Ian Keith Affleck
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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - Nov 2019)
In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively.In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire.In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.
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In this thesis we study two different one dimensional systems. The first project is on the transverse dynamical structure factors of the XXZ spin chain and the second project is on magnetism of zigzag edges of graphenenano-ribbons.In chapter 2, we apply field theory methods, first developed to study x-ray edge singularities, to interacting one-dimensional systems in order toinclude band curvature effects and study edge singularities at arbitrary momentum. We point out that spin chains with uniform Dzyaloshinskii-Moriyainteractions provide an opportunity to test these theories since these interactions may be exactly eliminated by a gauge transformation thatshifts the momentum. However, this requires an extension of these x-ray edge methods to the transverse spectral function of the XXZ spin chain ina magnetic field.In chapter 3, by considering the Hubbard model in the weak coupling limit, U
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The Kondo effect, wherein a local magnetic moment is screened via interactions with a continuum of quantum excitations, occurs in quantum dots with an odd number of electrons. By placing a quantum dot in an Aharanov-Bohm interferometer, one is able to probe the effects of electron interference on the manifestation of the Kondo effect. In this thesis, we present a theoretical study of the Kondo effect in a model system of a quantum dot embedded in an Aharanov-Bohm interferometer connected to two conducting leads. By transforming to the scattering basis of the direct inter-lead tunneling, we are able to describe precisely how the Kondo screening of the dot spin occurs. We calculate the Kondo temperature and zero-temperature conductance and find that both are influenced by the Aharanov-Bohm interferometer as well as the electron density in the leads. We also calculate the form of an additional potential scattering term that arises at low energies due to the breaking of particle-hole symmetry. In addition to these analytic results, a numerical renormalization group analysis of the system is presented. We fully describe the influence of the Aharanov-Bohm interferometer on the renormalization group flow of the quantum dot model and obtain strong support for the derived form of the Kondo temperature. A method for extracting the phase shifts of the strong-coupling fixed point from the numerical data is described. These phase shifts are compared with those derived analytically, providing further support for our conclusions.
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Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
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Master's Student Supervision (2010 - 2018)
A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified epsilon-expansion, confirming the previous conclusions.
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This thesis examines the Heisenberg antiferromagnetic spin chain in one dimension (1D) with a crystal field splitting term and applied magnetic field term. We use theoretical techniques from quantum field theory and conformal field theory (CFT) to make predictions about the excitation spectrum for our model. We then use Density Matrix Renormalization Group (DMRG) numerical techniques to simulate our spin chain and extract the energy spectrum as we vary our crystal field splitting and magnetic field terms. These results are compared and we examine where theoretical calculations accurately describe our system. This work is motivated by recent experimental work done on SrNi₂Vi₂O₈ by Bera et al. [1] which is a quasi-1D material with weakly coupled spin chains in the bulk. These 1D chains are expected to be described by the Hamiltonian we study in this thesis, and we neglect interchain coupling. We first consider our system where the crystal field splitting term is set to zero, which can be described theoretically using a mapping to the non linear sigma model (NLSM). Near the critical field, it undergoes a Bose condensation transition whose excitation spectrum can be mapped to non-interacting fermions in 1D. We then consider large negative crystal field splitting, and find that near small applied magnetic field we can describe some excited states using Landau-Ginsburg theory. Near critical field, we show that the transition is in the Ising universality, and use results from CFT to predict the spectrum for finite size systems. This allows us to make predictions about where the transition field would be for very large or infinite system size. Finally, we examine our crystal field splitting tuned to the value obtained in Ref. 1, which is a small, negative value. We observe qualitative elements in this spectrum from the spectra obtained at zero and large negative crystal field splitting.
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We study the magnetic structure of narrow graphene ribbons with patterned edges. Neglecting interactions, a broad class of edge terminations support zero-energy states localized at the edges of the ribbon. For the simplest (zigzag) ribbon supporting these edge states, electron-electron interactions have been shown to induce ferromagnetic ordering along the edges of the ribbon. We generalize this argument for such a magnetic edge state to carbon ribbons with more complex chiral edge terminations.
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Junctions of three quantum wires, or "Y junctions", are among the basic building blocks of circuits of quantum wires. In most previous work the wires are modeled as one-dimensional objects. A step towards reality takes into account their finite width. In this thesis, we study non-interacting two-dimensional Y junctions in the single-channel regime. The generic behaviors of zero-temperature conductance of two different types of Y junctions are discussed. Particular attention is given to the asymmetric line shapes, or Fano resonances, arising in the problem.
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