Topological quantum phases - Alexei Kitaev

first I should warn you it will be partly physics talk I'll try to be more precise as I usually do but it won't be exactly exactly mathematics and also it's a work in progress so it's unfinished right now it looks like a condensed matter inspired definition of case theory which is a bit disappointing but I'll try to present it in such a way that it is clear that there must be some generalization that is potentially very interesting so it's condensed matter inspired and this is the same picture it's supposed to present a sheet of graphene you know graphite is black material used to make the pencil core and it consists of sheets of atoms these are carbon atoms arranged in a honeycomb lattice and such sheets have some nice properties one of them the exhibit quantum Hall effect at low temperatures so electrons can move in this structure Hall between atoms and if we put this material in a magnetic field and also apply some magnetic field is perpendicular to the board and we may apply some electric field E and it's a whole material the current will flow in the perpendicular direction and the whole conductivity is the ratio of the current and electric field and under the right conditions since quantized in units of e squared over 2 pi H bar this is the Planck constant and that is called a integer quantum Hall effect n is an integer and of course the in physics nothing is precise but this formula is actually very good it's got very high precision and the temperature goes slowly the precision becomes better and in some cases one can find rational numbers and this is called fractional quantum Hall effect and the first is integer quantum Hall now of course both systems are very interesting and they have nice properties in addition to this quantization of whole conductivity one property is the existence of edge modes this system has an energy gap and I'll explain in a moment what energy gap is basically in the bulk at very low temperatures there are no quasiparticles but there are excitations like waves propagating along the edge of this graphene sheet in both cases in the frictional case there are also quasi particles called aliens however I want to exclude this kind of system because it's more complex and I'll focus on that kind of systems and the hope is to including the G quantum Hall effect and many other things into some nice mathematical theory and so far we have succeeded to do that for non interacting fermions for interacting fermions there are conjectures how the answer should look like but there is no proof and actually there are some analytic difficulties even defining the problem but I believe the problem can be defined and solved for so called invertible phases I'll explain what invertible means so first let me write an example of some Hamiltonian because we'll be considering quantum hamiltonians and their ground states and the goal is to understand there's ground States and construct the space of such ground States and find their homotopy types and this is an example it's called transverse a field isin model it's a Hamiltonian on a one-dimensional chain and one important property is locality the hilbert spaces detect the product of hilbert spaces of individual atoms in our spins in this case the Hilbert space is C 2 raised to the 10th power and there are n atoms and the Hamiltonian act in this Hilbert space it has two terms with two parameters H is one parameter and J's the other parameter now I wrote this some exactly from j2 n minus one meaning that this two spins a couple then those spins a couple till the very end and there are certain boundary conditions but later on I'll ignore boundary conditions and the philosophies that were studying materials and those materials can occupy some space and the one can cut accurately or not it doesn't matter and we want to construct the phase diagram of the system in terms of parameters a H and J so in this case we have a two-dimensional space of Immelt onehans but not all these Hamiltonians are equally good some of them are better at certain values of parameters the system his bad behavior so let's write H and J and put some points well which we will study this is point one this is point two you and there will be some point two Prime and one Prime and I'll draw this lines you'll see what they mean so at point one H is greater than zero J is zero and we can find the ground state the state with the lowest energy and the corresponding eigen state so the eigen value is minus n times H and the corresponding eigen vector is spin pointing to the right tender n so this is an eigen vector of Sigma act op oops this one no yeah yeah I use H but I actually will be no more Planck constant so all the spins are up in the ground state and there are some excited States so one can draw the spectrum like this this is energy this is the ground state and yeah yeah that's true also time goes up I don't know where the time goes horizontally in mathematics so e1 is 0 plus 2 H and E 1 corresponds to flip in one of the spins down so this is a degenerate state the ground state is unique and the spectrum has a gap between the ground state and the first excited state and that is a good situation the ground state is unique there is a gap we want such systems let's consider point 2 this was point 1 point 2 H 0 J is positive and the ground state energy is minus and minus and minus 1 J there will be two ground States and the spectrum looks like this there are two degenerate ground states