In this thesis the method of self-similar continuous unitary transformations (CUTs) in real space is developed. It is applied to various models from the field of strongly correlated solid state physics and optical lattices. The CUT method maps a given Hamiltonian to an effective Hamiltonian which is in some respect simpler. In a self-similar CUT the truncation scheme relies on the structure of the operators. The real space approach put forward in this thesis truncates the operators according to their extension in real space. We mainly use a MKU generator to derive an effective model which is block-diagonal in the number of quasiparticles. We give a detailed description of the method. The transformation of the Hamiltonian and of the observables is explained. The implementation on a computer and the optimization of the programs are extensively discussed. The first application of the method deals with the fermionic Hubbard model. We derive an effective Hamiltonian, a generalized t-J-model, which conserves the number of doubly occupied sites. The effective model includes beyond the well-known Heisenberg term also further two-spin and four-spin terms. In addition, the charge motion and interaction within the effective model is addressed. In the context of the Hubbard model we discuss the question if an effective model that conserves the number of double occupancies can be derived in principle. The antiferromagnetic spin ladder and spin chain are treated from the viewpoint of a dimerized system. For the spin ladder we calculate the dispersion, the two-particle energies including bound states and the multi-particle continua. The discussion of multi-particle continua shows that the order of the energy levels is important for the convergence of the transformation. For the spin chain the dispersion is calculated. In addition, the transformation of an observable yields the corresponding spectral weights. The spectral weights are compared to results from a perturbative CUT. For the bosonic Hubbard model we calculate the dispersion and determine the phase diagram. The spectral weight of a certain observable is calculated in order to explain the outcome of a recent experiment that measures the spectroscopic response of atoms confined in optical traps. The distribution of spectral weight is analyzed also at finite temperatures. We find evidence for a substantial temperature of the size of the microscopic parameters. This finding is supported by purely thermodynamic considerations.
In this thesis the method of self-similar continuous unitary transformations (CUTs) in real space is developed. It is applied to various models from the field of strongly correlated solid state physics and optical lattices. The CUT method maps a given Hamiltonian to an effective Hamiltonian which is in some respect simpler. In a self-similar CUT the truncation scheme relies on the structure of the operators. The real space approach put forward in this thesis truncates the operators according to their extension in real space. We mainly use a MKU generator to derive an effective model which is block-diagonal in the number of quasiparticles. We give a detailed description of the method. The transformation of the Hamiltonian and of the observables is explained. The implementation on a computer and the optimization of the programs are extensively discussed. The first application of the method deals with the fermionic Hubbard model. We derive an effective Hamiltonian, a generalized t-J-model, which conserves the number of doubly occupied sites. The effective model includes beyond the well-known Heisenberg term also further two-spin and four-spin terms. In addition, the charge motion and interaction within the effective model is addressed. In the context of the Hubbard model we discuss the question if an effective model that conserves the number of double occupancies can be derived in principle. The antiferromagnetic spin ladder and spin chain are treated from the viewpoint of a dimerized system. For the spin ladder we calculate the dispersion, the two-particle energies including bound states and the multi-particle continua. The discussion of multi-particle continua shows that the order of the energy levels is important for the convergence of the transformation. For the spin chain the dispersion is calculated. In addition, the transformation of an observable yields the corresponding spectral weights. The spectral weights are compared to results from a perturbative CUT. For the bosonic Hubbard model we calculate the dispersion and determine the phase diagram. The spectral weight of a certain observable is calculated in order to explain the outcome of a recent experiment that measures the spectroscopic response of atoms confined in optical traps. The distribution of spectral weight is analyzed also at finite temperatures. We find evidence for a substantial temperature of the size of the microscopic parameters. This finding is supported by purely thermodynamic considerations.
