In this work we will analyse the capabilities of several numerical techniques for the description of different physical systems. Thereby, the considered systems range from quantum over semiclassical to classical and from few- to many-particle systems. For each case we address an interesting, partly unsolved question. Despite the different topics we address in the individual chapters, the problems under study are somehow related because we focus on the time evolution of the system. In chapter 1 we investigate the behaviour of a single quantum particle in the presence of an external disordered background (static potentials). Starting from the quantum percolation problem, we address the fundamental question of a disorder induced (Anderson-) transition from extended to localised single-particle eigenstates. Distinguishing isolating from conducting states by applying a local distribution approach for the local density of states (LDOS), we detect the quantum percolation threshold in two- and three-dimensions. Extending the quantum percolation model to a quantum random resistor model, we comment on the possible relevance of our results to the influence of disorder on the conductivity in graphene sheets. Furthermore, we confirm the localisation properties of the 2D percolation model by calculating the full quantum time evolution of a given initial state. For the calculation of the LDOS as well as for the Chebyshev expansion of the time evolution operator, the kernel polynomial method (KPM) is the key numerical technique. In chapter 2 we examine how a single quantum particle is influenced by retarded bosonic fields that are inherent to the system. Within the Holstein model, these bosonic degrees of freedom (phonons) give rise to an infinite dimensional Hilbert space, posing a true many-particle problem. Constituting a minimal model for polaron formation, the Holstein model allows us to study the optical absorption and activated transport in polaronic systems. Using a two-dimensional variant of the KPM, we calculate for the first time quasi-exactly the optical absorption and dc-conductivity as a function of temperature. Concerning the numerical technique, the close relation to the time evolution in the other chapters get clear if we identify temperature with an imaginary time. In chapter 3 we come back to the time evolution of a quantum particle in an external, static potential and investigate the capability of semiclassical approximations to it. Considering various one-dimensional geometries, we address basic quantum effects as tunneling, interference and anharmonicity. The question is, to which extend and at which numerical costs, several semiclassical methods can reproduce the exact result for the quantum dynamics, calculated by Chebyshev expansion. To this end we consider the linearised semiclassical propagator method, the Wigner-Moyal approach and the recently proposed quantum tomography. A conceptually very interesting aspect of the compared semiclassical methods is their relation to different representations of quantum mechanics (wave function/density matrix, Wigner function, quantum tomogram). Finally, in chapter 4 we calculate the dynamics of a classical many-particle system under the influence of external fields. Considering a low-temperature rf-plasma, we investigate the interplay of the plasma dynamics and the motion of dust particles, immersed into the plasma for diagnostic reasons. In addition to the huge number of involved particles, the numerical description of this systems faces the challenge of a large range of involved time and length scales. Exploiting the mass differences of plasma constituents and dust particles allows for separating the PIC description of the plasma from the MD simulation of the dust particles in the effective surrounding plasma.
In this work we will analyse the capabilities of several numerical techniques for the description of different physical systems. Thereby, the considered systems range from quantum over semiclassical to classical and from few- to many-particle systems. For each case we address an interesting, partly unsolved question. Despite the different topics we address in the individual chapters, the problems under study are somehow related because we focus on the time evolution of the system. In chapter 1 we investigate the behaviour of a single quantum particle in the presence of an external disordered background (static potentials). Starting from the quantum percolation problem, we address the fundamental question of a disorder induced (Anderson-) transition from extended to localised single-particle eigenstates. Distinguishing isolating from conducting states by applying a local distribution approach for the local density of states (LDOS), we detect the quantum percolation threshold in two- and three-dimensions. Extending the quantum percolation model to a quantum random resistor model, we comment on the possible relevance of our results to the influence of disorder on the conductivity in graphene sheets. Furthermore, we confirm the localisation properties of the 2D percolation model by calculating the full quantum time evolution of a given initial state. For the calculation of the LDOS as well as for the Chebyshev expansion of the time evolution operator, the kernel polynomial method (KPM) is the key numerical technique. In chapter 2 we examine how a single quantum particle is influenced by retarded bosonic fields that are inherent to the system. Within the Holstein model, these bosonic degrees of freedom (phonons) give rise to an infinite dimensional Hilbert space, posing a true many-particle problem. Constituting a minimal model for polaron formation, the Holstein model allows us to study the optical absorption and activated transport in polaronic systems. Using a two-dimensional variant of the KPM, we calculate for the first time quasi-exactly the optical absorption and dc-conductivity as a function of temperature. Concerning the numerical technique, the close relation to the time evolution in the other chapters get clear if we identify temperature with an imaginary time. In chapter 3 we come back to the time evolution of a quantum particle in an external, static potential and investigate the capability of semiclassical approximations to it. Considering various one-dimensional geometries, we address basic quantum effects as tunneling, interference and anharmonicity. The question is, to which extend and at which numerical costs, several semiclassical methods can reproduce the exact result for the quantum dynamics, calculated by Chebyshev expansion. To this end we consider the linearised semiclassical propagator method, the Wigner-Moyal approach and the recently proposed quantum tomography. A conceptually very interesting aspect of the compared semiclassical methods is their relation to different representations of quantum mechanics (wave function/density matrix, Wigner function, quantum tomogram). Finally, in chapter 4 we calculate the dynamics of a classical many-particle system under the influence of external fields. Considering a low-temperature rf-plasma, we investigate the interplay of the plasma dynamics and the motion of dust particles, immersed into the plasma for diagnostic reasons. In addition to the huge number of involved particles, the numerical description of this systems faces the challenge of a large range of involved time and length scales. Exploiting the mass differences of plasma constituents and dust particles allows for separating the PIC description of the plasma from the MD simulation of the dust particles in the effective surrounding plasma.