plus 4j and this is multiply degenerate because we can flip any spin here and it will flip the spins at the boundary there are some spurious states with lower energy right between these two this will be ignored because we ignoring what happens at the boundary and also there is a notion of a local energy gap although these two states are degenerate we should not say that the energy gap is zero between those states because if we flip one spin in the middle will go up and to go from here to there we need to flip many spins we should apply a very big operator and part of this game is understanding what log local gap means and in the case of free fermions I'll give a precise definition of a local gap and the problem will be solved partly due to this consider this definition for many body Hamiltonians with interacting for spins of fermions it's hard to even define a local gap and this is one of the problems so we're looking for States or Hamiltonians with unique ground state and the local gap and in this picture this is good this is not good because the ground state is not unique regarding the gap as we go along this line the spectrum rearranges and starting from this picture it would go from 1 to 1 Prime those energy levels the first excited States will be will be split they will be banned and the gap between the ground state and the first excited state will remain for a while and it will close at this line it's a phase transition line and here to prime a winter a new phase and this phase is deemed bad but it can be fixed by adding some terms to the hamiltonian so in this picture it looks like the phase diagram consists of two good phases so this is good and that is good so there are two connected components of the space according to this phase diagram I mean connected components of the space of good Hamiltonians you add more parameters to the system of course the topology will change and I'll write down the answer for one-dimensional systems in principle one could define B sub D of good and all Toni ins on a lattice in D dimensions and will consider the unit disk in D dimensions and elaborate what the latest means good requires further qualification if the system has dimension higher than one but in dimension one one can get an answer at least no non rigorously as physicists do be one if we had more parameters is homotopy equivalent e K Z comma three they'll in Burma client space and I'll not explain that but let me explain the easy version b0 will be my topical into the alien Bert McLain space kz2 this is good Chimel tawny ins of spins or bosons and later we'll consider thermionic Hamiltonians so what is b0 it's a zero dimensional system and all these La Caleta conditions are not important here it was important that only neighbors can talk with each other here it's just a finite dimensional hilbert space and we can say that B 0 is the space of hamiltonians on the hilbert space C 2 to the capital n and such hamiltonians should have an energy gap the ground state should be separated now one can map the space of Hamiltonian to the space of the ground state and a ground state is just a vector in C to the N defined up to face factor no no no no it's just n-dimensional space capital and and a zero dimensional system means just the blob that consists of multiple spins and one may define n to be as 2 to the little n the number of spins but it doesn't matter whether the suspense or some other object yeah right so the ground state is a unit vector and so there is a map to CP and minus 1 this is the space of such states now this is actually a vibration and the fiber is contractible because we can take a convex combination of two Hamiltonians and get a Hamiltonian with the same ground state and so this is actually a homotopy equivalent and CP n minus n minus 1 as we take and going to infinity limit goes to CP infinity which is the same as kzkz to yeah that's the full story for zero-dimensional spin systems for zero-dimensional firm UNIX systems it's slightly more complicated and I'll have to spend some time on that particularly defining non-interacting systems so if we just think abstractly we can define FD this the same as BD but for fermions yeah they are connected we can add some perturbation for example we can add a poly matrix Sigma X here as a perturbation we may add yeah yeah this will work yeah that's a difficult question actually how precisely we define those Hamiltonian so we want certain properties of a Hamiltonian like I describe the interaction so local the ground state is unique and the excitation gap can be defined locally no no it's not translation invariant in general and furthermore I want to take a limit we consider a system in a disk on a lattice and we can consider fine and final lattices adding more and more atoms and there are two limits first we can simply add more atoms and there will be a high dimensional hilbert space but there's extra atoms may not be doing anything if the Hamiltonian is just fixes yeah yeah yeah yeah yeah it's possible yeah it's possible and eventually we want to add more atoms at the same time making interaction shorter range and take some sort of limit means why is it space of parameters to describe your Hamiltonian or it's a space of ground states it's by definition it's the space of Hamiltonian so their parameters but with fermions it becomes slightly more complicated even in zero dimensions if zero is from a topic Roland C P infinity cross Z 2 because the hilbert space of fermions is z2 abraded and the ground