Spin Liquid and Quantum Phase Transition Without Symmetry Breaking in a Frustrated Three-Dimensional Ising Model

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Spin liquid and quantum phase transition without symmetry breaking in a frustrated three-dimensional Ising model

Masterarbeit zur Erlangung des akademischen Grades Master of Science

vorgelegt von Julia Röchner geboren in Herne

Lehrstuhl für Theoretische Physik I Fakultät Physik Technische Universität Dortmund 2016 1. Gutachter : Prof. Dr. Kai P. Schmidt

2. Gutachter : Prof. Dr. Leon Balents

Datum des Einreichens der Arbeit: 30. September 2016 Abstract

In this thesis the transverse field Ising model on the three-dimensional pyrochlore lattice is investigated. This model is highly frustrated in the low-field limit which entails an extensively large ground-state degeneracy. Quantum fluctuations in the ground-state manifold at T = 0 are present and give rise to a Coulomb quantum spin liquid phase. Based on this knowledge the quantum phase transition without symmetry breaking between the polarized phase and the Coulomb quantum spin liquid phase is located quantitively. This is done by investigating the high- and low-field limit via perturbative continuous unitary transformation which allows to determine quantities exactly as series up to a certain order in a perturbation parameter. We show by extrapolating the bare series of the elementary excitation gap of the high-field limit that the model does not exhibit a second-order phase transition. Strong evidence for a first-order phase transition is gained by comparing and locating the crossing of the ground- state energies of both limits.

Kurzfassung

In dieser Arbeit wird das Ising-Modell im transversalen Magnetfeld auf dem dreidimensio- nalen Pyrochlorgitter untersucht. Dieses Modell ist im Limes kleiner Felder hoch frustriert, was eine extensiv große Grundzustandsentartung zur Folge hat. Quantenfluktuationen in der Grundzustandsmannigfaltigkeit bei T = 0 sind der Grund für die Existenz einer Coulomb Quanten-Spinflüssigkeitsphase. Basierend darauf wird der Quanten-Phasenübergang ohne Symmetriebrechung zwischen der polarisierten Phase und der Coulomb Quanten-Spinflüssig- keitsphase quantitativ bestimmt, indem der Hoch- und Niedrigfeldlimes mittels perturbativer kontinuierlicher unitärer Transformation untersucht wird, was eine exakte Reihenentwicklung von Größen bis zu einer bestimmten Ordnung im Perturbationsparameter ermöglicht. Durch Extrapolieren der nackten Reihe der elementaren Anregungslücke im Hochfeldlimes wird gezeigt, dass das Modell keinen Phasenübergang zweiter Ordnung aufweist. Stattdessen wird ein Phasenübergang erster Ordnung gefunden, der durch den Kreuzungspunkt der Grundzu- standsenergien im Hoch- und Niedrigfeldfall lokalisiert wird. IV Contents

Table of Contents V

1 Introduction 1

2 Quantum spin ice 5 2.1 Water ice ...... 5 2.2 Spin ice ...... 6 2.3 Quantum eﬀects ...... 10

3 Transverse Field Ising Model on the pyrochlore lattice 13 3.1 Model ...... 13 3.2 Particle representation ...... 18 3.2.1 High-ﬁeld limit ...... 18 3.2.2 Low-ﬁeld limit ...... 20

4 Series expansion methods 24 4.1 Perturbative Continuous Unitary Transformation ...... 24 4.1.1 Continuous Unitary Transformation ...... 24 4.1.2 Perturbative CUT ...... 27 4.1.3 Cluster decomposition and cluster additivity ...... 29 4.2 Extrapolation ...... 30 4.2.1 Pad´e-approximation ...... 30 4.2.2 DlogPad´e-extrapolation ...... 31

V 5 Full graph decomposition 32

5.1 Graph representation ...... 32

5.2 Graph generation ...... 33

5.3 Reduced energies ...... 36

5.4 Graph embedding in lattice ...... 38

5.4.1 Ground state ...... 38

5.4.2 Hopping elements ...... 40

5.5 Loop graphs ...... 41

6 Results 42

6.1 High-ﬁeld limit ...... 43

6.2 Low-ﬁeld limit ...... 51

6.3 Comparison of both limits ...... 53

6.4 Experimental realization ...... 54

7 Conclusion and outlook 56

A One-particle gap of antiferromagnetic TFIM on kagome lattice 60

B One-particle gap of ferromagnetic TFIM on kagome lattice 61

Bibliography 67

VI Chapter 1

Introduction

In condensed matter physics the study of phase transitions in strongly correlated systems displays an exciting field in theory as well as in experiment. Different phases in materials are distinguishable by their macroscopic properties. The likely most popular example of a phase transition is the one when water freezes to ice. In the fluid phase the H20 molecules ◦ are disordered, while for temperatures below the critical value Tc = 0 C, the system orders and the molecules form a crystalline structure. The terms "order" and "disorder" can also be defined in the realm of magnetism on a spin level, where "spin" corresponds to the orientation of magnetic moments caused by atoms in a compound. If we consider ferromagneticly coupled spins residing on the sites of a square lattice, the system is in the disordered paramagnetic phase for high temperatures. Below a critical value Tc, the system orders and takes one of the ground states, where all spins follow a parallel ordering according to the ferromagnetic coupling (see Fig. 1.1).

The ordered state spontaneously breaks the symmetry, since it has lower symmetry than the Hamiltonian. Note that the phase transitions described above are classical because they are induced by temperature change. In contrast to that, there can also occur phase transitions at T = 0, if materials show strong quantum effects. Those transitions are then called quantum phase transitions, which are for example generated by intrinsic or applied electric or magnetic fields. While phase transitions engendered by symmetry breaking are well known and extensively studied, phases without broken symmetries receive high attention, since they are prone to host fascinating and exotic properties. The most common examples are phase transitions generated by topological order [1–5]. Phase transitions without symmetry breaking in some other materials can be traced back to occurring frustration [6]. Frustration happens, if competing interactions in materials cannot be simultaneously satisfied usually due to the

1 2 Chapter 1. Introduction

Figure 1.1: Sketch of phase diagram of ferromagneticly coupled spins on square lattice. For low temperatures T < Tc the system is ordered while for high temperatures T > Tc the system is disordered and in a paramagnetic phase. geometric arrangement of the magnetic moments. This entails, that not only one or a few classical spin configurations in the material minimize the energy, but instead the number of ground states is extensively large inducing highly entangled states in case quantum effects play a significant role. Generically, frustration gives rise to exotic phenomena in materials and can lead, under certain conditions, to a classical or quantum spin liquid state, similar to the disorder of molecules in real liquids. Spin liquids are disordered states with fractional excitations which do not break any symmetries [7]. Due to the large ground-state degeneracy of classical spin liquids, fluctuations, mainly in the ground-state manifold, occur down to very low temperatures, which can be either classical or quantum. Especially if the origin lies in quantum effects, one expects to observe exotic properties.

The study of frustrated magnets therefore aroused great interest in the past. There has been much work on two-dimensional frustrated systems, for example antiferromagneticly coupled spins on the triangular or kagome lattice [8,10]. In turn, three-dimensional frustrated systems are much more complicated, since their dimensionality makes a microscopic investigation difficult, which is simultaneously the same reason why it has become an exciting and very challenging field of research. The most storied three-dimensional examples are the spin ice pyrochlores Ho2Ti2O7 or Dy2Ti2O7, where strong geometric frustration among coupled magnetic moments gives rise to classical spin liquids with defects that behave like magnetic monopoles [11,12]. However, quantum effects are essentially zero in these spin-ice compounds, 3

z which are described accurately by classical Ising models, i. e. only the (local) σi component of the spins appears in the Hamiltonian. However, theoretically, a quantum version of spin ice is highly desirable. One expects the presence of a so-called Coulomb quantum spin liquid (CQSL) [7, 13, 14] with gapped electric and magnetic excitations as well as an emergent gapless photon. Quantum fluctuations may be introduced by additional exchange interactions involving spin flips (e. g. XY or more complex couplings), which naturally occur in some other pyrochlores like Yb2Ti2O7 [15,16]. However, such quantum exchange models are quite complex, and their phase diagrams contain many other ground states in addition to the desired CQSL [18, 19]. At the model Hamiltonian level, a simpler route to "quantum-ize" classical spin ice is to add a transverse field. The low-energy physics of such systems is expected to be described to a good extent by the transverse-field Ising model (TFIM) on the three-dimensional pyrochlore lattice, which is depicted in Fig. 1.2.

Figure 1.2: Illustration of the pyrochlore lattice, which consists out of alternating up- and down– pointing corner-sharing tetrahedra.

The TFIM is one of the archetypal models used in various areas in physics and is known to host a plethora of interesting physical phenomena, especially on highly frustrated lattices [8–10,20]. At the same time the theoretical treatment of three-dimensional frustrated systems including TFIMs represents a notable challenge and quantitative results are hard to extract since common standard methods, for example exact diagonalization, are ineﬃcient due to the rapidly growing cluster. In turn, series expansions operating in the thermodynamic limit scale well with the dimension and are well suited to treat TFIMs on three-dimensional lattices. [20–22].

In this work we apply low- and high-field series expansions to determine the phase diagram of the three-dimensional pyrochlore TFIM quantitatively. More precisely, the quantum phase 4 Chapter 1. Introduction transition without symmetry breaking between the polarized phase and the CQSL phase is located and compared to the mean-field result (h/J)c = 0.7 in Ref. [17], where J is the nearest-neighbor coupling of spins and h is the transverse magnetic field.

The next chapter deals with spin ice models itself and how they were developed based on experimental discoveries. The subsequent chapter is about the TFIM on the pyrochlore lattice, which displays a proper model to describe some interesting features of quantum spin ice materials and which is used to locate the quantum phase transition between the polarized and the CQSL phase. The fourth chapter introduces the series expansion methods used in order to investigate the low- and high-field limit of the TFIM on the pyrochlore lattice. It turns out, that all calculations can be done on topological equivalent graphs, which makes it possible to reduce the computational effort and thus displays the cornerstone in order to establish an efficient computer program. This is explained in chapter five. Results are presented in chapter six and summarized in the last chapter. Chapter 2

Quantum spin ice

Quantum spin ices have been of great interest in the past few years because of their exotic phases and excitations [13, 23–25]. In particularly, we investigate in this thesis the quantum phase transition occurring between the polarized phase and the CQSL phase of the TFIM on the pyrochlore lattice. The TFIM model is suitable in order to study phase transitions in quantum spin ice. This chapter is about spin ice and later quantum spin ice in general, starting with water ice, which forms the origin of the nomenclature "spin ice". A more detailed description of water and spin ice can be found in [26].

2.1 Water ice

In the early 1930s, William Giauque and co-workers detected by speciﬁc heat measurements a non-vanishing entropy S0 = 0.82 ± 0.05 Cal/deg·mol in water ice in the low temperature regime [27, 28]. This was remarkable since it seemed to violate the third law of thermody- namics. The phenomenon was exposed by Linus Pauling, who traced it back to macroscopic proton (H+) conﬁgurations caused by the crystalline structure of water ice and the unsym- metric bonding of the hydrogen ions [29]. He calculated theoretically a zero point entropy of

S0 = 2N0kBln(3/2) = 0.81 Cal/deg·mol, where N0 is the number of water molecules, which is very close to the experimental value. In water ice, the oxygen ions O2− form a diamond lattice with H+ protons located in between the O – O bonds. The H+ protons itself reside on a pyrochlore lattice, which consists out of corner-sharing up- and down-pointing tetrahedra.

The structure of the H2O molecule in the solid phase is maintained because of their strong chemical binding energy of 221 kCal/mol. This entails that the H+ protons are not located

5 6 Chapter 2. Quantum spin ice in the middle of the O2− ions, but favor a position, in which they are closer to one of the O2− ions. This is depicted in Fig. 2.1.

Figure 2.1: Depiction of the position of the hydrogen ions H+ (red circles) in water ice. Two are located closer to the central O2− ion (gray circle), two favor a "far away" position.

The bond length between two distinct O2− ions is 2.76Å, the covalent O – H bond of the water molecule is 0.96Å [26]. This phenomenon is coined the Bernal-Fowler ice-rules which indicate that two H+ protons should be close to one O2− ions, while the other two favor a "far away" position. From an electrostatic point of view, the H+ protons want to be located as far as possible from each other resulting into strong frustration and an extensive ground-state degeneracy, which implicate the residual zero point entropy. A similar residual entropy was detected in so-called spin ice compounds, which is presented in the next section.

2.2 Spin ice

Spin ice has aroused great interest since it displays a novel class of frustrated ferromagnetic Ising systems. They exhibit intriguing properties at very low temperatures due to their frustration like a non-vanishing entropy [30, 31]. Spin ices are systems with spins located on a pyrochlore lattice (see Fig. 2.2). The ground state is formed by two spins pointing in and two spins pointing out of each tetrahedron and is therefore highly degenerate since six diﬀerent spin conﬁgurations per tetrahedron follow this constraint. This exponentially large number of degenerate ground states Ω0 entails an extensive residual, or zero point, entropy S0 = kBln(Ω0) = 0.81 Cal/deg·mol as it was the case for water ice. The name "spin 2.2. Spin ice 7

Figure 2.2: Illustration of spin ice taken from [35]. Spins are located on a pyrochlore lattice. The tetrahedra pointing upwards form an fcc Bravais lattice. White circles display spins that point into downward tetrahedra, black circles illustrate spins pointing out of downward tetrahedra. A ground state is formed by two spins pointing in and two spins pointing out of each tetrahedron. The gray bold line indicates a hexagon which contains alternating spins. ice" originates from the direct analogy between the H+ protons in water ice and the spin conﬁguration in spin ice itself.

