 Gwlad ni'n rhaid i gynnw'n argynnu'n ffordd y gähir o'r ffordd.umesu ar y bwrdd hyn sylwg modd Gwyrddiant Cymru. Gweithio'r gweithiau yn gyfer yr unrhyw d questão cyffrediant yn gweithio gyffrediant o'r cwmolion cyffrediant. Mae eraill hon y cyffrediant ynghylch, cyfrifod mladd ac ynwysig, mae'r llogol yn cael gandwll yn gydigol a'r gyflym ac yn gweithio. Jeffery has a particular interest in the emergence of exotic quasi-particles and topological logical effects that exhibit fractional behaviours such as electric charge and statistics. For my point of view it's worth just making a quick comment on why that's actually quite important. Discovering fractional particles or fractional excitations and condensed metaphysics is much like discovering a new particle in high energy physics. It's something with new quantum numbers and I spent my PhD working on that along with many others. At that time we considered that there wasn't really anything new to say about ordinary bands of electrons. This was a problem that was solved not so long after the advent of quantum mechanics and the textbooks said that's it, done. However they failed to think about the topology of those bands and it turns out that that really matters. It's not least because you can get states on the surface which are totally different in character from those in the bulk. So coming back to Jeffery and being a little bit more chronological, you are originally from Hong Kong I believe and you studied there. A BSc, I think significantly the mathematics comes before the physics. Double major mathematics physics, University of Hong Kong and you stayed on to do an MFIL which was in physics. Supervised by Xiaodong Wang on the geometrical phase and spin transport, so also known as the Berry phase. Then Jeffery moved to the University of Pennsylvania in 2006 where he pursued a PhD with Charlie Kane. Charlie Kane had at that point just written the paper that pointed out that we'd all been wrong about band structures in the context of the wonder material graphene. So it's probably not an exaggeration to say that Jeffery landed in the middle of a revolution. And there followed a string of papers by Jeffery, Charlie and collaborators dressing many aspects of topological insulators, topological superconductors, most of related semi metals and these have together accumulated several thousand citations. So pretty successful PhD. 2011, Jeffery moved on to a Simons funded postdoc at Champagne Urbana. And after three years in the cornfields, I can only say that because I worked in Wisconsin once. He took up his current faculty position in the University of Virginia. I only really got to know Jeffery personally very recently. I can tell you he also has a strong interest in diving and hasn't taken you very long to get in the waters. Anyway, enough of me talking. Jeffery, very pleased to have you here. We look forward to your talk. This is moving because I'm supposed to remind you to pick up one of the karaoke microphones every time you ask a question. This is because there are people on Zoom and a recording and the introduction. Sorry, your question will not go across unless the microphone. So I think I will take off my mask so that everyone can probably hear me better. So thank you next for the very, very nice introduction. And I'd like to also thank OIST and TSVP in particular in allowing me to visit for this semester. In the three weeks that I've been here already of making plenty of new friends, people here are really friendly and the view is really impeccable. So I really enjoy, have been enjoying so far. And today I'm hopefully I'll be able to tell you a story on recent developments in topological phases and in particular a new type of particle which I call universal fractional quasi-particles. So here's the outline of today's talk. So I will probably spend most of the time, maybe 30, 35 minutes on the background, on developments up to a few years ago. So this is basically some literature revealed in content that are relevant to the concepts that I'm going to introduce later. So one of the most important ingredients or toy model is the integer quantum toy effect. So I'll spend some time in this effect and in particular focus on the boundary edge channels, which carries all the electric and thermal response of this two-dimensional topological phase. So this is going to lead naturally to the idea of splitting one-dimensional channels. So this picture is going to come over many times. And I'll describe in different ways that one-dimensional channels can be split or fractionalised when you insert different topological phases in materials. So after that I'm going to switch key a little bit in introducing the vortex moana zero mode in topological superconductors or chiral P plus IP superconductors and its prospect in building a quantum computer. So the nice properties of these objects is that quantum informations are stored locally in space and therefore they are supposed to be robust against local decoherence. And lastly in the background section I'll introduce the fractional quasi-particles or more commonly known nowadays as anions in particular in fractional quantum Hall states which are topological phases that have long range and tangled ground states. And certain parts of fractional anions they can support universal topological braiding operations which are pretty powerful and have profound impact possibly in universal topological quantum computing. So here's the second part of the talk which can be decomposed into three parts. The first is I want to convince you that marana fermions which I have not defined yet is some sort of a self conjugate or self Hermitian fermion operator that exists in a topological phase with the presence of topological defects. And these known objects can actually be can be further split in half in both one dimensional channels and zero dimensional point particles. So in both cases I want to convince you that they can be split in half with a catch. The catch is you have to introduce many body strong interaction beyond the mean field paradigm. So the example I'm going to give the interacting superconducting topological service states. And finally I will end today's talk with certain superconducting patterns that in the future might have implications in universal quantum computing or braiding based universal quantum operations. So I'll start off with the individual quantum point effect. So the first thing we want to do here is maybe we just start off with a piece of two