicating the p orbital symmetry. Next we (anti)symmetrize these operators in Fourier space:

p±_kσ = 1 tk coskx 2 pxkσ±cos ky 2 pykσ , tk= q cos2 kx 2 + cos2 k2y, (1.5)

and transform them back to real space to obtain the Wannier orbitals: p±_lσ = √1 N X k eikrl_p± kσ. (1.6)

With these orbitals the previous ciσ orbitals can be represented as: ciσ= X l fi,lp+l,σ, fi,l= _2N1 X k eik(rl−ri)_t k. (1.7)

The fi,l coefficients decay rapidly [36], 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:

H_eff = 8t2_dp 1 Ud−∆ − 2V + 1 ∆ X i,l,l0 fi,lfi,l0 × 1 2 X σσ0 Si· p+†_lσσσσ0p+ l0_σ0 − 1 4 X σ nip+†_lσp+_l0_σ ! , (1.8) where σ is the vector of Pauli operators. The other terms are much more complicated since they involve also the antisymmetric operators p−

iσ [12], 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 √ 2

(21)

1.2. Cuprates 21

with + phase for triplet and − phase for singlet pairing. It is easy to calculate the energy of these four states using the effective Hamiltonian (1.8) for the special case i = l = l0_{, yielding:}

E_s+= −5.05 eV, E_t+= 1.12 eV, E_s−= 1.03 eV, E_t−= 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 (1.8). 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:

H_t=X l,l0

tl,l0(c†

lσcl0_,σ+ H.c.), (1.10) with the hopping constant

tl,l0 = t2_dp 1 Ud−∆ − 2V + 1 ∆ 1 4δhl,l0_i+ 6f_0,0f_l,l0 + . . . , (1.11) where δhl,l0_i 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 1.1, 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 [37, 38], represented by the terms (1.2) and (1.10). 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 (1.9). 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 [39, 40] as well as good agreement with arpes spectroscopic results even in the absence of transverse fluctuations [41].

Since calculating the hopping integral from eq. (1.11) is unreliable, it has been concluded that it would be more reasonable to determine it from nu-merical cluster calculations [42, 43] or by fitting the model to the Fermi

(22)

surface, obtained either from arpes measurements [44] or from band calcula-tions [45]. The currently widely accepted values of the t-J model parameters are t = 0.4 eV, J = 0.13 eV [46]. 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. (1.2) 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 scat-tering experiments, which found a Néel transition temperature TN = 195 K in lsco, although there is a long range af magnetic correlation in 2d as high as 300 K [47]. The low energy magnetic moment is strongly reduced (from 1.1 µB for a Cu2+ ion to 0.5 µB in lsco) because of quantum fluctuations [48].

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 [12].

At strong coupling, U → ∞, the Heisenberg model is equivalent to the Hubbard model. However, it has been showed with quantum Monte Carlo simulations [49] 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 de-scribed as an sdw state. Similar conclusions have been drawn from Lanczos calculations [50]. Furthermore, numerical studies based on the d-p model have yielded an antiferromagnetic ground state as well [51, 52].

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 [53]. 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. 1.4. 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.

(23)

1.2. Cuprates 23

Figure 1.4: 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 aclassical Néel state. Theexcited bondsare 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 [54]. However, their role is limited because of their high perturbative order—the simplest Trugman

loop is a 6th order process.∗ ∗

See also fig. 3.4, and the discussion in sec. 3.3 of the significance of Trugman loops for the methods used herein.

Consider now the action of the t-J model, restricted to the Ising part, on the state with a single hole added. If we denote by |li a state with a string of length l then, neglecting the possibility that the string might cut through

itself,† _{on a square lattice there are three ways in which the string can be}†_{Technically this means}

that the calculation is done on a four-fold Bethe lattice.

extended to l + 1, one in which it can be shortened to l − 1, and the Ising energy is the cost of the string, proportional to its length. This leads to the following algebraic equation:

H|li= J(3₂ + l)|li −√3t(|l − 1i + |l + 1i), (1.12) 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 E₀ ∝(J/t)2/3 and an average string length ¯l∝ (J/t)−1/3 _{[55]. 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 [56, 57].

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 [58, 59], variational [60], and numerical methods [61]. This result can be easily under-stood 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.

