Second-order phase transitions involve critical fluctuations just below and above the transition temperature. Macroscopically, they manifest in the power-law behaviour of many physical properties such as the susceptibility and the specific heat. The power-laws are predicted to be universal, i.e. the same exponents are expected for a certain class of transitions irrespective of the microscopic details of the system. The underlying commonality of such transitions is the divergence of the correlation length ξ and the correlation time ξ_τ of the critical fluctuations at the transition temperature. Both ξ and ξ_τ can be directly observed by neutron scattering experiments, making them an ideal tool for the study of critical phenomena. At classical phase transitions, the critical fluctuations will be thermal in nature. However, if a second-order transition occurs at T = 0, thermal fluctuations are frozen, and the transition is driven by quantum fluctuations instead. This is called a quantum critical point. The quantum nature of the fluctuations influences observable properties, also at finite temperatures, and causes unusual behaviour in the vicinity of the quantum critical point or the existence of exotic phases, e.g. unconventional superconductivity. Heavy-fermion compounds are a class of materials that is well suited for the study of quantum criticality. They frequently show second-order transitions into a magnetically ordered state at very low temperatures, which can easily be tuned to T = 0 by the application of pressure, magnetic fields or element substitution. In this thesis, fluctuations near a quantum critical point are investigated for three heavy-fermion systems. CeCu2Si2 shows unconventional superconductivity close to an antiferromagnetic quantum critical point. Results from single-crystal neutron spectroscopy and thermodynamic measurements are discussed and some details are also given about the synthesis of large single crystals. The focus of the study is the comparison of the inelastic response of magnetic and superconducting samples, which are found to be very similar for ΔE > 0.2 meV. CePdAl has an antiferromagnetic state with partial magnetic frustration. The ordering temperature can be suppressed by Ni substitution towards a quantum critical point. Single-crystal neutron diffraction experiments of three members of the substitution series were analysed. They revealed several unusual effects of the frustrated state in the pure sample, and show that magnetic order and frustration persist in the substituted samples. YbNi4P2 is a rare example of a compound with ferromagnetic quantum criticality, which has only been studied in the last few years. The aim of the powder neutron spectroscopy experiments presented here was to obtain an overview of the relevant energy scales, i.e. the crystal electric field, local magnetic fluctuations and ferromagnetic fluctuations. Simulations using the program McPhase were performed for a thorough understanding of the crystal electric field.

Second-order phase transitions involve critical fluctuations just below and above the transition temperature. Macroscopically, they manifest in the power-law behaviour of many physical properties such as the susceptibility and the specific heat. The power-laws are predicted to be universal, i.e. the same exponents are expected for a certain class of transitions irrespective of the microscopic details of the system. The underlying commonality of such transitions is the divergence of the correlation length ξ and the correlation time ξ_τ of the critical fluctuations at the transition temperature. Both ξ and ξ_τ can be directly observed by neutron scattering experiments, making them an ideal tool for the study of critical phenomena. At classical phase transitions, the critical fluctuations will be thermal in nature. However, if a second-order transition occurs at T = 0, thermal fluctuations are frozen, and the transition is driven by quantum fluctuations instead. This is called a quantum critical point. The quantum nature of the fluctuations influences observable properties, also at finite temperatures, and causes unusual behaviour in the vicinity of the quantum critical point or the existence of exotic phases, e.g. unconventional superconductivity. Heavy-fermion compounds are a class of materials that is well suited for the study of quantum criticality. They frequently show second-order transitions into a magnetically ordered state at very low temperatures, which can easily be tuned to T = 0 by the application of pressure, magnetic fields or element substitution. In this thesis, fluctuations near a quantum critical point are investigated for three heavy-fermion systems. CeCu2Si2 shows unconventional superconductivity close to an antiferromagnetic quantum critical point. Results from single-crystal neutron spectroscopy and thermodynamic measurements are discussed and some details are also given about the synthesis of large single crystals. The focus of the study is the comparison of the inelastic response of magnetic and superconducting samples, which are found to be very similar for ΔE > 0.2 meV. CePdAl has an antiferromagnetic state with partial magnetic frustration. The ordering temperature can be suppressed by Ni substitution towards a quantum critical point. Single-crystal neutron diffraction experiments of three members of the substitution series were analysed. They revealed several unusual effects of the frustrated state in the pure sample, and show that magnetic order and frustration persist in the substituted samples. YbNi4P2 is a rare example of a compound with ferromagnetic quantum criticality, which has only been studied in the last few years. The aim of the powder neutron spectroscopy experiments presented here was to obtain an overview of the relevant energy scales, i.e. the crystal electric field, local magnetic fluctuations and ferromagnetic fluctuations. Simulations using the program McPhase were performed for a thorough understanding of the crystal electric field.