state can belong to the even component or to the odd component and that's why we have two copies of CP infinity I'll use I will use my Ronnie fermions yeah I'll describe that now let's talk about three fermions is a pretty good approximation the integer quantum Hall effect in graphene is explained by non-interacting fergenson principle fermions can do two things that can hop from atom to atom and they can interact with each other repulsively if we ignore interaction it just happened and the hopping Hamiltonian can be written like this there is some hopping matrix an electron can hop from atom K to Adam J like this and this is hop in matrix a J dagger a K this is a narration in creation apparatus of electrons we remove an electron from atom K and put it on Adam J but it's just an accident of the world we live in that the electric charge is conserved and more generally we can consider fermions with non-conserved church and those fermions can be described by my own operators more concisely so I'll consider the my on the formalism and more general Hamiltonian than that so let me introduce the parrot algebra it will be the Clifford algebra it's Clifford of to and over complex numbers the generators are c1 / c2 l and c j c k+ c k CJ equals to Delta JK these are the commutation relations we can associate various objects with this algebra first we associate the real space of dimension to n such that each basis vector corresponds to one of the generators and that is called L it's r2d2 N and we can define a complex structure in that space and let me use a standard complex structure it's given by a matrix a knot that scores two minus one and more explicitly I'll just write down this matrix it consists of such 2x2 blocks so on this will be our imaginary unit and it makes this space into a complex space one can actually define two complex spaces L plus minus and one can define the spinor representation of this algebra abstractly like this it's easy to a graded space and it's the exterior power but being a physicist I prefer a more concrete definition so let's introduce apparatus corresponding to basis vectors of their spaces either with a placer with a minus and a J the annihilation operator is C 2 J minus 1 plus ic2 J / - and a Jade Agora yeah and the way we'll focus on this which correspond to a basis in L - and we built this Hilbert space like this we postulate the existence of a vacuum such that it is annihilated by the inhalation operator and we define the space index by zeros and ones and that is the the hilbert space it's z2 graded by the parity of K and we can construct the Hamiltonian acting in that Hilbert space that is a generalized hopping matrix it must be a real skew-symmetric and this matrix may be regarded as an element of the Lee algebra so2 n and this map from a to H of a it's a representation the spinor representation of little so2 and so we'll focus on such hamiltonians these are called free Permian Hamiltonians now we need to introduce more structure first we define a gap and then we define la College the gap is defined differently for a and for H because the spectrum of a it's imaginary and the spectrum of H one half so we can add up those numbers with different signs there are two to the n eigenvalues and two separate the ground state from the first excited state we need a lower bound on the spectrum of a so we will assume that there's numbers write it this way a and B will be constant and I want a a to be strictly smaller than than be in principle independent of n yeah independent of n yeah that's important now there is a reason why a equal to B is not convenient at least because we want a soft problem and softness will play an important role so we leave some gap here between a and B and also between 0 a and taken to infinity now how do we define a gap the this is already a definition which is a cruel interior let me put I a it's between - B - a union a B and that is actually too restrictive because we want to study systems like the integer quantum Hall effect and then they have gapless edge modes and we want to somehow express the idea that there is a gap in the boat but on the edge they maybe get less modes actually the conditions on the boundary are unimportant how do we define a local energy gap and to do this we'll massage this definition and then in the end we'll modify it in the right way so first let's write a is smaller minus a squared less wrinkles and B it's an operator inequality and now we can write it in this way - a squared minus a it was B squared over 2 a 3 2 norm is bounded by B squared minus a squared over 2 and that is equivalent to the original one it's too restrictive but let's look at this matrix and it's important that a is already squared and rather than trying to modify a will modify this big matrix so you consider a disk there will be atoms close to the boundary where an electron can help out and can hop in and hopping out means that something is missing that should be there we keep extending this system and that basically violates this condition it creates some gapless modes near the edge however if we consider a smaller disk the big disk is D to the D and the smaller disk called D tilde D tilde is a disk of radius 1 minus R D where R is the locality oh sorry I forgot to define I forgot to define the locality I should say a is our local if the hopping element between J and K separated by more than our vanishes so if is our local and we restrict this matrix not a itself to this smaller disc then this condition