In nature, rare earth pyrochlore oxides are candidates for realizing spin ice, such as Ho2Ti2O7 3+ 3+ and Dy2Ti2O7, where Ho and Dy are located on a pyrochlore lattice carrying the magnetic moments. Their f-electron spins are large and thus they behave classical. The unit cell can be chosen cubic with up-pointing (or down-pointing) tetrahedra forming an fcc Bravais lattice (see Fig. 2.2). Harris, Bramwell and collaborators [30] studied the material Ho2Ti2O7 and determined a positive Curie-Weiss temperature ΘCW ≈ +1,9K. Although this suggests overall ferromagnetic coupling, they could not prove any transition to long-range magnetic order down to 0.35K (and down to 0.05 K via muon spin relaxations [32]), which is surpris- ing since long-range order is expected when dealing with a three dimensional cubic system with ferromagnetic interactions below the Curie-Weiss temperature. This gives rise to the emergence of frustration preventing the system from ordering which, at a ﬁrst glance, seems to be inconsistent with the presence of ferromagnetic interactions. The term "spin liquid" was coined to describe the disorder in the temperature regime below ΘCW characterizing a state without conventional order that does not break any symmetries and which is smoothly connected to the paramagnetic phase [7].

Let us consider the main forces which play a role in spin ice compounds in order to explain the phenomenon of the occurring frustration. In rare earth ions the spin-orbit interaction is 8 Chapter 2. Quantum spin ice large and J = L + S represents a good quantum number. The physics can be described by

H = Hcf + HZ + Hint (2.1) where Hcf captures the crystal ﬁeld originating from electrostatic and covalent bonding eﬀects and which lifts the ion ground-state (2J + 1)-fold electronic degeneracy. The Zeeman energy

HZ describes the interaction of the rare earth magnetic moments with an applied magnetic

ﬁeld B and Hint is the inter-ion interaction Hamiltonian, which includes exchange coupling and long-range magnetostatic dipole-dipole interaction. As for spin ices, Hcf is the dominant term in H with an energy scale ∆ separating the ground state from the ﬁrst excited state.

Since the energy scale of Hcf is much larger than the energy scale of Hint and HZ, which means that the high-energy sector is well separated from the low-energy sector, we can find an effective Hamiltonian which describes the low-energy physics by neglecting the excited states of Hcf . Thus it appears convenient to describe the problem in the basis of the crystal-field eigenstates. We find from inelastic neutron measurements in the spin ice material Ho2Ti2O7, that magnetic moments can only point parallel or antiparallel to the local [111] direction assuming the prior mentioned cubic unit cell of the pyrochlore lattice [33]. Apparently, the crystal field forces the spins to point along the axis which connects two neighboring centers of tetrahedra (see Fig. 2.2). This means we are dealing with an h111i pyrochlore Ising model. In other words, the magnetic moments are Ising-like but each moment has its own local axis. We assume, that the low-energy physics can be described by the interaction term in (2.1) leading to the effective classical h111i pyrochlore Ising model or "dipolar spin ice model" (DSM) for the low-energy sector

J X zi zj D X (ˆzi · zˆj − 3ˆzi · rîjrîj · zˆj) zi zj HDSM = − (ˆzi · zˆj)σi σj + 3 σi σj (2.2) 2 2 (rij/rnn) hiji i,j where J is the exchange coupling between neighboring spins, D the dipolar interaction, zî the quantization axis of Ji pointing in or out of a tetrahedron, rij the distance between spin i and spin j and rnn denotes the distance between nearest-neighbor spins. J > 0 (J < 0) indicates ferromagnetic (antiferromagnetic) exchange coupling. If we set D = 0 for a moment, then

X zi zj Hnn = −J (ˆzi · zˆj)σi σj (2.3) hiji describes a material with no dipolar interactions. In pyrochlore it counts zˆi · zˆj = −1/3 and 2.2. Spin ice 9 we can therefore write (2.3) as

1 X z z H = −J(− ) σ i σ j (2.4) nn 3 i j hiji J X z z = σ i σ j . (2.5) 3 i j hiji

From this, we can extract a very interesting property. Obviously, a ferromagnetic nearest- neighbor h111i pyrochlore Ising model results into an effective antiferromagnetic Ising model on the pyrochlore lattice. A ground state is formed by two up and two down pointing spins per tetrahedron. The ground state is highly frustrated and degenerate since one can find an extensive large number of ground states fulfilling the "two-up-two-down" rule. A ferromagnetic h111i pyrochlore Ising model is therefore frustrated due to geometric arrangement of the spins while an antiferromagnetic nearest-neighbor h111i Ising model would form a conventional long-range Neel ordered ground state.

However, in the materials Ho2Ti2O7 and Dy2Ti2O7, D 6= 0 is of the same order of magnitude as J or even larger and thus not negligible. Monte-Carlo simulations revealed a positive value for the nearest-neighbor dipole-dipole interaction Dnn and a negative value for J, which implicates antiferromagnetic nearest-neighbor exchange coupling [34]. Still there was no transition to a long-range ordered state found down to 0.05K [32] and 0.35K [30]. Why do those materials behave like spin ices despite their antiferromagnetic exchange coupling?

In [34] they showed that it depends on the ratio J/Dnn. For values larger than J/Dnn ∼ −0.91, the ferromagnetic nearest-neighbor dipole-dipole interaction and the antiferromagnetic nearest-neighbor exchange coupling result into an eﬀective ferromagnetic nearest-neighbor coupling, which leads to frustration and spin-ice behavior.

Even though experimental data did not show any tendencies for an ordered state of spin-ice materials down to very low temperatures for a long time, Melko and co-workers [35] predicted an ordered state with zero total magnetization in spin ice below a critical value Tc and thus a vanishing residual entropy in contrast to water ice. They state, that long-range dipolar interactions, which have not been incorporated in this section so far, lift the degeneracy of the spin-ice ground-state manifold for suﬃciently low temperatures. The system orders and takes a unique ordered ground state. This was recently consolidated experimentally by Pomaranski and co-workers [36], who managed to verify the absence of a residual entropy in a thermally equilibrated Dy2Ti2O7 compound.

Summarized, it can be said, that classical spin ice is in a paramagnetic phase for T > ΘCW and in a spin liquid regime for Tc < T < TCW , which is not a separate phase but is smoothly 10 Chapter 2. Quantum spin ice connected to the paramagnetic phase. In the latter the system is frustrated due to an effective ferromagnetic nearest-neighbor coupling, preventing the system from ordering. Due to the low temperature, the system fluctuates almost entirely within the ice-rule manifold, but still spin flips, known as magnetic monopoles in spin-ice materials [11], can occur as temperature- induced excitations. As the temperature decreases, spin dynamics lower and it becomes more and more unlikely, that the system fluctuates between different spin-ice ground states due to the energy, which has to be brought up for flipping spins. At least six alternating spins resided on a hexagon within the pyrochlore lattice (see Fig. 2.2) need to be flipped in order to get from one ground state to another. The system freezes and takes one of the disordered ice-rule ground states. For even lower temperatures T < Tc, the system orders due to long range dipolar interactions which lift the degeneracy of the ground-state manifold.

Classical spin ice and their exotic excitations like magnetic monopoles have been investigated extensively [11,12]. Quantum effects are absent in these materials which is the reason why at sufficiently low temperatures, classical spin ices freeze because flipping spins is energetically too expensive and tunnelling between different ground states is not likely due to the weak quantum effects [7]. However, quantum spin ices are highly desirably, since they are prone to exhibit exotic quantum phenomena. Therefore quantum spin ices are introduced in the subsequent section.

2.3 Quantum eﬀects

It has recently been investigated, that in the spin ice compound Yb2Ti2O7 anisotropic exchange coupling predominates over long-range dipolar interactions [23]. Still, a ﬁrst-order phase transition for Yb2Ti2O7 at Tc ∼ 0.24mK could be detected [37]. Moreover, it was shown in Ref. [38] that there exists no long-range magnetic order below the critical value. In- stead the transition corresponds to a change of the ﬂuctuation rate of the Yb3+ spins. Above

Tc one observes thermal fluctuations whereas below Tc the spin fluctuation rate appears to be temperature-independent indicating the existence of quantum spin fluctuations. It seems like the anisotropic exchange coupling induces quantum phenomena and suppresses long-range magnetic ordering. Theoretically, these quantum spin ices can be investigated by using for example the S = 1/2 Heisenberg antiferromagnet on the pyrochlore lattice

X z z + − − + HH = JzzSi Sj − J± Si Sj + Si Sj (2.6) hiji 2.3. Quantum eﬀects 11

P P where t runs over all tetrahedra and hiji over all nearest-neighbor spins. Assuming J± Jzz allows a perturbative treatment of the low-energy physics. In Ref. [13] they investigated this model and it was found out, that those quantum spin ices enter a fractionalized quantum spin liquid phase for suﬃciently small J± with emergent U(1) gauge structure and exotic excitations like gapped "spinons", gapped "magnetic" monopoles and gapless "artiﬁ- cial photons". This phase is called Coulomb quantum spin liquid phase due to the similarities to conventional quantum electrodynamics.

However, for the sake of completeness, it should be emphasized, that this is only a theoretical model and that in real quantum spin ice materials even more couplings play a role leading to a Hamiltonian which comprises more terms than (2.6). In Ref. [15] they extracted all parameters of the model

X z z + − − + HH = JzzSi Sj − J± Si Sj + Si Sj hiji + + ∗ − − z + ∗ − + J±± γijSi Sj + γijSi Sj + Jz± Si ζijSj + ζijSj + i ↔ j (2.7) with matrices γij and ζij consisting of unimodular complex numbers, from high-ﬁeld inelastic neutron scattering experiments with a Yb2Ti2O7 sample. Results showed that J±± and

Jz± are non-negligible. Preliminary work actually suggests, that the Jz± interaction further stabilizes the quantum spin liquid phase. A schematic phase diagram has been derived as shown in Fig. 2.3.

In the scope of this thesis, we take the antiferromagnetic Ising model on the pyrochlore lattice at T = 0 and apply a transverse magnetic ﬁeld, which is an easier model in order to study quantum spin ice theoretically. The Hamiltonian reads

N X z z X x HTFIM = J σi σj + h σi (2.8) hi,ji i=1 where N denotes the number of sites of the lattice, J > 0 is the antiferromagnetic nearest- neighbor exchange coupling, h is an applied transverse magnetic field and σz and σx are Pauli matrices. As was shown earlier, the bare Ising term captures the frustration in spin ice. The transverse magnetic field induces quantum effects and lifts the extensive ground- state degeneracy in the system and thus makes it an appropriate model in order to expose some interesting properties of quantum spin ice. In case of high magnetic fields h, the system is in the conventional polarized phase, where all spins point into the direction of the transverse field. Since the model enters a CQSL phase for sufficiently small h, exactly the 12 Chapter 2. Quantum spin ice

Figure 2.3: Illustration of phase diagram of quantum spin ice taken from [15]. T denotes the temperature and H the transverse magnetic ﬁeld inducing quantum ﬂuctuations. one discussed for the antiferromagnetic S = 1/2 Heisenberg model (2.6), it has to undergo a phase transition. Note that this phase transition does not exist due to symmetry breaking. Instead the wave function of the polarized phase is a product state, whereas the CQSL phase is a disordered phase which cannot be described by such a product state. This gives rise to an existing quantum phase transition as a function of h.

This phase transition has only been investigated once in the past in [17] using mean-ﬁeld theory which can detect second-order phase transitions. They derived the critical value

(h/J)c = 0.7. In this thesis the critical point (h/J)c of the TFIM on the pyrochlore lattice is investigated via perturbative continuous unitary transformation (pCUT), which is a powerful method in order to find second-order phase transitions. Since it turns out, that this method does not provide any indications of a second-order transition, a first-order transition is found by comparing the perturbed ground-state energies from the low- and high-field regime. The next section therefore presents the TFIM on the pyrochlore lattice and provides necessary preliminary work in order to use pCUT. Chapter 3

Transverse Field Ising Model on the pyrochlore lattice

The antiferromagnetic TFIM is chosen to investigate spins on the pyrochlore lattice. It is used commonly to examine magnetism and quantum phase transitions in spin systems [39]. Although the TFIM represents one of the easier models in theoretical condensed matter physics, it can get very challenging when studying three-dimensional systems. In this chapter the model is introduced as well as suitable particle representations in the high- and low-ﬁeld limit, which makes it easier to understand the processes going on in the low-energy regime.

3.1 Model

The antiferromagnetic TFIM Hamiltonian reads

N X z z X x H = J σi σj + h σi (3.1) hi,ji i=1 where J > 0 is the coupling constant between two spins, h > 0 is the applied transverse magnetic ﬁeld and N the number of lattice sites. The ﬁrst sum runs over all nearest-neighbor spins which are located on sites of a lattice.

This model is only exactly solvable in one dimension with the help of the Jordan-Wigner- Transformation [40,41]. In all other cases it can only be approximately solved for instance it can be investigated in the high-ﬁeld limit (h J) and in the low-ﬁeld limit (h J) which makes a perturbative treatment possible as it is done in this thesis. As a starting point both

13 14 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice limits are discussed qualitatively. In general there are two kinds of lattices on which the antiferromagnetic TFIM can be investigated: Frustrated and unfrustrated systems. In case of a vanishing coupling constant J = 0, the only state which minimalizes the energy is the one where all spins point into the direction of the magnetic field h and thus forms the unique ground state independent of the underlying lattice (frustrated and unfrustrated). This phase is called polarized phase. An unfrustrated system undergoes a second-order phase transition at a critical value (h/J)c and is in the so-called ordered phase for sufficiently small fields h. In the ordered phase the system forms a classical antiferromagnet and the ground state for h = 0 is twofold degenerate which is also known as the Neel order. By selecting one of these ground states the symmetry of the Hamiltonian is broken which is the reason for the existence of the phase transition. An example for such an unfrustrated model is the antiferromagnetic TFIM on the bipartite square lattice. Frustrated systems in turn do not necessarily exhibit two phases or a phase transition, respectively. The Hilbert space of the ground state for h = 0 is extensively large and ground states are "disordered". By turning on an infinitesimal small field h some systems order ("order by disorder") and some stay disordered ("disorder by disorder"). Systems that follow the order by disorder phenomenon always have two phases and thus a phase transition as for instance the TFIM on the triangular lattice [42]. Some of the materials which are proved to stay disordered for infinitesimal applied fields h do not necessarily exhibit a phase transition as it was shown for instance in Ref. [10, 42], [43] and [8, 44] for the TFIM on the kagome lattice, on the diamond chain and on the sawtooth chain. The reason for the absence of a phase transition is that states for infinitesimal small h are adiabatically connected to the ground state of high fields h J so that these systems are therefore immediately in the polarized phase. In turn, the antiferromagnetic TFIM on the pyrochlore lattice, which is the topic of this thesis, is an example for a disorder by disorder scenario which exhibits a quantum phase transition not due to symmetry breaking but because of an emergent CQSL phase for low magnetic fields h.