dimensional metal so you can think of this as a slab, a thin slab where your two dimensional electron gas leaves. And then you can hook it up with a battery and then push your current through it. So that's not much interesting thing happens until maybe perhaps you put a magnetic field. So here I'm going to put a magnetic field in the perpendicular direction to the plane. And because of the Lorentz force, so originally the currents are moving horizontally, but now because of the magnetic field then the electron or the charge that is moving is going to experience a Lorentz force that want to push the electron or the charge in the vertical direction, so in the x direction. So in this case if the material is a finite piece of material then there's going to be charge accumulation on both sides and that is going to introduce an electric field that the electric force being counteracting with the magnetic Lorentz force. And after the balancing out and everything in equilibrium you can measure the potential difference across the x direction that are associated with the charge accumulation. So this voltage you can measure using a voltmeter and that tells you what is the whole voltage in this material. So there's going to be a potential currents relation. So that's the ohmic relation except the directions matters. So here we are relating the longitudinal currents with the transverse whole voltage. And we plug in the linear response equation and the constant we're interested in is called the whole resistance. So this is just an ohmic relation. It turns out that the whole resistance is just classical whole effect. At this point the classical whole resistance is exactly the magnetic field you apply to the system divided by the charge density per unit area. So typically what you can measure is the vertical axis here is the measured whole resistant RXY and the horizontal axis is the applied magnetic field. So as you can see if you increase the magnetic field then there's a linear relationship the bigger the magnetic field the bigger is the whole resistance. So there's a linear relationship. Up to a point where you increase the magnetic field strong enough and lower your temperature in your material low enough typically in the scale of one Kelvin then you start to see steps. So this is the onset of the integer quantum whole effect from classical to quantum. So we're trying to understand here maybe for the first few slides what's the origin of this step. And before I do so they follow the following relations. So these steps here they are quantized in integers. So at each of these steps so this is the first step is n equals one the second step is n equals two n equals three n equals four and for each step the resistance takes a quantized value sometimes it's more convenient to take the conductance which is the reciprocal of the resistance conductance g is one over the whole resistance. So the whole conductance would be integer multiple of a combination of universal constants e squared over h. So e is the charge of an electron h is the plan constant. So this experiment basically tells us a way to find out the numerical quantity of this combination of universal constants. So this h over e squared has units in ohms and using this technique you can measure the constant h over e squared up to nowadays 13 digits. So that's the implication of the integer quantum whole effect. So here's a question. I'll show you two pictures. Before I describe the picture there's already something strange you can extract from this formula. So n is an integer e and hd is a universal constant. None of these numbers on the right side depend on the material detail. So it really doesn't matter what's your 2D material. It can be a piece of aluminum or some gallium arsenide as long as you some 2D electron gas then you get this pattern of steps if you lower your temperature enough and if you have a big enough field. And the ratio of these steps are exactly going to be described by this relation regardless of what material you put in. So on the right side here is a potential local potential plot so the horizontal vertical axes are just x and y in real space and the color coding basically encodes the electric potential of the piece of material that we're measuring. So you can see that there's a lot of disorder a lot of peaks and troughs in the potential profile. So the material is dirty. And on the left hand side is a typical picture of what you would expect to use in the lab. This is some pictures taken in the 80s. So it's really a piece of junk. So this means that it's very surprising that you can actually use this. The only non-trivial thing in this experiment is low temperature and high field. And you can get H over E squared this constant up to 13 digits of accuracy. So this is really surprising because it's A not material dependence and B really doesn't depend on material detail and even with the presence of disorder you still get this good accuracy measurement. So why so accurate? So that's what I want to mostly focus on in the first part of this talk. So before I do that maybe I use a very crude model in describing the integer quantum Hall effect. So in this case it separates into the bulk and boundary degrees of freedom. So the red circles here these are the cyclotron orbits of the electrons when they are living deep inside the bulk of the material. So in the presence of the magnetic field which in this case is perhaps pointing out of the plane the magnetic field allow the electrons to sort of dance in a in this case a clockwise manner. So it gives a preferred handedness to the electrons and the electron orbit in the material. So the electrons in the material basically they are localized in cyclotron orbits and they just go round and round and round and round and they cannot go anywhere. So that's why the bulk of the material is like an insulator it doesn't allow electric transport or energy transport because the electrons cannot go anywhere except following a tight-binded orbit or orbital. On the other hand on the edge so these are the electron modes sketched by these yellow skipping orbits so you can see that on the edge then there's all still a preferred handedness you still go in the clockwise direction but after some time the electron has to bounce back because there's nowhere to go this is already a boundary of the system so somehow the electron has to be reflected and when it is reflected again it follows a clockwise trajectory so these are of course skipping orbits so you