# Research

Strongly correlated electron systems exhibit a fascinating and diverse range of physical phenomena.
Quantum many-body effects can result in collective, emergent behavior quite different from that of the
individual constituents, moving independently or treated classically. Experiment and theory advance complimentary aspects of our understanding of this
complexity. Indeed, computational studies make the link between new theoretical concepts, and experimental observations of new phenomena.

My work concerns two branches of condensed matter physics in which strong electronic interactions play a key role: correlated materials, and devices for nanotechnology. Both are united within a single theoretical framework by their common description in terms of generalized 'quantum impurities'. This highlights
important conceptual connections, and allows expertise and techniques developed in one field to be applied to the other.

The part of my work relating to nanotechnology builds upon recent interest in fabricated devices such as quantum dots, adatom clusters manipulated with STM, nanotubes and heterostructures. Investigation of systems that realize unusual physical behavior provide deeper insight into correlations at the nanoscale, and open the possibility of
novel applications.

Materials with strong electronic interactions can also realize new states of quantum matter with exotic physics that could find technological application. Here, we use dynamical mean field theory to investigate lattice models which capture the basic physics of real materials. Sophisticated renormalization group techniques, from the field of quantum impurities, can be applied in the materials context to obtain accurate results with predictive power

Below is a summary of some recent research projects. A full list of publications can be found here.

## Majorana fermions in nanowire heterostructures:

topological effects and interactions

Nanowire/superconductor heterostructures have been the focus of much attention recently, because they may host topologically-protected fractionalized particles called Majorana fermions. One fascinating consequence of this is that real fermionic modes can be reconstructed from spatially-separated Majoranas, producing non-local qubits which are impervious to decoherence from local perturbations. Thus, such systems might find important application within fault-tollerant quantum computation.However, the existence of Majoranas in these systems has yet to be demonstrated unambiguously. Indeed, the key role of interactions has also not been fully explored. In our recent paper Phys. Rev. B 89, 045143 (2014) we study a proposal realizing the novel 'topological Kondo effect', which arises due to the formation of a non-local quantum spin-1/2 degree of freedom. This object is then 'overscreened' by the nanowire conduction electrons, producing a robust non-Fermi liquid state. In this work, we calculate numerically exactly the full conductance lineshapes expected in experiment, showing that the topological Kondo effect provides a means to unambiguously identify the existence of Majorana fermions, and realize new physics.

## New method for solving multi-band impurity problems:

Generalized Wilson chain within NRG

The Numerical Renormalization Group is a powerful method for solving quantum impurity problems, which themselves describe nanostructures and correlated materials within DMFT. Thermodynamic and dynamical quantities can be obtained accurately and efficiently in the state-of-the-art implementation, on essentially any energy or temperature scale. However, the exponential scaling of the method with the number of conduction electron channels/bands has previously prevented application to problems involving three or more bands. In Phys. Rev. B 89, 121105(R) (2014) we developed a new method, based on a generalized Wilson chain. The effective single channel formulation brings a new range of multichannel quantum impurity problems within reach of NRG. This may be important for solving within DMFT multi-band lattice problems --- such as the Hubbard-Kanamori model of the open-shell transition metal oxides.
## Critical non-Fermi liquid physics in quantum dot devices

The most basic quantum impurity model exhibiting non-Fermi liquid (NFL) behavior is arguably the two-channel Kondo (2CK) model, describing the symmetric antiferromagnetic coupling of a local spin-1/2 impurity to two equivalent but independent conduction channels. The resulting ground state possesses various intriguing properties, including notably a residual entropy of (k_{B}/2) ln(2) and conductance that approaches its T=0 value as the square root of T.

Similar behavior is predicted at the critical point of the two-impurity Kondo (2IK) model. The tendency to form a trivial local singlet state is favored by an exchange coupling acting directly between the impurities; while the coupling of each impurity to its own metallic lead favors separate single-channel Kondo screening. The resulting competition gives rise to an unusual quantum critical point.

In Phys. Rev. Lett.

**108**, 086405 (2012), we study the NFL quantum critical state of the generalized spin-S 2IK model, and its potential realization in a quantum dot device. Using conformal field theory (CFT) and the numerical renormalization group (NRG), we show the critical point to be identical to that of the 2CK model with additional potential scattering, for any spin-S. Distinct conductance signatures are shown to arise as a function of device asymmetry; with the 'smoking gun' square-root behavior, commonly believed to arise at low-energies, dominant only in certain asymmetric regimes. Exact conductance lineshapes are calculated numerically for our proposed realization.

However, the central difficulty in observing experimentally the NFL physics of either 2CK or 2IK models is their extreme sensitivity to various symmetry-breaking perturbations. For example, channel asymmetry, magnetic field and inter-lead charge transfer processes destabilize the critical point and destroy NFL behavior in both cases. In Phys. Rev. Lett.

**106**, 147202 (2011) we study the T=0 crossover from NFL to more standard Fermi-liquid physics which inevitably results at low energies. We exploit a connection between the crossover arising in the 2IK model due to a detuning perturbation and the simpler 1d quantum critical boundary Ising model, to calculate the full crossover Green function analytically. A large SO(8) emergent symmetry at the critical point also allows us to generalize the results to other perturbations. Indeed, the same crossover must also appear in the 2CK model due to the connection between the 2IK and 2CK models. We thus obtain a single exact universal crossover Green function that applies for an arbitrary mixture of any relevant perturbations in either model. The results are in remarkable agreement with full NRG calculations.

