Strongly interacting many-body quantum systems constitute some of the most challenging problems in physics, and exact analytical solutions are rare. In this thesis, we will investigate the models of two such systems numerically using the unbiased worm-algorithm Monte Carlo method. The first model is the t-J model, which describes strongly correlated lattice electrons and is thought to capture the essential physics of high-temperature superconductors. Of particular interest to us is the emergent magnetic polaron studied in the first two papers. The microscopic pairing mechanism between these quasiparticles remains one of the most outstanding open questions in condensed matter physics. The second model is an extended multi-component Bose--Hubbard model, which describes lattice bosons of multiple species. This model is essential to the third paper, in which we demonstrate a novel superfluid phenomenon in the form of a dissipationless drag effect that couples the different superfluid components of a superfluid mixture in a noncollinear fashion. Dissipationless drag is a fundamental effect present in a diverse set of physical systems. In the fourth paper, we further use the Bose--Hubbard model to study other new superfluid phenomena that arise when the number of superfluid components is increased. A general introduction to the research field is first given in the thesis, after which some prerequisites are refreshed. Then essential concepts are introduced, and the numerical method is outlined and benchmarked. Following that is a summary of the research carried out.
Strongly interacting many-body quantum systems constitute some of the most challenging problems in physics, and exact analytical solutions are rare. In this thesis, we will investigate the models of two such systems numerically using the unbiased worm-algorithm Monte Carlo method. The first model is the t-J model, which describes strongly correlated lattice electrons and is thought to capture the essential physics of high-temperature superconductors. Of particular interest to us is the emergent magnetic polaron studied in the first two papers. The microscopic pairing mechanism between these quasiparticles remains one of the most outstanding open questions in condensed matter physics. The second model is an extended multi-component Bose--Hubbard model, which describes lattice bosons of multiple species. This model is essential to the third paper, in which we demonstrate a novel superfluid phenomenon in the form of a dissipationless drag effect that couples the different superfluid components of a superfluid mixture in a noncollinear fashion. Dissipationless drag is a fundamental effect present in a diverse set of physical systems. In the fourth paper, we further use the Bose--Hubbard model to study other new superfluid phenomena that arise when the number of superfluid components is increased. A general introduction to the research field is first given in the thesis, after which some prerequisites are refreshed. Then essential concepts are introduced, and the numerical method is outlined and benchmarked. Following that is a summary of the research carried out.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.
In multicomponent superfluids and superconductors, co- and counterflows of components have, in general, different properties. A. F. Andreev and E. P. Bashkin [Sov. Phys. JETP 42, 164 (1975)] discussed, in the context of He3/He4 superfluid mixtures, that interparticle interactions produce a dissipationless drag. The drag can be understood as a superflow of one component induced by phase gradients of the other component. Importantly, the drag can be both positive (entrainment) and negative (counterflow). The effect is known to have crucial importance for many properties of diverse physical systems ranging from the dynamics of neutron stars and rotational responses of Bose mixtures of ultracold atoms to magnetic responses of multicomponent superconductors. Although substantial literature exists that includes the drag interaction phenomenologically, only a few regimes are covered by quantitative studies of the microscopic origin of the drag and its dependence on microscopic parameters. Here we study the microscopic origin and strength of the drag interaction in a quantum system of two-component bosons on a lattice with short-range interaction. By performing quantum Monte Carlo simulations of a two-component Bose-Hubbard model we obtain dependencies of the drag strength on the boson-boson interactions and properties of the optical lattice. Of particular interest are the strongly correlated regimes where the ratio of coflow and counterflow superfluid stiffnesses can diverge, corresponding to the case of saturated drag.
In my dissertation, I will discuss unconventional superfluid states of various bosonic systems and present benchmark calculations for phase diagrams. The method of study is based on quantum Monte Carlo simulations by the Worm algorithm. The two qualitatively different systems to be discussed in the thesis are: (i) superfluidity of lower dimensional objects, i.e dislocations and defects, embedded in solid structures both in optical lattices and in realistic systems such as 4He . (ii) the two-component Bose-Hubbard model. Besides the interesting phases and phase diagrams that they exhibit, bosonic optical lattice systems are also interesting for studies of quantum magnetism where under certain conditions they can be mapped onto various spin Hamiltonians. On the other hand, superfluid dislocations in 4He is an important and interesting subject for it's relevance to supersolid behavior observed in solid 4 He . I will also present technical details of the path integral Monte Carlo and the Worm algorithm and generalization of the algorithm to the two-component bosonic systems.