can see that because of the handedness or the time we're also breaking from the magnetic field the electrons on both sides are now propagating in the reverse directions so on the left side, on the left edge the electron wants to propagate downward and on the right side the electron wants to propagate upward so these unidirectional channels are called chirodirach channels so this is a consequence of the magnetic field and quantum mechanics so there are some properties that are important and relevant to this talk so the evidence is about boundary correspondence so the bulk, we have this algorithm orbits on the boundary we have skipping orbits and therefore there's a gapless chirodirach modes that carries electric and energy transport on the edge of the material so these yellow channels are single directional electron channels single direction means they're chiral chiral means single directional and because the material is big so it's in a thermodynamic limit where the system size is big the yellow channels on the left side and the right side don't overlap so the quantum mechanical wave functions they have a wave packet and the wave function on the left side and the right side are exponentially localized on both sides so if the material is big enough so that the tail of the wave function don't overlap between the left and right modes then there's no tunneling so the electron cannot tunnel through the bulk of the material and therefore there's no back scattering so there's no way that the electron on the left side can go and hog to the right side if the system size is big enough and because there's no back scattering this means that energy and charge transport is dissipationless so there's no way you can have any loss in energy or charge transport so this is one of the hallmarks in topological materials and common features in topological materials so in ordinary materials I like to make the analog view inside a traffic jam so the vehicle can basically go any direction they want and in this case they are basically packed and jammed because there's too many back scattering between vehicles so they can change all the direction they want and sometimes they get stuck topological materials like highway or in this case you can see that left and right edge they support counter propagating channels and there's no back scattering and there's no tunneling between different channels it's like you have a multi-directional highway so on the left side then the vehicles are moving forward on the right side you have the vehicles moving the other way and there's really no way that the vehicles can go from one highway to another so that's one analog and another key ingredient is that these channels these 1D edge channels they must be supported as the boundary of a two-dimensional bulk there's no way that you can have a 1D channel exist purely in one dimension as a one-dimensional wire there's no way you can do that and so this page is for those who want to understand why that's the case why there must be a two-dimensional bulk that supports the 1D channel basically high-energy physicists want to use the term gauge and gravitation anomaly so that's something strange that violates some fundamental laws of physics so in this case for the chiral direct channel it violates charge conservation and the second law of thermodynamics so maybe I go through the first case first so this picture here represents the 1D chiral channel so now you have these electrons that only move in a single forward direction there's simply no backward moving mode so the electron can only move to the right side and therefore there must be a charge that is carried and a current that is carried by this wire so there's a relationship that when you increase the voltage so increase the potential of this wire then there's going to be an increase in the current that it carries and the relationship is exactly given by the e-square over H conductance so when you increase delta V then you get a delta I that is proportional to delta V up to the constant of e-square over H so now you can consider a system where I have a potential difference across so maybe I put on some electric field so that it provides me with an increment in the local potential so what it means is that maybe on the left side it carries a current but on the right side because of the difference electric potential it carries a different current so there's going to be an unbalanced incoming and outgoing current from this point so what it means is that because of the unbalanced current there must be charge accumulation so charge is going to accumulate more and more at this point because of the potential difference and because of the unbalanced current so at this point you can already see there's no way that you have an open edge an open line of chiral derogh fermions if you do then you keep on pumping charge from one end to another so what it means is that it cannot be a steady state because there's going to be charge accumulation on the left side and the right side indefinitely so there's no way you can have that in a steady state and as a matter of fact even if it's not steady it's not sustainable so there's no way you can have a 1D chiral channel on an open string you can ask what if I put it on a ring so now there is no end but if you put it on a ring I can imagine putting on a magnetic field through the ring and adiabatically turning up the magnetic field so the magnetic field becomes stronger and stronger and stronger as a function of time and according to the Faraday's law that introduced an electric multi-force around the ring basically an electric field around the ring and because of the electric field there's going to be a potential modulation so the potential is going to increase all over the ring so then you basically have charge accumulation everywhere because of the change in the increment of the magnetic flux you pass through so that's obviously violate charge conservation so you can put this ring if it exists on a pure 1D system in vacuum and if you pass the magnetic through it this theory tells you that you're going to get more and more and more charge so that obviously violate charge conservation so there's no way this can happen in nature so the way to rescue this paradox or anomaly is to allow this and allow this kind of derog mode to be present on the boundary edge of a 2D system so in this case the charge accumulation is actually supported by the two-dimensional bulk whose boundary is the 1D edge that holds the chiral