Das Auftreten von Supraleitung in Systemen mit schweren Fermionen, erstmals entdeckt in CeCu_2Si_2, wird mit der Nähe zu einem quantenkritischen Punkt in Verbindung gebracht. Daraus ergibt sich ein komplexes Zusammenspiel von Magnetismus und Supraleitung, das in der vorliegenden Arbeit durch Messungen der spezifischen Wärme, der Wechselfeldsuszeptibilität und durch inelastische Neutronenstreuexperimente an verschiedenen Einkristallen von CeCu_2(Si_{1-x}Ge_x)_2 untersucht wird. Der Schwerpunkt liegt auf der genauen Charakterisierung des magnetischen Anregungsspektrums von CeCu_2Si_2 des S-Typs. Die Ergebnisse der Neutronenstreumessungen implizieren stark, dass die Kopplung der supraleitenden Cooper-Paare durch überdämpfte Spinfluktuationen vermittelt wird, die in der Umgebung eines Quantenphasenübergangs gehäuft auftreten. Unter Substitution einiger Si- durch Ge-Atome in CeCu_2Si_2 stabilisiert sich die magnetische Ordnung, und die Supraleitung wird zunehmend unterdrückt. Neutronenstreumessungen ergeben, dass dies bei 2 % Ge-Substitution dazu führt, dass sich Magnetismus und Supraleitung gegenseitig verdrängen, während sie bei 10 % Ge-Substitution mikroskopisch koexistieren. - (Die Dissertation ist veröffentlicht im Logos Verlag Berlin GmbH, Berlin, Deutschland, http://www.logos-verlag.de, ISBN: 978-3-8325-2456-2) / The occurrence of superconductivity in systems with heavy fermions, discovered for the first time in CeCu_2Si_2, is often linked to the vicinity of a quantum critical point. This results in a complex interplay of magnetism and superconductivity, which is studied by means of specific heat and ac susceptibility measurements as well as neutron scattering experiments on different single crystals of CeCu_2(Si_{1-x}Ge_x)_2 in the present thesis. The focus is put on the detailed characterisation of the magnetic excitation spectrum in S-type CeCu_2Si_2. Neutron scattering results strongly imply that the coupling of superconducting Cooper pairs is mediated by overdamped spin fluctuations, which accumulate in the vicinity of a quantum phase transition. By substituting Si by Ge atoms in CeCu_2Si_2 magnetic order is stabilised and superconductivity successively suppressed. Neutron scattering experiments demonstrate that 2 % Ge substitution leads to magnetic order being displaced by superconductivity on decreasing temperature, whereas both coexist microscopically in the case of 10 % Ge substitution.

Das Auftreten von Supraleitung in Systemen mit schweren Fermionen, erstmals entdeckt in CeCu_2Si_2, wird mit der Nähe zu einem quantenkritischen Punkt in Verbindung gebracht. Daraus ergibt sich ein komplexes Zusammenspiel von Magnetismus und Supraleitung, das in der vorliegenden Arbeit durch Messungen der spezifischen Wärme, der Wechselfeldsuszeptibilität und durch inelastische Neutronenstreuexperimente an verschiedenen Einkristallen von CeCu_2(Si_{1-x}Ge_x)_2 untersucht wird. Der Schwerpunkt liegt auf der genauen Charakterisierung des magnetischen Anregungsspektrums von CeCu_2Si_2 des S-Typs. Die Ergebnisse der Neutronenstreumessungen implizieren stark, dass die Kopplung der supraleitenden Cooper-Paare durch überdämpfte Spinfluktuationen vermittelt wird, die in der Umgebung eines Quantenphasenübergangs gehäuft auftreten. Unter Substitution einiger Si- durch Ge-Atome in CeCu_2Si_2 stabilisiert sich die magnetische Ordnung, und die Supraleitung wird zunehmend unterdrückt. Neutronenstreumessungen ergeben, dass dies bei 2 % Ge-Substitution dazu führt, dass sich Magnetismus und Supraleitung gegenseitig verdrängen, während sie bei 10 % Ge-Substitution mikroskopisch koexistieren. - (Die Dissertation ist veröffentlicht im Logos Verlag Berlin GmbH, Berlin, Deutschland, http://www.logos-verlag.de, ISBN: 978-3-8325-2456-2) / The occurrence of superconductivity in systems with heavy fermions, discovered for the first time in CeCu_2Si_2, is often linked to the vicinity of a quantum critical point. This results in a complex interplay of magnetism and superconductivity, which is studied by means of specific heat and ac susceptibility measurements as well as neutron scattering experiments on different single crystals of CeCu_2(Si_{1-x}Ge_x)_2 in the present thesis. The focus is put on the detailed characterisation of the magnetic excitation spectrum in S-type CeCu_2Si_2. Neutron scattering results strongly imply that the coupling of superconducting Cooper pairs is mediated by overdamped spin fluctuations, which accumulate in the vicinity of a quantum phase transition. By substituting Si by Ge atoms in CeCu_2Si_2 magnetic order is stabilised and superconductivity successively suppressed. Neutron scattering experiments demonstrate that 2 % Ge substitution leads to magnetic order being displaced by superconductivity on decreasing temperature, whereas both coexist microscopically in the case of 10 % Ge substitution.