We will now specify the phenomenon of frustration in greater detail which occurs in case of h = 0. As already mentioned, applying the antiferromagnetic Ising model on the square lattice does not lead to frustration because spins are able to arrange in a way such that all neighboring spins follow an antiparallel orientation. If we now try to arrange the spins analogously on a triangle, we will observe that we cannot satisfy every bond because there will always be two nearest-neighbor spins that point into the same direction and thus cause a ferromagnetic bond. This is called frustration and is depicted in Fig. 3.1 (a). Within the scope of this thesis, the spins are located on a pyrochlore lattice which consists out of corner-sharing tetrahedra. If we consider four spins occupying the corners of a tetrahedron and interacting via an antiferromagnetic nearest-neighbor Ising coupling, one will notice that 3.1. Model 15 this arrangement is highly frustrated because only four out of six bonds can be satisﬁed and two end up being ferromagnetic due to frustration (Fig. 3.1 (b)). ?

(a) Frustrated spins on triangle (b) Frustrated spins on tetrahedron

Figure 3.1: Figure (a) outlines frustrated spins on a triangle. In case the uncertain spin on the top points upwards the red colored bond is ferromagnetic and thus not satisﬁed. Figure (b) visualizes one arrangement of frustrated spins on a tetrahedron. The minimal number of ferromagnetic bonds is achieved if two spins point upwards and two downwards. This way there exist two ferromagnetic bonds which are marked red.

One can also extract from Fig. 3.1 (b) that this arrangement is obtained by two up and two down pointing spins per tetrahedron. Hence the ground state is formed if every tetrahedron on the lattice follows the rule "two spins up, to spins down" which enables us to draw a connection to spin ice where the ground state in the low-ﬁeld limit was determined by the ice-rules "two-in-two-out". The ground state for the Ising model with no applied magnetic ﬁeld ends up having the same extensive degeneracy as the ground state in spin ice materials and therefore represents a suitable approximation to describe spin ice materials. To show qualitatively that the ground state is formed by two up and two down pointing spins per tetrahedron one can rewrite the Ising Hamiltonian in the following way:

X z z X X z z HIsing = J σi σj = J σi σj (3.2) hi,ji t hi,ji ! J X X X = σzσz − (σz)2 (3.3) 2 i j i t i,j ∈ t i ∈ t | {z } = const = 4 2 ! J X X = σz − 4 (3.4) 2 i t i ∈ t J X 2 = −JN + σz (3.5) 2 tot,t t 16 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice

P P z z where t runs over all tetrahedra and N is the number of spins. The sum i ∈ t σi = σtot,t calculates the total z-component of the spin of one tetrahedron. Thus the energy is minimized, z if σtot,t of all tetrahedra is zero. Taking into account that there are 1/2 N tetrahedra in a pyrochlore lattice, one gains the unperturbed ground-state energy per site E0/N = −J.

Inducing quantum fluctuations, which means that we now consider the full Hamiltonian (3.1) on the pyrochlore lattice, leads to exotic phases and excitations in case of low magnetic fields. Before this scenario is discussed we will have a short look at the high-field case. As already mentioned above, the ground state is not degenerate and all spins are aligned in direction of the applied magnetic field h. As for the ground-state energy every tetrahedron takes the energy −4h. Every spin flip acts as an excitation and lifts the energy of a tetrahedron by +2h. Thus the energy spectrum is equidistant with the energy levels −4h, −2h, 0, +2h and +4h for each tetrahedron. Note that this property is one reason why pCUT, which is presented in section 4.1, is a suitable and efficient method to describe the low energy physics of the system in case of large magnetic fields.

Contrarily, in the low-field regime, where the ground state in case of h = 0 is formed by two spins up and two spins down per tetrahedron which results into an infinitely large ground- state Hilbertspace in the thermodynamic limit, it was stated in [13] that the system is in a CQSL phase. This phase exhibits intriguing and exotic excitations like gapless photons, magnetic monopoles or "spinons" which carry fractional quantum numbers. However, as for the perturbative treatment of this phase, single spin flips that are caused by the transverse magnetic field are of prior importance within the scope of this thesis. Tetrahedra can be in one of 24 = 16 different states with three distinct energies. In the ground state every tetrahedron takes the energy −2J which derives from four antiferromagnetic and two ferromagnetic bonds z z z z z and is equivalent to a total spin |σtot| = |σ1 + σ2 + σ3 + σ4| = 0 where the indices 1 to 4 enumerate the unit cell sites. The elementary excitation is again a spin flip and lifts the energy of two corner-sharing tetrahedra to zero. In this state these tetrahedra have three z spins pointing in one direction and one spin pointing in the opposite direction (|σtot| = 2) which leads to three antiferromagnetic and three ferromagnetic bonds per tetrahedron and thus energy zero. A second spin flip can either transfer a tetrahedron back in a ground state again by flipping one of the three spins that are aligned equally or place it in an even higher excited state by flipping the only left spin with a different orientation which would lead to z a total spin |σtot| = 4 and exhibits the energy +6J because of emerging six ferromagnetic bonds. Possible energy states of a single tetrahedron are depicted in Fig. 3.2. Thus the energy spectrum is not equidistant but holds the energy states −2J, 0 and +6J. We will later see that even in this case pCUT can be a useful method to investigate low energy physics in this regime by artificially adding the energy levels +2J and +4J. 3.1. Model 17

E 0 = -2J E1 = 0 E2 = +6J

Figure 3.2: Depiction of some representative spin configurations for each energy state of a tetrahedron in the low-field Ising limit. The red lines illustrate ferromagnetic bonds. All states which exhibit z z |σtot| = 0 are in the energy state E0 = −2J. A total spin |σtot| = 2 results into the energy E1 = 0. z Every configuration with |σtot| = 4 has the energy E2 = +6J.

One fact that makes the perturbative treatment of this phase very interesting and not trivial is the tunnelling between different ground states which is permitted by quantum fluctuations. If we restrict the series expansion to order 6 for a moment, one will notice that only tunnelling between ground states which differ only by a flipped 6-link loop are permitted. The requirement to the 6-link loop is that spins are arranged in alternating orientation. This way flipping each spin on the loop does not change the total spin of any tetrahedron and thus does not lift the energy because two neighboring loop spins always belong to the same tetrahedron leaving the ground-state configuration of "two spins up, two spins down" per tetrahedron unchanged. This scenario is shown in Fig. 3.3.

(a) 6-link loop in pyrochlore (b) Tunnelling of ground states via 6-link loop

Figure 3.3: Figure (a) shows a 6-link loop (marked as a bold green line) in the pyrochlore lattice. In case up- and down-spins are alternating on the sites of the loop, tunnelling between these two ground states, that only differ by one flipped 6-link loop, is permitted in order 6 and higher. This is depicted in figure (b) where the two colors green and red stand for either up pointing or down pointing spins.

Using degenerate perturbation theory to calculate the ground-state energy one gains an 18 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice eﬀective Hamiltonian of the form

O(6) X + − + − + − Heﬀ = e0 + Jring σ1 σ2 σ3 σ4 σ5 σ6 + h.c. (3.6)

7 where the sum runs over all hexagons in the pyrochlore lattice, E0 represents the diagonal part of the ground-state energy until order 6 and thus gives the energy of the system in case of no tunnelling. The non-diagonal term, the so-called ring exchange model, is investigated in more detail in [13, 14, 17] and was numerically solved in [14] based on quantum Monte Carlo simulations. Note that this eﬀective Hamiltonian is valid until order 8 because there do not occur any additional processes. We will later determine explicit expression for E0 and

Jring up to order 8 in h/J.

One can now ﬁnd particle representations of the Hamiltonian (3.1) in both limits to describe the low energy physics which makes the treatment of these regimes easier and serves as a preparation for using pCUT to ﬁnd series expansions in the perturbation parameter of the ground-state energy and the dispersion.

3.2 Particle representation

The particle representations of both limits is equivalent to the spin representation but will make it easier to describe the low energy physics and provides a more conceivable picture of the underlying low-energy physics in the system. In the first section the high-field limit is discussed whereas in the second section the low-field limit is represented. For both limits we will introduce appropriate quasi particles which describe the low-energy physics.

3.2.1 High-ﬁeld limit

The quasi particles that are used to describe the polarized phase are hardcore-bosons that correspond to spin ﬂips in the original spin language, which behave like fermions on-site but are bosonic on diﬀerent sites. As a starting point we take the Hamiltonian (3.1) and rotate the basis with the help of a unitary transformation so that the new z-axis lies on the old x-axis and the new x-axis points into the direction of the old z-axis which makes the description more convenient since everything can be described in the eigenbasis of σz. The Hamiltonian 3.2. Particle representation 19 then reads

N X x x X z H = J σi σj + h σi (3.7) hi,ji i=1

In the ground state for J = 0 all spins point downwards because of σz |↓i = − |↓i which we will now interpret as a state where no particles exist. Every spin ﬂip is taken as a particle on the site where the spin is located. This can be summarized by

|↓ii ˆ= |0ii und |↑ii ˆ= |1ii (3.8) where i can take the values 1 to N and stands for the lattice site. This way the Hamiltonian † † can be expressed with the help of creation and annihilation operators bi and bi , where bi creates a particle on the site i and bi annihilates a particle on the site i. Therefore the following counts:

bi |0ii = 0 and bi |1ii = |0ii , (3.9) † † bi |0ii = |1ii and bi |1ii = 0 . (3.10)

x z Based on the knowledge that σi engenders a spin ﬂip and σ is diagonal in the spin and particle representation with the eigenvalues +1 of the eigen state |1i and -1 belonging to the † eigen state |0i, one can also rewrite these operators and replace them by bi and bi according to the Matsubara-Matsuda transformation as introduced in [45] in the following way:

z † x † σi = 2bi bi − 1 = 2ni − 1 and σi = bi + bi (3.11)

P where Q ≡ i ni is the operator that counts the number of quasi particles in the system.

Applied on one single site i it thus follows the equations ni |0ii = 0 and ni |1ii = |1ii. Hence the Hamiltonian (3.1) reads

N H 1 X J X Hhf := = (2n − 1) + (b + b†)(b + b†) (3.12) QP 2h 2 i 2h i i j j i=1 hi,ji N N X J X = − + n + (b†b + b b† + b b + b†b† ) (3.13) 2 i 2h i j i j i j i j i=1 |{z} hi,ji | {z } |{z} |{z} x T0,hi,ji T−2,hi,ji T+2,hi,ji N = − + Q +x (T + T + T ) := H + xV . (3.14) 2 0 −2 +2 0 | {z } | {z } H0 V 20 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice

in the particle representation. The operator T0 annihilates a particle, if one is there, and creates one on the neighboring site and thus is responsible for particle hopping. T−2 annihilates while T+2 creates two particles on neighboring sites which is the reason why they are not particle-conserving opposed to T0. From (3.14) we gain, that the unperturbed part H0 has an equidistant ladder energy spectrum where every creation of a particle costs energy 1.

The T -operators follow the commutator relation [Q,Tn] = nTn.

3.2.2 Low-ﬁeld limit

As for the low-field limit one can also find an appropriate particle representation which is a bit more complex and not as straight forward as the representation in the high-field limit. Let us start with the Hamiltonian (3.1) where the Ising term is rewritten as shown in (3.5)

N J X 2 X H = −JN + σz + h σx (3.15) 2 tot,t i t i=1 N J X 2 X = E + σz + h σx . (3.16) 0 2 tot,t i t i=1

As already discussed in section 3.1, the energy levels of one tetrahedron are -2J, 0 and +6J. In case the states +2J and +4J existed as well, dividing by 2J would lead to an energy spectrum without units where the diﬀerence between energy levels is 1. This is a subtle approach in order to be able to use the pCUT method. Lifting the whole energy spectrum of every tetrahedron by one does not change the physics, but puts the ground-state energy to zero and thus makes it more convenient. The renormalized Hamiltonian then reads

N H N 1 X 2 h X Hlf := + = σz + σx (3.17) 2J 2 4 tot,t 2J i t i=1 where the ﬁrst sum represents the unperturbed part and the second sum describes the perturbation. The energy spectrum is depicted in Fig. 3.4 which shows the three renormalized energy levels of one tetrahedron. Note that the energy spectrum of the whole system can of course be in a state with energy 2 or 3 if there are two respectively three tetrahedra with energy state 1.

We do now introduce three kinds of hardcore bosons g, a and b with different flavors α, which are located in the center of the tetrahedra and which characterize the 24 = 16 different states the tetrahedra can take. Particle g has six flavors and is present if a tetrahedron is in the ground state. The a particles exhibits eight different flavors which cost energy 1, whereas 3.2. Particle representation 21

Figure 3.4: Equidistant energy spectrum of one tetrahedron. Energy levels differ by 1. Dashed lines indicate that these levels do actually not exist. particle b has energy 4 and two flavors. Therefore particles a and b exist if a tetrahedron is z z in an excited state and correspond to |σtot| = 2 and |σtot| = 4 in the spin language. The different particles, energy states and their degeneracy are depicted in Fig. 3.5.

