Quantum spin liquids and valence bond solids

hello everyone and welcome to this virtual lecture course on quantum condensed metaphysics i'm dr andrew mitchell and in this lecture we'll be discussing spin systems with anti-ferromagnetic interactions the classical nail state with a staggered magnetization up down up down and so on we will see it's actually not an exact eigenstate of the heisenberg model describing the exchange interaction between quantum spins in fact however we can describe these systems in a rather intuitive way using so-called resonating valence bonds we imagine that between a pair of sites we have a spin singlet which is favored by the anti-ferromagnetic interaction an up down minus down up kind of state however um systems cannot necessarily uh satisfy all of these spin singlets simultaneously so we imagine that the system is fluctuating between states with these resonating valence bonds on different pairs of sites in particular we'll look at magnetically frustrated systems for example the heisenberg model on a triangular lattice here it's not possible to satisfy all of the antiferromagnetic interactions simultaneously we'll be talking about that and explaining why that's the case and then we'll see that this gives rise to a macroscopic degeneracy of the ground state in such systems which suppresses the magnetic order in these systems down to very low temperatures we'll see that we can discuss these systems and describe these systems in terms of this resonating valence bond theory and this gives rise to this concept of a quantum spin liquid we imagine that we have these valence bonds these singlet states between neighboring sites but that's uh the the sites involved in these singular states are fluctuating and resonating throughout the system so we have a kind of liquid of these spin singlet states in the second part of lecture we're going to consider anti-ferromagnetic spin systems which have exactly solvable hamiltonians giving precise and exactly solvable ground states in particular we'll see that there's a class of these basically given by the generalized aklt construction in which we have valence bond solid ground states this is where we have a product state of these valence bond singlets on neighbouring sites we'll see that there's a whole class of hamiltonians that have these as exact ground states and try to understand something about magnetic ordering from these systems in particular we'll see that at the boundary of these systems we have the possibility of topological defects and fractionalization well we'll give a perspective towards quantum computation and how this might be actually useful as a resource in quantum information so in this lecture we're going to discuss um two categories of systems first the resonating values bond and secondly the valence bond solid so let's get down to work in this lecture we're going to be considering antiferromagnetic spin systems in the previous lectures we've been focusing more on the ferromagnetic side in this lecture we'll see that for antiferromagnetic coupling we can get a different kind of magnetic ordering resulting in a staggered magnetization another important difference between antiferromagnets and ferromagnets is that in the antiferromagnetic case we have the possibility of magnetic frustration in the second part of the lecture we will discuss a class of exactly solvable models which are referred to as valence bond solids we will discuss how to construct those models how to construct the exact ground states and learn something about fractionalization of excitations we will start as usual with the paradigmatic model for describing uh spin systems which is the heisenberg exchange hamiltonian i've written it out here we have a sum over sights on the lattice i and j and then a spin spin interaction si vector hat dot sj vector hat and this has a particular exchange coupling strength which we label here j i j so as we discussed in the previous lectures these jijs here basically encode the dimensionality the connectivity and the geometry of our lattice these operators here these spin operators are of course quantum mechanical operators for a spin s and we will talk in the first part about spin half but later on we'll talk about higher s as well in this lecture we'll be talking about anti-ferromagnetic spin systems and therefore we'll assume that the exchange couplings j will be greater than zero this favors the anti-parallel alignment of neighboring spins as an example let's consider the one-dimensional spin half chain here i'm depicting a particular state it's got anti-ferromagnetic ordering in the sense that i have this staggered magnetization up down up down up down and so on this is referred to as their nail state however as explored in the previous lectures this nail state is not an exact eigenstate of the heisenberg hamiltonian it's not the ground state or any eigenstate that's because contained within this heisenberg exchange hamiltonian are spin flip terms these terms for example lower the sz projection of a particular spin and is compensated for by raising the asset projection of a neighbouring spin for example as i've depicted here the strict up down up down up down and so on order of the nail state is disturbed by this spin flip process the true ground state of the system will be one in which we have a superposition of various states that look a bit like this in fact we can get these domains