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Quantum states can be described equivalently by density matrices, Wigner functions, or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance, and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.
In this thesis I present contributions to the development, analysis and application of tensor network state methods for numerically exact quantum dynamics in one and two-dimensional lattice systems. The setting of numerically exact quantum dynamics is introduced in Chapter 2. This includes a discussion of exact diagonalization approaches and massively parallel implementations thereof as well as a brief introduction of tensor network states. In Chapter 3, I perform a detailed analysis of the performance of n-ary tree tensor network states for simulating the dynamics of two-dimensional lattices. This constitutes the first application of this class of tensor network to dynamics in two spatial dimensions, a long-standing challenge, and the method is found to perform on par with existing state-of-the-art approaches. Chapter 4 showcases the efficacy of a novel tensor network format I developed, tailored to electron-phonon coupled problems in their single-electron sector, through an application to the Holstein model. The applicability of the approach to a broad range of parameters of the model allows to reveal the strong influence of the spread of the electron distribution on the initial state of the phonons at the site where the electron is introduced, for which a simple physical picture is offered. I depart from method development in Chapter 5 and analyse the prospects of using tensor network states evolved using the time-dependent variational principle as an approximate approach to determine asymptotic transport properties with a finite, moderate computational effort. The method is shown to not yield the correct asymptotics in a clean, non-integrable system and can thus not be expected to work in generic systems, outside of finely tuned parameter regimes of certain models. Chapters 6 and 7 are concerned with studies of spin transport in long-range interacting systems using tensor network state methods. For the clean case, discussed in Chapter 6, we find that for sufficiently short-ranged interactions, the spreading of the bulk of the excitation is diffusive and thus dominated by the local part of the interaction, while the tail of the excitation decays with a powerlaw that is twice as large as the powerlaw of the interaction. Similarly, in the disordered case, analysed in Chapter 7, we find subdiffusive transport of spin and sub-linear growth of entanglement entropy. This behaviour is in agreement with the behaviour of systems with local interactions at intermediate disorder strength, but provides evidence against the phenomelogical Griffith picture of rare, strongly disordered insulating regions. We generalize the latter to long-ranged interactions and show that it predicts to diffusion, in contrast to the local case where it results in subdiffusive behaviour.
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
In this second part we present a set of methods, analytical and numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. The key points demonstrating advantages of this approach are: (i) effects of localization of possible quantum states; (ii) effects of non-perturbative multiscales which cannot be calculated by means of perturbation approaches; (iii) effects of formation of complex/collective quantum patterns from localized modes and classification and possible control of the full zoo of quantum states, including (meta) stable localized patterns (waveletons). We demonstrate the appearance of nontrivial localized (meta) stable states/patterns in a number of collective models covered by the (quantum)/(master) hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of "wignerization" procedure (Weyl-Wigner-Moyal quantization) of classical BBGKY kinetic hierarchy, and present the explicit constructions for exact analytical/numerical computations. Our fast and efficient approach is based on variational and multiresolution representations in the bases of polynomial tensor algebras of generalized localized states (fast convergent variational-wavelet representation). We construct the representations for hierarchy/algebra of observables(symbols)/distribution functions via the complete multiscale decompositions, which allow to consider the polynomial and rational type of nonlinearities. The solutions are represented via the exact decomposition in nonlinear high-localized eigenmodes, which correspond to the full multiresolution expansion in all underlying hidden time/space or phase space scales. In contrast with different approaches we do not use perturbation technique or linearization procedures. Numerical modeling shows the creation of different internal structures from localized modes, which are related to the localized (meta) stable patterns (waveletons), entangled ensembles (with subsequent decoherence) and/or chaotic-like type of behaviour.