The vicinity of continuous quantum phase transitions displays unique properties such as scaling behavior and incoherent excitation spectra which are not found in any stable phase of matter. This fascinating quantum critical regime is crucial for progress on key problems of modern condensed matter physics. The three research projects of this thesis challenge and refine our understanding of quantum criticality in different ways. Part I concerns unexpected quantum critical behavior near the Mott transition. The bandwidth-controlled Mott transition in the half-filled one-band Hubbard model is one of the most paradigmatic phenomena of strongly correlated physics. Within the approximation of dynamical mean-field theory (DMFT) this metal-insulator transition is of first order at low temperatures, with the transition line ending at a critical temperature. Surprisingly, numerical calculations with DMFT and experiments in organic salts consistently found quantum critical scaling of the resistivity above the critical temperature. The aim of this project is to explain this unexpected scaling in the absence of a quantum critical point in the phase diagram. To this end, we perform extensive DMFT simulations with the numerical renormalization group as a state-of-the-art impurity solver. We find that the quantum critical scaling can be traced back to the metastable insulator at the boundary of the coexistence region at T = 0 which exhibits previously unknown scale-invariance on the frequency axis. In Part II we study how magnetic quantum criticality is affected by the coupling to additional non-critical degrees of freedom. Considering typical electronic energy scales the study of quantum critical phenomena in magnets requires very low temperatures in the sub-100mK range. In this regime additional effects which are typically neglected in the theoretical modeling may become important. Here we focus on one particular example, which is the hyperfine coupling to nuclear spins. We investigate the fate of the quantum critical behavior at lowest temperatures and determine crossover scales below which a purely electronic description is no longer sufficient. Explicit calculations for paradigmatic models on the level of mean-field theory plus Gaussian fluctuations reveal that the quantum phase transition can be shifted or smeared in the presence of nuclear spins. More exotic effects of nuclear spins, e.g. in spin liquids, are discussed on a qualitative level. Part III is devoted to the discussion of domain-driven phase transitions in easy-axis ferromagnets.This work is motivated by an experimental study of LiHoF4, a dipolar easy-axis ferromagnet that displays a well-studied quantum phase transition from a ferromagnetic to a paramagnetic phase as function of a transverse field. Measurements of the ac susceptibility found a well-defined phase transition even in tilted fields where the Ising symmetry is explicitly broken and Landau theory of the microscopic order parameter predicts a crossover. We are able to explain and model the transition in tilted fields by the inclusion of domain effects, i.e., by taking into account the spontaneous breaking of translational symmetry by mesoscale pattern formation in the ferromagnetic phase. The modeling of stray-field energies as effective antiferromagnetic couplings between magnetization components in different domains is in excellent quantitative agreement with the experimental results.:1 Phases and their transitions . . . . . . . . . . . . . . . . . . . . 4 1.1 Thermal and quantum phase transitions . . . . . . . . . . . . . . . . . . . . 4 1.2 Theoretical description of phase transitions . . . . . . . . . . . . . . . . . . 8 1.3 Project overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I Mott quantum criticality in the one-band Hubbard model . . . . . . . . . . .15 2 Introduction to the Mott transition . . . . . . . . . . . . . . . . . . . . 16 2.1 Metal-insulator transitions and the Hubbard model . . . . . . . . . . . . . . 16 2.2 A local perspective: the idea of dynamical mean-field theory . . . . . . . . . 19 2.3 Quantum critical scaling near the Mott transition . . . . . . . . . . . . . . . 21 3 Dynamical mean-field theory (DMFT) . . . . . . . . . . . . . . . . . . . . 25 3.1 Single-impurity Anderson model . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Theoretical foundations of DMFT . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Wilson's numerical renormalization group . . . . . . . . . . . . . . . . . . . 32 3.4 Implementation and choice of parameters . . . . . . . . . . . . . . . . . . . 36 4 Power-law spectra and quantum critical scaling . . . . . . . . . . . . . . . . . . 38 4.1 Scale-invariant solutions of DMFT . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Spectral power laws at T=0 in the metastable insulator . . . . . . . . . . . 