To generalize these results to finite temperature T = 1/beta, we require exact results for the 1d quantum critical boundary Ising model at this temperature. The model is equivalent to a classical 2d Ising model on a semi-infinite cylindrical geometry, with a magnetic field applied at the circular boundary of circumference proportional to beta. Using CFT methods, we re-examine this system in J. Phys. Stat. Mech. P04006 (2012), obtaining the full scaling function for the local magnetization crossover analytically in the continuum limit. The validity of our result as the continuum limit of the 1d lattice model is confirmed numerically, exploiting a modified Jordan-Wigner representation.

In Phys. Rev. B

**85**, 235127 (2012), we then proceed to derive an exact expression for the electron Green function in two-channel Kondo models with one and two impurities, describing the crossover from NFL behavior at intermediate temperatures to standard Fermi liquid physics at low temperatures. Symmetry-breaking perturbations generically present in experiment ensure the standard low-energy FL description, but the full crossover is shown to be wholly characteristic of the unstable NFL state. Distinctive conductance lineshapes in quantum dot devices are shown to result. Excellent agreement is demonstrated between exact results and full NRG calculations.

Finally, in Phys. Rev. B

**84**, 035119 (2011) and Phys. Rev. B

**81**, 075126 (2010), we consider emergent 2CK physics in generic odd-membered quantum dot chains and rings. In chains, two overscreening mechanisms are found to occur depending on coupling strength, with distinct signatures in physical properties. 2CK physics is shown to be wholly robust to variable dot filling; in particular the single-particle spectrum at the Fermi level, and hence the low-temperature zero-bias conductance, is always pinned to half-unitarity. We derive a Friedel-Luttinger sum rule and from it show that, in contrast to a Fermi liquid, the Luttinger integral is non-zero and determined solely by the 'excess' dot charge as controlled by gate voltage. In rings, a frustration-induced quantum phase transition between non-Fermi liquid phases is supported.

## 3D topological insulators

In Phys. Rev. B**87**, 075430 (2013), we investigate the scattering off dilute magnetic impurities on the surface of three-dimensional topological insulators such as Bi

_{2}Te

_{3}. Despite the exotic locking of spin and momentum in the protected surface metal and the effect of hexagonal warping at higher energies due to the underlying crystal structure of the material, a regular Anderson impurity model is shown to result. The complexity of the local problem enters through the unusual conduction electron density of states hybridizing with the impurity. We consider the special case where the chemical potential is tuned precisely to the Dirac point, and show that Kondo screening is precluded. But in the generic case relevant to experiment, the impurity is always Kondo screened on the lowest temperature/energy scales. To make contact with recent experiments, we determine signatures of this Kondo effect appearing in quasiparticle interference (QPI) patterns as recorded by scanning tunneling spectroscopy, taking into account the full energy dependence of the t-matrix as well as the hexagonal warping of the surface Dirac cones. We identify a universal energy dependence of the QPI signal at low scanning energies as the fingerprint of Kondo physics, markedly different from the signal due to non-magnetic or static magnetic impurities.

## Kondo physics of reconstructed vacancies in graphene

Defects in the honeycomb lattice of graphene are known to induce local moments and strong correlation effects. Distortions due to structural reconstruction around vacancies in graphene were studied recently by Cazalilla et al, who formulated an effective model consisting of a localized sigma-level hybridized with the pi-band. In Phys. Rev. B 88, 075104 (2013) we analyze the rich quantum impurity physics of this system using a combination of numerical renormalization group and analytical techniques, focusing on the special role played by the unusual local density of states, which is enhanced at low energies due to potential scattering. Depending on microscopic parameters, the model hosts both exactly-screened spin-1/2 (doublet) Kondo or underscreened spin-1 (triplet) Kondo phases, and we study the quantum phase transition separating them. Although the effective Kondo models also support new stable phases characterized by strong renormalized particle-hole asymmetry, such phases cannot in fact be accessed in the full Andersonian model describing the vacancy. We show that distinctive signatures of the modified powerlaw Kondo effect thus always appear at low energies in thermodynamic quantities and the scattering t-matrix.## Kondo correlations in real-space

Most fundamental aspects of the Kondo effect are by now very well understood, with various detailed theoretical predictions having been confirmed directly by experiments on impurity systems or quantum dot devices. Key insights into the underlying physics have been provided by the renormalization group (RG) concept, where progressive reduction of the temperature or energy scale results in RG flow between 'fixed points' (FPs) that can be easily identified for a given model. Surprisingly however, the basic physics in real-space is still somewhat controversial, although the emergence of a fundamental length-scale R_{K}~ 1/T

_{K}(with T

_{K}the Kondo temperature) has long been predicted. At low temperatures T << T

_{K}, the standard interpretation is that a spin-1/2 impurity is screened by a surrounding 'Kondo cloud' of spatial extent R

_{K}. We argue in Phys. Rev. B

**84**, 115120 (2011) that RG flow between any two FPs results in a characteristic length-scale, observed in real-space as a crossover between physical behavior typical of each FP. We show explicitly that 'free orbital', 'local moment' and 'strong coupling' regions of space can be identified for a single Anderson impurity embedded in a 1d host. These distinct regions are separated by two crossover length-scales R

_{LM}and R

_{K}, with the latter diverging as the Kondo effect is destroyed on increasing temperature through T

_{K}. One implication is that moment formation occurs inside the Kondo cloud', while the screening process itself occurs on flowing to the strong coupling FP at distances ~R

_{K}. In ongoing work, we hope to generalize these results to more complex impurity systems and 2d or 3d hosts.