In this thesis we describe a novel perturbative approach to low-dimensional quantum many-particle systems, which is based on continuous unitary transformations. We consider systems, which are defined on a lattice and allow a perturbative decomposition. The unperturbed part must have an equidistant spectrum -- the difference between two successive levels is called a quasi-particle. In this case the perturbative transformation leads to an effective Hamiltonian, which conserves the number of particles. The same transformation is used to also derive the effective counterparts of other, experimentally relevant observables. The effective operators are obtained as series expansions in the (small) perturbation parameter. In each order we find a set of products of ladder-operators, which act on states uniquely defined by the number of quasi-particles and their position in the lattice. Thus all calculations can be done in real space. The mathematical structure of the effective operators is extensively analysed. We additionally give all details necessary to implement the method on a computer, which allows the calculation of the effective quantities up to high orders. The method facilitates quantitative calculations of multi-particle excitations and spectral densities of experimentally relevant observables. We include a comprehensive application of the method to the two-dimensional Shastry-Sutherland model, a strongly frustrated quantum spin system. The model has experience an experimental realization by SrCu2(BO3)2. The limit of strong dimerization serves as starting point. Here the ground state is given by singlets on all dimers -- a single triplet constitutes an elementary excitation, i.e. quasi-particle. We quantitatively calculate the one- and two-triplet energies as well as the spectral densities of the Raman and neutron scattering operators. The findings for the spectral densities in particular represent new results. Comparing them to experimental data leads to interesting insights into the spectral properties of low-dimensional quantum spin systems.
The emergence of complex behavior out of simple elementary building blocks is a typical property of condensed matter systems. While the individual constituents may be well-studied, strongly interacting many-particle systems can give rise to novel collective behavior that can not be understood by considering the individual properties alone. Effective models are a powerful tool to handle this complexity: Instead of aiming at a full description a complex many-particle system on the high-energy scale, an effective model covers only those degrees of freedom which are essential for the low-energy physics. While the high-energy degrees of freedom are integrated out, they influence the low-energy degrees of freedom due to a renormalization of the effective coupling constants. In particular, the concept of the quasi-particle allows for the description of a strongly-interacting many-particle excitations in terms of a simple objects with similar degrees of freedom as a dispersive elementary particle, which gives rise to phenomena like quasi-particle bound states or quasi-particle decay. This thesis aims at the application and methodological improvement of a systematic and well-controlled framework to derive effective low-energy models, denoted as Continuous Unitary Transformations (CUT). In this method, a Hamiltonian is mapped continuously to an effective basis, in which the low-energy degrees of freedom can be interpreted as quasi-particles. Mathematically, the transformation is parametrized by a flow parameter and is represented as a differential equation, denoted as flow equation, while the direction of the transformation is controlled by the generator scheme. As foundation of our studies, we use the sCUT method, in which the types of possible interaction terms are fixed by a truncation scheme. In a common project, we established a new flavor of CUT, called epCUT. It allows us to derive the effective Hamiltonian in a perturbation series. In contrast to the already existent pCUT method, epCUT can handle system with a non-equidistant spectrum as well. Moreover, we discovered the deepCUT method, the non-perturbative twin of epCUT. While it bears large similarities to sCUT, it replaces the cumbersome definition of an appropriate truncation scheme by the order of the flow equation system. Compared to sCUT, deepCUT offers a larger robustness while the ambiguities of the truncation scheme are eliminated. My contribution comprises the computational, parallelized construction of the flow equation system as well. In another project, we investigated the breakdown of the conventional, particle-sorting generator schemes in system with strongly overlapping quasi-particle sub-spaces and identified the emergence of a spectrum unbounded from below due to truncation errors as origin of the divergence. As a testing ground, we used a system of two coupled bosonic degrees of freedom. To overcome these problems, we developed a family of three novel variational generator schemes, denoted as scalar, vectorial and tensorial optimization. After this, we investigated the dimerized, two-dimensional S=1/2 Heisenberg model with an emphasis on the quantum phase transition between a magnetically disordered and a magnetically ordered phase using sCUT. Parametrizing the starting point by a variational parameter, we were able to describe the spontaneous breaking of the SU(2) spin symmetry by CUT for the first time. We were able to calculate the ground state energy and the dispersion of triplons, and magnons, respectively, in both phases. However, we were not able to provide quantitative results for the spontaneous magnetization. Furthermore, we found a way to derive an optimal starting point for the integration of the flow equations that is independent on the particularities of the model and does not require a parametrization by a variational parameter. As a further application, we analyzed the S=1 Heisenberg chain by means of deepCUT. To this end, we mapped the system to an extended S=1/2 Heisenberg ladder with the help of a variational parameter. Using the variational generator schemes developed before, we were able to decouple the low-energy spectrum and to calculate spectral densities. In this investigation, we found an indication for a S=0 bound state. Furthermore, we were able to find some signs of spontaneous quasi-particle decay, but a quantitatively correct description is pending.
