The following is intended to give a guide to what is going on in the group at right now, and some of the new directions we are looking at. However, prospective students and postdocs with interests and ideas in other directions are always welcome!

These are systems in which interactions (eg., between electrons) play a key role, and can in fact change the physics in fundamental ways.

A1. SUPERFLUIDITY and SUPERCONDUCTIVITY: Two key problems in this field concern the nature of the 'normal' state underlying superconductivity in high-Tc superconductors (is it a Fermi Liquid, or some more peculiar state?), and the physics of the quantum vortices that exist in the superfluid state.

• (i) Our most important recent work in this area has been to give the solution to a 60-year old fundamental problem, to find the correct equation of motion for a quantum vortex [1]. The results show that the standard Hall-Vinen-Iordanski (HVI) equations are valid in a 'classical regime' of low frequency and high temperature, provided inertial and noise fluctuation terms are added. However at low T and/or higher frequency of vortex motion, one enters a 'quantum regime' where the physics is very different. These results open up very new physics for superfluids and superconductors, which we have begun to explore [2,3]. This will involve looking at this new physics in cold Bose-Einstein condensed gases, in superfluid He-4 and He-3, and in a variety of superconductors - notably in high-Tc superconductors [4].

• (ii) Our recent work on the normal state has focussed on the theory of quantum oscillations in 2d and quasi-2d systems: the key problem has been to understand the role of interactions on these oscillations [5]. Such theory is essential to the interpretation of dHvA and SdH experiments, which provide the best way of currently probing the normal state of, eg., high-Tc superconductors.

Other related work has been on room-temperature Bose-Einstein condensation of magnons in magnetic insulators (see below).

Future Work: This will undoubtedly focus on vortices (see above).

A2. QUANTUM MAGNETISM: The main question of current interest in this field concerns the new kinds of order that can exist in magnetic systems where interactions can frustrate classical ordering; and the quantum dynamics of these systems. This field strongly overlaps with 'quantum nanomagnetism', the field concerned with spin qubits, tunneling spins, etc. (see A3. below).

• (i) Our main recent work in this area has looked at Quantum Spin Glasses. The archetypal quantum spin glass system is the famous 'Quantum Ising' magnetic insulator LiHo_xY_{1-x}F_4, wherein spin-8 Ho ions interact via strong magnetic dipolar interactions, which try to frustrate conventional magnetic ordering when the spins are disordered. Our key advance here [1-3] was to show how in this system (and many like it), the ordering and dynamics are controlled to a great extent by the hyperfine coupling of the electronic spins to the nuclear 'spin bath'; this completely changes the phase diagram. The story here remains to be finished, as does the link to the physics of quantum dielctric glasses [4]. There is an even more important link to quantum information and the dynamics of spin qubits (see A3. below).

• (ii) Room-temperature Bose-Einstein condensation of magnons was recently discovered by experimentalists in magnetic insulators, and we immediately showed [5] that '4-magnon' interactions between magnons played a crucial role in such systems, and that they could be tuned by external fields. We await experimental confirmation of the initial discovery - if supefluiduty can be demonstrated, this promises to be exciting.

A3. QUANTUM SPIN NETS and SPIN QUBITS: One of the most exciting challenges in physics is to devise networks of 'qubits' (ie., quantum 2-level systems) which can behave as a quantum information processing system. In our view the most promising candidates for the qubits are electronic and/or nuclear spins, provided the fundamental problem of decoherence (see also XXXX below) can be brought under control. This problem has various aspects, as follows:

• (i) The dynamics of such quantum spin nets is the central problem. The problem of this dynamics in the 'incoherent tunneling' regime was already solved by us in the period 1993-2004; there is a fascinating interplay between the electron-nuclear hyperfine coupling, and the long-range dipolar interaction between electronic spins [1]. But the key has always been to understand how decoherence works in the 'quantum coherent regime', and we have recently succeeded [2] in demonstrating (in a collaboration with an experimental group) that we finally have a proper predictive theory of the decoherence mechanisms (this work verified theoretical predictions given [3] in 2006). A key role was played, in all this work, by the 'spin bath' theory [1] of dissipation and decoherence from localized modes in the environment (see XXXX below). It will be interesting to see what other 'spin net' systems can be explored - spin chains are one possibility [4].

• (ii) The physics of the spin qubits themselves is also important - here we have concentrated on 'chemistry-based' molecular magnets (ie., the 'bottom-up' approach to nanofabrication of spin qubits). The microscopic physics and dynamics of these molecules is of key interest - and here again work with experimenters is important [5,6]. We expect that such chemistry-based approaches may be quite competitive in the search for solid-state quantum computation schemes [7,8].

Future Directions: We expect to be looking at decoherence and spin dynamics for spins in semiconducting systems, and also in hybrid quantum optical/solid-state spin systems.

