The effect of a local Kondo coupling and Hubbard interaction on the topological phase of the one-dimensional topological Kondo insulator (TKI) is numerically investigated using the infinite matrix-product state density-matrix renormalization group algorithm. The ground state of the TKI is a symmetry-protected topological (SPT) phase protected by inversion symmetry. It is found that on its own, the Hubbard interaction that tends to force fermions into a one-charge per site order is insufficient to destroy the SPT phase. However, when the local Kondo Hamiltonian term that favors a topologically trivial ground state with a one-charge per site order is introduced, the Hubbard interaction assists in the destruction of the SPT phase. This topological phase transition occurs in the charge sector where the correlation length of the charge excitation diverges while the correlation length of the spin excitation remains finite. The critical exponents, central charge, and the phase diagram separating the SPT phase from the topologically trivial phase are presented.
The effect of a local Kondo coupling and Hubbard interaction on the topological phase of the one-dimensional topological Kondo insulator (TKI) is numerically investigated using the infinite matrix-product state density-matrix renormalization group algorithm. The ground state of the TKI is a symmetry-protected topological (SPT) phase protected by inversion symmetry. It is found that on its own, the Hubbard interaction that tends to force fermions into a one-charge per site order is insufficient to destroy the SPT phase. However, when the local Kondo Hamiltonian term that favors a topologically trivial ground state with a one-charge per site order is introduced, the Hubbard interaction assists in the destruction of the SPT phase. This topological phase transition occurs in the charge sector where the correlation length of the charge excitation diverges while the correlation length of the spin excitation remains finite. The critical exponents, central charge, and the phase diagram separating the SPT phase from the topologically trivial phase are presented.
We propose that ultracold alkaline-earth-like atoms confined in one-dimensional optical lattice can realize a Kondo lattice model which hosts a symmetry-protected topological (SPT) phase and an associated quantum phase transition in a controllable manner. The symmetry protection of the phase transition is discussed from two different viewpoints: topological properties related to spatial patterns of Kondo singlets and symmetry eigenvalues of the spin states. We uncover the role of various symmetries in the phase diagram of this system by combining a weak-coupling approach by Abelian bosonization and strong-coupling pictures of ground states. Furthermore, we show that the bosonization approach correctly describes a crossover from a fermionic SPT phase to a bosonic SPT phase and an associated change of protecting symmetries as the charge degrees of freedom are frozen by the Hubbard repulsion.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.
We investigate interplay of topological and Kondo effects in a one-dimensional Kondo–Heisenberg model with nontrivial conduction band using the density matrix renormalization group method. By analyzing the density profile, the local hybridization, and the spin/charge gap, we find that the Kondo effect can be destructed at edges of the chain by the topological end state below a finite critical Kondo coupling <?CDATA JKc?> J K c . We construct a phase diagram characterizing the transition of the end states.
We investigate the ground-state properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the p-wave Kondo-Heisenberg lattice model) by means of the density-matrix renormalization-group method. The nonstandard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the ``p-wave'' spin density of conduction electrons, inducing a topologically nontrivial insulating ground state. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin profile in the ground state, we show that, at half filling and low energies, the system is in the Haldane phase and hosts topologically protected spin-1/2 end states. Beyond its intrinsic interest as a useful ``toy model'' to understand the effects of strong correlations on topological insulators, we show that the p-wave Kondo-Heisenberg model could be experimentally implemented in $p\text{\ensuremath{-}}\mathrm{band}$ optical lattices loaded with ultracold Fermi gases.
We investigate, by means of a field-theory analysis combined with the density-matrix renormalization group (DMRG) method, a theoretical model for a strongly correlated quantum system in one dimension realizing a topologically-ordered Haldane phase ground state. The model consists of a spin-1/2 Heisenberg chain coupled to a tight-binding chain via two competing Kondo exchange couplings of different type: a ``s-wave'' Kondo coupling (JKs), and a less common ``p-wave'' (JKp) Kondo coupling. While the first coupling is the standard Kondo interaction studied in many condensed-matter systems, the latter has been recently introduced by Alexandrov and Coleman [Phys. Rev. B 90, 115147 (2014)] as a possible mechanism leading to a topological Kondo-insulating ground state in one dimension. As a result of this competition, a topological quantum phase transition (TQPT) occurs in the system for a critical value of the ratio JKs/JKp, separating a (Haldane-type) topological phase from a topologically trivial ground state where the system can be essentially described as a product of local singlets. We study and characterize the TQPT by means of the magnetization profile, the entanglement entropy and the full entanglement spectrum of the ground state. Our results might be relevant to understand how topologically-ordered phases of fermions emerge in strongly interacting quantum systems. Received: 1 July 2016, Accepted: 30 August 2016; Edited by: K. Hallberg; Reviewed by: D. C. Cabra, Instituto de Física La Plata, La Plata, Argentina; DOI: http://dx.doi.org/10.4279/PIP.080005Cite as: F T Lisandrini, A M Lobos, A O Dobry, C J Gazza, Papers in Physics 8, 080005 (2016)This paper, by F T Lisandrini, A M Lobos, A O Dobry, C J Gazza, is licensed under the Creative Commons Attribution License 3.0.
We study, by means of the density-matrix renormalization group (DMRG) technique, the evolution of the ground state in a one-dimensional topological insulator, from the noninteracting to the strongly interacting limit, where the system can be mapped onto a topological Kondo-insulator model. We focus on a toy model Hamiltonian (i.e., the interacting ``sp-ladder'' model), which could be experimentally realized in optical lattices with higher orbitals loaded with ultracold fermionic atoms. Our goal is to shed light on the emergence of the strongly interacting ground state and its topological classification as the Hubbard U interaction parameter of the model is increased. Our numerical results show that the ground state can be generically classified as a symmetry-protected topological phase of the Haldane type, even in the noninteracting case U=0 where the system can be additionally classified as a time-reversal Z2-topological insulator, and evolves adiabatically between the noninteracting and strongly interacting limits.