(24)

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 back-ground. This has been analyzed in the linear spin wave approximation using the Holstein-Primakoff transformation and 1/S expansion of spin waves [58]. This approach has also proven fruitful in the dominant pole approximation, assuming the weight beyond the first pole of the Green’s function was inco-herent, to obtain analytical results for a single hole [62]. Later studies of this problem applied the same principle in a numerical setting, the so-called self consistent Born approximation,‡_{to obtain the spectral function in a wide range}

‡_{This method is one of}

the techniques used in this thesis. Its de-tails are discussed at length in sec. 3.2.

of (k, ω) points [63–65]. These results are in surprising agreement with exact diagonalization calculations [56, 66]. The general findings for a single hole are the scaling of the quasiparticle bandwidth W ∼ 1.5J0.79 _{[64], 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 kxcos ky+ 0.13(cos 2kx+ cos 2ky), (1.13) indicating that the holes travel at doubled momenta, i.e., over sites belong-ing to a sbelong-ingle sublattice and avoidbelong-ing the other ones [67]. 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 [68] are very vulnerable both to temperatures and to doping rates.

The Green’s function method also allows for the calculation of the dynam-ical 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 cor-responds to a nearly localized hole with a large mass, and the higher peaks are associated with short string states of 1 and 2. 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 spec-tral 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 ∝ Jα_{, however the results remain inconclusive, with the}

(25)

1.3. Orbital Ordered Systems and KCuF₃ 25

exponent varying from α = 1/2 obtained from exact diagonalization [69, 70] to α = 2/3 extracted from Born approximation [64, 71].

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 [72]. 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 [73– 75]. Subsequent numerical studies have confirmed its existence [76, 77] and semi-analytical studies based on scba indicate, that this feature cannot be explained without employing the three site terms [78].

1.3 Orbital Ordered Systems and KCuF

3

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 [79]. 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 occupa-tion number, leads to the emergence of different spin and orbital degrees of freedom [80]. 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,

yz, and zx) and the eg orbitals (3z2− r2 and x2− y2). 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: HU = U X iα niα↑niα↓+ X i,α<β

(Uαβ−1₂Jαβ)niαniβ

+ X

i,α<β

Jαβ(d†_iα↑d†_iα↓d_iβ↓d_iβ↑+ H.c.) − 2 X i,α<β

JαβSiα· Siβ, (1.14) where the Greek indices run over the d orbitals; the intraorbital Coulomb repulsion Uαβ and the exchange integral Jαβ are generally anisotropic and

(26)

Table 1.2: On-site interorbital exchange elements Jαβ expressed in terms of the Racah parameters. Adapted from [82], for more details see ref. [81].

dorbital xy yz zx x2− y2 3z2− r2 xy 0 3B + C 3B + C C 4B + C yz 3B + C 0 3B + C 3B + C B+ C zx 3B + C 3B + C 0 3B + C B+ C x2− y2 C 3B + C 3B + C 0 4B + C 3z2_{− r}2 _{4B + C B + C} _B_{+ 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 paramet-ers [81]; their expressions have been tabulated in table 1.2. Furthermore, the intraorbital Coulomb integral is expressed as:

U = A + 4B + 3C, (1.15)

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

Uαβ = U − 2Jαβ. (1.16)

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 [83–90]. 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 [82] and for detailed derivations of con-crete models one should consult the relevant literature on manganites [91–93], vanadates [86, 94, 95], copper fluorides [96, 97] and other systems [85, 98]. The effective model relevant for our case (a single charge in an eg orbital system, such as KCuF3) will be derived in sec. 2.2.

The copper fluoride perovskite KCuF3is a pseudocubic (actually tetragonal)

system that has attracted attention since the 1960s, 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 [99]. Two basic polytype structures distinguished

(27)

1.3. Orbital Ordered Systems and KCuF₃ 27

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 con-vention seems to be ref. [99], where the au-thor denoted the poly-types as a and c, with

b and d being their

re-spective versions with stacking disorder. The notation for c and d was later reversed in an erratum.

persisting up to the structural phase transition at Ts ≈800 K at which the Jahn-Teller effect leads to the distortions of the CuF6 octahedra. Below this

temperature the structure is tetragonal, with the in-plane Cu-Cu distance d⊥= 4.14 Å and the c direction distance shortened to dc= 3.93 Å [100, 101].

The distance disproportion of dc/d⊥≈0.95 indicates that the system is very

close to cubic and such an approximation, at least for the sake of model calculations, is justified.

The polytype structures of KCuF3 differ by the orbital order, namely the

a type exhibits alternating orbital order of the g type (g-ao), with the occu-pied 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-2d 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 or-bitally 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 [102, 103]. 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 [104] and the stabilization of GdFeO3-type rotations

of the CuF6 octahedra [105].

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 ferromagantiferromag-netically in the ab plane [101, 106], in accordance with the Goodenough-Kanamori rules [107, 108]. 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 [101]. A number of neutron scattering studies have shown that magnetic structure of KCuF3 is composed of quasi-1d af

chains which are weakly coupled along the in-plane bonds, with the ratio of exchange couplings J⊥/Jc ≈0.01 [109]. The quantum spin fluctuations are strong and the system can be viewed as a textbook example of a 1d Heisenberg antiferromagnet. The magnetic excitations have been confirmed by neutron scattering to have spinon character [110].