Figure 3.5: Summary of the three energy levels, their degeneracy and corresponding hardcore bosons with ﬂavor α which map the spin model onto a quasi particle representation.

The usual hardcore boson constraints count which also means that at most only one particle g, a or b with a specific flavor can exist per tetrahedron center. With the help of the new P x particles the perturbation h/(2J) i σi in (3.17) can now be expressed with T-operators as in the high-field case. We remind ourselves that σx flips a spin that belongs to two tetrahedra and hence it always changes the energy state of both of them which means that particles can only be created or annihilated in attached tetrahedra. Some of the possible states that can be 22 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice achieved by applying σx and thus flipping a spin are shown in Fig. 3.6. The numbers inside the tetrahedra specify the energy of the tetrahedron. Therefore the number in the indices of the T -operators relates to the change of energy and can be interpreted as an effective number of a particles that were created or annihilated because they carry energy 1. One will notice that only transitions where an even number of a particles net are created respectively annihilated exist, as Fig. 3.6 already suggests. The Hamiltonian (3.17) can now be expressed in second quantization and is given by

lf X † X † HQP = at,α at,α + 4 bt,αbt,α t,α t,α | {z } Q h X h X 0 X 0 + U ββ a† g† g a + V ββ b† a† a b 2J αα0 t,β t0,β0 t,α t0,α0 αα0 t,β t0,β0 t,α t0,α0 ht,t0i α,α0 α,α0 β,β0 β,β0 | {z } T0,ht,t0i X ββ0 † † X ββ0 † † + Wαα0 at,βat0,β0 gt,αgt0,α0 + Xαα0 bt,βgt0,β0 at,αat0,α0 α,α0 α,α0 β,β0 β,β0 | {z } T+2,ht,t0i X ββ0 † † X ββ0 † † i + Yαα0 at,βbt0,β0 gt,αat0,α0 + Zαα0 bt,βat0,β0 at,αbt0,α0 +H.c. α,α0 α,α0 β,β0 β,β0 | {z } | {z } T+4,ht,t0i T+6,ht,t0i h ≡ Q + (T + T + T + T + T + T + T ) (3.18) 2J 0 +2 −2 +4 −4 +6 −6 |{z} |{z} | {z } H0 x V

= H0 + xV . (3.19)

This Hamiltonian now acts on the center of the tetrahedra that form a diamond lattice. The T-operators comprise the sum over all corner-sharing tetrahedra which is meant by ht,t0i as ββ0 ββ0 well as the ﬂavor α. The transition elements Uαα0 to Zαα0 take the values 1, if the transition α,α0 → β,β0 is possible or 0, if the transition cannot take place according to the original physics described in the spin language. The introduction of a ﬂavor α is essential in order to maintain the number of states and thus not to reduce the size of the Hilbert space of the spin system. We will see in the next chapter that the ability to express the perturbed part of the Hamiltonian by particle changing T-operators is one important property which allows us to calculate energies to high orders in the perturbation parameter based on the pCUT method. 3.2. Particle representation 23

Figure 3.6: This ﬁgure illustrates some possible changes of energy states of two corner-sharing tetrahedra by applying σx to the spin which connects these tetrahedra. The numbers in the center of each tetrahedron represent the energy. Particles a and b are represented as light blue and orange balls. Chapter 4

Series expansion methods

This chapter gives an introduction about series expansion methods, that are used in this thesis, and particularly focuses on pCUT which is a powerful method in order to calculate high order series expansions of energies (or other physical quantities) in a perturbation parameter as for instance high or low magnetic fields in the TFIM. It additionally provides a better and more physical picture of the underlying physics. The first section therefore deals with pCUT and how the effective pCUT Hamiltonian can be obtained based on continuous unitary transformation (CUT). The second part is about extrapolation schemes, which is especially useful when working with series expansions, because they often provide a smoother and more realistic curve compared to the bare series.

4.1 Perturbative Continuous Unitary Transformation

The pCUT method [46,47] has already been applied to numerous models of condensed matter problems like in [10] or [21]. It allows us to calculate quantities in the thermodynamic limit to high orders in the perturbation parameter which serves as a truncation. One scope of application is to detect second-order quantum phase transitions by determining the ground- state energy and the ﬁrst excitation and thus the elementary excitation gap. The pCUT method is based on CUT, which is introduced in the next section. A more detailed description can be found in the PhD thesis [48] of Christian Knetter.

4.1.1 Continuous Unitary Transformation

The CUT method was ﬁrst proposed by Wegner [49] and simultaneously by Glazek and Wilson [50,51] and is based on ﬁnding a suitable basis for the physical many-body problem in which

24 4.1. Perturbative Continuous Unitary Transformation 25 the Hamiltonian is diagonal and which preferably provides a better understanding of the underlying low-energy physics. One cannot always find an appropriate unitary transformation for physical problems that comply with the previous mentioned points. However, let us first consider a trivial example, where it is possible to determine a unitary transformation without great effort, i.e. the spin-1/2-system in z-basis with the spin states |↑i and |↓i. The operator σz is diagonal, whereas σx induces spin flips. By applying the unitary transformation ! 1 1 1 U = √ (4.1) 2 1 −1 √ to the spin-flip operator, σx becomes diagonal with the eigenstates (|↑i + |↓i)/ 2 and (|↑i − √ |↓i)/ 2 which would be a suitable basis in order to describe a spin-1/2-system that is coupled to a magnetic field in x-direction. In case of more complicated systems with larger Hilbert spaces, it can get really challenging to find a unitary transformation so that the problem becomes diagonal, but still there are methods that can be used to diagonalize a matrix step by step by applying a series of unitary transformations, i.e. the Householder or Jacobi algorithm [52]. These methods have two disadvantages. First, we cannot diagonalize an infinitely large Hilbert space and we thus need to limit the diagonalization scheme to a finite subspace. The second disadvantage is, that the transformation to a new basis is uncontrolled which might not provide any better understanding of physical processes. Because of this the main idea of CUT is to rotate the basis in a continuous and controlled fashion. We therefore define our Hamiltonian to be dependent on the so-called flow parameter l

H → H(l) (4.2) with l ∈ [0,∞]. Furthermore, we set

H(0) = H and H(∞) = Heff (4.3) which means that in case of l = 0 the Hamiltonian equals the initial Hamiltonian H and for l → ∞ one gains an effective Hamiltonian which is (block-) diagonal in a desired basis. We now introduce a unitary transformation U(l) also dependent on the flow parameter. The Hamiltonian thus transforms locally like

H(l) = U(l)H(0)U †(l) . (4.4) 26 Chapter 4. Series expansion methods

The evolution of U(l) itself is governed by an inﬁnitesimal anti-hermitian generator η = −η† and reads

dU(l) = η(l)U(l) . (4.5) dl

One can now derive the so-called ﬂow equation which deﬁnes the evolution of H(l)

dH(l) = [η(l),H(l)] . (4.6) dl

In order to obtain an eﬀective Hamiltonian, the aim is that the ﬂow of H(l) vanishes and thus does not change anymore in the limit l → ∞. This entails that in this limit

[η(l → ∞),Heﬀ ] = 0 and therefore η(l → ∞) = 0. Wegner ﬁrst proposed the generator

η(l) = [Hd(l),H(l)] (4.7) where Hd denotes the diagonal part of the Hamiltonian. In this case a vanishing

η(∞) = [Hd(∞),Heff ] would either indicate, that Heff became zero during the flow, the un- interesting case, or that Heff and Hd commute with each other and thus a common basis can be found. In case the elements of Hd are sorted according their value, this means that Heff reached a block-diagonal structure which was desired in the first place. The drawback of the generator (4.7) is that zero elements in the initial Hamiltonian do not necessarily stay zero during the flow, which significantly enhances the computational effort.

Therefore a diﬀerent generator is proposed which is based on the fact that physics of many condensed matter models can be described by quasi particles where creation of a particle corresponds to an excitation of the system. The number of quasi particles qi in a system can be counted by the operator Q that has the eigenstates |ii and thus follows the eigenvalue equation Q |ii = qi |ii. The aim is to obtain a block-diagonal Hamiltonian which acts only in subspaces of the Hilbert space where states exhibit the same number of quasi particles and which therefore conserves the number of particles in the system (see right illustration in Fig. 6.2) . The so-called quasi-particle generator in an eigenbasis of Q which fulﬁls this is given by

ηij(l) = sgn(qi − qj)hij (4.8) and was independently introduced by Uhrig and Knetter [53] for second quantization and Mielke [54] for band-matrices. The advantage of this generator is, that initial non-diagonal zero elements stay zero during the flow. The only restriction which has to be made is, that 4.1. Perturbative Continuous Unitary Transformation 27 the energy spectrum of H is bounded from below which applies for most physical problems anyway. The resulting effective Hamiltonian is particle-conserving and therefore enables us to investigate the zero and one particle channel separately which corresponds to the ground-state energy and the first excitation.

The flow equation (4.6) in general provides an infinite set of recursive first-order differential equations. Hence, we need to find an appropriate truncation scheme to solve the flow equation. Various truncation schemes have already been proposed and applied like self-similar CUT [55], graph based CUT [56] or the directed evaluated CUT [57]. In this thesis we will use a perturbative ansatz in order to derive the effective Hamiltonian which we can use for calculating quantities exactly up to a certain order n in the perturbation parameter. This is the pCUT method presented in the next section.

4.1.2 Perturbative CUT

To this point, we just presumed the existence of quasi particles, which describe the excitation of the system, and that the energy spectrum of the Hamiltonian is bounded from below. We will make further restrictions to our Hamiltonian in order to be able to use a perturbative ansatz which solves the ﬂow equation (4.6) model independently.

1. The Hamiltonian can be decomposed into H = H0+xV for a suﬃciently small perturba-

tion parameter x. H0 denotes the unperturbed part whereas V acts as the perturbation.

2. The unperturbed part H0 has an equidistant energy spectrum bounded from below. The diﬀerence between energy levels corresponds to the energy of one quasi particle.

PN 3. V can be written as V = n=−N Tn, where the Tn-operators, that follow the commutator relation [Tn,H0] = nTn, increment or decrement the number of particles in the system by n. N ∈ N is the ﬁnite maximum number of particles that can be created respectively annihilated by H.

Even if the energy spectrum is not equidistant, restriction 2 can often be achieved by rewriting the Hamiltonian without changing the underlying physics as it was done in case of the low- ﬁeld limit of the TFIM in section 3.2.2. The more essential restriction is number 3 which only allows a band structured quasi particle Hamiltonian which means that only subspaces of the Hilbert space with a particle diﬀerence of at most N are coupled. This is depicted for the case V = T0 + T+2 + T−2 in the left illustration in Fig. 6.2. As a perturbative ansatz, one 28 Chapter 4. Series expansion methods takes

∞ X n X H(x; l) = H0 + xV (l) = H0 + x F (l; m)T (m) (4.9) n=1 |m|=n with

m = (m1,m2,m3,...,mn) with (4.10)

mi ∈ 0, ± 1, ± 2,..., ± N (4.11) |m| = n (4.12)

T (m) = Tm1 Tm2 Tm3 ...Tmn (4.13) n X M(m) = mi . (4.14) i=1

The unperturbed part is stated to be already diagonal, which is the reason why it does not depend on the ﬂow parameter l in order to maintain it. The ansatz for the unperturbed part contains a sum over the power n of the perturbation parameter x and over all possible T -sequences weighted by real functions F (l; m). Inserting this ansatz into (4.8) leads to the particle-conserving generator

∞ X X η(x; l) = xn sgn(M(m))F (l; m)T (m) . (4.15) n=1 |m|=n

Solving the differential equations that are gained by plugging (4.9) and (4.15) into the flow equation yields the real functions F (l; m). The highest order nmax to which the effective Hamiltonian is calculated is restricted by the computation time. The effective Hamiltonian reads

nmax X n X Heff (x) = H0 + x C(m)T (m) (4.16) n=1 |m|=n, M(m)=0 where the C(m) ≡ F (∞; m) are rational numbers that are obtained by the real functions F (l; m) in the limit l → ∞. Determining the coefficients C(m) shows, that all coefficients in front of T -sequences that are not particle-conserving vanish, which is indicated by M(m) = 0 in (4.16). Therefore (4.16) is block-diagonal in the particle representation and thus is particle-conserving because it only acts in subspaces with a constant number of particles. We emphasize that no model dependent properties have been incorporated to far. It was 4.1. Perturbative Continuous Unitary Transformation 29

PN just presumed that the Hamiltonian can be written as H = H0 + xV with V = n=−N Tn which exhibits an equidistant energy spectrum bounded from below. This makes the pCUT method extremely versatile. The particle-conserving Hamiltonian can be written as

X Heﬀ = Hn (4.17) n where Hn is the operator that acts in the n particle subspace. Depending on the model, the n particle energy can be determined by diagonalizing Hn.

Another essential property of the Hamiltonian is the cluster additivity, which is explained in the next section.

4.1.3 Cluster decomposition and cluster additivity

As already mentioned, the perturbed part V can be written as a sum over diﬀerent number of particle changing T -operators. The T -operators act on the whole lattice indeed, but still they are changing the number of particles only in a certain region. This is the reason, why they can be further split into a sum over τ-operators, that only act on ﬁnite clusters

X Tm = τm,k . (4.18) k

The index k specifies the region, on which the τ-operator acts and m still denotes the number of particles that are created or annihilated. In case of nearest-neighbor Ising coupling, the τ-operators act on bonds and thus k can be replaced by hi,ji. The effective Hamiltonian can now be decomposed into sums that act on finite clusters Cn

nmax X n X X Heﬀ = H0 + x C(m) τm1,k1 τm2,k2 ...τmn,kn (4.19) n |m|=n, k1k2...kn M(m)=0

nmax X n X = H0 + x T (Cn) (4.20)

n {Cn} with

X X T (Cn) = C(m) τm1,k1 τm2,k2 ...τmn,kn . (4.21) |m|=n, k1k2...kn=Cn M(m)=0 30 Chapter 4. Series expansion methods

It can be shown that τ-sequences acting on disconnected clusters vanish so that only the sequences that act on connected clusters remain. Thus the whole energy can be calculated on finite clusters in the thermodynamic limit only truncated by the order of the perturbation parameter. This is called cluster additivity which lays the foundation for the full graph decomposition presented in chapter 5. Before discussing the calculation on graphs, two extrapolation schemes, Padé- and dlogPadé-approximation, are introduced.