derog formula so the charge accumulation comes from a bulk insulator, a topological insulator that supports this chiral channel so the second violation is the violation of the second law so if this 1D system exists then I can connect using this 1D chiral derog channel to connect a cold reservoir to a hot reservoir so in this case because the electron only moves in the fourth direction it carries also energy so this is going to be a case that directs heat currents from a cold reservoir to a hot reservoir and therefore it violates the second law of thermodynamics so there's no way that this can happen unless you can violate a second law so the way to rescue this is to realise that the 1D edge channel must be present on the edge of a topological bulk and in this case if you insist on connecting with a cold and a hot reservoir the edge channel must come in pairs because the topological bulk must have two edges that connect the hot and cold with counter propagating directions so one with four and the other one the opposite edge is going to have a counter propagating direction so overall it will rescue the second law so all these basically are very solid arguments in convincing us that these 1D chiral channels must live on the boundary of a 2D topological bulk so that's one of the key idea in relating the boundary and the bulk of a topological insulator so then you can ask what kind of topology are encoded in the bulk so here I'd like to make an analogy to the Gauss-Bonnet theorem so here the picture here shows you two geometric surfaces on the left side you have a sphere with no holes and on the right side you have a torus which is the surface of a donut so the Gauss-Bonnet theorem is an equation that relates the topological quantity on the left side called a genus, so zero for the sphere one for the torus two an integral or sum over a geometrical quantity called a Gaussian curvature so if you sum over the curvature over the entire surface then you get a topological quantity which tells you how many holes was the genus of this surface so you can verify this by seeing that everywhere on the sphere you have a positively curved point so when you sum over the curvature the quantity should be positive and indeed it is because the genus of a sphere is zero so the left side of this equation is positive and so is the right side because the curvature is everywhere positive on the other hand for the torus depending on what point you're looking at on the torus standing on the edge of the torus is positively curved but if you're sitting inside on the inner rim of the torus then it's negatively curved so you have as many positively curved points as negatively curved ones and therefore when you sum them up they cancel it's zero and then you put in a number G equals one for a torus then on the left side of the equation indeed you get zero so that's the Gauss-Bonnet theorem that relates topological quantities with some geometrical ones so you can also see that the topological quantity will not change if you do deformation so unless you use a pair of scissors and punch holes into your surface if you don't do that no matter what deformation you do the genus quantity on the left side is going to stay unchanged the curvature, the quantity in the integrand on the right hand side that is going to change quite a lot if you deform your surface but the sum of all the curvatures will not change so if you integrate the curvature although the curvature can change drastically from point to point if you do deformation the sum does not change so applying this idea then the operator used the Kubo formula to basically a linear response theory to compute what is the conductance in the materials and this is the formula that they derived and proved so this is called the TKN formula or mathematician called the first churn number so basically this number here is a analog of the right hand side of the Gauss-Bonnet theorem the integrand here is not the Gaussian curvature but rather it's called a Berry curvature it's like a momentum space version of curvature and it encodes the information about the ground state of your system so the lowest energy level in the system and the whole integrand here encodes quantum entanglement if this integral is not managing so there are more recent generalisation on topological phases in two and three dimensions so in 2D integer quantum Hall effects requires time reversal breaking and the presence of magnetic field or magnetisation, you can have topological insulators that preserve time reversal symmetry so in this case the example would be quantum Hall, a spin Hall insulator so in this case you have a tree field not tree, you have a topological insulating bulk whose boundary carries a pair of counter propagating channels so you have spin up electron move in one way and spin down electron move in the opposite directions so these channels again they cannot exist alone as a pure 1D system with time reversal symmetry, they must exist and supported by a two-dimensional topological bulk so this also the same similar idea can be present in three dimensions so you can have three-dimensional topological insulators which support mesless surface fermions which is already a fractional version of graphene so for those who are familiar with graphene which is one atomic layer of graphite of carbon atoms in that material you have four diracons one at each k point you have spin degeneracy so you have two diracons per each k point you have four diracons in general so this system supports one fourth of the degrees of freedom of graphene so that's fractionalisation due to the presence of topology in the bulk so in fact you can measure this using ARPA's angle resolve photo emission spectroscopy to map out the energy to momentum dispersion relation so you can even do spin resolve office then you can see that there's going to be spin polarization on the Fermi surface along the surface dirac fermion and the spin orientation and the momentum is locked in a helical manner so these are revealed on topological insulators in two and three dimensions so let me go back to the one-dimensional channels that focus of this talk so the simplest truly one-dimensional degrees of freedom that you can write down the simplest lowest degrees of freedom you can write down with time real symmetry and all that in one dimension is the following so you basically just take a one-dimensional system with p square over 2m energy dispersion and this parabolic dispersion has to be spin degenerate