can be satisfied even for integer quantum Hall the system the gap is defined differently for a it's a gap around zero for H it's a gap near the Gretna near the bottom right now the exact definition will look like this yeah yeah the the parameter space is a so that is the projector on to the disk D tilde and we require that the normal this restricted separator is B squared minus a squared over two and to see that this definition is reasonable let me give some lemma first let's call this metric matrix M and M is to our local and it's norm is bounded by a constant and the lemma will be formulated like this for our local matrices and suppose that the norm of Pi be M PI B is bounded by one for all disks B of radius R if this hypothesis is satisfied then the norm of M itself is bounded by 1 plus o of little R squared over capital R squared and the implicit constant in capital o depends on the dimensionality of the space I'm functions in yeah these are numbers yeah pi times this I pray the times that I prayed yeah yeah yeah your capital is an operator so the conclusion is we can test this condition locally to some accuracy and that is what we want and we can choose a suitable R capital R that depends on the little R and those parameters little a little B such that this condition is meaningful now let me define the space of good hamiltonians or good metrics is a we say that capital a and leave some room here but I forgot to define this thing that is just a formality it's a a set of atoms and mathematically an atom is characterized by position in the disk and a finite dimensional hilbert space and it will be a real Hilbert space so and also we have Euclidean space l sub J for each element of s or maybe I'll erase this D too many symbols so this separator belongs to this space if a is our local and a comedy locally kept and locally kept basically means this with a suitable choice of capital R now we need to take some limits and we want to get rid of all those parameters one limit is pretty easy because this is a directed sort of filtered category this asshat so we can take the limit and it would be nice to take a limit over our like this and there is all should not depend on a and B yeah yeah exactly right so it's on this board we began with this condition actually it's a squared we began with this condition so the spectrum is here's a gap around zero and that grant East that the spectrum of H has a gap around the bottom because the ground state corresponds to the choice of minuses everywhere in the first excited state we change one minus two plus so this is what we want ideally but we cannot do that it's too restrictive so the idea is to write this condition in terms of a squared and then modify it so those numbers basically come from this sequence of transformations yeah yeah Oh actually hear it it's better to say that M is the whole operator here and when we multiply by a projector onto a smaller disk it's okay because then we can ignore this PI D so now I it would be nice to take our 2-0 first we add in more and more atoms and the electrons can help it's very small distances almost continuously and this in principle should allow us to reduce the hopping range so I don't know how to do this and maybe it's hard to do but I can do something slightly weaker so I can define I can define this guy the idea is to define a spectrum this a sub D is supposed to be a spectrum it should define a generalized ecology theory and we first defined the ecology groups and then use they represent abilities um so the definition looks like this that is a pair of spaces and this guy is by definition a relative gross indie group which is defined like this equivalence classes of pairs which agree on why not and the coolants is defined in terms of paths so we say that yeah I want to understand the topology of AD which is not defined explicitly we first define this homotopy theoretic object and reconstruct ad because we we have this set of atoms in the D dimensional disk the hopping matrix has a locality restriction electrons cannot help further than are in this space so this is the restriction yeah yeah I'll write down the answer right so now let me forget about this limit procedure although it's also a bit interesting and let me pretend that ad is defined directly then I'll list some high-level properties that Greinke that it is a spectrum and the first principle is invertibility and invertibility means that we can take two systems a and a minus a and the such a pair of systems together who will not depend on a up to homotopy so we have the following thing there is a path from here to this matrix and then to another matrix I can write down the path explicitly yeah yeah it lies it lies in this space a sub D that is defined by the limit so let me call it X 0 maybe x and y and we define X of T let me consider the path from this matrix to that matrix now x and y anticommute and therefore the square of this matrix is just cosine squared i t over 2 x squared and if x and y are gapped then X of T is also get with by the same balance so this is the first principle be can be shown in the same spirit a plus B which is the same as the block diagonal matrix with blocks a and B is connected by a path to you B plus a and and the last thing is softness and more concretely let me call it diagonal restriction and suppose we have some function u from the disk two local matrices and we want to construct one local matrix using using this whole function and it's done like this we construct a