of antiferromagnetism separated from other bits by these domain walls where we have two parallel spins all of these kind of configurations will enter into this superposition for the ground state of this system just to emphasize the ground state of this system is not a pure product state with this pristine anti-ferromagnetic ordering we can also consider systems with a different geometry or dimensionality another classic example being the 2d square lattice here we can again imagine having a nail type state with the pristine anti-ferromagnetic ordering every spin for example take this one is surrounded by other spins its nearest neighbors of the opposite s said here too the pristine anti-ferromagnetic ordering is not present in the exact ground state of this system for example if i flip this pair of spins here then i see i create a number of domain walls where we have parallel spins here here here here here and here the true ground state of the system is one that's much more complex we can however still talk about the propensity for magnetic ordering in such systems by looking at spin spin correlators for example if we look at the expectation value of s i vector hat dot sj vector hat for a pair of sites on this lattice i and j then we can see how that scales with the distance between those two sites in general what we would see is that there is an antiferromagnetic ordering but the strength of those correlations die off as some power law here parametrized by this gamma if we had pristine anti-ferromagnetic ordering then i could say for example if this spin here is up then i know with absolute certainty that this spin over here is up and this one over here is down for example in the true ground state um it's not so simple and what we could say is that there is a correlation between the spin on this site and this site but that correlation decays with increasing separation as one might expect these correlation functions and in particular these power laws really typify and characterize these systems the situation is somewhat different on the triangular lattice because here we have the possibility of magnetic frustration suppose i imagine spin half particles on the vertices of this lattice imagine that i have some anti-ferromagnetic ordering along the top row here up down up how then do i arrange the spins on the other sides to maximize the number of anti-parallel bonds let's say in the second row i start off with down up down up in this configuration i have anti-parallel bonds running along each horizontal line and also we see anti-parallel arrangements of the spins on these bonds however if i look on these bonds i see parallel spins in fact it's easy to see that however i try to tile my triangular lattice with up and down spins i cannot simultaneously satisfy all of the anti-ferromagnetic bonds meaning the spins like to be in an anti-parallel configuration but i can't simultaneously satisfy that condition for all of these uh these bombs denoted by the blue lines here however i try to arrange the spins there will always be these pairs of neighboring sites in which i have the spins in a parallel alignment we saw that the situation was rather different on the 2d square lattice there i could write down an exact product state um in which all of the pairs of neighbors had the anti-parallel configuration on the 2d triangular lattice that's not possible this is referred to as magnetic frustration and it would also be an issue even in the classical system of spins the magnetic frustration is a result therefore of the competing interactions to understand this a little more detail let's consider a single melody on this lattice a single triangle let's consider the following hamiltonian we have a constant j between our different bonds here i will also assume that j is greater than zero to look at the anti-ferromagnetic case and then we have spin spin interactions between the three sites s1.s2 s1.s3 and s2.s3 thereby forming this equilateral triangle of spins we can consider a possible spin configuration like this one in which spins 1 and 2 are in an anti-parallel alignment but what do we do with the third spin do we choose it to be up there by making these two anti-parallel but these two parallel or down in which case these two would be anti-parallel and these two will be parallel energetically you can see that this remaining spin can be either up or down and it doesn't make any difference so there's some freedom there but there's even more configurations imagine that i have spins two and three in an anti-parallel alignment then what do i do with spin number one again if i choose either up or down spin i cannot simultaneously satisfy all of the anti-ferromagnetic interactions and finally i can write this spin configuration it's the same deal all of these three configurations are basically equivalent and all of them have this ambiguity as to what to do on the third site of course i'm simplifying the situation a bit here by talking about the simple classical product state configurations of the spins in reality something like this up down here or this up down here are really spin singlets in the quantum mechanical sense these would be a superposition of up down minus down up that still leaves an ambiguity as to what to do on the third site however in both the classical and the quantum case the magnetic frustration leads to a ground state degeneracy this is following from the inherent