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 mag- netic 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 tran- sitions 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