4.2 Extrapolation

Energies in this thesis are calculated as series expansions to high orders in a perturbation parameter. The drawback of series expansions is, that the radius of convergence can be small and strongly dependent on the degree of the polynomial. Beyond this radius, the function will diverge to ±∞ depending on the sign in front of the highest power of the perturbation parameter and is thus not reliable anymore at this point. However, the interesting physics like a quantum phase transition can occur for larger values of the perturbation parameter. In order to derive a smoother function which predicts a more realistic development of the curve beyond the radius of convergence than the bare series expansion does, one can use extrapolation schemes. The Pad´e- as well as the DlogPad´e-extrapolation, which are used in this thesis are presented in this section. A more detailed description can be found in [59].

4.2.1 Pade-approximation´

The Pad´e-approximation is named after the French mathematician Pad´e and is based on the idea of extrapolating a series expansion

nmax X n F (x) = anx (4.22) n=0 of order nmax by ﬁnding a rational function whose Taylor expansion until this order is equal to the initial series (4.22). The Pad´e-approximant is a quotient of two polynomials PL and

QM of degree L and M and reads

L PL(x) p0 + p1x + ... + pLx [L/M]F ≡ ≡ M . (4.23) QM (x) q0 + q1x + ... + qM x

It can be calculated by solving two systems of linear equations. The coefficients pi and qi are unique which means that [L/M]F is unambiguous. The first L + M + 1 coefficients of the 4.2. Extrapolation 31

Taylor expansion of the Pad´e-approximant [L/M]F is equal to the series F (x). Thus

L + M + 1 ≤ nmax + 1 ⇔ L + M ≤ nmax (4.24) must be adhered to when choosing L and M. Padé-extrapolants with L+M sufficiently close to nmax usually give reliable results, but still can exhibit poles which might not always be of physical nature. In general it is said, that the more Padé-approximants lie on or at least almost on top of each other, the more secure it is that these extrapolants represent the real physical nature of the system. The drawback of this extrapolation scheme is, that no critical behavior of quantities can be detected. We therefore introduce the DlogPadé-extrapolation in the following section.

4.2.2 DlogPade-extrapolation´

As already mentioned, Padé-extrapolation cannot detect critical behavior which occurs phys- ically close to phase transitions. We therefore introduce the DlogPadé-extrapolants which are useful to examine critical exponents and which are based on taking the Padé-approximant to the logarithmic derivative of F (x)

d F 0(x) ln F (x) = . (4.25) dx F (x)

Since the derivative of F (x) is taken, L + M now has to fulﬁl L + M ≤ nmax − 1 in order to extrapolate F (x) accurately. The DlogPad´e-extrapolant is given by

Z x 0 Dlog[L/M]F (x) = exp [L/M](F 0/F )dx . (4.26) 0

The reason why critical points and exponents can be detected with DlogPad´e originates from the assumption, that the series expansion F (x) exhibits a pole xc with critical exponent α

nmax α X x F (x) = a xn ∼ A(x) 1 − + B(x) . (4.27) n x n=0 c

One then observes that in the limit x → xc the logarithmic derivative of F (x) (4.25) shows the same critical behavior as the original series expansion

d α ln F (x) = − [1 + O(x − xc)] (x → xc) . (4.28) dx x − xc Chapter 5

Full graph decomposition

In this chapter the full graph decomposition is introduced which is an important method that improves the computation time significantly of series expansions described by Hamiltonians which meet the linked cluster theorem as introduced in section 4.1.3. We will later make use of this graph expansion for the high-field limit in the TFIM in order to calculate the ground-state energy and the dispersion to high orders in the perturbation parameter. Graph expansions for the low-field limit are much more difficult which is the reason why the ground- state energy for this regime will not be calculated on graphs. For reasons of clarity the whole graph decomposition will be explained with the help of the square lattice. Graph expansion for the pyrochlore lattice is then straight forward.

The main idea of graph decomposition is, that topological equivalent structures in the cluster lead to the same contribution eventually. Some of these structures are presented in Fig. 5.1. The next section deals with graphs, their deﬁnition and how to represent them mathematically.

5.1 Graph representation

In graph theory a graph consists of vertices which are connected by edges or bonds. Within the scope of this thesis we will use simple undirected graphs, where undirected means that edges do not have any direction and simple indicates that vertices are connected at most by one single edge. We call a graph disconnected if there exist two vertices such that no path in that graph contains those two vertices, otherwise it is said to be connected. Fig. 5.2 shows examples of simple undirected connected and disconnected graphs. Mathematically those

32 5.2. Graph generation 33

Figure 5.1: Illustration of some topological equal graphs on a cluster. Same colors indicate that these graphs will give the same contribution to the energy.

(a) connected graph (b) disconnected graph

Figure 5.2: Depiction of some graphs. Figure (a) shows a simple undirected connected graph, ﬁgure (b) depicts a simple undirected disconnected graph. graphs can be described by adjacency matrices A. Therefore one has to enumerate the n vertices of a graph from 1 to n. The elements of those matrices are either zero or one, where

Aij = 1 (Aij = 0) means that vertex i and j are connected (disconnected). The diagonal part

Aii is set to zero by deﬁnition. To minimize the memory allocation as well as to ﬁgure out and generate topological equivalent graphs it is convenient to interpret the upper triangular matrix as a binary number which will further be explained in the next section.

5.2 Graph generation

One convenient way when working with graphs is to assign an integer number which is gained by interpreting the upper diagonal adjacency matrix as a binary number. It will turn out that one graph has more than one of those integer numbers depending on the enumeration 34 Chapter 5. Full graph decomposition of the vertices which will now be called graph keys. The highest graph key is deﬁned as the graph number. As an example consider Fig. 5.3 which sketches the calculation of the graph number. Permuting the enumeration of the graph vertices leads to diﬀerent keys. In order to identify the highest key one has to calculate this integer number for all possible enumerations of the vertices. 0 1 1 0 1 1 0 1 1 4 2 |1 |1 |0 |1 |1 |0| = 54 3 1 1 0 0 0 1 0 0

0 1 1 1 2 1 0 1 0 4 1 |1 |1 |1 |1 |0 |0| = 60 3 1 1 0 0 1 0 0 0

0 1 1 1 3 1 0 1 0 4 1 |1 |1 |1 |1 |0 |0| = 60 2 1 1 0 0 1 0 0 0

Figure 5.3: Sketch of how to calculate the graph keys. The adjacency matrix can be constructed by enumerating the vertices. The graph key can be calculated by interpreting the upper triangular matrix as a binary number.

Fig. 5.3 also suggests that there are some enumerations which end up having the same graph key. The number of permutations with an equivalent graph key is called the symmetry factor which will be important later when all the graph contributions are merged with the embedding factors in order to compute the accurate energy. Finding the graph number can be very expensive especially if one has to go through all permutations of vertex numbers. One fact which lowers the computation time significantly is the property, that only enumerations where the vertex with the highest number of neighbor vertices is labeled as vertex number one. This way the first row of the upper triangular matrix contains many ones which results into a binary number where the bits with a dominant contribution to the corresponding integer number are one as well. An efficient algorithm of how to compute those graph numbers including their symmetry factor can be found in [58]. Another fact which makes the representation of graphs by their graph numbers even more convenient is, that topological equivalent graphs have the same graph number. Hence it is possible to identify all topological 5.2. Graph generation 35 distinct graphs and avoid redundant calculations.

Let us assume we want to calculate all relevant graphs with k vertices in order n of the perturbation parameter. As we learned from section 4.1, only connected graphs can have a contribution as well as the maximum number of τ-operators in the τ-sequences in the resulting effective Hamiltonian is equal to the order n. This and the additional fact that we only allow nearest-neighbor interaction entails that the maximal number of bonds within a graph cannot be higher than the order n, otherwise the graph will not have a contribution. Therefore in order to generate all the relevant topological distinct graphs with k vertices, one has to connect k vertices with (k - 1) to n bonds in all possible ways and check, if the graph is connected and if it can be embedded into the underlying lattice. This involves an enormous computational effort because a lot of redundant graphs are generated. A more efficient algorithm which avoids the generation of disconnected graphs and even more redundant calculations can be summed up in the following way.

Loop over all the graphs with k − 1 vertices (we assume we already calculated them). In every loop proceed as follows:

• Add a bond without adding an additional vertex by connecting vertices of the graph which already exist and which are not linked yet.

• Calculate the graph number and check the subsequent requirements:

1. Is the graph number unique respectively was a topological equivalent graph not yet calculated? 2. Is the maximum number of neighbors of vertices smaller or equal to the coordination number of the underlying lattice? 3. Can the graph be embedded into the lattice?

Only if all the statements are true, save the graph.

• When all possible vertices have been linked once, add another vertex and connect it once to every already existing vertex. In every step repeat the previous point.

This way a lot of redundant calculations are avoided. Another restriction to contributing graphs is the so called double touch property, which is explained in the next section in more detail. Once all the graphs are calculated it is about gaining the contribution of those graphs. 36 Chapter 5. Full graph decomposition

5.3 Reduced energies

This section presents an efficient way to calculate the contributions of each graph to the resulting energy. We get those contributions by applying the effective pCUT Hamiltonian (4.20) to each graph. One can save a lot of computation time by identifying those τ-sequences which cannot contribute and discarding them right away. Let us consider the ground state for a moment where no particles are present. The only τ-operator which can be applied at the very beginning is τ+2 because τ0 and τ−2 need particles to have a contribution. Thus Q P check after every τ-operator in a sequence m τm if m m ≥ 0, where the sum goes over all indices of τ-operators that have already been applied. In case the sum is zero make sure that the next τ-operator is τ+2. Extending this to the first excitation where one particle exists is straight forward. To additionally save computation time the τ-sequences can be sorted in a way, that no redundant calculations are made.

τ0τ0τ0τ0τ−2τ+2

τ−2τ+2τ0τ0τ−2τ+2

τ0τ−2τ+2τ0τ−2τ+2 and so on...

This way it is not necessary to start over the whole calculation for each τ-sequence. After applying the eﬀective Hamiltonian under consideration of the previous mentioned time saving arrangements to a graph one gains an energy that contains contributions of subgraphs. In order to obtain the correct contribution one has to subtract all contributions of subgraphs which is visualized in Fig. 5.4. The result is then called reduced energy of the graph and reads

X 0 0 (g) = h0| Heﬀ |0i − m(g )(g ) (5.1) g0∈g as for the ground-state energy where g0 denotes subgraphs of g and m(g0) reﬂects how often subgraph g0 is contained in graph g. Reduced energies of particle hoppings on graph g from site i to j are given by

X 0 0 tij(g) = hi| Heﬀ |ji − m(g )tij(g ) − δij(g) . (5.2) g0∈g

Note that (g) has to be subtracted in case of the local hopping i = j in order to avoid double counting. 5.3. Reduced energies 37

_ _ _ _ _ = 2

_ _ = 3 3

Figure 5.4: Illustration of how to gain the reduced energy of a graph. Graphs in red color stand for the energy which is calculated by applying the full Hamiltonian. Blue colored graphs represent the reduced energies. To attain the reduced energy of a graph, one has to subtract all the contributions of subgraphs.

As one can imagine calculating the reduced energies this way leads to a huge amount of redundant steps because applying the whole eﬀective Hamiltonian (4.20) includes a lot of calculations on subgraphs which have already been done in previous steps and which are eventually subtracted anyway. In order to eﬃciently compute the reduced energies of graphs, it is essential to stop calculations as soon as it is certain that this calculation belongs to a contribution of a subgraph. The following so-called "double touch" property is build on the fact that τ-sequences applied on a graph with n bonds which act only on m < n bonds cannot be part of the reduced energy because they obviously acted not on the whole graph but on a subgraph.

Let us consider the ground state where no particles exist in the system. We only have operators that create/annihilate two particles on neighboring vertices and induce a hopping of a particle to a nearest-neighbor site. Thus τ+2 is the only operator that can be applied in the beginning. To end up in a ground state again, we have to act on the same link with

τ−2 to annihilate the two particles again, hence every bond has to be touched at least twice which is a property that does not change by taking τ0 also into account. Note that the double touch property does not apply for loop graphs.

As for the ﬁrst excitation where one particle exists in the system a light version of the double touch property holds. Let us assume we consider a graph with no loops. If we want to calculate the contribution of the one particle hopping from site A to B, it is possible to get to the ﬁnal vertex by touching every bonds once on the way. All the links that have not been touched during this procedure have to be touched twice for reasons that are explained above for the ground-state energy in order to end up in a one particle state again and to contribute 38 Chapter 5. Full graph decomposition to the reduced energy of this graph. This is depicted in Fig. 5.5. Here again this rule is not valid for loops graphs.

Figure 5.5: Depiction of identifying reduced energies considering one particle states. Blue bonds indicate that a τ-operator already acted on them once. Black bonds have not been touched yet. The particle hopping from vertex A to B can at most contribute to the reduced energy in order ten, because all black bonds have to touched twice in order to end up in the one particle state where the particle is located on vertex B.

With these rules we are capable of identifying operations as soon as possible that cannot contribute to the reduced energy by keeping track of already touched bonds. Once all the reduced energies are calculated one has to embed them accurately into the underlying lattice. This is what the next section is about.

