CORDIS - EU research results

Quantum Integrability, Conformal Field Theory and Topological Quantum Computation

Final Report Summary - QICFT (Quantum Integrability, Conformal Field Theory and Topological Quantum Computation)

Recent years have seen fascinating convergences between many branches of physics and mathematics never experienced before. Topics as quantum field theory, conformal invariance, integrable models, statistical physics, quantum spin chains, anyons, cold atoms -- from one side -- and infinite-dimensional algebras, combinatorics, topology and geometry -- from another side -- have witnessed an impressive development thanks to their mutual cross-fertilization. The exact methods of low dimensional quantum systems have been crucial for promoting new research activities in emerging fields such as quantum phase transitions, cold atoms, entanglement, quantum computation and off-equilibrium quantum statistical systems. The increasing number of experimental data coming either from neutron scattering, interferometry and clever optical lattice devices, have led to the exciting possibility of comparing abstract and beautiful theoretical ideas on new phases of matter with precise experimental results. The unifying concepts that have come increasingly to the fore by these new developments are both conformal invariance and quantum integrability. The implications and scope of these concepts have grown dramatically since their first appearance in physics in the late Sixties and have been now extended to many fields, in particular in the context of many-body quantum systems. More significantly, conformal field theory (CFT) and integrable models (IM) have recently shown their high potentiality in describing very precisely several physical set-ups, as for instance, the quantum Ising model.

Integrability and conformal field theory, methods that were the pillars of the grant, have allowed us to reach full control and understanding of many differenta phenomena, among which: effects of thermal phase fluctuations in new materials, universal ratios of critical systems, the determination of the so-called Artic curve in the six-vertex model, new topological realization of quantum gates, exact correlation functions after a quantum quench and several measures of entanglement -- all quantities which are extremely useful from the experimental point of view. We fully exploited the integrable Sine-Gordon model proves for investigating quantum Luttinger liquid behavior, and nicely matching it with experiment. In one-dimensional cold atoms, the integrability of a quantum field theory has permitted us to compute all the one-point correlation functions of the Bose interacting gas and also to determine its finite temperature recombination rate.

There were also very exciting theoretical developments in pure CFT, where the so-called AGT conjecture reveals a deep connection between two-dimensional CFT and N=2 supersymmetric four-dimensional gauge field theories. This correspondence provides a remarkable explicit representation for the so-called conformal block coefficients in terms of the Nekrasov partition function that was not known in the framework of two-dimensional CFT. Equally important was the investigation of all physical consequences of the duality relations (the so-called AdS/CFT correspondence), which disclosed new perspectives on strongly correlated systems.

We made also impressive progress on new phenomena of low-dimensional physics, clarfyfing the key concepts of quantum entanglement and
quantum computation, acquiring new knowledgment about the breaking of integrability, in particular to address a central idea in modern physics such as that of confinement, observed in condensed matter experiments.

Finally there was a very remarkable advance in the physics out of equilibrium of extended quantum systems, clarifying concepts such as thermalisation, ergodicity, generalised Gibbs ensemble, etc.

In summary, through the international collaborations which involves several leading groups and experts in Europe to non-EU countries we made progress on (a) physical consequences of quantum integrability and conformal invariance; (b) the new perspective in quantum field theory disclosed by recent correspondence between gauge theories and CFT; (c) pairing up theory with experiments performed both in cold-atom physics and condensed matter systems.