In this work a perturbative realization of particle conserving continuous unitary transformations is applied to study the energies and the spectral properties of quasi one-dimensional quantum antiferromagnets. The systems considered are defined on a lattice and they allow for a perturbative decomposition. The unperturbed part is chosen to be a fully dimerized state. The groundstate is a product-state of singlets and the excitation spectrum is equidistant. The related energy quantum is called a triplon becaues it has total spin one. The continous unitary transformation leads to an effective triplon-conserving Hamiltonian and effective, experimentally relevant observables. The effective operators are obtained in a high-order series expansion in the perturbation parameters. All calculations are performed on finite clusters in real space and yield exact results in the thermodynamic limit due to the linked cluster theorem. The results are exact in the given order. In order to improve the representation of the results extrapolation techniques are used. A detailed description of extrapolation tools like standard Pade and dlogPade extrapolation, optimized perturbation theory and the use of internal parameters is given. The dimerized and frustrated spin-chain is analysed first. At zero frustration, a detailed investigation of the spectral weights shows that even in the limit of vanishing dimerization, the one-dimensional Heisenberg model, almost the total spectral weight is situated in the two-triplon sector. So, besides spinons, triplons may be used as elementary excitations for the one-dimensional Heisenberg model. The case of strong frustration is not yet settled. An extensive review of one- and two-triplon spectral densities at large and intermediate value of the dimerization for various values of the frustration is presented. The findings are compared with field theoretical results. In addition, the Raman response and the infrared absorption are investigated. Second, the antiferromagnetic two-leg Heisenberg ladder plus additional four-spin interaction is investigated. The transformation starts from the limit of isolated rung dimers. The excitations are rung-triplons. The relative energies of one-triplon states, the two-triplon bound states and the multi-triplon continua are given for various couplings. Optical observables are discussed in detail. The extent of the rung-singlet phase is calculated in the whole parameter space. It is shown that the experimental realizations of two-leg ladder systems are always situated in the rung-singlet phase. In the experimentally relevant regime, most of the spectral weight is captured by the one- and the two-triplon sector, but also three- and four-triplon contributions become sizable. The current understanding of the spectroscopic signatures of magnetic excitations in cuprate ladders measured with inelastic neutron scattering, Raman spectroscopy and infrared absorption is presented. The results obtained are compared with experimental findings. The first experimental evidence of a triplon-triplon bound state in a ladder system is found.
