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Superfluid and Mott insulating phases of magnons in an easy-plane ferromagnetic insulator

V.I. Dagnelie

Published Mar 6, 2015• 0 citations
OpenAlex

Abstract

In this thesis we will construct a phase diagram showing the transitions to the superfluid and Mott insulating states of magnons in an easy-plane ferromagnetic insulator with an external magnetic field applied. First we transform the Hamiltonian describing this system in several ways, to explore the basic behaviour and to obtain a boundary for the superfluid phase of these magnons. We then use a mean-field approach with the Bose-Hubbard Hamiltonian, done by van Oosten et al., "Quantum Phases in an Optical Lattice", Physical Review A 63, 053601 (2001), to find the Mott insulating regions. Finally, we scrutinize the validity of using this Hamiltonian for our system, by showing some of the different results it yields as compared to the original spin-Hamiltonian. We conclude that the obtained phase diagram is inaccurate for low spins (S=2 or lower) but significantly more reliable for high spins (S=6 and higher).

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•Jan 1, 2014•1 citation
In this thesis we will construct a phase diagram showing the transitions to the superfluid and Mott insulating states of magnons in an easy-plane ferromagnetic insulator with an external magnetic field applied. First we transform the Hamiltonian describing this system in several ways, to explore the basic behaviour and to obtain a boundary for the superfluid phase of these magnons. We then use a mean-field approach with the Bose-Hubbard Hamiltonian, done by van Oosten et al., "Quantum Phases in an Optical Lattice", Physical Review A 63, 053601 (2001), to find the Mott insulating regions. Finally, we scrutinize the validity of using this Hamiltonian for our system, by showing some of the different results it yields as compared to the original spin-Hamiltonian. We conclude that the obtained phase diagram is inaccurate for low spins (S=2 or lower) but significantly more reliable for high spins (S=6 and higher).
Research on the phase transition and phase diffusion between the Mott-insulator and superfluid phase has not yet been done extensively for the case of magnons. In this thesis we start from a model Hamiltonian for an easy-plane magnetic insulator and construct the phase diagram describing the transition between the Mott insulator and superfluid phase. Finally, we consider phase diffusion of the Bose-Einstein condensate in the superfluid phase and find that phase diffusion indeed occurs in a finite-sized system.
We study the superfluid–Mott insulator (SF–MI) transition of a spin-1 boson system in an optical lattice, especially the responce of the system to an external magnetic field. Using the mean field theory, we obtain the phase diagram of SF–MI transition at zero temperature. For antiferromagnetic bosons, the Mott insulator region is stabilized when the number of particles per site is even. We find that only by changing the external magnetic field, the SF–MI transition is realized. On the other hand, the ferromagnetic case perfectly recovers the spin frozen case, that is already performed in experiments.
We describe the ground state of a gas of bosonic atoms with two coherently coupled internal levels in a deep optical lattice in a one-dimensional geometry. In the single-band approximation this system is described by a Bose-Hubbard Hamiltonian. The system has a superfluid and a Mott insulating phase that can be either paramagnetic or ferromagnetic. We characterize the quantum phase transitions at unit filling by means of a density-matrix renormalization-group technique and compare the results with a mean-field approach and an effective spin Hamiltonian. The presence of the ferromagnetic Ising-like transition modifies the Mott lobes. In the Mott insulating region the system maps to the ferromagnetic spin-1/2 XXZXXZXXZ model in a transverse field and the numerical results compare very well with the analytical results obtained from the spin model. In the superfluid regime quantum fluctuations strongly modify the phase transition with respect to the well-established mean-field three-dimensional classical bifurcation.
We generalize the mean-field theory for the spinless Bose-Hubbard model to account for the different types of superfluid phases that can arise in the spin-1 case. In particular, our mean-field theory can distinguish polar and ferromagnetic superfluids, Mott insulator, that arise at integer fillings at zero temperature, and normal Bose liquids into which the Mott insulators evolve at finite temperatures. We find, in contrast to the spinless case, that several of the superfluid-Mott insulator transitions are of first order at finite temperatures. Our systematic study yields rich phase diagrams that include first-order and second-order transitions and a variety of tricritical points. We discuss the possibility of realizing such phase diagrams in experimental systems.
The superfluid--Mott-insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean-field theory, the Mott ground states and phase diagrams of the superfluid--Mott-insulator transition at zero temperature are revealed. An applied magnetic field leads to splitting of some phase boundaries. For all the initial Mott ground states containing multiple spin components, different spin components take on different phase boundaries. It is found that in this system phase boundaries with different magnetization can be moved in different ways by changing only the intensity of the applied magnetic field.
In recent years the quantum simulation of condensed-matter physics problems has resulted from ex- citing experimental progress in the realm of ultracold atoms and molecules in optical lattices. In this thesis we analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons with a linear Zeeman effect, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. To this end we add artificial symmetry- breaking currents to the spin-1 Bose-Hubbard Hamiltonian in order to break the global U (1) symmetry. With this we determine a diagrammatic expansion of the grand-canonical free energy up to fourth order in the symmetry-breaking currents and up to the leading non-trivial order in the hopping strength which is of first order. As a cross-check we demonstrate that the resulting grand- canonical free energy allows to recover the mean-field theory. Applying a Legendre transfor- mation to the grand-canonical free energy, where the symmetry-breaking currents are transformed to order parameters, we obtain the effective Ginzburg-Landau action. With this we calculate in detail at zero temperature the Mott insulator-superfluid quantum phase boundary as well as condensate and particle number density in the superfluid phase. We find that both mean-field and Ginzburg-Landau theory yield the same quantum phase transition between the Mott insulator and superfluid phases, but the range of validity of the mean-field theory turns out to be smaller than that of the Ginzburg-Landau theory. Due to this finding we expect that the Ginzburg-Landau theory gives better results for the superfluid phase and, thus, we restrict ourselves to extremize only the effective Ginzburg-Landau action with respect to the order parameters. With- out external magnetic field the superfluid phase is a polar (ferromagnetic) state for anti-ferromagnetic (ferromagnetic) interactions, i.e. only the hyperfine spin 0 (1) is macroscopically occupied, in accor- dance with previous mean-field results. On the other hand, in the presence of the external magnetic field for ferromagnetic interaction, the superfluid phase does not change as the minimization of the energy implies the maximal spin value. However, when an anti-ferromagnetic interaction competes with the linear Zeeman effect, we can distinguish various ferromagnetic and anti-ferromagnetic su- perfluid phases within the range of validity of the Ginzburg-Landau theory. Increasing the external magnetic field yields a breaking of spin singlet pairs and a subsequent alignment of spins, thus anti- ferromagnetic phases decrease until only a ferromagnetic superfluid phase prevails. In addition, we find that the superfluid-Mott insulator phase transition is always of second order for both ferromag- netic and anti- ferromagnetic interactions. However, the transitions between different superfluid phases for an anti-ferromagnetic interaction can be both of first and second order depending on whether the respective macroscopic occupation of hyperfine spin states changes discontinuously or continuously. The established Ginzburg-Landau theory for spin-1 bosons in optical lattices will certainly be the basis for many further applications as, for instance, time-of-flight absorption pictures or collective excitations, which are of experimental importance.