### Topological Protection and Non-Equilibrium States in Strongly Correlated Electron Systems

TOPNES is short for “Topological Protection and Non-Equilibrium States in Strongly Correlated Electron Systems”.  So what’s that all about?

The idea of topological protection is easy to explain.  Imagine that you want to send a message to a friend, and you have at your disposal some strips of paper.  “This is easy!” you think.  “I’ll just write my message on the strips of paper and send them to my friend.”  But this isn’t a very safe method: the message could easily get partly erased during transit, or altered so that it doesn’t say what you wanted.  What can you do to protect your message against this kind of “environmental noise”?

One interesting option is the following.  Suppose you take your message, convert it to numbers, and convert those numbers to binary numbers, i.e. strings of 1s and 0s.  (For details of how to do this, see for example http://www.teach-ict.com/gcse_computing/ocr/214_representing_data/character/miniweb/pg2.htm.)  Now you get your strips of paper: each strip is going to represent one digit of your binary number.  If the digit is a “0”, you glue the strip into a loop.  If the digit is a “1”, you glue the strip into a loop, but including a twist – i.e. you make it into a Moebius strip, as shown in the image below.  Then you send these loops of paper, in order, to your friend.

Picture credit - berserk/www.instructables.com

Why does this help protect your message against environmental noise?  Because there is no way to remove the twist from a Moebius strip, or add a twist to an ordinary loop, without completely cutting through the paper.  Small defects in the paper, or marks on it, or whatever, don’t change whether there’s a twist or not, and so they won’t change the message that your friend receives.  Whether there is a twist or not is called a topological property of the loop.  What you have done is to transmit your message robustly by encoding the information in topological (rather than local) properties of the paper.

This idea has many applications.  In particular, it may be important for developing the famous “quantum computer”.  In order to work, quantum computers need to put their component bits into a delicate quantum superposition state.  Such states are very vulnerable to environmental noise.  But if we could use topological properties of matter (rather than local ones) to encode our quantum information, this would improve our chances of getting the computation done before the message gets spoiled.

What we’re working on at TOPNES is how to do this in practice.  We’re investigating what materials could be used to encode quantum information in a topological way, and we’re trying to predict how robust that quantum information would then be to external noise.  We’re looking particularly at low-temperature materials whose electrons exhibit unusual collective behaviour.  Several research groups have already published suggestions for how – if we can find the right type of collective behaviour in these materials – we can use it to encode information topologically.  We’re not building a topological quantum computer; but we are laying the foundations by identifying suitable materials and methods for the topological processing of quantum information.

#### Scientific details

Five key areas have naturally emerged from the original project portfolio which will be the focus of research work over the next three years:

### Search for p+ip Condensates and New Materials

A considerable joint research effort is focussed on identifying and characterising spin-triplet superconductivity. Of particular interest is spin-triplet superconductivity in two dimensions, which has been predicted to host Majorana states in vortex cores and which can potentially be exploited as a platform for topological quantum computation.

Promising candidate materials for triplet superconductivity are Sr2RuO4 and UPt3, for which we aim to establish a detailed understanding of the superconducting order parameter, adopting a multi-faceted approach combining experiment and theory. The only fully established p+ip condensate in nature is superfluid 3He in its so-called B-phase. TOPNES researchers are exploring the 2-dimensional gas of Majorana states on the surface of the 3-dimensional topological superfluid phase 3He-B. New theoretical and experimental approaches are designed to confirm their existence, and to detect the spectrum of Majorana excitations through their effects in damping third sound. We have an ongoing effort to search for new materials which might host p-wave superconductivity.

### Topologically Protected States and Textures

We are experimentally and theoretically investigating a range of topologically protected states and textures. A major focus to date has been on the chiral surface or edge states that are a hallmark of topologically non-trivial phases, where spin-momentum locking of the quasiparticles is thought to offer protection from backscattering. We have built up a detailed picture of the realistic surface electronic structure of exemplar topological insulators, which often show striking but potentially useful deviations from idealized theories. We are investigating the fundamental nature of topological phase transitions, how these can be controlled through structure-property relations controlled by external tuning parameters, and how local inversion symmetry breaking can be used to generate large spin polarisations of electronic states in systems with strong spin-orbit coupling.

