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Oxford University Press eBooks

Theoretical Studies of Superconductor–Insulator Transitions

Yen Loh Lee, Nandini Trivedi

Published Jun 21, 2012• 2 citations
Publisher

Abstract

This chapter presents a theoretical treatment of two types of superconductor-insulator transitions — the disorder-tuned transition and the parallel-magnetic-field-tuned transition. This is performed within the framework of the attractive Hubbard model, which is a ‘minimal’ lattice model that nevertheless captures much of the essential physics. The effects of hopping, attraction, disorder, and parallel magnetic field are taken into account one by one in proressively more refined approximations, from pairing-of-exact-eigenstates to Bogoliubov-de Gennes to determinant quantum Monte Carlo. By examining the successes and failures of each approach, the chapter elucidates the role of amplitude and phase fluctuations. This pedagogical approach provides considerable details of the calculation of thermodynamic, transport, and spectral properties, such that a suitably inclined reader should be able to reproduce many of the results.

DOI
10.1093/acprof:oso/9780199592593.003.0017

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This chapter presents a theoretical treatment of two types of superconductor-insulator transitions — the disorder-tuned transition and the parallel-magnetic-field-tuned transition. This is performed within the framework of the attractive Hubbard model, which is a ‘minimal’ lattice model that nevertheless captures much of the essential physics. The effects of hopping, attraction, disorder, and parallel magnetic field are taken into account one by one in proressively more refined approximations, from pairing-of-exact-eigenstates to Bogoliubov-de Gennes to determinant quantum Monte Carlo. By examining the successes and failures of each approach, the chapter elucidates the role of amplitude and phase fluctuations. This pedagogical approach provides considerable details of the calculation of thermodynamic, transport, and spectral properties, such that a suitably inclined reader should be able to reproduce many of the results.
In this article we study superconductor-insulator transitions within the general framework of an attractive Hubbard model. This is a well-defined model of s-wave superconductivity which permits different tuning parameters (disorder and field). Furthermore, it allows a comparison of various analytical and computational approaches in order to gain a complete understanding of the various effects of amplitude and phase fluctuations. We present a systematic pedagogical approach, aiming to equip the lay reader with enough apparatus to be able to understand the numerical calculations, reproduce some of the simpler results, and be able to tackle future problems related to inhomogeneous phases. We go into considerable detail on mean-field theory (MFT) and the Bogoliubov-de Gennes (BdG) approach, as these are a first line of attack which can capture much of the physics, but we also outline cases where this fails to capture phase fluctuations and more sophisticated Quantum Monte Carlo (QMC) calculations are necessary. We discuss the behavior of many observables, including densities of states, superfluid stiffness, and dynamical conductivity, for the disorder-tuned superconductor-insulator transition. We also discuss SITs tuned by parallel magnetic field, which are quite different due to pairbreaking.
We investigate the disorder-driven superconductor to insulator quantum phase transition (SIT) in an interacting fermion model using determinantal quantum Monte Carlo (QMC) methods. The disordered superconductor is modeled by an attractive Hubbard model with site disorder chosen randomly from a uniform distribution. The superconducting state which exists for small disorder is shown to evolve into an insulating phase beyond a critical disorder. The transition is tracked by the vanishing of (a) the superfluid stiffness, and (b) the charge stiffness or the delta function peak in the optical conductivity at zero frequency. We also show the behavior of the charge, spin, pair, and current correlations in the presence of disorder. Results for the temperature dependence of the dc conductivity, obtained by an approximate analytic continuation technique, are also presented both in the metallic phase above Tc{T}_{c}Tc​ and the insulating phase. We discuss some of the complications in extracting the resistance at the transition 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 investigate the superconductor-insulator phase transition in the two-dimensional boson Hubbard system with short-range interactions. Fluctuations of both the phase and the amplitude of the superfluid order parameter are included in the determination of the phase diagram at zero and finite temperatures. The mean-field phase boundary is compared to quantum Monte Carlo results. We also calculate the frequency-dependent conductivity in the vicinity of the phase transition and find it universal at the multicritical point.
Superconductor-insulator transitions, especially in thin films, can provide the simplest examples of the continuous quantum phase transition paradigm. Quantum phase transitions differ from thermal phase transitions in that they occur at zero temperature when the ground state of a system is changed in response to a variation of an external parameter of the Hamiltonian. In the example of the superconductor-insulator transition, this control parameter could be the parallel or perpendicular magnetic field, disorder, or charge density. Quantum phase transitions are studied through measurements at nonzero temperature of physical behaviors influenced by the quantum fluctuations associated with the transition. This review will focus on experimental aspects of superconductor-insulator transition in disordered films that are effectively two-dimensional. In particular, the evidence for quantum critical behavior in the various types of transitions will be presented. The various theoretical scenarios for the transitions will also be discussed along with the extent to which they are supported by experiment. Open questions relating to the nature of the very puzzling insulating regime and whether there are many different types of superconductor-insulator transitions will be presented. Although this research area is more than 20 years old, rather central issues are not resolved.
We calculate the dynamical conductivity σðωÞ and the bosonic (pair) spectral function PðωÞ from quantum Monte Carlo simulations across clean and disorder-driven superconductor-insulator transitions (SITs).We identify characteristic energy scales in the superconducting and insulating phases that vanish at the transition due to enhanced quantum fluctuations, despite the persistence of a robust fermionic gap across the SIT.Disorder leads to enhanced absorption in σðωÞ at low frequencies compared to the SIT in a clean system.Disorder also expands the quantum critical region, due to a change in the universality class, with an underlying T ¼ 0 critical point with a universal low-frequency conductivity σ Ã ≃ 0.5ð4e 2 =hÞ.