In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and $epsilon$-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-$d$ expansion method, $d$ being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-$d$ formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of $1/d$. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of $1/d$ leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=$1/2$, we have calculated the $1/d$ expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the $1/d$ expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-$1/2$ in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same $1/d$ expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In $d=2$, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-$1/2$ model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-$d$ are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.

In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and $epsilon$-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-$d$ expansion method, $d$ being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-$d$ formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of $1/d$. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of $1/d$ leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=$1/2$, we have calculated the $1/d$ expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the $1/d$ expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-$1/2$ in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same $1/d$ expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In $d=2$, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-$1/2$ model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-$d$ are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.