In the first part of this thesis, a stochastic adaptation of the microcanonical simulation method is applied to the numerical simulation of the Su-Schrieffer-Heeger Hamiltonian for polyacetylene, a one-dimensional polymer where fermion-boson interactions play a dominant role in the dynamics of the system. The pure microcanonical simulation method fails in the marginally ergodic case and a stochastic adaptation, the hybrid microcanonical method, is employed to resolve problems with ergodicity. The hybrid method is shown to be an efficient method for higher dimensional fermionic quantum systems. In the second part of this thesis, a numerical simulation of the evolution of a network of global cosmic strings in an expanding Robertson-Walker universe is carried out. The system is quenched through an order-disorder phase transition and the nature of the string distribution is examined. While the string distribution observed at the phase transition is in good agreement with earlier estimates, the simulation reveals that the dynamics of the strings are suppressed by interactions with the Goldstone field. The network decays by topological annihilation and no spatial correlations are observed at any point in the simulation.
In the first part of this thesis, a stochastic adaptation of the microcanonical simulation method is applied to the numerical simulation of the Su-Schrieffer-Heeger Hamiltonian for polyacetylene, a one-dimensional polymer where fermion-boson interactions play a dominant role in the dynamics of the system. The pure microcanonical simulation method fails in the marginally ergodic case and a stochastic adaptation, the hybrid microcanonical method, is employed to resolve problems with ergodicity. The hybrid method is shown to be an efficient method for higher dimensional fermionic quantum systems. In the second part of this thesis, a numerical simulation of the evolution of a network of global cosmic strings in an expanding Robertson-Walker universe is carried out. The system is quenched through an order-disorder phase transition and the nature of the string distribution is examined. While the string distribution observed at the phase transition is in good agreement with earlier estimates, the simulation reveals that the dynamics of the strings are suppressed by interactions with the Goldstone field. The network decays by topological annihilation and no spatial correlations are observed at any point in the simulation.
This thesis applies techniques of non-perturbative quantum field theory for solving both bosonic and fermionic systems dynamically on a lattice. The methods are first implemented in a bosonic system to examine the quantum decay of a scalar field oscillon in 2+1D. These configurations are a class of very long-lived, quasi-periodic, non-topological soliton. Classically, they last much longer than the natural timescales in the system, but gradually emit energy to eventually decay. Taking the oscillon to be the inhomogeneous, (quantum) mean field of a self-interacting scalar field enables an examination of the changes to the classical evolution in the presence of quantum fluctuations. The evolution is implemented through applying the Hartree approximation to the quantum dynamics. A statistical ensemble of fields replaces the quantum mode functions to calculate the quantum correlators in the dynamics. This offers the possibility for a reduction in the computational resources required to numerically evolve the system. The application of this method in determining the oscillon lifetimes, though, provides only a negligible gain in computational efficiency: likely due to the lack of any space or time averaging in measuring the lifetimes, and the low dimensionality. Evolving a Gaussian parameter-space of initial conditions enables comparing the classical and quantum evolution. The quantum fluctuations significantly reduce the lifetime compared to the classical case. Examining the evolution in the oscillatory frequency demonstrates the decay in the quantum system occurs gradually. This markedly contrasts the classical evolution where the oscillon frequency has been demonstrated to evolve to a critical frequency when the structure abruptly collapses. Despite the distinctly different evolution and lifetime, a similar range of the Gaussian initial conditions in both cases generates oscillons. This indicates the classical effects dominate the early evolution, and the quantum fluctuations most significantly alter the later decay. The methods are next implemented in a fermionic system to examine "tunnelling of the 3rd kind". This phenomenon is examined in the case where a uniform magnetic field propagates through a classical barrier by pair creation of fermions: these cross unimpeded through the barrier and annihilate to (re-)create the magnetic field in the classically shielded region. A statistical ensemble of fields, similarly to the oscillon simulations, is initially constructed for evaluating the fermionic contribution in the gauge field dynamics. This ensemble, importantly and in contrast to the bosonic case, involves two sets of fields to reproduce the anti-commuting nature of the fermion operator. The ensemble method, again, offers the possibility for a reduction in the computational resources required to evolve the system numerically. A test case indicates the method for the tunnelling system, though, requires impracticable computational resources. Using the symmetries in the system to construct an ansatz for the fields provides an alternative method to evolve the dynamics on a lattice. This procedure effectively reduces the system to a 1+1 dimensional problem with the fermion mode functions summed over the three-dimensional momentum space. The significant decrease in the real-time for the evolution (and quite attainable computational resources) on applying the ansatz provides a practical technique to examine the tunnelling. Measuring the magnetic field in the classically shielded region confirms the analytic estimates. These (qualitatively) reproduced the exponential decrease estimated in the classical transmission on varying the interaction strength between the barrier and the magnetic field. The observed tunnelling signal, moreover, matches the perturbative, analytic estimate within the expected correction in the lattice configuration. These bosonic and fermionic quantum, dynamical simulations demonstrate limitations to the benefits in applying the ensemble method. The highly practical and successful tunnelling computations, in contrast, indicate the potential power of a suitable ansatz to significantly reduce the computational times in simulations on a lattice.