ambiguity of the spin configurations and not being able to simultaneously satisfy all of the anti-ferromagnetic interactions so let's now see the magnetic frustration and the resulting ground state degeneracy in this simple model comprising three spins for this simple system we can label our basis states and indeed our eigenstates by the total spin and s said quantum numbers first of all let's define the total spin operator s vector hat simply as the sum of the spin operators for the three sites s1 s2 and s3 and likewise for the s z operator it's then easy to see that the hamiltonian commutes with both s squared and s said s and s said are therefore good quantum numbers and we can label both our basis states and the eigenstates of h according to these quantum numbers in fact by considering the usual rules of angular momentum addition here we're talking of course about the spin angular momentum we can see that this system of three spin half particles will give us two spin half states and one spin three halves state the multiplicity of a multiplet with spin s is of course two s plus one so we can easily see that this s equals a half has two states and this s three half state has four states and so overall we have eight states and that is the correct dimensionality of the overall hilbert space here we have three spins one half two to the three equals eight with anti-ferromagnetic interactions the two spin half states are the ground states the hamiltonian can be diagonalized to find the eigen states of the system this is of course equivalent to solving the schrodinger equation i'll leave you to have a go at that yourself but let me write down two of the eigen states here what i've written down are the two spin half states with s said plus a half i've introduced a label or an index here k equals one and k equals two to distinguish between our distinct spin half doublet states you can easily check that both of these are eigenstates of the hamiltonian they're also eigenstates of the s squared operator and the s said operator both of them are spin half states and both of them have s z plus a half they're also degenerate with common eigenvalues minus three quarters j you can also check that these states are orthogonal and normalized i can write that in terms of the overlap or inner product of these two states which gives me a delta function delta k k primed when looking at a state k in the bra and a state k primed in the cat here again you can confirm that yourselves just to check that it's correct okay so how to interpret these two eigenstates what do they mean and what has it got to do with this magnetic frustration well let's have a look at this first state that i've written here i can expand this out in the basis of the spin configurations for sites number one two and three and you'll see that both of the two cats in this expression have spin number one pointing up so let me factorize that out in terms of this direct product notation we see then that spins two and three are actually in a single combination one over root two up down minus down up we can understand the structure of this state in the following cartoon fashion imagine that we have our three spins in the triangle like this we see that the spin configuration of spin number one up here is in the up configuration whereas spins two and three are in a singlet here i'm using the up down with a red circle around it to mean a singlet combination rather than literally just this product state this thing is supposed to denote the up down minus down up configuration okay so what about our other state this seems to be a little more complicated because i can't factorize out a single configuration of an individual spin in this uh in this full eigenstate here actually i could write the states in the following way imagine that i have a single spin with the uh with the upspin configuration let's say this time on the two side and then i arrange the remaining spins in this case the one and threes into a singlet in the second term i do the same thing but this time i have the spin number three fixed in its up position and spins one and two in the singlet i have a superposition of these two states and then overall i have to normalize it by some factor it's easy to check that if i add up all of these terms and write them in terms of these single cats normalize it properly then i'll obtain this eigenstate we can therefore interpret the second eigenstate as a superposition of the following two configurations the first one with one and three spins in a singlet configuration and the second one with spins one and two in the singlet configuration so we see here how the idea of magnetic frustration comes into play we can arrange this singlet bond here on any of the three axes but we can't simultaneously make a singlet on all three sides of our triangle now you might be wondering why these uh two states look rather asymmetrical here why for example in this first eigenstate do i just have a single spin configuration with um the two and three sites in a singlet whereas in the second i can state i have this superposition of these two well the answer is actually that it's pretty arbitrary these two icon states are degenerate they have the same energy that means that any linear combination of these two eigenstates that i've written down will also be an eigenstate in particular i can write down an eigenstate that will be any linear combination of these