40 4.3 Finite-temperature crossovers in the spectral function . . . . . . . . . . . . 47 4.4 Resistivity scaling driven by spectral power laws . . . . . . . . . . . . . . . 50 4.5 Scaling analysis of the dynamic susceptibility . . . . . . . . . . . . . . . . . 58 4.6 Ideas and obstacles towards an analytical understanding . . . . . . . . . . . 62 4.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 II Limits on magnetic quantum criticality from nuclear spins . . . . . . . . . . . . .65 5 Stability of magnetic transitions to hyperfine coupling . . . . . . . . . . . . . . . .66 5.1 Nuclear spins near quantum criticality . . . . . . . . . . . . . . . . . . . . . 66 5.2 Introduction to nuclear spins and hyperfine coupling . . . . . . . . . . . . . 67 5.3 Magnetic phases in the presence of nuclear spins . . . . . . . . . . . . . . . 69 5.4 Two scenarios for magnetic quantum criticality plus nuclear spins . . . . . . 70 6 Paradigmatic models for magnetic quantum phase transitions . . . . . . . . . 73 6.1 Transverse-field Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Coupled-dimer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.3 Frustrated spin models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7 Crossover scales introduced by nuclear spins . . . . . . . . . . . . . . . . . . .83 7.1 Shifted transitions: transverse-field Ising magnets . . . . . . . . . . . . . . . 83 7.2 Smeared transitions: coupled-dimer magnets . . . . . . . . . . . . . . . . . . 90 7.3 Additional transitions due to nuclear spins . . . . . . . . . . . . . . . . . . . 98 7.4 Exotic magnetic quantum phase transitions plus nuclear spins . . . . . . . . 101 7.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 III Domain-driven phase transitions in easy-axis ferromagnets . . . . . . . . 105 8 Easy-axis ferromagnet LiHoF4 . . . . . . . . . . . . . . . . . . . . 106 8.1 Easy-axis ferromagnets in tilted fields . . . . . . . . . . . . . . . . . . . . . 106 8.2 LiHoF4 and its phase transitions . . . . . . . . . . . . . . . . . . . . . . . . 109 9 Modeling of microscopic degrees of freedom in LiHoF4 . . . . . . . . . . . . 112 9.1 Landau theory in tilted fields . . . . . . . . . . . . . . . . . . . . . . . . . . 112 9.2 Crystal field effects and microscopic Hamiltonian . . . . . . . . . . . . . . . 113 9.3 Crossovers in the microscopic model . . . . . . . . . . . . . . . . . . . . . . 118 10 Modeling of mesoscopic degrees of freedom in LiHoF4 . . . . . . . . . . . . . . .123 10.1 Domains in ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.2 Modeling of domain effects as effective interactions . . . . . . . . . . . . . . 127 10.3 Combined mean-field Hamiltonian and domain optimization . . . . . . . . . 130 10.4 Nature of the phase transition in tilted fields . . . . . . . . . . . . . . . . . 132 10.5 Domain-driven phase transition at T = 0 . . . . . . . . . . . . . . . . . . . . 135 10.6 Domain-driven phase transition at finite temperatures . . . . . . . . . . . . 141 10.7 Comparison with experimental results . . . . . . . . . . . . . . . . . . . . . 146 10.8 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 IV Summary & Outlook . . . . . . . . . . . . . . . . . . . . 151 V Appendices . . . . . . . . . . . . . . . . . . . . 155 A Part I: NRG level spectra . . . . . . . . . . . . . . . . . . . . 156 B Part I: Analytical properties of scale-invariant DMFT solutions . . . . . . . . . . .159 B.1 Kondo perturbation theory as an impurity solver . . . . . . . . . . . . . . . 159 B.2 Analytical properties of a power-law self-energy . . . . . . . . . . . . . . . . 166 C Part I: Scaling analysis of the resistivity . . . . . . . . . . . . . . . . . . 168 D Part II: Solution of the transverse-field Ising model with nuclear spins . . . . . . 172 D.1 Holstein-Primakoff representation of the electronic and nuclear spins . . . . 172 D.2 Determination of the classical reference state . . . . . . . . . . . . . . . . . 174 D.3 Excitation spectrum of the coupled nuclear-electronic model . . . . . . . . . 175 D.4 Magnetization, susceptibility, and heat capacity . . . . . . . . . . . . . . . . 177 E Part II: Solution of the coupled-dimer model with nuclear spins . . . . . . . . . . . 181 E.1 Bond-operator description of the electronic spins . . . . . . . . . . . . . . . 181 E.2 Determination to the electronic ground state . . . . . . . . . . . . . . . . . 185 E.3 Holstein-Primakoff representation of the nuclear spins . . . . . . . . . . . . 188 E.4 Excitation spectrum of the coupled nuclear-electronic model . . . . . . . . . 189 E.5 Staggered magnetization and susceptibility . . . . . . . . . . . . . . . . . . 192 F Part III: Calculation of domain-induced effective interactions . . . . . . . . . . . . . 198 Bibliography . . . . . . . . . . . . . . . . . . . . 203