In this study, we simulated a quantum rotor model describing a Josephson junction array (JJA) in the phase representation at zero temperature in a perpendicular magnetic field [Formula: see text] (in units of [Formula: see text]) on a [Formula: see text] square lattice with spacing a for [Formula: see text]. The superconductor-insulator transition (SIT) is tuned by the ratio of charging energy to Josephson coupling, U/J. Abrupt drops in the magnetization values were observed in the bigger lattices at certain values of B and U/J caused by the formation of vortices. Increasing U/J at a fixed B field causes quantum vortex melting. The magnetization drops to zero around [Formula: see text] indicating SIT. For B = 0.1 the SIT occurs without an intermediate vortex state and the magnetization scales as [Formula: see text], whereas for B = 0.4 the scaling is [Formula: see text] during the vortex melting. For B between 0.1 and 0.4 the scaling is not clear. We used the diffusion Monte Carlo (DMC) method with a guiding wavefunction optimized using the variational Monte Carlo (VMC) method. The ground state energy is calculated easily in DMC and its error estimates were generally smaller than [Formula: see text], both with and without the guiding wavefunction. Quantities like magnetization and vorticity that do not commute with the Hamiltonian were calculated using an efficient forward walking algorithm. Their estimates are affected severely in absence of the guiding wavefunction. With the guiding wavefunction, errors for the magnetization were generally less than [Formula: see text] and going up to [Formula: see text] percent around the phase transition from the Meissner to the vortex state, and without the guiding wavefunction errors were generally higher than [Formula: see text] and going up to [Formula: see text] around the critical point.
The first chapter of this book provides a brief history of the important developments in superconductivity. After a general introduction, the superconductor-insulator transition is discussed in regards to open questions in the field and some of the questions tackled in this dissertation. Also, a brief introduction to ferromagnetism and phase transition in ferromagnetic systems is presented. In the second chapter, the results of simulations of three types of random inductor-capacitor (LC) networks on square lattices are presented [1,2]. The dynamical conductivity was calculated using an equation-of-motion method. The critical exponent was extracted at low frequencies. The results suggest that there are three different universality classes and that classical percolative 2D superconductor-insulator transitions (SITs) generically have sigma(omega)->infinity as omega->0. The third chapter presents results of simulations of a quantum rotor model describing a Josephson junction array (JJA) in a perpendicular magnetic field B on a square lattice [3]. The SIT is tuned by the ratio of charging energy to Josephson coupling, U/J. Abrupt drops in the magnetization values were observed in the bigger lattices at certain values of B and U/J caused by the formation of vortices. Increasing U/J at a fixed B field causes quantum vortex lattice melting. The magnetization drops to zero around U/J ~ 5 indicating SIT. In the fourth chapter, results from simulations of anisotropic Ising models are presented. These simulations were performed for a Hf2MnC2O2 monolayer under uniaxial strain [4]. The Curie temperature increases with the increasing strain, which means magnetic ordering survives up to higher temperatures under strain. In the fifth and final chapter, important results accrued over the whole dissertation are presented.
We numerically study the superconductor-insulator phase transition in a model disordered two-dimensional (2D) superconductor as a function of applied magnetic field. The calculation involves quantum Monte Carlo calculations of the
(2+1)D XY model in the presence of both disorder and magnetic field. The
XY coupling is assumed to have the form $\ensuremath{-}J\text{ }\text{cos}({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j}\ensuremath{-}{A}_{ij})$, where
Aij has a mean of zero and a standard deviation $\ensuremath{\Delta}{A}_{ij}$. In a real system, such a model would be approximately realized by a 2D array of small Josephson-coupled grains with slight spatial disorder and a uniform applied magnetic field. The different values $\ensuremath{\Delta}{A}_{ij}$ then correspond to an applied field such that the average number of flux quanta per plaquette has various integer values
N: larger
N corresponds to larger $\ensuremath{\Delta}{A}_{ij}$. For any value of $\ensuremath{\Delta}{A}_{ij}$, there appears to be a critical coupling constant ${K}_{c}(\ensuremath{\Delta}{A}_{ij})=\sqrt{{[J/(2U)]}_{c}}$, where
U is the charging energy, below which the system is a Mott insulator; there is also a corresponding critical conductivity ${\ensuremath{\sigma}}^{\ensuremath{\ast}}(\ensuremath{\Delta}{A}_{ij})$ at the transition. For $\ensuremath{\Delta}{A}_{ij}=\ensuremath{\infty}$, the order parameter of the transition is a renormalized coupling constant
g. Using a numerical technique appropriate for disordered systems, we show that the transition at this value of $\ensuremath{\Delta}{A}_{ij}$ takes place from an insulating (I) phase to a Bose glass (BG) phase, and that the dynamical critical exponent characterizing this transition is $z\ensuremath{\sim}1.3$. By contrast,
z=1 for this model at $\ensuremath{\Delta}{A}_{ij}=0$. We suggest that the superconductor-to-insulator transition is actually of this I to BG class at all nonzero $\ensuremath{\Delta}{A}_{ij}$'s, and we support this interpretation by both numerical evidence and an analytical argument based on the Harris criterion [A. B. Harris, J. Phys. C 7, 1671 (1974)].