5.4 Graph embedding in lattice

In order to calculate perturbed energies in the thermodynamic limit it is inevitable to know how often we can ﬁnd those graphs, to which we applied the pCUT Hamiltonian, in relation to the number of sites in the lattice. This is called the embedding factor. In case of the ground state respectively zero particle state every graph has one embedding factor whereas for the one particle state there exist multiple embedding factors depending on which hopping is regarded.

5.4.1 Ground state

Let us investigate the ground state ﬁrst. The ground-state energy per site is composed of the reduced energies of all relevant graphs and their embedding factors. One has

E0 X = c(g)(g) (5.3) N g 5.4. Graph embedding in lattice 39 where the sum runs over all relevant graphs g, (g) is the reduced energy of graph g and c(g) represents the corresponding embedding factor into the lattice. This section deals with how to derive the embedding factor c(g).

We keep on explaining everything with the help of the square lattice which is easier to visualize. For simple graphs it can be rather easy to ﬁgure out the embedding factor. If we consider the graph with two vertices and one bond the embedding factor is 2 because in the square lattice with N sites there are 2N bonds and thus the relation of bonds to sites is 2. Calculating the embedding factor can get extremely sophisticated for more complex graphs. Therefore a general algorithm is explained in this section. Note that this algorithm has to be slightly modiﬁed for loop graphs, which is presented in section 5.5.

In the very beginning one has to create a cluster which is big enough and contains all the graphs in a certain order to avoid ﬁnite size eﬀects. This cluster is now called master cluster. Choose one site close to the center as the center site. The algorithm works as follows:

1. Choose an arbitrary site of the graph as a pivot.

2. Enumerate all the bonds in a fashion that the numbers ascend the further the bonds are away from the pivot (Fig. 5.6 (a)). This counts for every branch of the graph

3. Fix the pivot to the center site of the master cluster.

4. Assign a counter qi which goes from 0 to (coordination number − 1) to every bond i. Every counter value stands for a speciﬁc orientation within the lattice. (In case of the square lattice the counter goes from 0 to 3.)

5. Start with the ﬁrst bond and set the counter q0 to 0.

6. After every time a bond i is set to a value, check if the site of the master cluster, that the graph site has been ﬁxed to, is already occupied by another graph site. If yes,

enhance qi by 1 and check again. If not, go to the next bond i + 1 and set qi+1 to 0.

7. If any qi exceeds (coordination number − 1), go back to bond i − 1 and enhance qi−1 by 1. Proceed with the previous point.

8. Every time when qM , where M is the number of the last bond, is ﬁxed and approved, a possible embedding of the graph is found.

9. The algorithm ends when the ﬁrst counter q0 is supposed to be set to the coordination number, which terminates the calculation because it exceeds the maximum number the counter is allowed to take. 40 Chapter 5. Full graph decomposition

10. Counting all the approved embeddings and ﬁnally dividing by the symmetry factor yields the embedding factor.

The correct enumeration of the bonds as well as the embedding scheme is sketched in Fig. 5.6. If the unit cell consists of more than one site, one has to ﬁx the pivot to each site of the unit cell once and apply the algorithm. Eventually in order to get the energy per site the result has to be divided by the number of sites in the unit cell. The embedding scheme for one-particle states is similar but worth to be shortly discussed which is done in the next section.

0 q0 = 2 8 3 1 q 1 = 2 5 2 7 q = 3 6 2 6 7 5 q3 = 1 4 3 4 q4 = 1 0 q = 0 0 5 q = 3 3 1 6 1 2 q7 = 0 2 q 8 = 3

(a) Enumeration of bonds (b) Embedding

Figure 5.6: Figure (a) gives an example of an accurate enumeration of the bonds. Figure (b) shows one possible embedding of the chain graph with ten sites and the corresponding values of the counters. The site in lighter gray represents the pivot of the graph.

5.4.2 Hopping elements

As already mentioned above there is not only one embedding factor in case of one-particle states. Instead there are multiple embedding factors depending on which hopping on the graph as well as on the master cluster is considered.

Let us assume we calculated the reduced energy of the amplitude tij(g) of a particle hopping from site i to j on a certain graph g and we want to know how this amplitude contributes to the total hopping element dαβ(δ) of the lattice, where α,β ∈ {1,2,3,4} denote the number of 5.5. Loop graphs 41 the site in the unit cell and δ is the vector that connects the unit cells which contain sites α and β. The total hopping amplitude from site α located in unit cell r to site β, which is part of unit cell r + δ, is then gained by

X X dαβ(δ) = hr,α| H1 |r + δ,βi = cij(g)tij(g) . (5.4) g i,j where H1 denotes the part of the Hamiltonian which acts in the one-particle subspace. In order to calculate the embedding factor cij(g) of a graph g, ﬁx the initial site i to the center of the master cluster {r,α} and use the algorithm described in the previous section with the addition only to count the embedding, if the master cluster site {r+δ,β} is occupied by graph site j. Merging the number of successful embeddings with the corresponding amplitude tij(g) yields the contribution of this hopping on this graph to the hopping element dαβ(δ).

So far we neglected graphs that contain loops. The next section therefore deals with the embedding of loop graphs.

5.5 Loop graphs

The whole embedding scheme becomes more complicated if graphs contain loops. The problem can be solved by decomposing the loop graph into a suitable tree graph which is deﬁned as a graph where two vertices are connected by exactly one path. Fig. 5.6 (a) shows one of those tree graphs. This enables us to use a slight modiﬁcation of the algorithm described in section 5.4.1.

While decomposing a loop graph into a tree graph, it will happen that some graph sites of the loop graph will occur in the tree graph multiple times. This is an important additional information one has to incorporate when representing a loop graph as a tree graph, otherwise the conversion back from the tree to the loop graph is ambiguous. Note that a loop graph can have more than one accurate tree graph, but it is unambiguous to which loop graph a tree graph belongs to. Those sites that occur multiple times in the tree graphs are called doppelganger. A more detailed description of how to gain suitable tree graphs out of loops graphs can be found in the master’s thesis [42] by Michael Powalski. Once a loop graph is decomposed into a tree graph, one can use the embedding algorithm in section 5.4.1 with the slight modiﬁcation that doppelganger sites have to occupy the same master cluster site for a valid embedding. Chapter 6

Results

This section presents the results we gained by applying pCUT for the TFIM on the three- dimensional pyrochlore lattice. Fig. 6.1 shows the pyrochlore lattice and the enumeration of the unit cell sites used in this thesis.

3 1 2

e3 4

3 3 1 2 1 2 3 e1 1 2 e2 4 4 unit cell 4

Figure 6.1: Illustration of unit cell and unit cell vectors (bold black arrows) in the pyrochlore lattice.

As a byproduct, we also reproduced the one-particle gap of the TFIM on the kagome lattice, a sublattice of pyrochlore, which has already been investigated in [10]. We found out, that in

42 6.1. High-ﬁeld limit 43 case of antiferromagnetic nearest-neighbor coupling the elementary excitation gap in Ref. [10] is only accurate up to order 11 and shows slight deviations in order 12 and 13 from the correct value. Therefore the exact one-particle gap up to order 13 can be found in appendix A. The one-particle gap for the ferromagnetic TFIM on the kagome lattice as well as the location of the phase transition is summarized in appendix B. A detailed analysis of the pCUT results for the antiferromagnetic TFIM on the kagome lattice can be found in the Master’s thesis of Michael Powalski [42].

In the first section we calculate the ground-state energy as well as the one-particle dispersion in the high-field limit using the full graph decomposition which allows us to reach order 11 in the perturbation parameter x. We show, that investigating the dispersion does not provide any evidence in favor of a second-order phase transition. Therefore in the second part of the chapter the calculation of the ground-state energy in the low-field limit is discussed which is determined up to order 8, so that in the last section the ground-state energies from both limits are compared. The comparison suggests a first-order phase transition which is located quantitatively at the crossing point of these two quantities.

6.1 High-ﬁeld limit

One observes that the TFIM Hamiltonian (3.14) presented in section 3.2.1 complies with the requirements for pCUT listed in 4.1.2, since the unperturbed part has an equidistant energy spectrum bounded from below as well as particles serving as excitations which carry energy one. If this model exhibits a second-order phase transition, we expect to ﬁnd the critical value by tackling the problem using pCUT and calculating the one-particle energy in the high-ﬁeld limit. A phase transition is found when the gap closes at a critical value (h/J)c, which can be detected by proper extrapolations.

The eﬀective pCUT Hamiltonian based on (3.14) is given by

nmax hf X n X X Heﬀ = H0 + x C(m) τm1,hi1,j1iτm2,hi2,j2i...τmn,hin,jni (6.1) n=1 |m|=n, {hi1,j1i,...,hin,jni} M(m)=0 J with m ∈ {0, ± 2} , x = i 2h N N X and H = − + n . 0 2 i i=1

The initial TFIM Hamiltonian is not particle-conserving but couples subspaces which diﬀer 44 Chapter 6. Results

by ∆N = 2 particles due to the perturbation V = T0 + T−2 + T+2. The resulting eﬀective pCUT Hamiltonian preserves the number of quasi particles in the system and thus acts only in subspaces with a constant number of particles. This is illustrated in Fig. 6.2. The pCUT therefore maps a many-particle problem on a few-body problem. In this thesis the focus is laid on the zero- and one-particle subspace.

Figure 6.2: Sketch of transformation of band structured TFIM Hamiltonian via pCUT. Numbers indicate the number of quasi particles in corresponding subspaces. Left illustration: Band structured TFIM Hamiltonian which couples subspaces that diﬀer by ∆N = 2 particles. Right illustration: Block-diagonal pCUT Hamiltonian that preserves the number of particles.

Exploiting the linked cluster additivity of the eﬀective pCUT Hamiltonian (6.1), the whole pCUT calculation is done on topological distinct graphs, as it was described in the previous chapter. This way computation time and memory usage can be reduced signiﬁcantly which makes a calculation of zero- and one-particle energies up to order nmax = 11 in the perturbation parameter possible. In order 11 one has 1056 relevant topological distinct graphs for the pyrochlore lattice. In comparison to that, there are only 431 relevant graphs for the kagome sublattice, which is the reason why quantities of the kagome lattice can be calculated to higher orders, as it was done in [10] to order 13.

In the zero-particle block one receives the ground-state energy, which reads

hf 1 3 57 93 867 5235 589953 0 = − − x2 + 3x3 − x4 + x5 − x6 + x7 − x8 2h 2 2 8 4 8 8 128 8660373 252903465 3696508953 + x9 − x10 + x11 (6.2) 256 1024 2048 in order 11 and is depicted in Fig. 6.3. From the plot one can guess that up to x ≈ 0.17 the bare series converges and represents the ground-state energy accurately. For larger values one has to resort to Pad´e-extrapolation as discussed in section 4.2.

The one-particle energy is calculated as described in section 5.4.2. Therefore the reduced 6.1. High-ﬁeld limit 45

hf ϵ0 /2h

0.0 x=J/2h 0.1 0.2 0.3 0.4

O(4) -0.5 O(5) O(6) O(7) -1.0 O(8) O(9) O(10) -1.5 O(11)

hf Figure 6.3: Plots of ground-state energy 0 /2h in different orders in x of the TFIM on pyrochlore lattice calculated with pCUT. energy of every hopping amplitude on all 1056 different graphs is determined. After that the graphs are embedded into the pyrochlore lattice to figure out to which hopping elements they contribute. For the sake of clarity, let us consider the chain graph with four sites and assume we want to calculate the hopping amplitude from one end site, now called site 1, to the other end site, defined as site 4, in order 3. Results from the pCUT calculation show, that only the

τ-sequences 1/4 τ−2τ0τ+2, −1/8 τ−2τ+2τ0, −1/8 τ0τ−2τ+2, −1/8 τ0τ+2τ−2, −1/8 τ+2τ−2τ0 and 1/4 τ+2τ0τ−2 survive. The last three sequences cannot have a contribution, because τ−2 applied on a one-particle state yields zero. Thus the ﬁrst three sequences remain. When applying them to the chain graph, one will notice, that these τ-sequences give the following contributions to the reduced energy of the hopping from site 1 to 4

1 3 h1| τ τ τ |4i = 4 −2 0 +2 4 1 1 h1| − τ τ τ |4i = − 8 −2 +2 0 8 1 1 h1| − τ τ τ |4i = − . (6.3) 8 0 −2 +2 8

O(3) State |ii means, that one particle is present on site i. The hopping amplitude t14 is then gained by the sum of all the single contributions of the τ-sequences and takes the value 3 1 1 1 4 − 8 − 8 = 2 in this case. This scenario is illustrated in Fig. 6.4 as well as a few possible embeddings into the pyrochlore lattice in Fig. 6.5.

This procedure is done in all orders up to 11 with all relevant graphs. Following the equation 46 Chapter 6. Results

O(3) Figure 6.4: Illustration of the composition of the hopping amplitude t14 on the chain graph with 4 sites. The given τ-sequences are the only sequences that can have a contribution for the one-particle hopping on the chain graph in order 3. Green circles represent particles. Red lines indicate bonds, on which τ-operators have already been acted. Only τ-sequences that act on every bond of the graph can contribute to the reduced energy of the hopping amplitude.

Figure 6.5: Example of possible embeddings of the chain graph with four sites into the pyrochlore lattice which contribute to the one-particle hopping from the site {r,1}, presented as a black filled circle, to site {r + e1 − e2,1}, illustrated as a green filled circle. Blue shaped tetrahedron indicates the center unit cell. The black circle is chosen as the pivot of the pyrochlore cluster. Different colored lines stand for the four possible embeddings, when fixing site 1 of the graph to the pivot and calculating the hopping from lattice site {r,1} to {r + e1 − e2,1}. The hopping amplitude t14 of this chain graph thus contributes four times to the hopping element d11(e1 − e2).