because electron is a fermion, it comes with spin so it comes with spin up and spin down species and because of the parabolic dispersion principle and the Fermi statistics of the electrons theory tells us that at zero temperature you occupied your electron states up to a Fermi level below which all the electron states are occupied above which all the electron states are empty so that defines you with a Fermi level across this energy band, this p square over 2m band or k square over 2m band and depending on the slope of this dispersion if the slope is positive then you get right moving channels if the slope is negative you get left moving channels and for each Fermi points you get a spin up and spin down electron and therefore there are four channels in total two spin up, two spin down, a pair moving to the left and a pair moving to the right so that's the lowest degrees of freedom you can have in a true 1D system you can fractionalise or split this 1D degrees of freedom by inserting a quantum spin-haul insulator so if you do that then this channel got decomposed spatially so this is really in real space so if you have a mercury cadmium tellerite the first quantum spin-haul material or you can take indium arsenide gallium and timonite heterostructures or monolayer tungsten dipellerite so these are good candidates for quantum spin-haul insulators and if you do that then you can split the fundamental 1D electron channels into two pairs of helical each helical pair contains spin up electron moving in one way and spin down electron moving in the other further you can insert a quantum-haul insulator so that's the same quantum-haul insulator that we have discussed previously or more contemporarily instead of apply a magnetic field you can apply a magnetisation without the need of a magnetic field so in this case that's a quantum-anomalous-haul insulator that allows you to further decompose these modes so now here you lower the symmetry in the quantum spin-haul case you have time-resolution symmetry in the anomalous-haul case you violate time-resolution symmetry by either putting on a magnetic field or a magnetisation and the consequence of that is it allows you to further split the helical direct channels into chiral ones so chiral means electrons only want to move in a single forward direction without a counter-propogating channel so you can see that now on the top edge for example the electron only wants to move in a single forward direction that's not the end of the story so it turns out if you lower symmetry enough so here at the middle you destroy time-resolution symmetry on the far right you can further destroy the charged conservation symmetry by putting on a superconductor so there's a couple of ways to do this one is you insert a chiral paper as an IP superconductor the other way is you start with this manganese-dope business satellite and you put on a strongly coupled superconductor and induce proximity superconducting pairing to the material so if that's the case you can further decompose this chiral Dirac fermions into two maranas so what are maranas fermions? so if you write down an electron theory so electron is described by a complex Dirac fermions complex means for the electron wave function you can apply a complex phase e-to-the-i-fi because it's a complex wave function this is complex operators and you can decompose a complex Dirac electron operator into real and imaginary components so this called maranas components of Dirac fermions so each Dirac fermions comes with two maranas fermions one is real parts, the other one is imaginary parts so each one their own self-partner or anti-partners so this means that they are self-conjugated anti-particles so they satisfy this reality condition and it's exactly this type of fermions that is propagating along the edge of this chiral superconductor which carries half of the degrees of freedom of the edge of the quantum hole so to see this effect of one half exactly describe the response so this equation here basically tells you roughly how much thermal current heat current is carried by the edge channel so the C number here would be one for the edge mode of a quantum hole the chiral channel it is one half for the chiral superconductor which supports the maranas fermions maranas fermions is c equals one half carried half of the thermal heat current as the chiral fermions so this is a little bit of detail that is necessary in this talk about chiral superconductors so superconductor in general exists because of electron pairing so here the electron in minus k, k is momentum I want to pair up with the electron in positive k and these electron pairs called couper pairs condense in the ground state so the order parameter which is the ground state expectation value of the couper pair is encoding a complex phase it can be a complex phase because the electron operator is a complex operator so it has a complex phase associated with this and this phase five here is called a superconducting phase so this superconducting phase in general can modulate slowly without destroying the superconductor so here each arrow here represents the value of this complex phase five so this on the diagram of the left hand side is plot in real space so the arrow here at different point can take different directions and that represents phi, the value of phi being slowly modulated or wind around the vortex in two dimensions so at the centre of this vortex here is a magnetic vortex core so there's a magnetic flux with value hc over 2e passing across this region so what it's going to do to the system is it's going to introduce this winding pattern of the magnetic, of the superconducting pairing phase so this phi value, this superconducting pairing phase value is going to wind by 2 pi as you go once around the vortex core and the consequence of that in the chiral hypercypher subconductor in general is that there's going to trap a marana zero mode so there's a zero energy zero momentum marana fermion mode that is trapped inside close to the centre of this vortex so the combination of these two effects and magnetic flux together with a zero energy marana bound state is called an ison quasi particle so this is one of the basis of the simplest topodriol quantum computer you can have the reason for that is that we can count the quantum information that is stored in this system so each pair of zero mode forms a two level system because you can combine each pair of maranas to form a complex Dirac fermion and that fermion state can