Kc is found to be a monotonically increasing function of $\ensuremath{\Delta}{A}_{ij}$. For certain values of
K, a disordered Josephson array may undergo a transition from an ordered, Bose glass phase to an insulator with increasing $\ensuremath{\Delta}{A}_{ij}$.
We study the zero-temperature phase diagram of a dissipationless and disorder-free Josephson junction chain. Namely, we determine the critical Josephson energy below which the chain becomes insulating as a function of the ratio of two capacitances: the capacitance of each Josephson junction and the capacitance between each superconducting island and the ground. We develop an imaginary-time path integral quantum Monte Carlo algorithm in the charge representation, which enables us to efficiently handle the electrostatic part of the chain Hamiltonian. We find that a large part of the phase diagram is determined by anharmonic corrections which are not captured by the standard Kosterlitz-Thouless renormalization group description of the transition.
We study the differential resistivity transition of two-dimensional superconducting arrays induced by an external driving current, in the presence of thermal fluctuations and a magnetic field corresponding to
f flux quantum per plaquette. Recent experiments have identified this transition as a dynamic vortex Mott insulator transition at vortex densities near rational values of
f. The critical behavior is determined from a scaling analysis of the current-voltage relation near the transition, obtained by Monte Carlo simulations of a Josephson-junction array model in the vortex representation. For a square-lattice array, the critical exponents obtained near
f=1/2 are consistent with the experimental observations. The same scaling behavior is observed near
f=1/3. For a honeycomb array, although similar results are obtained for
f=1/3, the transition is absent for
f=1/2, consistent with an incommensurate vortex phase.
Vortices and vortex lattices play a major role in determining the transport properties of type-II superconductors[1–3], and enable a platform to investigate exotic superconducting physics[4,5]. The study of vortex matter has generally focused on novel states in 2D films and structures, and has recently moved to investigating systems with constrained dimensions and smaller vortex numbers[6–11]. Vortices are responsible, for example, for some electrical transport regimes in superconducting films, as well as the Berezinkskii-Kosterlitz-Thouless phase transition in superconducting films[12]. Unconventional forms of superconductivity, such as the spin triplet pairing predicted in Sr2RuO4, or in topological insulators paired to s-wave superconductors, contain two condensates that may support two vortex lattices, and may display Majorana modes, signatures of which may have been seen in other superconducting systems[13–17]. The vortex-vortex interactions, or inter and intra-condensate couplings in multicondensate systems, are important parameters that characterize the behavior of the systems that display such phenomena[18–20]. In investigating these parameters, a technique that can both probe the energies in a system, as well as manipulate the vortices therein, has long been desired. In this work, we report on progress in determining the energy scales of vortex systems, as well as limited control over the vortex motion. Using a technique based on magnetic force microscopy, we can directly measure the resonant motion of vortices present in a superconducting lattice. We use a scanning magnetic tip to trap a small number of vortices in a superconducting Josephson junction array near the tip. By observing the resonant motion of the configuration of vortices, a map of the location of energy degeneracies between different stable configurations is generated. From this data, we use a simulation to extract the relative strengths of the characteristic energy scales for the system, including the vortex-magnetic field interaction, the vortex-vortex interaction strength, and the chemical potential for the vortices. The simulations for small numbers of vortices fits the data well for multiple field profiles and lattice spacings. The ability to tune the vortex number and configurations by changing the magnetic field profile from the tip, as well as the lattice parameters of the superconducting surface, are key portions of this technique. We demonstrate that the relative strengths of the chemical potential and vortex-vortex interactions can be tuned relative to the vortex-magnetic field energy by changing the lattice spacing of the array. We also show that by moving the tip farther from or closer to the surface, which changes the potential well from the tip, that the configurations of vortices can be modified. From the experiments, we show that this technique can be used to both extract the strengths of the relative energy scales in this system and other superconducting systems, as well as for manipulating the vortex configurations for quantum computation applications.