(5.4) yields the total hopping element dαβ(δ) from unit cell site α in a center unit cell r to site β in unit cell r + δ and thus the one-particle sector of the eﬀective Hamiltonian. In 6.1. High-ﬁeld limit 47 order to diagonalize the one-particle subspace, one exploits the translational symmetry of the Hamiltonian and uses the Fourier transformation

1 X |k,αi = √ eikr |r,αi (6.4) N r of the basis states |r,αi in the spatial domain with α ∈ {1,2,3,4} denoting the number of the site in the unit cell located at r and N, the number of unit cells. Due to the four site unit cell, a Fourier transformation leads to a 4 by 4 matrix. The matrix elements of the Fourier transformed Hamiltonian H1, which acts in the one-particle subspace, are then given by

Ωαβ = hk,β| H1 |k,αi 1 X = e−ikr’eikr hr’,β| H |r,αi , r’ ≡ r + δ N 1 r,r’ 1 X = e−ik(r+δ)eikr hr + δ,β| H |r,αi N 1 r,δ | {z } dαβ (δ) 1 X = eikδd (δ) N αβ r,δ X ikδ = e dαβ(δ) (6.5) δ

It is suﬃcient to determine Ω11 and Ω12 because all other elements can be gained from the symmetry relations

Ω22(k1,k2,k3) = Ω11(k2,k1,k3)

Ω33(k1,k2,k3) = Ω11(−k2,k1 − k2,k3 − k2)

Ω44(k1,k2,k3) = Ω11(k3,k1,k2)

Ω31(k1,k2,k3) = Ω12(−k2,k1 − k2,k3 − k2)

Ω23(k1,k2,k3) = Ω12(k2 − k1, − k1,k3 − k1)

Ω41(k1,k2,k3) = Ω12(k3,k1,k2)

Ω24(k1,k2,k3) = Ω12(k2,k3,k1)

Ω34(k1,k2,k3) = Ω31(k3,k1,k2) ∗ Ωαβ = (Ωβα) .

Diagonalizing this matrix yields the four dispersion bands, which are depicted for x = 0.1 in Fig. 6.6 (a). The entire 4 by 4 matrix is provided in form of a Mathematica sheet on the 48 Chapter 6. Results

CD ﬁxed to the end of this thesis. The two lowest one-particle energy bands remain ﬂat and

1.5

1.4

1.3 0.8343795 )  ( k )

1.2 1,2  ω ( k n 0.8343760 ω 1.1 (0,0,0) (0,π,0)

1.0

0.9

0.8 (0,0,0) (π,π,π) (0,π,π) (0,0,0) (0,π,0) (π,π,0) (a) Dispersion bands (b) Brillouin zone

Figure 6.6: Plot (a) shows the four one-particle energy bands calculated as bare O(11) series of the pyrochlore lattice through specific points of the Brillouin zone. The inset illustrates the lowest two one-particle energy bands which are almost degenerate. Illustration (b) shows the simplified depiction of the Brillouin zone as a cube as well as the path chosen through it. degenerate until order 7 and get a finite dispersion in order 8 which is similar to the TFIM on the kagome lattice [10]. Up to order 7, these energies indicate two localized one-particle modes in the pyrochlore TFIM. Details about those local modes in the kagome lattice, which are related to the one in the pyrochlore lattice, can be found in Ref. [10]. As Fig. 6.6 already suggests, the minimum of the dispersion, the one-particle gap ∆hf , is T located at kmin = (0,0,0) . This is consistent with the existence of a quantum spin liquid phase for low magnetic fields, since this disordered phase breaks no translational symmetry of the Hamiltonian. Thus the minimum of the dispersion is expected to be located at T hf kmin = (0,0,0) which means that the gap ∆ would close at exactly this value of k when a second-order quantum phase transition occurs. If the low-field phase was an ordered phase, the dispersion minimum would be located at kmin 6= 0 and therefore the gap would vanish for this specific kmin. An example is the antiferromagnetic TFIM on the square lattice, which T takes the Neel order with kmin = (π,π) in the low-field regime.

T Inserting kmin = (0,0,0) into (6.5) leads to a highly symmetric matrix of the form   ABBB   BABB   (6.6) BBAB   BBBA 6.1. High-ﬁeld limit 49 which exhibits the eigenvalues

λ1,2,3 = A − B (6.7)

λ4 = A + 3B (6.8)

T independent of the physical problem. The eigenvalues at kmin = (0,0,0) can thus be extracted analytically. The lowest eigenvalue of H(k = 0) is threefold degenerate and is shown in Fig. 6.7 together with the forth eigenvalue as a function of x.

ω1 =E 1-E0 2.0

1.5 Eigenvalues

1.0

0.5

0.0 x 0.00 0.05 0.10 0.15 0.20

T Figure 6.7: Energies for kmin = (0,0,0) . The lowest eigenvalue is the one-particle gap and is threefold degenerate.

The eigenvalues up to order 11 in the perturbation parameter read

∆hf 901 82783 458339 ω = = 1 − 2x + 4x2 − 6x3 + 6x4 − 25x5 + x6 − x7 + x8 1,2,3 2h 4 32 16 17244199 872369819 1361632112501 − x9 + x10 − x11 (6.9) 64 384 73728 61959 3904437 ω =1 + 6x − 12x2 + 42x3 − 282x4 + 2049x5 − x6 + x7 4 4 32 64680777 8705145273 4772705701191 − 982614x8 + x9 − x10 + x11 , (6.10) 8 128 8192

hf where ∆ /2h is the gap. Note that the highest eigenvalue ω4 represents the gap for the ferromagnetic TFIM on the pyrochlore lattice (x → −x). A phase transition occurs at the point xc where the gap closes and thus where excitations do not cost any energy in the system. In this case the bare series of the gap indeed closes at x ≈ 0.22, but this is due to the divergence of the series expansion and not due to physical nature. The plot 6.7 suggests, 50 Chapter 6. Results that the bare series of the gap converges up to x ≈ 0.17 and is not reliable for larger x. Because we are interested in a possible gap closing for larger x, we extrapolate the gap using dlogPad´e as described in section 4.2.2. In order to scan the whole x parameter axis, we use the Euler transformation

u x = (6.11) 1 − u which maps x ∈ [0,∞) to u ∈ [0,1). The gap and the dlogPad´e-extrapolants are shown in Fig. 6.8.

Bare O(10) 1.4 Bare O(11) DLog[8,2] 1.2 DLog[7,2] DLog[6,2] DLog[7,3] 1.0 DLog[6,3] DLog[5,3] DLog 6,4 0.8 [ ] / 2h DLog[3,2]

hf DLog[4,3] Δ 0.6 DLog[5,4] DLog[2,2] DLog[3,3] 0.4 DLog[4,4] DLog[2,3] DLog[4,5] 0.2 DLog[2,7] DLog[2,8] 0.0 0.0 0.2 0.4 0.6 0.8 1.0 u

Figure 6.8: Plot of the gap ∆hf /2h in the polarized phase as a function of u. Dashed lines show the bare series in order 10 and 11 based on pCUT calculations. Solid lines are dlogPad´e-extrapolants.

Apparently the dlogPadé-extrapolants do not show any tendencies for a gap closing. Even the extrapolants, which approach zero, do not possess any poles. However, a CQSL is expected to be present for small fields and therefore at least one phase transition must exist in the TFIM on the pyrochlore lattice, which is different to the disorder by disorder scenario in the two-dimensional kagome TFIM [8, 10]. The analysis of the gap in the polarized phase therefore suggests that the quantum phase transition in the TFIM on the pyrochlore lattice is first order. In order to detect a first-order quantum phase transition, the ground-state energies from both limits are compared in a next step. The crossing of these two energies corresponds to the location of the first-order quantum phase transition. Since the high-field limit was already discussed in this section, the subsequent section deals with the perturbative 6.2. Low-field limit 51 treatment of the ground-state energy in the low-field limit.

6.2 Low-ﬁeld limit

We showed in section 3.2.2, that we can find a suitable particle representation in the low-field limit, such that it meets the requirements for using pCUT listed in 4.1.2. Summarized, we created two artificial energy levels in order to obtain an equidistant energy spectrum where levels differ by energy one. This supplement enables us to use pCUT and leads to the effective particle-conserving Hamiltonian based on (3.18) in the low-field limit

nmax lf X n X X Heﬀ = H0 + x C(m) τm1,hi1,j1iτm2,hi2,j2i...τmn,hin,jni (6.12) n=1 |m|=n, {hi1,j1i,...,hin,jni} M(m)=0 h with m ∈ {0, ± 2, ± 4, ± 6} and x = (6.13) i 2J

and H0 = Q (6.14)

In the low-field limit, graph expansion is highly not trivial because of the 16 different kinds of quasi particles in the system (see section 3.2.2). Therefore the calculation is done on a periodic cluster. Due to this, order nmax = 8 is the highest order that can be reached which can be boiled down to two main reasons. First, the calculations are restricted by memory usage. The size of the periodic cluster is barely big enough to avoid finite size effects in order 8 and at the same time not to exceed memory capacity. The second reason is related to what was already presented in section 3.1. The perturbative calculation of the ground-state energy in the low-field regime is much more complex than in the high-field regime due to the extensive ground-state degeneracy. From order 6 on, quantum fluctuations enable tunnelling between different ground states which lead to the effective Hamiltonian (3.6)

X + − + − + − Heﬀ = e0 + Jring σ1 σ2 σ3 σ4 σ5 σ6 + h.c. . (6.15)

7 In general, tunnelling between ground states in a certain order n is only possible if one can find a loop with m ≤ n sites in total, which always comprises an even number of sites per tetrahedron, such that by flipping all spins of this loop, an even number of spins in the z tetrahedra are inverted and thus σtot is not changed. One observes then that the Hamiltonian (3.6) is also valid up to order 8 because no other fluctuations besides this flipped hexagon can couple different ground states. This changes in order 10 where we can find such loops that z do not change σtot of any tetrahedron by flipping it. One of these loops is shown in Fig. 6.9. 52 Chapter 6. Results

Figure 6.9: Illustration of loops that induce tunnelling between diﬀerent ground states in order 10 in the TFIM low-ﬁeld limit. The green bold line marks a hexagon which enables tunnelling already in order 6. The red bold line emphasizes a 10-site loop which would have to be considered when calculating the ground-state energy in order 10.

The quantities e0 and Jring are determined via pCUT using the eﬀective Hamiltonian (6.12) and are rewritten based on (3.17) in order to yield the original TFIM energy. They read

1 h2 7 h4 893 h6 209966173 h8 e = − J − − − − 0 4 J 192 J 3 34560 J 5 6967296000 J 7 63 h6 33833 h8 J = − − . (6.16) ring 256 J 5 165888 J 7

The calculations were done with the pCUT coeﬃcients for τm-operators with mi ∈ {0, ± 1, ± 2, ± 3}. The coeﬃcients then have to be adjusted by multiplying order−1 (1/2) . The leading order-6 contribution of Jring has already been derived in [17].

The ground state energy of the ring exchange model

X + − + − + − Hring = σ1 σ2 σ3 σ4 σ5 σ6 + h.c. (6.17)

7 can be gained from Shannon, Sikora, Pollmann and Fulde [14] who solved it by using quantum ring Monte Carlo simulations. They derived the ground-state energy per site e0 = 0.189078. No calculations have been done in the past yet on an eﬀective order 10 Hamiltonian, that includes 10-site loops which is another reason why our calculations are restricted to order 8. Note that odd order contributions vanish for this limit. 6.3. Comparison of both limits 53

Merging all quantities results into the ground-state energy per site

lf ring 0 = e0 + Jringe0 1 h2 7 h4 h6 h8 = −J − − − 0.0723700023 − 0.0686985566 (6.18) 4 J 192 J 3 J 5 J 7

This result is now compared with the ground-state energy of the high-ﬁeld limit in the preliminary section.

6.3 Comparison of both limits

The two ground-state energies

3 J 2 3 J 3 57 J 4 93 J 5 867 J 6 5235 J 7 589953 J 8 hf = −h − + − + − + − 0 4 h 4 h2 64 h3 64 h4 256 h5 512 h6 16384 h7 8660373 J 9 252903465 J 10 3696508953 J 11 + − + (6.19) 65536 h8 524288 h9 2097152 h10 1 h2 7 h4 h6 h8 lf = −J − − − 0.0723700023 − 0.0686985566 (6.20) 0 4 J 192 J 3 J 5 J 7 are now compared by using the replacements

h = sin ϕ (6.21) J = cos ϕ (6.22) and plotting the energies against ϕ. This way ϕ = 0 corresponds to h = 0 whereas ϕ = π/2 equals the case J = 0. Fig. 6.10 shows the result. The dashed green line represents the ground-state energy in the high-field limit up to order 11. The orange lines show Padé- extrapolants of the high-field energy which are necessary because the crossing point of the mf two quantities is expected to be located in the vicinity of the gray dashed line at ϕc ≈ 0.611 (h/J ≈ 0.7) which was calculated in [17] and corresponds to the phase transition using mean-field calculations. The Padé-extrapolants seem to be reliable since they lie almost on top of each other in a large parameter regime up to ϕ < ϕc. The blue lines are the low- field series in different orders. In comparison to the high-field limit, the bare series of the low-field limit is sufficient, since the crossing between both energies is at rather small values of ϕ. Therefore an extrapolation is redundant. The crossing at ϕc ≈ 0.542 (h/J ≈ 0.602) corresponds to the location of the first-order quantum phase transition between the CQSL and the polarized phase. This is consistent with the mean-field result which is expected to overestimate the CQSL phase. Although our theoretical calculations predict a quantum 54 Chapter 6. Results

Low field O(2) -0.6 Low field O(4) Low field O(6) Low field O(8) -0.8 High field O(11) Pade high field 0

ϵ -1.0

-1.2

-1.4

0 π/8 π/4 3π/8 π/2 (h=0) (J=0) φ

Figure 6.10: Comparison of the ground-state energies 0 from the low- and high-field expansions as a function of ϕ setting J = cos ϕ and h = sin ϕ. Solid orange lines are the Padé-extrapolants of the bare O(11) series of the high-field phase. The dashed vertical line refers to the mean-field result from [17] and the solid vertical line corresponds to location of the first-order quantum phase transition as obtained from the crossing of the series expansions. phase transition in quantum spin ice, it is very challenging to proof it experimentally. A short outlook is presented in the next section.

