Senior Research Fellow
Senior Research Investigator
Department of Mathematics,
Imperial College London,
180 Queen's Gate,
London SW7 2AZ,
, to find the entrance to the Huxley Building. Then take the lift in the Huxley foyer to level 6M, on arriving at level 6M you go through the double doors and take the corridor to the right, then at the junction take the left corridor and you will arrive at room 6M41.
My research interest is in the field of condensed matter theory , particularly with systems where the electrons are subject to strong local interactions, such as in heavy fermion compounds and high temperature superconductors. Strong local interactions are important in impurity systems, such as transition metal or rare earth ions in a host metal, where they induce local magnetic moments which in turn scatter the conduction electrons of the host causing an anomalous temperature dependent resistivity, the Kondo effect (Scholarpedia article). The study of model Hamiltonians of these impurity systems (Anderson model, Coqblin-Schrieffer model etc) has led to an understanding of the effects of strong electron correlation in impurity systems. The many-body methods which have been developed to predict the behaviour of the impurity models are reviewed in my book, 'The Kondo Problem to Heavy Fermions', (now available as a paperback). Strong correlation effects for heavy fermion systems and high temperature superconductors are actively being investigated using similar models to the impurity systems but generalized to a lattice so that there are strong local interactions at all sites (e.g. periodic Anderson model, Hubbard model). Similar models have also been used to describe electron transport in mesoscopic systems, such as quantum dots.
I am particularly interested in developing and applying renormalization group methods to tackle these types of problems.
Recent Projects :
Strongly Coupled Electron-Phonon Systems
(In collaboration with Dietrich Meyer, Winfried Koller, Ralf Bulla and David Edwards)
The scattering of electrons with the phonons is important in all metals as it gives the dominant contribution to the temperature dependence of the resistivity. It is the virtual phonon scattering processes that also lead to the effective attraction between electrons which induces BCS superconductivity in many metals at low temperatures. The coupling between the electrons and the phonons in some materials can be so strong that it leads to small polaron behaviour, where the mobility of the electrons is reduced considerably due to the energy cost in perturbing the lattice as they move from site to site.
The effects of the electron-phonon coupling can be expected to be important in strongly correlated systems but this topic has not received so much attention because of the difficulty in taking this interaction into account at the same time as the strong local inter-electron interaction. With the development of the non-perturbative techniques some of these challenging problems can now be tackled. In our work in this field we have applied a combination of the numerical renormalisation group (NRG) techniques, which we developed earlier for the calculation of spectral densities for impurity models, to lattice models via the dynamical mean field theory (DMFT). In the DMFT the lattice problem gets mapped into an impurity one with a self-consistency condition, which can then be tackled using the NRG.
Our work has been based on the Hubbard-Holstein (HH) model which has a coupling g of the electron charge to a local phonon mode (Holstein model) as well as an on-site electron-electron interaction U (Hubbard model). We did calculations for the impurity Anderson model version first using the NRG, and then extended it to map out the phase diagram for the half-filled HH model which has two different types of metal-to-insulator transition. For large U and small g there is a metal-to-insulator transition of the Mott-Hubbard type. For large g and small U there is a metal to insulator transition when a gap opens up due to bipolaron formation. For the lattice model away from half-filling, for large g and U, we get for the first time a description of small polaron behaviour for a many electron system. A narrow renormalised polaronic band develops at the Fermi level. The dispersion in this band has a kink which correlates with the renormalized phonon frequency.