three configurations involving spin singlets on any two of the three bonds the ones i've written down here are just a specific choice but they're a choice where we have an orthonormal set it's important that whatever choice i make the states are orthogonal to each other and normalized to one but just to emphasize it would be a perfectly valid state to consider a superposition of this one and this one for example or this one with this one or some other superposition of all three i've basically made a gauge choice about how to set this up uh in these two eigenstates the specific choice is unimportant to the underlying physics because these two things are degenerate so the punch line of all of this is that because of the magnetic frustration we end up with two degenerate ground states we have a degeneracy in the ground state because we cannot fulfill all of our anti-ferromagnetic interactions simultaneously the take-home message is that frustration leads to degeneracy the system cannot choose a unique ground state this also implies that there will be enhanced quantum fluctuations and we'll be exploring the consequence of that in this lecture this idea of writing our eigenstates as superpositions of spin configurations where we have singlets on a given bond is actually useful when we go to the lattice generalization so let's have a look now again at the magnetically frustrated triangular lattice let's imagine that i have a spin singlet somewhere on this lattice connecting a pair of neighboring sites here this cartoon is has a specific meaning i'm taking this moety here to indicate that we have a spin singlet which means one over root two of up down minus down up for the spin configurations of these two neighboring sites now i can construct a spin configuration for the whole lattice by simply tiling the lattice using these spin singler bonds on the infinite lattice i can arrange these singlet bonds on all possible configurations in this finite patch you see here i have one left over but i'm assuming that's going to be connected to some other state up here that i've just not drawn of course this tiling of the lattice in terms of these spin singlet bonds is very far from being unique i can choose all sorts of different configurations for example i could choose one like this and indeed you can probably convince yourself that there's a macroscopically large number of these different configurations here i've just illustrated a few of them the true ground state of these systems will be some linear combination of all of these different configurations we will have a quantum superposition of these kind of states another way of viewing this is as fluctuations in time we can imagine that the system occupies a particular configuration like this at a given instant in time and then it quickly evolves to this state and then this state and this state and so on we have fluctuations of these various bonds between these various configurations and we imagine that this is happening in time in this sense the magnetic frustration is leading to the enhanced fluctuations that i mentioned on the previous slide here the fluctuations are between which pair of bonds are in a singlet state the eigenstates of our system can therefore be viewed as so-called resonating valence bonds the idea is that this singular state here denoted by this red bubble containing a pair of black sites is a so-called valence bond it's like having a chemical bond in the molecule hence a valence bond the term resonating valence bonds is meant to indicate these enhanced fluctuations i imagine that these different spin configurations are in resonance meaning that we interconvert very rapidly between them so i imagine that i interconvert between all of these things they're in resonance and therefore the valence bonds are themselves resonating the specific configuration of the valence bonds is changing from this configuration to this configuration this class of system is often referred to as a spin liquid this is because we imagine that these uh valence bonds here these singlet bonds are flowing around from position to position they're moving around each other and flowing just as one would have with molecules in a liquid of course this is just an analogy here we're really talking about a quantum superposition of these states but we can imagine it as a kind of spin liquid in this very precise sense also notice that in each of these configurations i actually have zero net magnetic moment all of the bonds in the lattice are in the singlet configurations this would suggest in the thermodynamic limit at least that these different resonating covalent bond configurations are suppressing the magnetic ordering and giving us a disordered state rather than a magnetically ordered state and this is exactly what happens in real materials so let's discuss these quantum spin liquids in a little more detail the heisenberg model on the triangular lattice that we've just discussed is a good example of such a quantum spin liquid it's locally and globally magnetically frustrated it has a macroscopic ground state degeneracy and what i mean here by a macroscopic degeneracy is that the degeneracy is extensive in the lattice size as i increase the number of sites in my lattice the ground state degeneracy grows the enhanced quantum fluctuations between the degenerate ground states also suppresses the magnetic ordering in particular quantum spin liquids remain