6.4 Experimental realization

In the presented theory we describe quantum spin ice within the antiferromagnetic TFIM, so we apply a magnetic field transverse to the Ising spin component, which induces quantum fluctuations in a spin ice model. This way we are able to locate the phase transition between the polarized and the CQSL phase. The experimental realization of verifying a quantum phase transition in quantum spin ices turns out to be very complicated since real materials exhibit much more couplings than only one coupling transverse to the Ising spin orientation [15]. These additional couplings are uncontrollable and most likely will modify the CQSL 6.4. Experimental realization 55 phase essentially. It is therefore strongly desirable to find a material without any additional interactions which only exhibits a coupling equivalent to the transverse magnetic field in the TFIM. Naively applying a transverse magnetic field as the theory might suggest is not an option, since it couples most strongly to the Ising spin component which rather quenches the development of a CQSL than enhancing it [60]. In Ref. [17] it was suggested to use non- magnetic disorder in spin ice materials with non-Kramer’s rare earth ions Ho3+ or Pr3+, ions with an even number of electrons, to induce random transverse fields, where non-magnetic refers to no addition or removal of spins from the system. This disorder lowers the symmetry of the system and thus induces quantum fluctuations. They showed, that disorder engenders (electrostatic) crystal fields, which act as transverse magnetic fields in the spin Hamiltonian

X z z X x H = J Si Sj − hiSi . (6.23) hiji i

The magnetic field has then a random magnitude on every lattice site. Here we propose that generating a transverse field can be easier realized by straining a non-Kramer’s spin ice material and thus creating a uniform transverse field on each site (see also Ref. [60]). Strains can be achieved in thin films [61], which allow to lower the symmetry in a controlled manner. Chapter 7

Conclusion and outlook

In this thesis, we determine the location of the phase transition occurring in quantum spin ice, which is a novel class of frustrated quantum ferromagnetic systems exhibiting intriguing properties. This is realized by investigating the antiferromagnetic transverse-field Ising model on the pyrochlore lattice which is able to capture some interesting physics of quantum spin ice. The transverse-field Ising model is in the conventional polarized phase for large transverse magnetic fields h and in a Coulomb quantum spin liquid phase for large nearest-neighbor coupling J of two spins. States in the Coulomb quantum spin liquid phase are disordered and exhibit exotic excitations such as gapped spinons, gapped magnetic monopoles and gapless artificial photons [13]. Since two phases in the system are present, there needs to be a quantum phase transition at a critical value (h/J)c. This phase transition is first sought by calculating the first excitation gap in the high-field limit via perturbative continuous unitary transformation. The results show no indications of an existing second-order quantum phase transition suggesting the transition to be first order. Therefore the series expansions of the two ground-state energies of both limits are compared, which reveals a first-order phase transition at (h/J)c ≈ 0.602.

The starting point is the investigation of the high-field phase of the transverse-field Ising model on the pyrochlore lattice. The condition of a small magnetic field h J justifies a perturbative treatment with perturbative continuous unitary transformation, a powerful method in order to determine quantities exactly in the thermodynamic limit only truncated by the order of a perturbation parameter. The ground state in the high-field regime is formed by all spins pointing into the direction of the magnetic field whereas a spin flip enhances the energy and thus acts as an elementary excitation. The problem is expressed with the help of quasi-particles, which correspond to dressed flipped spins or excitations,

56 57 respectively. The vacuum state with no present quasi-particles therefore is the ground state. Using perturbative continuous unitary transformation maps the many-particle problem onto a few-body problem providing an effective particle-conserving Hamiltonian. In the course of this thesis, the zero- and one-particle sector is investigated. This is done by exploiting the cluster additivity of the perturbative continuous unitary transformation Hamiltonian. In this regime one can even go further and perform all calculations on topologically equivalent graphs, since they give the same contribution to the resulting energy. Each contribution needs to be merged with an embedding factor, which gives the number of embeddings of the graph in relation to sites in the lattice. With the help of this full graph decomposition we are capable of calculating the ground state energy and the one-particle dispersion up to order 11 in the perturbation parameter x = J/2h. Applying the perturbative continuous unitary transformation Hamiltonian on the zero-particle state leads to a constant ground state energy as a function of x since the ground state is unique. The Hilbert space of the one- particle states is extensively large and can be boiled down to a 4 by 4 matrix exploiting the translational symmetry of the lattice and using the Fourier-transformation. The dimension 4 corresponds to the 4-site unit cell of the pyrochlore lattice. Diagonalization of this matrix hence yields four one-particle dispersion bands. It turns out that the minimum (the gap) T is located at kmin = (0,0,0) . Since H(k = 0) is highly symmetric, the eigenvalues can be extracted analytically as series in x. However, the phase transition is expected beyond the radius of convergence of the bare series, so that the gap is extrapolated via dlogPadé. The Euler-transformation maps x ∈ [0,∞) onto u ∈ [0,1) and thus the whole parameter axis of x can be displayed within one plot in u from 0 to 1. If the gap vanished at a critical point xc, this would identify the location of a second-order phase transition. We find that none of the extrapolants show any tendencies for a gap closing though, which indicates that the phase transition is first order. We therefore additionally calculate the ground state energy in the low-field limit for the purpose of comparing it with the ground state energy of the high-field limit. The crossing of both energies corresponds to the first-order quantum phase transition.

The investigation of the zero-particle sector in the low-field limit is much more complex, since high frustration leads to an extensive ground-state degeneracy. In other words, the zero- particle Hilbert space is extensively large, which is different in comparison to the high-field limit. We can also find a quasi-particle representation in this limit, which is more complicated since one has to introduce 16 different kinds of particles. Applying the perturbative continuous unitary transformation Hamiltonian does not provide a constant ground-state energy, but leads to an effective low-energy Hamiltonian. This matrix is diagonal up to order 4 in the perturbation parameter h, but non-diagonal terms appear in order 6, since one can get from one ground state to another by flipping a 6-link loop containing alternating spins. This leads 58 Chapter 7. Conclusion and outlook

Figure 7.1: Phase diagram of the transverse-field Ising model on the pyrochlore lattice as a function of h/J consisting of the low-field Coulomb quantum spin liquid phase (Coulomb quantum spin liquid) and the high-field polarized phase separated by a first-order phase transition at hc/Jc ≈ 0.602 shown as a black filled square. Left illustration: Green hexagon exemplifies a hexagon on which the ring exchange Jring-term in equation (3.6) acts on alternating spins. Right illustration: Moving elementary spin flip excitation above the high-field polarized state. to an effective ring exchange Hamiltonian [17] comprising a sum over all hexagons in the pyrochlore lattice, which is also valid up to order 8 because no additional new loops occur. Note that odd orders in h vanish because it is not possible to get to a different or back to the initial ground state by flipping an odd number of spins. Since the pure ring exchange model has been treated quantitatively in Ref. [14], we are able to gain the ground state energy as a series expansion up to order 8 in the perturbation h of the transverse-field Ising model using perturbative continuous unitary transformation. Graph expansion is not trivial in the low-field regime, which is the reason why calculations are performed on a sufficiently large periodic three-dimensional pyrochlore cluster.

The comparison of both energies yields the location of the first-order quantum phase transition (h/J)c ≈ 0.602 between the polarized phase and the Coulomb quantum spin liquid phase and is in agreement with the critical point (h/J)c = 0.7 calculated via mean-field theory in Ref. [17] which is expected to overestimate the Coulomb quantum spin liquid phase. The intriguing result can be summarized with the help of a phase diagram of the transverse-field Ising model on the pyrochlore lattice shown in Fig. 7.1.

An interesting point for the future is to calculate series expansions in more expansion parameters than just one, that is to add more interactions to the Hamiltonian than just the transverse 59 magnetic field and to study how these additional couplings renormalize the one-particle energy bands. A model with additional couplings is expected to describe conventional quantum spin ice compounds more accurately. Also interesting is, to compare the results presented in this thesis with Monte-Carlo simulations, which has not been done so far. Experimentally, this quantum phase transition could be verified by straining a non-Kramer’s spin ice material instead of inducing quantum fluctuations by using disorder [17]. Straining a material represents a different way to lower the symmetry of the system and thus to generate quantum effects. This way it is possible to engender a uniform transverse field on every site of the lattice and simultaneously achieve absent additional interactions which are present in real quantum spin ice materials like in Yb2Ti2O7 and quench the Coulomb quantum spin liquid phase. Appendix A

One-particle gap of antiferromagnetic TFIM on kagome lattice

The one-particle gap of the antiferromagnetic TFIM on the kagome lattice in the high-ﬁeld limit as a bare series in the perturbation parameter x = J/2h up to order 13 reads

∆antiferro 243 2609 28531 = 1 − 2x + 2x2 − 3x5 + x6 − x7 + x8 2h 8 8 16 3448807 149125883 1340557475 − x9 + x10 − x11 512 6144 12288 172295453855 33156075124357 + x12 − x13 . (A.1) 294912 10616832

60 Appendix B

One-particle gap of ferromagnetic TFIM on kagome lattice

The one-particle gap of the antiferromagnetic TFIM on the kagome lattice in the high-ﬁeld limit as a bare series in the perturbation parameter x = J/2h up to order 13 reads

∆ferro 261 2523 41891 = 1 − 4x − 4x2 − 6x3 − 36x4 − x5 − x6 − x7 2h 2 4 16 760415 14668135 450811297 18337562173 − x8 − x9 − x10 − x11 64 256 1536 12288 560368254859 206526192099709 − x12 − x13 (B.1) 73728 5308416 and is shown in Fig. B.1 with some dlogPad´e-extrapolants. The gap closes at (J/2h)c ≈ 0.170.

61 62 Appendix B. One-particle gap of ferromagnetic TFIM on kagome lattice

Δferro/2h 1.0 bare O(13) 0.8 dlog[5,6] dlog[4,5] 0.6 dlog[6,6] dlog[5,5] 0.4 dlog[4,4] dlog[6,5] 0.2 dlog[5,4]

x=J/2h 0.00 0.05 0.10 0.15 0.20

Figure B.1: One-particle gap ∆ferro/2h of ferromagnetic TFIM on kagome lattice in the polarized phase as well as some dlogPad´e-extrapolants. Bibliography

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and S. T. Bramwell. Dy2Ti2O7 Spin Ice Thin-Films. Nature communications, 5, 2014. Danksagung

Mein besonderer Dank gilt Kai Schmidt für die Vergabe des spannenden Themas und die intensive Betreuung. Vor allem aber bin ich sehr dankbar, dass ich meine Masterarbeit in Kollaboration mit der University of California Santa Barbara erstellen durfte, was ohne seine Unterstützung und Hilfsbereitschaft niemals m¨glich gewesen wäre. Daher geht eine weiteres großes Dankeschön an Leon Balents vom KITP aus Santa Barbara, der sich auf diese Kollaboration eingelassen hat und während meiner Zeit in Santa Barbara und auch darüber hinaus als Betreuer immer zur Verfügung stand. Ich habe in dieser Zeit viel gelernt und viele Einblicke in weitere Forschungsgebiete gewonnen.

Ich danke an dieser Stelle auch der Artur- und Lieselotte-Dumcke-Stiftung, die mich im Rahmen des Deutschlandstipendiums seit 2013 ﬁnanziell unterstützt hat und mir während meiner Zeit in Santa Barbara eine außerplanmäßige Förderungssumme gewährt hat.

Danke an Kris Cöster und Michael Powalski für die Hilfe bei der Einarbeitung in mein Thema, sowie an Kai Schmidts Arbeitsgruppe für das tolle Arbeitsklima. Besonderer Dank geht an Jonas Haarz, Dominik Ixert und David Schneider, die mir besonders bei technischen Fragen immer zur Seite standen.

Zuletzt danke ich noch Sascha Heußen für das Korrekturlesen meiner Arbeit. Eidesstattliche Versicherung

Ich versichere hiermit an Eides statt, dass ich die vorliegende Masterarbeit mit dem Titel ”Spin liquid and quantum phase transition without symmetry breaking in a frustrated three- dimensional Ising model” selbständig und ohne unzulässige fremde Hilfe erbracht habe. Ich habe keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie wörtliche und sinngemäße Zitate kenntlich gemacht. Die Arbeit hat in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen.

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Belehrung

Wer vorsätzlich gegen eine die Täuschung über Prüfungsleistungen betreﬀende Regelung einer Hochschulprüfungsordnung verstößt handelt ordnungswidrig. Die Ordnungswidrigkeit kann mit einer Geldbuße von bis zu 50.000,00 e geahndet werden. Zuständige Verwaltungsbehörde für die Verfolgung und Ahndung von Ordnungswidrigkeiten ist der Kanzler/die Kanzlerin der Technischen Universität Dortmund. Im Falle eines mehrfachen oder sonstigen schwer- wiegenden Täuschungsversuches kann der Prüﬂing zudem exmatrikuliert werden (§ 63 Abs. 5 Hochschulgesetz - HG - ).

Die Abgabe einer falschen Versicherung an Eides statt wird mit Freiheitsstrafe bis zu 3 Jahren oder mit Geldstrafe bestraft.

Die Technische Universität Dortmund wird ggf. elektronische Vergleichswerkzeuge (wie z.B. die Software ”turnitin”) zur Überprüfung von Ordnungswidrigkeiten in Prüfungsverfahren nutzen. 70 BIBLIOGRAPHY

Die oben stehende Belehrung habe ich zur Kenntnis genommen.

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