More recent work with Johannes Bauer has been to look antiferromagnetism and charge order in this model. There is a competition between these two types of ordering. A measure of the attraction term due to phonons, which favours charge order is $\lambda=2g^2/\omega_0$. A DMFT-NRG calculation for the half-filled model gave a transition from antiferromagnetism when $U>\lambda$, to charge order for U<\lambda$. The transition was found to be discontinous for small phonon frequency and strong coupling. Detailed calculations were also made of the spectral density of the electron and the phonon Green's functions in the different phases. A study was also made of the probability distribution function for the local lattice fluctuations in this model, and the Anderson-Holstein model, exploiting the fact the the NRG method can now be applied to the calculation of reduced density matrices. Recent papers on this topic
Development of Renormalised Perturbation Theory(In collaboration with Johannes Bauer, Akira Oguri, Khan Edwards, Yunori Nishikawa and Dan Crow)
Renormalised perturbation theory (RPT) was developed originally in the field theory of electrodynamics as a way of avoiding the singularities, which arise due to the lack of an upper energy cut-off. These singularities plagued the earlier attempts at developing a theory using the conventional perturbational theory methods. The renormalised theory is basically a reorganisation of perturbation theory so that one works with renormalised parameters rather than the original bare parameters. Counter terms are introduced to prevent overcounting. In condensed matter there is always an upper cut-off so singularities of this type do not develop, and so such a reorganisation of perturbation theory is not necessary. However, where there are very large renormalisation effects, such as in heavy fermion materials where the effective mass of the electrons can be 1000 times greater than that of the "bare" mass, it makes sense to develop a perturbation theory in terms of the "fully dressed particles" (quasiparticles).
We first applied the RPT approach to the Anderson model, and showed that the exact low temperature results for this model in all regimes, including the Kondo regime, could be derived by working to only second order in the renormalised interaction. Subsequently we have linked this approach to NRG and shown how to calculate the renormalised parameters in terms of the original or bare parameters from the NRG fixed point. By taking the effects of repeated quasiparticle scattering into account, we have shown that it gives remarkably accurate results for the dynamic spin susceptibilities over the whole relevant energy scale. These calculations have been extended to include an arbitrary magnetic field H, such that the renormalised parameters becomes field-dependent. We can follow the de-renormalisation effects of a magnetic field in suppressing the spin fluctuations, following the complete transition from strongly renormalised quasiparticles to bare quasiparticles as the field strength is increased.
In further calculations we have extended this approach to lattice models using DMFT, and in particular to the Hubbard model in a magnetic field (see next section), and to non-equilibrium processes in quantum dots (see section below).
More recent developments have been the application of the RPT to an n-channel Anderson model with a Hund's rule coupling. Exact relations can be established between the renormalized parameters of the effective model in the Kondo regime, so that they can all be expressed in terms of a single quantity the Kondo temperature. NRG calculations of these parameters verified the predicted relations for the case n=2 (due to the size of the matrices involved NRG calculations for larger n are not possible).
A very significant recent development has been to show that the renormalized parameters can be calculated entirely within the RPT method. The approach was demonstrated for the Anderson model in the strong correlation regime. The method depended on using mean field theory and RPA to calculate the renormalized parameters in the presence of a large magnetic field where the spin fluctuations are depressed. These were when used as an input to a RPT calculation for a reduced magnetic field, allowing the renormalized parameters to be recalculated at this small field value. This leads to a scaling equation allowing the parameters to be calculated for small and smaller values of the applied magnetic field. The renormalized parameters obtained in this way were in very good agreement with those deduced using the NRG, so established the viability of the method. The results for the magnetic susceptibility in zero field were found to be in excellent agreement with the exact Bethe Ansatz results in the strong interaction regime, and the magnetization as a function of magnetic field deduced in this way is in excellent agreement with the results of a direct NRG calculation. The generality of the approach suggests that this method can be applied to a wide class of models which have been put forward to describe the behaviour of strongly correlated electrons.
Further recent work has been on a two impurity model which has two types of quantum critical points. The RPT calculations predict that the renormalized parameters can be expressed in terms of a single energy scale T* on the approach to the local quantum critical points. These predictions have been confirmed by direct numerical renormalization group calculations. We put forward the conjecture that the emergence of a single energy scale on the approach to a quantum critical point provides a natural explanation of the w/T scaling which has been observed at the quantum critical points in some heavy fermion systems.
Recent papers on this topic
A recent talk (2012) on this topic at the Institute of Theoretical Physics, Frankfurt, as power point presentation or as a pdf file .