disordered even at very low temperatures as discussed on the previous slides quantum spin liquids can be viewed as resonating valence bonds each pair is in a singlet but the pairs are constantly switching all of the different but equivalent degenerate ground states are in resonance with each other and although these systems are disordered in some sense quantum spin liquids actually still have long range spin correlations to understand this feature let's again consider our triangular lattice of spin halves consider the following situation i can actually make up the spin singlets constituting our resonance valence bonds using spins that are not on neighboring sites for example in this case we have two spins in a singlet but they're not on neighbouring sites the singlet state is extended in space in this example i have an even more extended singlet state there's nothing to stop us making our singlet states from sites that are actually well separated in space all of these things contribute to the superposition of states in the quantum spin liquid ground state in the end this actually gives us long-range spin correlations and these are actually a typical feature of quantum spin liquids in fact there are many real materials which exemplify this quantum spin liquid behavior for example tantalum sulfide can be understood as a spin system on the triangular lattice like the ones that we've been discussing whereas this compound which is known as herbert smithite is something that's defined on the kagame lattice that's a different kind of magnetically frustrated letters and similarly with sodium irritate we have a system on the hyperkagame lattice and there are many many other examples so these systems really exist and some of the smoke and gun signatures of quantum spin liquids is their resistance to magnetic ordering down to low temperatures and the long-range spin correlations as well as the macroscopic ground state degeneraces these phenomena can all be observed in experiments on these materials finally i want to talk about the spin-on excitations in quantum spin liquids these so-called spin-ons are the elementary collective excitations above the resonance valence bond ground states let's take another look at our favorite example of the triangular lattice suppose that i tile my lattice as usual using these spin singlet bonds but i leave one site unpaired this site in the middle here has no companion it's not in a single state with any other site in the system and i'm denoting it here just with this residual spin up state this single unpaired spin can be regarded as a defect in the resonating valen bond state this defect obviously costs a finite energy to produce and therefore it represents an excited state the finite energy cost for this state arises because the maximal anti-ferromagnetic pairing of the spins is here not satisfied we would always lower the energy by arranging this spin to be paired with some other spin in the lattice if we really have an unpaired spin like this it's obviously a higher energy state these spin-ons have interesting properties every side of the lattice is occupied by a fermion which has a charge and a spin so relative to the resonating valen bond ground state where we have a single particle located on every site this spin on excitation has zero charge because there's still a spin sitting here but it has a net spin half remember the resonating valen bond state is one where all of the spins are locked up into singlet states albeit resonating amongst different configurations and therefore they have no net spin so these excitations have a net spin half as you can see here i've depicted with an up arrow but they have zero charge of course both of those statements are relative to the resonating valen bond ground state so these spin-on excitations are actually neither fermions nor bosons they're kind of new emergent excitations in the system they have spin but no charge another important property is these spin-on excitations are said to be deconfined although the excitations require a finite energy to produce once they're produced they can actually move freely throughout the lattice with zero energy cost we have resistanceless spin transport in particular the resistanceless spin currents allow these systems to be useful for spintronics applications these are where we can transport the spin from one side of the structure to the other and use it as a quantum information resource but this transport occurs without resistance it's very fast and it happens without energy loss spintronics devices based on this kind of deconfinement of spinons is something that's at the forefront of current research but why is it that these spin-ons are deconfined why is it that we have resistance-less transport of spins across a structure well imagine that i have a specific configuration like the one i've depicted here with this free spin located in the center of the lattice by simply changing which of the states are in a singlet and which states are free we can actually move this spin onto a neighboring site this is another completely equivalent configuration where now i have a spin singlet here and the free spin has moved over one site both of those configurations have exactly the same energy so it costs no energy to move that spin there's also no kinetic barrier because all of these configurations are in resonance with each other we imagine that the system is fluctuating