either be empty or occupied so each pair of zero mode or each pair of vortices corresponds to a single two levels quantum system but the key point is the location of these two vortices can be very far apart and the consequence of that is it stores quantum information locally and therefore coherently so if you imagine that some local perturbation or accidental measurements so here I introduce a local scalar potential perturbation so I just put on some electric perturbation so there's no way that it can couple to the charge that is associated or the quantum state associated to this pair of zero mode because the zero modes are so far apart so that the wave function they don't overlap when you put on a local perturbation which is represented by this Gaussian packet of vx so that can only overlap one of the two zero mode wave packets if at all overlapping with any of them so this means that the tunneling or the perturbation has to be exactly vanishing unless you put these two zero mode close enough so that you can measure it so if they're not close enough they're very well separated in space there's no way, there's no local perturbation that can destroy the quantum state on the other hand this system is still this quantum state is still successful to fermion poisoning so that for example if you stick an STM tube tunnel electrons which is a fermion in and out of one of the two marana zero mode then it can change the parity of the quantum state so this state is still not robust against fermion poisoning so it can form a universal not universal but still braiding operations because you have non-Abelian exchange operations for example if I put in a closed system four vortices so it associates with a two-dimensional huber space and for the pure state I can enumerate this on the block sphere and if you do exchange operations basically in this case you can either have operation that rotates nine degrees on the block sphere in two orthogonal axes and if you do random braiding operation basically you can not cover the block sphere you can only generically get 24 states so if you start with any arbitrary or generic quantum states and apply any operation braiding operations you want you are going to create a new state but you are not going to cover the entire block sphere you can only generically create 24 states out of any given state, initial state so that is not very good because you really want for braiding operation to be uniform to be able to cover all states on the block sphere so to do that we move on to fractional systems so in fractional quantum hall states if you lower the temperature further to millicelvin then you start to see fractional filling numbers so this n number becomes a fraction so the simplest example is the Lafflin one-third state where the n number is one-third and in this case electrons are further fractionalised into e over three quasi-particle excitations called anions so no longer they not only carry fractional charge but also fractional statistics so for normal electrons if you do exchange you get minus one exchange phase so there are different types of anions and depending on what you want you might want the more exotic ones to support braiding-based universal operations so for beating anions like Lafflin anions these are no good in quantum computing or quantum information storage so if you want a complex phase and you want a complex phase in quantum computing or quantum information storage because quantum states are still local for non-abiding anions these are now better anions that you can store information on locally in space pretty much like the icing quasi-particles we discussed in previous pages so in this case you can have icing anions in filling five-half fractional quantum hall states these are good in storing quantum information so they are so good in manipulation in unitary braiding operations because the braiding operations are not universal so in the previous page we have seen that it can only cover 24 points on the Bloch sphere so it's not universal at all there are better anions, one of the good ones called the Fibonacci anions so they are proposed to exist in the read-resize fractional quantum hall states in doing universal quantum operations so I'm going to skip over the counting of the quantum states so these are called Fibonacci anions because if you have more and more Fibonacci anions than the dimension of your hubris space grows according to the Fibonacci sequence so that's why they are called Fibonacci anions more importantly what I want to show you is that it indeed is universal so if I have a closed system with four Fibonacci then it forms a two-level system I have to do a little bit of explaining before I can convince you that but believe me it's going to form a two-level system and therefore I can describe the quantum states on the Bloch sphere so I can do different types of exchanges maybe I can exchange the first two Fibonacci or I can exchange the second and third Fibonacci and they generate a bunch of braiding operations they can represent it by 3 pi over 5 rotations on the Bloch sphere and the funny thing is that the axis of the rotation is some irrational angle so if I do random braiding operations so what I'm doing here is I'm starting with some generic states and then I'm going to do some random braiding operation maybe I do B1, B2, B2, B1, B1, B1, B2, B2 some random combinations so the simulation goes as the following so if you increase the number of random braiding operations you can see that now you get more and more and more states and eventually it covers the Bloch sphere densely what it means is that you can start off with any initial generic quantum states and you can approximate any other arbitrary quantum states to arbitrary accuracy in a finite step braiding process so that's what it means by having a universal braiding operation a set of universal braiding operations so the problem about that is these are actually the confined excitations so they are prone to thermal excitations and thermal fluctuations so even if you cool it down into mili kelvin you will never be able to actually go to zero kelvin so you'll always have thermal fluctuations some temperature profile on your sample and your sample is always a thermodynamic limit so you always have a big sample with a big like 10 to the 23 number of electrons so you have many, it's a medibody system so in this thermodynamic limit you must, because of thermal fluctuation cannot avoid the presence of accidental spontaneous creation