A fundamental principle in condensed matter physics is the effective description in terms of quasiparticles. The high-energy part of the system is accounted for by renormalized properties of the quasiparticles, providing both, an accurate description and interpretation of the relevant low-energy physics. We consider two classes of methods suited to provide effective quasiparticle descriptions in the thermodynamic limit: (non-perturbative) linked-cluster expansions ((N)LCEs) and continuous unitary transformations (CUTs). We focus specifically on the combination of both methods, providing a different perspective and solutions to existing challenges. LCEs provide high-order series expansions in a perturbation parameter, by combining effective Hamiltonians determined on finite clusters. We introduce a white-graph expansion for the method of perturbative continuous unitary transformations when implemented as an LCE. The essential idea behind an expansion in white graphs is to perform an optimized bookkeeping during the calculation by exploiting the model-independent effective Hamiltonian in second quantization and the associated inherent cluster additivity. This approach is shown to be especially well suited for microscopic models with many coupling constants, since the total number of relevant graphs is drastically reduced. The white-graph expansion is exemplified for a two-dimensional quantum spin model, illustrating its efficiency. In NLCEs, the perturbative treatment of finite clusters is replaced by a numerical exact (block) diagonalization. While LCEs are restricted due to their perturbative nature, the non-perturbative variant is not. In graph-based continuous unitary transformations (gCUTs), the block diagonalization is achieved by a non-perturbative CUT performed on finite clusters. The central objective of this thesis is a modification of the gCUT scheme, allowing to treat two kinds of major issues, denoted by pseudo and genuine decay, occuring for NLCEs due to the non-perturbative treatment of clusters. Indeed, one finds surprising effects caused by the non-perturbative renormalization. In particular, we identify a fundamental challenge for any non-perturbative approach based on finite clusters resulting from the reduced symmetry on graphs, most importantly the breaking of translational symmetry when targeting the properties of excited states. This can be traced back to the appearance of intruder states in the low-energy spectrum, which represent a major obstacle in quasi-degenerate perturbation theory. Here, a generalized notion of cluster additivity is introduced, which is used to formulate an optimized scheme of gCUTs, allowing to solve and to physically understand this major issue. Most remarkably, our improved scheme demands to go beyond the paradigm of using the exact eigenvectors on graphs. We demonstrate that the modified scheme is correct in the non-perturbative regime. Even at quantum criticality, the scheme gives valid results. To determine the critical behavior, one must rely on extrapolation schemes. We introduce a generic approach to extract critical properties from sequences of numerical data which is directly relevant for NLCEs. The scheme is applied to the quantum phase transition between the dimerized and the isotropic spin 1/2 Heisenberg chain. Finally, we investigate a scenario where the gapped quasiparticle excitation is not stable for all momenta, i.e., the one-particle mode merges with the continuum for certain momenta and one observes quasiparticle decay. In this case, intruder states merging with the low-energy spectrum on finite clusters represent genuine physics of the system. Again, a proper renormalization on finite clusters satisfies the generalized cluster additivity. Our adjusted renormalization approach performed on finite clusters does not necessarily lead to a full decoupling of the quasiparticle subspaces and the remaining interactions are part of the effective description. The resulting effective Hamiltonian can be analyzed to describe quasiparticle decay in the thermodynamic limit. The modified scheme is applied to four-leg spin 1/2 Heisenberg ladders, providing insights into the spectral properties. Overall, this thesis presents several important developments for the derivation of effective quasiparticle pictures in quantum spin models via perturbative and non-perturbative LCEs, opening various opportunities for future investigations.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
The Hubbard model is a simple but profound theoretical tool for understanding the physics underlying strongly correlated systems. This phenomenology includes, but is not limited to, metal-insulating transitions, charge and spin order (charge density/spin density waves), superconducting phases, non-Fermi liquid physics, among others. Examining one dimensional physical models described by the Hubbard Hamiltonian can teach us much about the related physics of higher dimensional systems. Applying advanced numerical computational techniques such as density matrix numerical renormalization group (DMRG) to one-dimensional manypbody model Hamiltonians allows for the calculation of ground state energies and dynamical spectral functions with extremely high precision and accuracy. This thesis explores the application of DMRG numerical techniques to one-dimensional physical systems described by Hamiltonians beyond the conventional Hubbard model, and the nuanced physical effects observed for these systems across various regimes and phases. A Wannier Hubbard chain composed of non-orthogonal overlapping atomic orbitals was transformed to an orthogonal atomic basis using a non-unitary Lowdin transformation. The calculation of energies, dynamical spectral functions and the photoemission spectrum for this Wannier Hubbard chain revealed phases not previously observed for Hubbard chains at half-filling. Additional analysis and calculations were performed for a CuO chain described by the Luther-Emery Hamiltonian that epitomizes charge-transfer physics. The motivation for studying this CuO chain was previous literature on transition metal oxides (TMOs) and the mapping of multiband Hamiltonians to single band models in certain energy regimes. Finally, we performed computational analysis for a spin-3/2 generalized Hubbard chain. By spanning the full phase space of our spin-3/2 Hubbard chain in terms of chemical potential μ and external magnetic field h we observed non-typical Cooper pairing states and novel phases for different symmetries of our Hamiltonian.--Author's abstract