between all of these configurations and of course this can happen again and again to move the spin arbitrarily far from its original position for example now the spin has hopped onto this position and i'm forming a singlet valence bond between this pair of sides doing this over and over again allows the defect to move freely throughout the lattice so it's not only the ground state but also the excited states they're interesting in quantum spin liquid systems next i want to consider the opposite limit of so-called valence bond solids in particular here i want to discuss a class of exactly solvable models where we can construct the exact ground states of the system as we'll see these ground states will be so called valence bond solid states as you might be able to guess from the name here we'll be using some of the same kind of ideas as in the case of the resonating valence bonds but instead of now having a liquid of these different states in resonance we now consider a fixed configuration of these valence bonds the analogy here is that we have a crystalline solid of valence bonds as a concrete example let's again consider a spin system with frustrated magnetism for simplicity we'll consider a 1d spin half chain and we will frustrate this system by using both nearest neighbor and next nearest neighbor anti-ferromagnetic exchange interactions this is the hamiltonian that i will consider i have a 1d chain where i'm summing over the sites labelled by i i is running basically from minus infinity to plus infinity here then we have two terms in hamiltonian the first parameterized by an exchange coupling of strength j is the nearest neighbor interaction between a pair of spins s i dot si 1 the second term parametrized by the exchange coupling j primed is the next nearest neighbor interaction between sites i and sites i plus 2. if both j and j primed are greater than zero the system is magnetically frustrated why is that well this first term obviously favors an anti-parallel alignment of the spins we would like something like up down up down up down and so on however the anti-parallel alignment of neighboring spins from this term favors a parallel alignment of the next neighboring spins as you can see here indicated in blue however the second term here likes to see next nearest neighbors anti-parallel this term therefore has nearest neighbors in a parallel configuration so these two terms are actually working antagonistically against each other this first term wants nearest neighbors anti-parallel and next nearest neighbor's parallel and the second term has it the other way around the system is maximally frustrated if both of these two terms are competing equally with each other and have the same magnitude obviously in that case we cannot satisfy all of the interactions simultaneously however instead of considering the point of maximum frustration let's consider a different and special point where j primed is equal to one-half of j the system is still somewhat frustrated because all of the interactions cannot be perfectly fulfilled but this point is actually an exactly solvable point of the model by looking at the exactly solvable point and the ground state of the system in terms of these valence bond solid states we will actually learn something about these magnetic systems the frustrated one d spin half heisenberg chain with nearest neighbor and next nearest neighbor terms in exactly this ratio goes by the name of the majumder gauche model here is the hamiltonian written out at this special point so how can we solve this model well actually there's a simple trick that will allow us to rewrite this in a different form the first step is to separate this out into two separate sums the second step is to split up this first term into two pieces we simply copy this term and then put a factor of a half before each one of them now here's the trick because we have an infinite sum over all lattice sides i here i can shift this sum by one to the right this will not affect the sum because i'm summing over all i anyway this then gives me the following for the second of the two terms these two terms are actually completely equivalent because we're summing over all i and i is basically a dummy index now you'll see that i have one half j and the sum over i in each of the three terms in this system with in the first term the coupling between i and i plus one in the second term between i plus one nice two and in the third term between i and i plus two i can therefore write the majumder gauche model in this alternative form the final trick is to write this expression in the following form where we have now the square of the sum of spin operators on three neighboring sites s i vector hat plus s i plus one vector hat plus s i plus two vector hat all squared the coefficient here is one quarter j and we're subtracting off some constant where n here is the total number of sites in the system which is going to infinity so how did i obtain this expression well imagine expanding out this square here we will get s i squared plus s i plus 1 squared plus s i plus 2 squared we'll also get twice all of the cross terms these cross terms will therefore give us one half j the sum over i and then exactly these terms that are in in the original model but what about s i squared well we're talking about spin half particles and so s i squared it gives me an eigenvalue of s into s plus one which is three quarters i get