In this thesis we consider the hybrid quantum Monte Carlo method for simulations of the Hubbard and Su-Schrieffer-Heeger model. In the first instance, we discuss the hybrid quantum Monte Carlo method for the Hubbard model on a square lattice. We point out potential ergodicity issues and provide a way to circumvent them by a complexification of the method. Furthermore, we compare the efficiency of the hybrid quantum Monte Carlo method with a well established determinantal quantum Monte Carlo method for simulations of the half-filled Hubbard model on square lattices. One reason why the hybrid quantum Monte Carlo method loses the comparison is that we do not observe the desired sub-quadratic scaling of the numerical effort. Afterwards we present a formulation of the hybrid quantum Monte Carlo method for the Su-Schrieffer-Heeger model in two dimensions. Electron-phonon models like this are in general very hard to simulate using other Monte Carlo methods in more than one dimensions. It turns out that the hybrid quantum Monte Carlo method is much better suited for this model . We achieve favorable scaling properties and provide a proof of concept. Subsequently, we use the hybrid quantum Monte Carlo method to investigate the Su-Schrieffer-Heeger model in detail at half-filling in two dimensions. We present numerical data for staggered valence bond order at small phonon frequencies and an antiferromagnetic order at high frequencies. Due to an O(4) symmetry the antiferromagnetic order is connected to a superconducting charge density wave. Considering the Su-Schrieffer-Heeger model without tight-binding hopping reveals an additional unconstrained Z_2 gauge theory. In this case, we find indications for π-fluxes and a possible Z_2 Dirac deconfined phase as well as for a columnar valence bond ordered state at low phonon energies. In our investigations of the several phase transitions we discuss the different possibilities for the underlying mechanisms and reveal first insights into a rich phase diagram.
In the first part of this thesis we study the Aubry-André model for interacting fermions. We numerically describe its phase diagram at half filling, performing both DMRG and QMC simulations. We show the existence of a localized phase and other three regimes: luttinger liquid, charge density wave and productstate. We study the properties of the excited states of the Hamiltonian, looking for a many-body mobility edge in the spectrum, i.e. an energy threshold that separates localized from ergodic states. Analyzing many indicators we prove its existence. Finally we propose a quench-spectroscopy method for detecting the mobility edge dynamically. In the second part we study the expansion dynamics of two bosons in a one-dimensional lattice as ruled by the Bose-Hubbard model Hamiltonian, both in the attractive and repulsive regime. Using the Bethe Ansatz we identify the bound states effects and how the two-particles state evolves in time. We show that, independently from the initial state, there exists a strong relation between the expansion velocity and the presence of bound states in the spectrum. Moreover, we discuss the role of the lattice in the system expansion. In the third part we study the time evolution of the entanglement entropy in the Ising model, when it is dynamically driven across a quantum phase transition with different velocities. We computed the time-evolution of the half chain entanglement entropy and we found that, depending on the velocity at which the critical point is reached, it displays different regimes: an adiabatic one when the system evolves according to the instantaneous ground state; a a sudden quench regime when the system remains frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations. Moreover, we discuss the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase.
In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
This thesis presents our work on two separate subfields and has for this reason been divided into two parts: (I) the development and implementation of a numerical technique for non-equilibrium dynamics and a (II) detailed investigation of the dirty-boson problem. In part (I), we optimize and develop the quantum phase space method known as the positive-P representation (PPR) specifically for the simulations of the real time quench dynamics of quantum spin systems. The main benefit of this approach is that the dynamics of the density operator is mapped onto ∼N stochastic variables (where N is the size of the system) that obey Langevin-type stochastic differential equations, thereby greatly reducing the complexity of the problem. The first publication presents our initial use of the PPR on spin systems by using a Schwinger mapping for spin operators on to Bosonic operators since the underlying basis comprises of Bosonic coherent states. We simulate the quench dynamics of the generalized transverse-field spin-1/2 XXZ model in 1d showing that simulations of up to 100 spins are possible, albeit for relatively short simulation times. In our second publication, we reformulate the PPR using SU(2) spin coherent states and further optimize simulation lifetimes by implementing an extrapolation scheme, to achieve several-fold improvements over the results of our first publication. We focus solely on the transverse Ising model and simulate its quench dynamics in 1d and 2d using up to 104 spins, while significantly extending simulation lifetimes. The second part of this thesis is a numerical study of the universality class of the Superfluid-Bose Glass transition of the dirty-boson system. Recently the longstanding exact result z=d, where z is the dynamic critical exponent and d is the dimensionality of the system, has been challenged by a series of numerical studies, suggesting the alternative scenario that z should be instead unconstrained. To address this controversy, we use large scale quantum Monte-Carlo simulations on two independent quantum models, and average over 5×104−105 disorder realizations to numerically determine the universality class of the dirty-boson transition, paying particular attention to the dynamic critical exponent, z and without any biasing assumptions.