of these Fibonacci anions and the spontaneous creation of them will eventually cause error in your braiding operations and also cause error in measuring your quantum state due to anion poisoning so these are undesirable properties that you cannot avoid if you use anions as building blocks of quantum operations so let me skip the quick summary and maybe give you the punchline so to that's the best way to manipulate topological properties and topological defects is to introduce a new types of quasi-particle which I call universal fractional quasi-particles that bypass all these challenging, undesirable properties so the definition is that they better be not a billion so the quantum state should be able to store non-locally in space so that it is not prone to local decoherence second, the braiding operation should be universal so in a two-level system it covers the block sphere densely and third, the point particle better be confined so they should not be the confined quantum excitations meaning that they should not be able to just spawn spontaneously due to thermal fluctuation so one way to do this is to perhaps realise them in interacting moirana fermions so I go back to this picture and this is actually a relate to the following question so in this picture I've shown you by inserting different topological states you can continuously, sort of not continuous but in steps, splitting your fundamental channel into smaller and smaller and smaller components up to this point you have a single moirana channel on the edge of a topological superconductor so these are all materials that are topological but their short range and tango in the sense that they can be all understood in mean field single-body theory so the question is can you further split this in half can you further split a moirana fermion in half so I want to convince you that this is indeed possible but there's a catch, the catch is you have to introduce many-body interaction to make this happen so there's no way this can be done if you are in the single-body mean field paradigm so the way to do this is to insert on the surface of a topological insulator a topological ordered service state which preserves time resolution symmetry so here in a bulk, 3D bulk, I have a topological superconductor whose service is the moirana version of the service de racon of a topological insulator so on that service you can break time resolution in two different manners and on the domain war the junction is going to carry a single chiral moirana fermion and this fermion can be split in half if you insert a third domain that allows you to get out the service state of a superconductor, a topological superconductor by many-body strong interaction so using that, there's a way you can write down the interaction precisely in an exactly-solvable manner that perhaps I do not have time to go through in this talk and if you tweak it a little bit then you can actually even reduce the system to avoid having a topologically ordered service state but instead only use a scheme that involves single-body mean field theory almost everywhere except on this dash line on the superconducting pattern to get fractional cosy particle which bi-partition the ison cosy particle so I'm going to skip this but the most important concept is what we have discussed so far is the electron is a fermion is a Dirac fermion you can decompose it into the real and imaginary parts so those are the moirana components of a fermion and we can use interaction to further decompose the moirana components into fractional components which are called universal fractional cosy particles so this cosy particle can be done in an interaction manner so I'm going to skip this page but the start-up is the following so you start with an insulator that is topological and Fu and Kane in 2008 show us that if I pass a magnetic flux of H02E through the hetero structure then there's going to be a zero energy moirana bound state so on the superconducting pattern it can be realised in a following manner so the bulk is a topological insulator considered a business satellite and on the surface you put on proximity in Dirac's topological service state so you put on different plates you put on some superconductor and they can have different phases because of the Josephson effect so you can pass a supercurrent to change the mutual phase in between adjacent superconducting plates so if you do that depending on your phase configuration sometimes you get moirana zero mo at the tri junction and in situations where you have a pi junction where the red and green superconducting plates are off by a superconducting phase of exactly pi so in this case the moirana zero mo is going to be delocalised along this 1D junction and that's the end of the story on the single-body non-minfield non-interacting setting with interaction on the other hand there's a way, there's a third axis that you can get out and introduce a many-body mass on this 1D junction, the pi junction and the result of that is going to lead you to fractional universal quasi-particles which are labelled by alpha and beta on these two tri-junctions point defects so I'll show you a final slide so this is a detailed slide I just want to show you that there's some calculation that can be done and the result is I'll have a block sphere with different brating operations so in this case I should mention I put four of these quasi-particles on a closed system and that's going to form me a two-level system and then I can do mutual exchange between different pairs of quasi-particles and that corresponds to some powerful rotation in the block sphere that is off their axis is off by some irrational angle and pretty much like the Fibonacci case if I do random sequence of random operation then you can see that it pretty much cover the block sphere densely pretty quickly so this means that again if you start with any arbitrary initial state generic initial state you can approach and approximate any other arbitrary state to arbitrary accuracy in a finite step brating process and even better unlike the Ison-Causie particle which is prone to fermion poisoning these type of Causie particle can encode quantum information that is even robust against fermion poisoning and the reason is because now the zero energy-mariner bound state split fractionalize not locally in space so that these alpha and beta Causie particles can be well separated and if you put an STM tip or any source of the fermion tunneling there's no way that you can cover both