Broken Symmetry States in Strongly Correlated Systems
(In collaboration with Johannes Bauer, Nicolas Dupuis and Winfried Koller)
In compounds and alloys with strongly correlated electrons there are competing interactions on low energy scales that can lead to phase transitions to different broken symmetry states. In many heavy fermion materials there are low temperature transitions to some form of antiferromagnetic order, and in some cases transitions to a superconducting state. Through application of pressure or alloying, it is possible to lower the magnetic phase transition to zero in some compounds resulting a quantum critical point (QCP). Anomalous temperature dependence of transport and thermodynamic properties have been found in the region of a QCP. Modifications of the classical renormalisation group methods to include the quantum, as well as the thermal fluctuations of the order parameter, do not appear to provide the explanation of the behaviour observed at the QCP. How to explain the singular behaviour at a QCP is one of the outstanding questions to be answered in this field.
A magnetic field induces a form of broken symmetry, and it is also a useful probe of strong correlation behaviour as it couples directly to the spin fluctuations, which are enhanced by the strong local interactions. We have studied the Hubbard model in a uniform magnetic field. Using a combination of the DMFT and NRG, we have calculated the spectral densities and response functions for a full range of magnetic field values from weak to very strong, and found that the low energy behaviour can be well explained in terms of quasiparticles with field dependent parameters. We found that, using these parameters in a RPT calculation which takes into account repeated quasiparticle scattering, that we could obtain a good description of the DMFT-NRG results for the local dynamic longitudinal and transverse spin susceptibilities. This work has been followed up with a similar study of antiferromagnetism in the Hubbard model.
More recent work has involved looking at superconductivity in the Hubbard model with an attractive interaction. The interesting question here is how the superconducting state and response functions vary as the strength of the interaction is increased. In the weak interaction case the superconducting state can be described by the standard BCS mean field theory. In the very strong interaction limit the electrons form tightly bound pairs and the superconductivity is viewed in this regime as a condensation of these pairs (BEC). The region of special interest is the crossover from the BCS to the BEC picture. We have used the NRG to together with the DMFT to calculate response functions over the full range from weak to strong coupling.
The competition between antiferromagnetic order and charge order in the Hubbard-Holstein model has also been studied recently using a combination of dynamical mean field theory and the NRG method.
Recent papers on this topic
Transport through Quantum Dots
(In collaboration with Akira Oguri, Johannes Bauer, Y. Tanaka and Y. Nisikawa)
Nanoscale materials have been cleverly constructed so as to create small, almost isolated, islands of electrons, known as quantum dots, which are linked to one dimensional leads. Electronic transport through these dots can be controlled by gate voltages so that they behave like single electrons transistors (SETs). The theoretical models for quantum dots are similar to those of magnetic impurities, and many of the effects associated with magnetic impurities, such as a sharp many-body resonance (Kondo resonance) induced at the Fermi level at low temperatures, can be observed in these systems. We have been able to apply the NRG and RPT techniques developed for impurity models to quantum dot situations. In particular we have used the RPT to look at the on-set of the double peak structure observed in differential conductance as a function of bias voltage in the presence of a magnetic field. This approach is currently being developed further to look at non-equilibrium behaviour for higher bias voltages to compare with the experimental results.
More recent work has involved extending these calculations to different geometrical arrangements of quantum dots, in particular to a triangular arrangement of three quantum dots. There is a very large parameter space to explore in this case, with different types of Kondo effects in different regimes which can be traversed experimentally via changing the gate voltage.
We have also calculated the non-equilibrium current fluctuations in an orbitally degenerate quantum dot model, which includes a Hund's coupling term, using a combination of renormalized perturbation theory and full counting statistics.
Recently we have considered the transport through a double quantum dot when the excitations between the dots are degenerate with spin excitations on each dot to answer the question as to whether the low energy behaviour ever corresponds to that of a localized Kondo model with SU(4) symmetry. We show that recent experimental results on a double dot system in this regime can be interpreted in terms of quasiparticles described by temperature dependent renormalized parameters.
Recent papers on this topic
Link to Scholarpedia