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:

|θi= cosθ

(28)

(a) (b)

Figure 1.5: 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-gested that the single hole might occupy orbitals corresponding to cos θ = 1

3

(θ ≈ 0.4π), which follows from the local Jahn-Teller lattice distortions [106]. On the other hand, based on the electronic interactions in the magnetic phase, the Kugel-Khomskii type orbitals |y2_{− z}2_i_/|z2_{− x}2_i _{were suggested [83],}

cor-responding to θ = π/3. Over the years, experimental data [102, 103, 110] as well as various lda+u calculations have come to support this hypothesis [111– 113]. 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_{− r}2_{i ± |x}2_{− y}2_i_,

corresponding to θ = π/2 ± φ, with a small detuning angle φ ≈ 0.012π [105]. 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 [114], 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. 1.5 we have rendered the orbital order on a unit plaquette of the ab-plane for the two extreme values of θ.

(29)

CHAPTER

2

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 47 of “How to be a Detective in 10 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 (1946) 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 elec-tronic operators are decoupled into spinless fermions and bosons describing excitations in the polarized background.

2.1 The t-J Model

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

HH = −t X hijiσ (d† iσdjσ+ H.c.) + U X i ni↑ni↓, (2.1) 29

(30)

where d†

i/di is the electron creation/annihilation operator acting on site Ri, tis 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 [115, 116]: Ht= −t X hijiσ h (1 − ni¯σ)d † iσdjσ(1 − nj¯σ) + H.c. i , (2.2a) HJ = 4J X hiji h (Sz iSjz+1₄) +α₂(Si+S − j + S − i Sj+) i , (2.2b) where J = t2

U is the exchange constant. The electron operators in the kinetic term above are projected onto the space with singly occupied sites, thus form-ally ensuring the Hamiltonian applies to the lower Hubbard band. Henceforth we will omit this notation for simplicity, although it is always implicitly under-stood that the t-J Hamiltonian refers to the projected space. Equation (2.2b) 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-Jz _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 antifer-romagnetic (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

(31)

2.1. The t-J Model 31

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:

H_3s= −J X iδσσ0

d†_i_+,σ0d_iσ0d†_iσd_i_+δ,σ(1 − δδ), (2.3) 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:

T_3s =X kσ

εkd†kσdkσ, (2.4)

where εk = −4J(4γ2

k −1) is the second order free electron dispersion, and γk =Pδeikδ/4 is the structure factor for the case of a 2d 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 ad-vantage 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 [64].

The transformation is a standard rotation of the spin basis, parametrized by the lattice dependent rotation angle θi:

|θii |¯θii ! = cosθ2i −sinθ2i sinθi 2 cosθ2i ! |↑i |↓i ! , (2.5)

which corresponds to the following transformation of the spin operators: ˜ S_iz ˜ Sx i !

= ₋cos θi_{sin θi} _{cos θi}sin θi !

Sz_i S_ix

!

, (2.6)

where θi= Q · Ri [117], and Q = (π, π) is the so-called ordering wave vector for the case of a 2d af. Clearly if Ri points to the sublattice a, on which the spins are oriented upwards, the rotation angle θA= 0, whereas on sublattice

(32)

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 (2.6) for the af ordering vector leads to the following transformations of the Hamiltonian:

d†_jσ → d†_j_¯σ, S_jz→ −Sz

j, Sj±→ −S

∓

j , (2.7)

which finally produces the fm representation of the t-J model: H_t= −tX hijiσ h d†_iσd_j_¯σ+ H.c.i, (2.8a) HJ = 4J X hiji h (1 4 − SizSjz) −α₂(Si+Sj++ S − i S − j ) i , (2.8b)

while the three-site term (2.3) 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:

d_i↑= f_i, d_i↓= f_iS_i+, (2.9) and represent the spins with bosons using the Holstein-Primakoff transforma-tion: S_i+= q 2S − b† ibibi ≈ √ 2Sbi, (2.10) S_i−= b†_i q 2S − b† ibi ≈ √ 2Sb† i, (2.11) S_iz = S − b†_ib_i, (2.12)

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:

f_i†f_i+ b†_ib_i ≤2S. (2.13) Following these steps, the t-J Hamiltonian (2.8) can be expressed in the linear spin wave (lsw) formalism:

Ht= −t X iδ h f_i†f_i_+δb†_i_+δ+ H.c.i, (2.14a) H_J = JX iδ h b†_ib_i+ b†_i_+δb_i_+δ− α(b†_ib†_i_+δ+ bib_i_+δ)i, (2.14b)

(33)

2.2. The Spin-Orbital Model 33

and the three-site terms take the form: H_3s= −JX

iδ

f_i†₊f_if_i†f_i_+δh1 + b†_ib_i+ b†_i_+δb_i₊+ b†_i_+δb†_i + bib_i₊i(1 − δδ). (2.15) Assuming there is only one electron doped into the system, we can simplify it further by substituting fif

†

i = 1. The first term above is the free fermion term, which can be diagonalized exactly in Fourier space:

T_3s =X k

εkfk†fk, (2.16)

which is the only free hopping term in the Hamiltonian, while the coupling to magnons is always higher order than in (2.14a).