matrix a as it depends on you with in this is J and K and now we have to assume something about this function who assumed that it's Lipschitz constant has a certain band and this bell depends on the dimensionality and the constants a and B and under this assumption we can show that a sub u belongs to a D with slightly different parameters and the idea can be illustrated no no suppose we have such a map Lipsius bounded and we construct a upper you this way then a upper you is also locally gapped with different parameters for each no s is a configuration let's fix it we have one configuration s and when we have two sides J and K this sites or atoms are close to each other so the function U does not change significantly significantly from J to K and that basically means that this guy and that guy are always the same and we can kind of interpolate between between different different values of the function U let me illustrate it geometrically in the one dimensional case and instead of local matrices let's consider planks make made of wood with some grain and the function you will show the dependence of the grain on some parameter on the homotopy parameter so we're in in the one dimensional case year and I'll do straighten this things a 1-up with wood grain so this is the parameter T that goes from minus 1 to 1 and here the grain is vertical then at some point in between stilted here it's parallel tilt it this way and vertical again so the softness condition says that we can interpolate between all these patterns and build a single pattern that looks like this yeah maybe I don't know I I'm not good at the thing but this three conditions basically guarantee that we get a spectrum and we can construct maps FD from ad to the loop space of ad plus one I recall that ad comes with a base point given by a node this standard complex structure so we can consider a loops based at that point and GD going in the other direction and one can show that this maps are homotopy inverse of each other how do we construct those Maps F sub D uses the invertibility and G sub D uses this construction so f sub D goes like this we have some system in dimension D and we want to construct a family of systems in one dimension higher so let's call our original system a and we start alternating a a not the base point and - a node a not - a note then this is a homotopy primary key it goes from 0 to 1/2 to 1 and at 1/2 it will be some non-trivial a and at 1 it will be a not again and one can continue indefinitely in both directions but the idea is that we first apply this homotopy going from a note - a note - a - a and then we apply the other homotopy in an alternating pattern and to make this look better let me just show the analogs of there's intermediate states first there is a state a note and this is this little dot then we go up using this homotopy and in the middle we would create this thick dot corresponding to a and empty empty circle corresponding to minus a so there will be this alternating pattern at the middle and going from the trivial system to this non-trivial system there will be some system that doesn't split into individual dots like like this one because there is some hot in between those two sites and above it there is an alternating pattern now we run the second construction this diagonal restriction we'll get a pattern that looks like this we synthesize the following pattern so there is a bunch of dimers and one single dot when we cut across this section corresponding to this middle middle thing and we can actually remove all those dimers by pushing them down or up in this diagram and this will be a homotopy to the identity map just a rough idea well this is this picture is it or that picture is a description of F sub D G sub D is described by this procedure and this procedure in principle gives a pattern in the same dimension however since we beginning and with a knot which is diagonal or block diagonal it's like those grains of wood that are perpendicular and we can cut at any place so the starting point and the end point can be cut and this whole section will be considered as a system of the original dimension not d plus 1 but dimension D so we just ignore the locality restriction in the horizontal direction and that's how we construct a G sub D and now one can easily show that the composition GD FD is from a topic : to the identity map on ad and that requires some non-trivial argument basically we need to track this patterns and it's actually a two-sided inverse one can compose them in and in the other way and it's also the identity this is this makes a sub D into a spectrum it it means that a sub D is homotopy equivalent to the loop space of a sub D plus 1 and that means that ground States are free from in Hamiltonian so the Hamiltonian himself himself form a spectrum is the dimension changes from such abstract arguments one cannot go further but since everything is built out of matrices one can use more information and in this particular case it's pretty easy the answer is a sub D is K Oh d- - let me check [Applause] yeah so so we get a shifted ko spectrum but then it's basically as I say condensed matter inspired message definition of KCRA and the interesting part of this story is that it looks like everything can be generalized to interacting systems with suitable conditions [Applause]

Channel: Institute for Advanced Study

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Id: W2vUbTR2RWQ

Length: 82min 7sec (4927 seconds)

Published: Tue Nov 26 2019