that contribution for each of these three terms and that gives me nine quarters multiplying by this factor of one quarter j out the front i get 9 16 of j and i get that term for every single i in our sum here and if we have n terms in the sum the total contribution is 9 16 j n this term of course doesn't feature in our original model so here i subtract it off again furthermore we know that constants don't affect the eigenstates of our system they don't affect the dynamics this is just a constant and so we'll actually ignore it in the following furthermore the term in the hamiltonian here which is squared is basically this combined spin of three neighboring sites on our chain this is actually sufficient information to now determine the exact ground state of the system and to show that it's actually of a valence bond solid type so how does writing the hamiltonian in this special form give us some insight into the ground state of the system well here let's remind ourselves that we want to pick j greater than zero which is the anti-ferromagnetic case where we have a partially frustrated system with j greater than zero the energy is clearly minimized if the total spin on any three neighboring sites is the smallest it can possibly be since we're talking about spin half particles what's the smallest spin we can have for a collection of three spins we actually discussed this at the beginning of the lecture and we saw there that from the rules of angular momentum addition if we're adding three spin half particles the smallest spin that we can get is a spin half so the ground state will be the one in which any three neighboring sites are in a spin half configuration how is this possible so let's depict our one-dimensional chain of spins in the following way the blue lines here are indicating the exchange couplings between the spin half particles which are themselves denoted by these black circles let's again use our valence bond notation where a red bubble encapsulating a pair of these black sites represents a spin singlet state meaning one over root two up down minus down up for the spin configuration on this pair of sites this is of course a spin zero configuration for those two sites let's now tile our one dimensional chain using these valence bonds the lattice would then look something like this in fact it's now clear that this is in fact the exact ground state of the montgomery digest hamiltonian that i've written here we can immediately see that by considering any three sides on the lattice for example here i'm considering these three consecutive spins two of them will be in a singlet state and one of them will be left over and therefore will give a spin half contribution this is exactly what we want to minimize the total energy for three neighboring spins we'll have always a spin half state that's because two of them will be locked up into a singlet and there'll always be one left over as a free spin for example if i consider these three neighboring spins i have exactly the same situation two of them are in a singlet state and one of them is left over that will always be the case no matter which pair of three consecutive spins i choose this configuration of valence bonds therefore clearly gives me the lowest possible energy because the total spin on any three sites is the smallest it can be which is spin one half therefore the total energy of this hamiltonian will be the smallest that it can be and the state with the lowest energy is the ground state the exact ground state is therefore a product state of these neighboring singlets these are the valence bonds of our system and the chain therefore becomes dimerized in terms of these bonds this is the so-called valence bond solid because we basically have a frozen configuration of the valence bonds of these singlets on neighboring sites each singlet involves entangled spins on neighboring sites this is definitely a quantum mechanical effect but they are not resonating and fluctuating with other valence bonds as in the spin liquid the correlations in such a system are inherently very short-ranged the expectation value of the spin spin correlation between sites i and j is precisely equal to zero unless i and j are nearest neighbors on the one dimensional chain we can now write our ground state in the following exact product state form we have a singlet state for sites one and two with the direct product of the singlet states on sites three and four with a direct product of singlet states on five and six and so on finally i want to discuss localized edge states in the montgomery just chain here we're going to focus on the boundary effects in a finite chain so imagine that i just have a finite bit of the chain here i've drawn an illustration with 10 sites in the chain but we can have this with an arbitrary number of sites in general on the previous slide you might have been wondering whether it matters where i put my valence bonds in the infinite chain it doesn't matter however if one has a finite strip of the chain then clearly it does make a difference i can imagine two distinct tilings for example in this upper case we have all of the five pairs in valence bonds in the lower case we have only four of the five pairs in valence bonds and there are two left over at the ends of the chain in the second example the states at the end are basically three spin half excitations they're not coupled into a valence bond and therefore they