A4. QUANTUM COHERENCE PHENOMENA IN BIOLOGY: In the last few years experiments have shown that some key biological processes rely on room-temperature quantum coherence. Key examples are photosynthesis, and the use of 'magnetoreception molecules' for navigation by birds and some mammals. Our work is focussing on 2 aspects of this:

• (i) The underlying microscopic description of the interactions causing decoherence in electron transport in light-harvesting photosynthesis molecules is not clear. We think that the 'non-diagonal' coupling to both bulk and quasi-localized phonons is important, and we recently clarified this, by solving the key problem of the dynamics of polarons coupled non-diagonally to phonons [1]. We can also map the problem a polaron, or exciton, coupled to localized phonons to one of a hopping particle coupled to a 'spin bath' environment [2]. This model can also be used to describe entangled free radicals, coupled to nuclear spins, hopping around a molecule (the key problem in bird navigation).

• (ii) The biological function of these preocesses depends on the competition between coherent electron spin dynamics and decoherence from phonons and nuclear spins. This dynamics is a complex many-body problem, wich we can solve by mapping to a spin bath model [3,4]. We hope to be able to to test this kind of theory very soon on some real biological molecules, working in collaboation with experimenters.

Future: There is a possibility that we are on the verge of a revolutionary new development in biology, whereby large-scale quantum coherence is found to be crucial to many biological processes. If so, then this is likely to be a key focus of future work in our group.

A5. QUANTUM GLASSES: The main problem here (described by both PW Anderson and AJ Leggett as a really central problem in physics) is to understand the dynamics of a set of localised modes (defects) in a disordered solid, coupled by strain fields, electric dipole interactions, phonons, etc. A key mystery is the existence of low-T 'universality' in this dynamics.

In our work so far on this problem we have tried to quantify the key interactions in these systems [1], and their role in both controlling the effecive random field Hamiltonian and the distribution of these fields [2]. We have more recently given a new theory of the universality properties [3], which hypothesizes two kinds of defect, differing in their inversion symmetry and in their coupling to phonons - universality arises because level repulsion forces the collective levels of the inversion symmetric levels to very low energies.

Future Work: A key problem is to give a scaling theory of the crossover to the low-T regime. We are developing a field theory for this purpose. All new ideas are welcome.

A6. GRAPHENE: The discovery of graphene in 2004, the demonstration that its 'Dirac electron' properties were those predicted by Semenoff in 1984 using topological field theory, and the promise of a new generation of graphene-based electronic devices, has led to an explosion of activity.

We have just begun a collaboration with GW Semenoff, to try and solve a key problem, viz., the role of electron-electron interactions in this system. We will be using a combination of condensed matter many-body theory and string theory methods to do this.

Some of the condensed matter questions described above lead to much more general problems in physics. A key question is to understand decoherence in Nature, and also in quantum information processing We are dealing here not just with conventional environmental decoherence, caused by coupling to spin and oscillator bath modes, but also other possible 'intrinsic decoherence' mechanisms. One such mechanism is suggested by the clash between Quantum Mechanics and General Relativity, the 2 foundational pillars of 20th century physics; this mechanism is sometimes called 'gravitational decoherence'. We are thus led to the most important unsolved problem in physics, viz., how to go beyond these 2 central theories, to find a new theory embracing both.

B1. MECHANISMS of ENVIRONMENTAL DECOHERENCE: There are two standard models which describe quantum environments: the 'oscillator bath' model, for extended environmental modes (phonons, photons, spin waves, electron-hole pairs, etc), and the 'spin bath' model [1], developed in the 1990's by Prokof'ev and Stamp, which describes localized modes (defects, nuclear and paramegnetic spins, local phonons, etc.). Our more recent work has focussed on the application of variants of these models to different physical systems (see A3,A4 above, and B2 below). However, there are still general questions remaining: notably, what other environmental decoherence mechanisms exist in Nature. Two of these we have focussed on [2] are '3rd party decoherence' (where system-environment interactions are mediated by a 3rd system) and intrinsic decoherence, a process which, if it exists, would amount to a breakdown of quantum mechanics. The specific example of graviational decoherence is discussed in B3 below.

B2. DECOHERENCE and QUANTUM COMPUTING: Decoherence is the key obstacle facing efforts to make a quantum computer. Our main work has focussed on decoherence caused by spin bath systems, with the quantum computer represented by a quantum walk Hamiltonian [1]. Some remarkable results are found, including the simultaneous existence of ballistic and anomalous diffusion in the quantum diffusion [2]. A much more through investigation (which requires numerical work in the intermediate regime [3]) shows just how different the resuts are from oscillator bath decoherence.

Many questions about the dynamics of decoherence remain to be answered; some of the mathemtical problems are rather severe, and we have found that methods imported from string theory can be very useful [4].

B3. GRAVITATIONAL DECOHERENCE and QUANTUM GRAVITY: The basic idea here, which has been discussed by Diosi, Penrose, and others, is to modify quantum mechanics so as to accommodate superpositions of different spacetimes, in a way which doesn't do too much to damage to General Relativity. This idea, of intrinsic decoherence [1], has yet to find a precise theoretical formulation. Our idea [2], still in its early stages, is to introduce correlations between paths in a path integral formulation of quantum mechanics coming from gravitational interaction between these paths.

Future work: The further development of ideas about gravitational decoherence will be a major focus of future work. Some of this will be done in a joint project with W,G Unruh.