We are particularly interested in understanding the role that spin-orbit interactions have on the fundamental excitations of strongly-correlated electron systems, and the especially rich interplay of spin, orbital, and lattice degrees of freedom in these systems. Studies to date have ranged from iridates to ruthenates and low-dimensional electron systems in tantalates and titanates. Through ongoing efforts in this area, we aim to better understand conditions necessary to realise topologically non-trivial phases in strongly-correlated systems including transition-metal oxides and Kondo insulators, and to theoretically and experimentally understand scattering processes of possible topological states in such systems. We are also investigating topologically non-trivial textured phases near to quantum criticality, including joint theoretical and experimental studies aimed at identifying and understanding spatially modulated states near to ferromagnetic criticality. Our recent observation of atomic-scale magnetism in iron chalcogenides by spin-polarised STM opens new avenues for exploring such real space magnetic textures in correlated electron materials.back to top

### Quantum Non-equilibrium Physics

Quantum physics out of equilibrium is a very broad area, covering a wide range of possible problems.  Our current focus s covers a number of these areas, including non-equilibrium steady states, quenches, and non-equilibrium transport problems, and covers both theoretical and experimental projects, as well as joint theory-experiment projects.  Experimental work on electrical transport in Delafossite materials has shown evidence for phonon drag.  In addition, there has been theoretical work using AdS/CFT to study transport near quantum critical poits, and related experimental work uncovering the existence of a striking universality of metallic resistivity in a broad range of materials.  Future work on electron transport will include work on the biased Kondo problem in quantum dots.

Driven quantum systems, and in particular, coupled matter light systems are of both theoretical and experimental interest; joint projects here have studied, and continue to study, superfluidity and phase coherence in polariton condensates.  Polaritons are superpositions of excitons and photons, and thus combine a low effective mass with relatively strong thermalisation rates.  They however have a finite lifetime, and so questions arise of what superfluidity means when the particles involved are continuously replaced.  Similar questions arise in other systems, such as superconducting qubits in microwave cavities, cold atoms in optical cavities, and photons in dye filled cavities.  Several theoretical projects are addressing these systems, asking how one can engineer dissipation and driving to control and stabilise non-trivial quantum states of matter and light.back to top

### Entanglement Structures

Since the early days of quantum mechanics, the concept of entanglement has been one of the most intriguing aspects of modern physics.  However, it has only been in the last few years that entanglement has become a central unifying feature of condensed matter physics.   The concept appears in several different and inter-related guises.   On the one hand, the nature of entanglement can be used as to characterize a state of matter.  For example, topological phases of matter have long-range entanglement whereas topologically trivial matter can be described with short ranged entanglement only.   Indeed, characterizing the possible types of entanglement is believed to be equivalent to characterizing all types of matter.

A main thrust of the TOPNES programme is the exploration of topologically nontrivial matter, which, in the modern era we now understand as being characterized by its interesting (and long ranged) type of entanglement.  On the other hand, since matter can be described in terms of its entanglement, computationally it is extremely advantageous to represent matter via this concept.  By directly describing the spectrum of entanglement when a system is partitioned,  powerful “Density Matrix Renormalization Group” or “Matrix Product State” techniques have made essentially any gapped one-dimensional system computationally tractable.   Theorists in the TOPNES programme have been actively using these techniques to describe topological and non-equilibrium matter more effectively.   back to top

### Innovative Techniques for Topological Quantum Matter

There are plenty of good ideas on how to exploit the properties of topological structures to perform a quantum computation, but do real materials in the laboratory have the required properties?  For example, suppose we discover a superconductor with an order parameter of the right type to support Majorana vortices crossing its boundaries, how will we know the vortices actually form?

We are working on new techniques to detect properties such as Majorana vortices, spin currents and complex magnetic structures experimentally. Our main focus is on imaging surface properties.  This includes the development of  STM and ARPES to detect electron spin and to work with samples under the special conditions (e.g strain) to tease out signatures of their topological properties.   In parallel we are developing a low-temperature magneto-Kerr microscope.  The magneto-Kerr effect is the rotation of the polarisation of light on reflection. This rotation is a very sensitive probe of time reversal symmetry breaking in zero field. In finite field it provides a microscopic probe of the Hall-conductance a property intimately connected with the Berry phase and topology.  At the simplest level measuring the magnetic flux through a vortex can determine if a vortex is indeed a half-quantum vortex.

In parallel with the above programme we are also developing and improving our routine measurement capabilities to measure macroscopic properties.  These are needed to better understand the physics driving the formation of different superconducting and magnetic states, linking with our work to find new topological materials.   back to top