## Overview: Quantum phases of matter

My current research includes novel phases of liquid helium films and ultracold atomic systems, exciton condensation in quantum Hall systems, topological states of matter and nanothermodynamics.

## Exciton condensation

### INTRODUCTION

Excitons are bound pairs of electrons and holes which obey Bose statistics. The Bose-Einstein condensation of these objects has been theoretically studied nearly fifty years ago. Since these pairs are electrically neutral, the system loses its ability to conduct net charge. In other words, it becomes an insulator. The first clear observation of exciton condensation was achieved in **quantum Hall bilayers **which are double semiconductor quantum wells in a strong magnetic field. In the exciton experiments, two layers of electrons are separated by only ten nanometres. Coulomb interactions lock together the motion of the charges in the two layers, and excitons are formed from particles in one layer and holes in the other layer.

The signature of exciton condensation is the superfluid flow of excitons. This corresponds dissipationless counterpropating currents in the two layers. This has been confirmed in a remarkable series of experiments by the Eisenstein group (Bell Labs/now Caltech) who have been able to measure electron conduction in each layer separately. Although this has generated much theoretical attention, the theoretical calculations and experimental observations disagree by several orders of magnitude. In particular, the experiments see counterflow currents propagating across the whole sample while theory predicts that these currents would have decayed due to particle-hole recombination by interlayer tunnelling.

My work in this area involves a collaboration with Robert Jack (Nottingham), Paul Eastham (Trinity College Dublin) and Nigel Cooper (Cambridge). We have shown that disorder plays an important role in the counterflow. In a theory invoking the nucleation of topological defects in the exciton condensate, we discovered a new characteristic lengthscale emerges from the interplay of disorder and superfluidity which explains the anomalously persistent counterflow. Our theory resolves the quantitative discrepancy between theory and experiments.

We have also developed a theory of coherent interlayer tunnelling across the bilayer, explaining an anomalous current-voltage characteristic for interlayer tunnelling which shows a strong peak in the current at nearly zero voltage. While this is not a dc Josephson effect (seen in superconductors), this does imply strong coherent tunnelling across the layers, as an indirect signature of exciton condensation.

*Why are electrons in high magnetic fields good candidates for exciton formation?*

Each quantum wall confines the electrons into two-dimensional sheets. In the presence of a strong perpendicular magnetic field, the electron states (Landau levels) consists of individual stationary cyclotron orbits. The orbits can be placed anywhere on the sheet. This means that the Landau level states have a *macroscopic* degeneracy that is proportional to the area of the system. This means that the system is highly sensitive to perturbations such as Coulomb interactions and disorder. As a result, exotic grounds states emerge from the manifold of degenerate states in the lowest Landau level.

In this example of the formation of particle-hole pairs, the attractive energy must compensate for the increase in kinetic energy for confining the particles together. Since the Landau level states are electrons in stationary orbits, they have effectively zero kinetic energy and so the pairing of electrons and holes in high magnetic fields is easier to achieve that in zero field.

## Nanothermodynamics

It is an everyday observation that a small object warms up or cools down to the temperature of its environment. However, what happens in nanoscale systems where the whole system is isolated and can be prepared and evolve coherently as a quantum-mechanical pure state? How does the equilibrium âmixed stateâ of the subsystem arise from the out-of-equilibrium pure state of the overall system? How is quantum entanglement between the subsystem and its surroundings lost?

These are fundamental questions about the basis of thermodynamics. They have become all the more relevant with the advent of ultracold atomic systems. These are laser-trapped atomic clouds that are essentially isolated from the rest of the world over the duration of the experiments. Moreover, experimental techniques that can address local portions of the system are becoming available.

In collaboration with Andrew Ho (Royal Holloway University of London) and Sam Genway (Nottingham), I have studied the quantum dynamics of a non-equilibrium quantum system as it approaches a thermalised state. We have developed a generic description of the thermalisation and decoherence dynamics using a combination of numerical results for a small lattice system and a model of quantum chaotic bath dynamics using a random matrix approach. This provides an analytical explanation for ubiquitous numerical observations of a Gaussian relaxation towards thermal equilibrium for systems that are coupled non-perturbatively to its environment.

In addition to asking basic questions about thermodynamics, there is much interest to understand and control decoherence dynamics in the field of quantum information processing. Also, in the context of strongly correlated quantum systems, there is increasing interest in analysing the dynamics of a strongly interacting system by its interaction with a small probe.

## The Many Faces of Bosons

### SUPERFLUIDS

Atomic ^{4}He is the classic example of an ultraclean bosonic system whose low-temperature phases are dictated by quantum mechanics. At ambient pressure, bulk ^{4}He becomes superfluid below 2K. This phase is the result of the Bose-Einstein condensation of these interacting atoms. Superfluid helium has zero viscosity and its vortices have quantized circulation. Bose condensation has also been observed in optically trapped ultracold atoms. These quantum optics systems are particularly attractive for researchers because of the unprecedented control we have over the behaviour of these atoms such as their interatomic interactions.

### SUPERSOLIDS

There has been intense interest in other forms of quantum ground states for bosonic systems, including supersolid, normal and insulating phases. Much excitement was generated by the observation of a small superfluid signal by Kim and Chan in solidified ^{4}He in 2004. Although careful analysis by the same group has since discounted the signal as an intrinsic property of the solid, the search for supersolidity continues in helium and other systems. The possibility of the coexistence of superfluidity and density wave order has been revived by recent experiments on helium films on a graphite substrate. In collaboration with the experimentalists, I have been working on the implications of the complex interplay of two-dimensional superfluidity, density wave order and a quantum phase transition to an insulating solid.

### BOSE GLASS

Superfluidity can be destroyed by disorder. The disorder-induced insulating phase has been termed a Bose glass. The properties of the Bose glass remain a puzzle. By analogy with the Anderson localisation for disordered fermions, theory expects a system with gapless excitations. However, experiments on helium in Vycor suggests a system with an energy gap.

### BOSE METAL

It is generally expected that the only homogenous fluid ground state of a Bose fluid is a superfluid with a nodeless ground state wavefunction. However, theorists have attempted to find non-superfluid states. In a collaboration with Patrick Lee (MIT), we proposed such a phase as part of a theory for the non-Fermi-liquid normal state of the cuprate superconductors. This scenario makes specific predictions for the dc and ac Hall effects and remains consistent with the latest measurements of optical conductivity in these systems.

## Collaborators

Nigel Cooper, University of Cambridge, Strongly correlated quantum systems: quantum Hall physics

Sam Genway, University of Nottingham, Thermalisation and decoherence in closed quantum systems

Andrew Ho, Royal Holloway University of London, Strongly correlated quantum systems

Paul Eastham, Trinity College Dublin, Strongly correlated quantum systems: quantum Hall physics

## Research Student Supervision

Bradley,C, Warm dense matter

Malone,F, The density matrix and configuration interaction quantum Monte Carlo methods

Varley,J, Exciton condensation

West,, Quantum impurity in Bose condensates

Yates,T, Effects of electronic temperature on the forces between atoms