alpha and beta simultaneously because now they're separated spatially so the wave function they do not overlap and therefore the first fermion poisoning channel is exactly vanishing so there's no way you can destroy the quantum states even by adding or subtracting fermions locally so this means that a quantum state is really really protected so I think because of time I'm going to skip the details of the interactions so the only thing I want to say is that this interaction as a model can be written down exactly precisely as a four fermion two-body interaction and instead of single-body mariner mass there's a three-body mariner mass that you can write down to describe the ground states of this system and depending on the mutual signs the pi junction take different phases and at the domain wall that is going to trap these universal Causie particles so these are all unpublished work so hopefully by the end of my stay in July I can have a more complete story that I can tell you after publishing this perhaps so I'll end today's talk with a summary so I've baked a connection and correspondence between two logical phases and fractionalisation in both 1D channels and zero-dimensional Causie particles and hopefully I can not really prove but convince you that mariner splitting is possible in the presence of many-body interactions and in particular in this particular superconducting pattern using topological insulator and here's a couple of directions so first is how do we actually realise this type of interactions in a real system so that requires strong correlation a strong correlated system and is in theory a numerical work and a few theoretical problems and the second is I only make suggestions that these are universal braiding operations but there's still a second step in promoting these into building a topological quantum computer that is universal so the protocol in making this happen is still unknown so that's two important future directions in my opinion Thank you very much, Geoff Questions? Sir Yes, please do use the microphone Hello? OK Thank you, that was really cool so one way with icing anions to get them confined is you can simulate them using defects in the torrid code phase at then points of a domain wall Yes Is this something similar? Yes, so I should make a distinction if I have time between the icing anion in the morrid farfion state versus the seal energy marana bound state that for example you mentioned in the dislocation of the torrid code so the former the icing anion excitations in the farfion state is a de-confined excitation so it's a dynamical excitation in the quantum Hamiltonian on the other hand the vortex state I presented like these guys so you can think of this as a vortex on a Carol P wave superconductor that is analogous to the dislocation case that you mentioned in the torrid code both being externally introduced to the system so themselves they are classical objects the quantum mechanics comes from the electronic degrees of freedom within so it's a mixture between a classical object and a quantum object so the classical one being for example the vortex configuration so that's a classical configuration and the quantum mechanics is the fermionic degrees of freedom on the electrons so it's a mixture between the two and that makes them confined because they cannot be freely flowing around not like the confined icing anion in the farfion state so once they arise for example from thermal fluctuations they can be freely moved around without any energy caused but for the vortex state or the dislocation state in the torrid code there's always a confining energy if you want to pull apart a pair of dislocations or vortices so those are actually desirable for the purpose of brating operation because it allows you to avoid them to introduce any errors in your brating operations so I guess my question is for the fractionalised ones that you have here it looks like these are also living at the intersections of the main walls between the superconducting phases so is there like an interpretation of these could I define some kind of error correcting code or I don't know that's the thing that I skipped so if time allow I could actually describe more on this page so basically that's the page this is the picture that relates to the dislocation in torrid code I've worked on these before so then the issue is for twist defects it turns out that brating alone is not universal there's no way you can get universal brating operation from twist defect alone if you want you can do measurement and if you have a measurement based quantum computer that can be universal but I would say that's still undesirable because you are somewhat throwing dice and wait for what to happen because there's no way you can control the stage the outcome that you measure so if you do some quantum measurements of some quantum states then it either can be 0 or 1 and the outcome is always randomised and you have to there's no way you can have a well controlled number of operations if you want to go from A to B so it's always a random process so that's why I would argue that it's somewhat undesirable on the other hand you want some brating operation from confined defects that are universal so they must follow these definitions so there has to be confined the brating has to be universal and they have to store information on locally and these cannot be in typical twist defects so it has to be something else and one of these examples is defects of this sort so how does this differ from the twist defect? yeah so there's no group associated with it so in the twist defect case for example for this location let me go back to this diagram there's a kitabwana called weak symmetry brating basically it's a symmetry group so the anion structure they are symmetric under some symmetry group so if you have that then you can prove that if your underlying topological phase is a billion all you can get is a fusion rule that is similar to this and this can never be universal so for example if I go back to the fusion rule for Fibonacci maybe I don't need the board maybe I just use the equation I have so if you look at the Fibonacci fusion rule which is a the Fibonacci is a universal object this fusion rule cannot be satisfied by any group relation so the distinction is here I have a Z2 group so the defects is represented by the non-trivial elements so Z2 is just one and minus one so I can associate a Z2 number to these objects in the equation so sigma has the quantum number of minus one