2.2 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 t2gstates and the upper egstates. 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 be equivalently described as e3

g in terms of electron occupation, or e1g in terms of hole occupation.

The description of the charge dynamics starts with defining the kin-etic Hamiltonian. We consider electron hopping between Cu(d9_{) sites along}

|zγi= (3zγ2− r2)/ √

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 orthogonal orbitals |¯zγi = (x2γ− yγ2)/

√

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 [87]:

H_t= −tX γ X hiji⊥γ d†_iz_γd_jz_γ+ H.c.. (2.17)

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

Ht= −t h X hijikc d†_izd_jz+1 4 X hiji⊥c (d† iz ∓ √ 3d† i¯z)(djz∓ √ 3dj_¯z)i + H.c., (2.18)

(34)

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 (1.14), which for the case of an exclusively eg orbital system simplifies to: HU = U X iα niα↑niα↓+ (U − 5₂JH) X i,α<β niαniβ + JH X i,α<β (d† iα↑d †

iα↓diβ↓diβ↑+ H.c.) − 2JH X

i,α<β

Siα· Siβ, (2.19) where U is the Coulomb repulsion and JH is the Hund exchange element. For the limit U t, and considering the d9_d9 _d8_d10 _{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}

from (2.19), consists of four spectroscopic terms 3_A₂_, 1_E

θ and 1E, and 1E1, with energies U − 3JH, U − JH (double), and U + JH, respectively. This leads to the following four superexchange terms:

Hγ₁ = −2Jr₁ X hijikγ (Si· Sj+ 3₄)(₄1 − τ_iγτ_jγ), (2.20a) Hγ₂ = 2Jr₂ X hijikγ (Si· Sj−1₄)(1₄− τiγτ γ j), (2.20b) Hγ₃ = 2Jr₃ X hijikγ (Si· Sj−1₄)(1₂− τiγ)(1₂ − τ γ j), (2.20c) Hγ₄ = 2Jr₄ X hijikγ (Si· S_j−1₄)(1₂− τ_iγ)(1₂ − τ_jγ), (2.20d)

where the ri coefficients serve to impose the Hund rule and follow from the above-mentioned multiplet structure of the e2

g configuration: r₁ = 1 1 − 3η, r2 = r3 = 1 1 − η, r4 = 1 1 + η, (2.21) where η = JH/U, and τ_iγ are direction dependent orbital operators which can be expressed in terms of pseudospin operators:

τ_ia/b= −1₂(T_iz∓√3T_ix), τ_ic= T_iz, (2.22) under the standard convention

|¯zi ≡ |↑i, |zi ≡ |↓i. (2.23)

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

(35)

2.2. The Spin-Orbital Model 35

just one state of energy U and the coefficients ri = 1. Equations (2.20) can then be simplified to just two terms:

Hγ₁ = 4J X hijikγ (Si· S_j +1₄)(τ_iγτ_jγ+1₄), (2.24a) Hγ₂ = 2J X hijikγ (1 4 − Si· Sj)(τiγ+ τ γ j). (2.24b)

Although the orbital ground state is known to be alternating, the actual occupied states do not necessarily have to be the basis (2.23). 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:

H_z = −EzX i

T_iz. (2.25)

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 |¯zi 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 (2.24b) 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 (2.23) in the same way as in (2.5). 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 = (|¯zi ± |zi)/√2, which corresponds to a rotation by an angle θ = π/2 [96]. Therefore, to incorporate the orbital field, we will parametrize the sublattice rotations with respect to this angle, θi = π

2 + φi,

where φi = eiQRi_φis 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 (2.24) according to the spin transformations given in (2.6) with the above rotation angle, we arrive at the exchange Hamiltonian dependent on the orbital field:

H⊥₁ = J X hiji⊥c (Si· Sj+ 1₄) h 1 + (2 cos 2φ + 1)Tz iTjz+ (2 cos 2φ − 1)TixTjx +2 sin 2φ(Tx i Tjz− TizTjx) ± √ 3(Tx i Tjz+ TizTjx) i , (2.26a)

Spin and Orbital Polarons in Strongly Correlated Electron Systems