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Density Matrix Renormalization Group Studies of Interacting Dipoles in a Zigzag Chain

Niraj Ghimire

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this dissertation, we study the many-body effects of dipoles in a quasi-one-dimensional zigzag optical lattice, which we call the zigzag chain. We study this system using the Density Matrix Renormalization Group (DMRG) method, which has established itself as the most powerful numerical method to simulate one-dimensional lattices. For the implementation of DMRG, we have used the open-source package Intelligent Tensor, or simply ITensor. The first part of this research focuses on dipoles polarized in the plane of the zigzag chain by an external electric field with the condition that the dipoles can hop around and interact with their first and second neighbors and the number of dipoles is exactly a half of the number of lattice sites. With the interactions much stronger than the hopping, the system comprises of frustrated and non-frustrated regimes owing to the combination of attractive and repulsive interactions in different directions. The consequence is a complex phase diagram featuring the trivial ferromagnetic and antiferromagnetic phases as well as the non-trivial dimerized and superfluid phases. The second part of the research deals with randomly oriented dipoles in the zigzag chain such that each lattice site has exactly one dipole and each dipole interacts with its first and second neighbors but in the absence of any hopping. We study the second system classically allowing many different orientations for each dipole in a lattice site as well as quantum mechanically by allowing a few quantized degrees of freedom to each dipole. The result is a phase diagram showing ferromagnetic, antiferromagnetic and paramagnetic phases.

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In this dissertation, we study the many-body effects of dipoles in a quasi-one-dimensional zigzag optical lattice, which we call the zigzag chain. We study this system using the Density Matrix Renormalization Group (DMRG) method, which has established itself as the most powerful numerical method to simulate one-dimensional lattices. For the implementation of DMRG, we have used the open-source package Intelligent Tensor, or simply ITensor. The first part of this research focuses on dipoles polarized in the plane of the zigzag chain by an external electric field with the condition that the dipoles can hop around and interact with their first and second neighbors and the number of dipoles is exactly a half of the number of lattice sites. With the interactions much stronger than the hopping, the system comprises of frustrated and non-frustrated regimes owing to the combination of attractive and repulsive interactions in different directions. The consequence is a complex phase diagram featuring the trivial ferromagnetic and antiferromagnetic phases as well as the non-trivial dimerized and superfluid phases. The second part of the research deals with randomly oriented dipoles in the zigzag chain such that each lattice site has exactly one dipole and each dipole interacts with its first and second neighbors but in the absence of any hopping. We study the second system classically allowing many different orientations for each dipole in a lattice site as well as quantum mechanically by allowing a few quantized degrees of freedom to each dipole. The result is a phase diagram showing ferromagnetic, antiferromagnetic and paramagnetic phases.
We investigate the zero-temperature quantum phases of a quasi-one-dimensional zigzag chain of dipoles that are polarized in a plane by an external electric field. Since the Hamiltonian contains nearest-neighbor and next-nearest-neighbor hopping and interaction terms, this model allows frustration, which induces phases that can be interesting and unusual. By using the density matrix renormalization group algorithm, we produce a complex phase diagram. This is an extension of an earlier work by Wang et al. [Phys. Rev. A 96, 043615 (2017)].
A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-12\frac{1}{2}21​ chain with frustrated antiferromagnetic exchange J1{J}_{1}J1​ and J2{J}_{2}J2​ between first and second neighbors. The modified algorithm yields accurate results up to ${J}_{2}/{J}_{1}\ensuremath{\approx}4$ for the magnetic gap $\ensuremath{\Delta}$ to the lowest triplet state, the amplitude BBB of the bond order wave phase, the wavelength $\ensuremath{\lambda}$ of the spiral phase, and the spin correlation length $\ensuremath{\xi}$. The J2/J1{J}_{2}/{J}_{1}J2​/J1​ dependences of $\ensuremath{\Delta}$, BBB, $\ensuremath{\lambda}$, and $\ensuremath{\xi}$ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.
A system with externally polarized dipole molecules at half-filling moving along a one-dimensional zigzag chain is studied, including ground-state phase diagrams. The dipoles are oriented in-plane. Together with the geometry of the chain, this gives rise to a bond-alternating nearest-neighbor interaction due to simultaneous attractive and repulsive interactions. By tuning the ratio between the nearest-neighbor interaction and hopping, various phases can be accessed by controlling the polarization angle. In the ultrastrong coupling limit, the system simplifies to a frustrated extended axial Ising model. For the small coupling limit, qualitative discussion of the ordering behavior using effective field theory arguments is provided. We show that when the chain angle is small, the system mostly exhibits BKT-type phase transitions, whereasa large chain angle would drive the system into a gapped (Ising) dimerized phase, where the hopping strength is closely related to the orientation of dimerized pairs.
The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.