A major focus in condensed matter physics is to study the origin of exotic quantum phases such as coexistent and inhomogeneous phases, quantum criticality, and secondary ordered phases close to quantum critical points. Exotic phenomena in strongly correlated systems occur due to competing complex interactions of spin, charge, lattice, and orbital degrees of freedom. Complex quantum phases in strongly correlated systems are challenging, but they might be very useful due to their possible functionality to make advance devices. In order to understand, utilize, and optimize such behaviors, we need to improve our understanding of these systems. Studies in cold atom systems are of interest since cold atom experiments provide a control on model parameters. In this thesis, we use novel analytical and computational techniques to treat strongly interacting bosonic systems. Taking advantage of the very versatile quantum Monte Carlo Stochastic Green Function algorithm, we studied several interesting problems. First, we explore a recently developed new confining method for cold atoms on optical lattices. Atoms are confined via a hopping integral that decreases as a function of the distance from the center of the lattice. This method might lead to lower temperatures than existing diagonal confinement methods. Next, we study the ground state phase diagram of interacting bosons on a ring-shape lattice with a region of weak hopping integrals. The model, an extension of the well known Bose-Hubbard model, develops a novel local Mott phase in addition to the usual Mott and superfluid phases in the homogeneous system. This might provide a new insight to the description of atomtronics applications. Finally, we study the two species Bose Hubbard model in a two-dimensional lattice. This model presents novel phases due to the complexity associated with multiple species. Its phase diagram shows ordered and coexistence phases including a ferromagnetic phase separated phase with high entropy. This phase might be accessible experimentally. The novel phases found from our studies are linked to experiments on ultra-cold atoms trapped by laser beams.
The present Thesis aims to deepen the understanding of the physics of strongly-correlated systems along diverse directions with a common denominator, namely a focus on the physical role played by the collective modes shaping quantum many-body correlations. This investigation has been mostly led through the heuristic application of the Quantum Gutzwiller Approach, a recently developed theory of the quantised excitations above the Gutzwiller approximation for strongly-interacting systems on a lattice. In the first place, the intertwining between quantum correlations and collective effects is explored in the context of Bose-Hubbard models, intended both as simple yet highly non-trivial realisations of complex many-body scenarios, and as controllable environments for the quantum simulation of impurity problems. In particular, we are able to provide a comprehensive and semi-analytical description of quantum fluctuations across the critical regimes induced by the proximity of a Mott transition (Mottness), ranging from accurate predictions for the superfluid density and pair correlations across quantum critical regimes to the Andreev-Bashkin effect in a two-component bosonic mixture. As case studies for the effectiveness and flexibility of our approach, we give a detailed account of the rich behaviours of a quantum impurity, either fixed or itinerant, immersed in a Bose-Hubbard environment, showing that both the decoherence dynamics of spinful particle and the resulting Bose polaron in the second case are extremely sensitive probes of the type of critical fluctuations experienced by the surrounding system. Additionally, we extend our analysis of quantum fluctuations to the Fermi-Hubbard model, where we uncover the interplay between low-energy quasiparticles and bosonic elementary excitations in determining two-particle correlations. A more complex physical scenario addressed in this Thesis is represented by driven-dissipative interacting systems, which find a state-of-the-art technological implementation in circuit QED platforms, among others. Inspired by the experimental interest in these systems as a potential toolbox for the generation of exotic quantum states of light, we consider a strongly-correlated fluid of photons on a lattice stabilised via incoherent drive and dissipation, and examine its collective fluctuations across the out-of-equilibrium insulator-to-superfluid transition of the system. Specifically, establishing a conceptual parallelism with our approach to fluctuations at equilibrium, we develop a linear-response theory of the excitations on top of the Gutzwiller stationary state of the Lindblad dynamics, and characterise their physical role in quantum observables and response functions. Taking advantage of our formalism, we describe also dynamical instabilities and novel physical regimes arising in the ultrastrong coupling limit of the superfluid phase. Among the most remarkable results, we bring to light the paradoxical nature of the Mott-like insulating phase of the system, whose peculiar one-body dynamical fluctuations are shown to echo the behaviour of a lasing state. Encompassing diverse modern domains of many-body physics, our investigation goes in the direction of solidifying the interpretation of collective modes as fundamental actors in correlation-induced phenomena under a unified physical picture.