remain a free spin half in fact these two states are not energetically equivalent the upper one with all pairs in valence bonds is actually of a lower energy the first excited states on top of that ground state are the ones where we have these excitations these spin half particles lying at the ends of the chain in the next example i'll actually show you a system where we have a fractionalization of the fundamental particles in our system and that there are zero energy states perfectly localized at the boundary of the system even in the ground state this will be our first encounter with topology in quantum matter and it's a topic that we'll return to later on in the course the next example i want to talk about in this lecture are the so-called aklt chains this is an acronym named after the inventors of this model affleck lib kennedy and tasaki these actually constitute a whole class of systems with exact valence bond solid ground states it's a formulation that puts the previous analysis on a more general footing there's a whole family of these models but i'll just consider a specific one that's particularly insightful this is the hamiltonian that i want to study on the next few slides it looks like a rather weird object it's a one-dimensional chain with a constant exchange coupling j which is assumed to be anti-ferromagnetic j greater than zero and we have a sum over sides i of our chain running from minus infinity to plus infinity then we have a bunch of terms there's a constant which we can basically ignore but i'll keep it in for now we have a nearest neighbor exchange coupling and then we have this funny term which is a nearest neighbour exchange coupling squared and the coefficients here a third one half and one sixth are actually crucial to obtaining the exact valence bond solid ground state finally it's important to mention that these si vector hat operators are actually spin 1 operators in this model that means that the local hilbert space for a single spin is actually three dimensional the z projection of the spin can take values of minus one zero or plus one on every site in the system notice then that the combined spin on a neighboring pair of these spin one sites can be either zero one or two of course this hamiltonian is a bit of a contrivance why would we look at this particular thing does it relate to a particular system in uh in the real world well the answer is that it doesn't really relate to a particular material as such but it does teach us something interesting about the nature of these magnetically frustrated spin systems so we'll study it as an abstract model it will teach us something useful i hope furthermore even though i will discuss the exact valence bond solid ground state of this system one can consider more general systems that really might model real materials as simply perturbations to the exact state given by this system so first let's consider two neighboring spin one sites let's say we're looking at sites number one and two we can label these by the combined spin of these two sites which is just given as the sum of s1 vector hat plus s2 vector hat i'll denote this combined spin operator s12 vector hat let's label the states of our two neighbouring spins by their combined spin s12 i'll denote that with this ket if i now act on that ket with my s12 squared operator it should return the combined spin eigenvalue s12 into s12 plus one now for our two neighboring spin one particles the combined spin can either be zero one or two by the usual rules of addition of spin angular momentum let's now define some projectors onto the states with a definite spin for the pair of sights one and two in particular i'll define p0 as the projector onto the spin zero state here i just have the bra for s12 equals zero state and the ket for the s12 equals zero state and similarly for the projector onto the spin one state which looks like this and the projector onto the spin 2 state which looks like this so let me remind you what i mean by these projectors imagine that i act with the projector p0 onto the state with the combined spin equals zero the result of that due to the orthonormality condition of these kets is equal of course to the s12 equals zero state however if i act with the p0 operator on the s12 equals one state i will get zero and if i act with the p0 operator on the s12 equals 2 state i also get 0. i can of course repeat the procedure by acting with the p1 projector onto the states and acting with the p2 projector onto the states and i would see something very similar in fact let me write out a table that summarizes this information let's say on the left here i have the states with s one two equals zero one and two this is defining all the possible spin states for our neighboring pair of sites and then on the top here i'm describing these various operators there's s12 squared and then the projectors p naught p1 and p2 in the table here i'm giving the eigenvalues of these operators acting upon these states for example s12 squared acting on the spin 1 state gives me an eigenvalue s into s plus 1 and for s equals 1 that gives me an eigenvalue of 2. likewise if i act with the p2 operator on the state with spin 1 i get 0. only the state with spin 2 gives me an eigenvalue of 1 of the p2 operator okay so what's the point of all of this and how does it help us so with this table at hand let's now try to express our p2 projection operator onto the combined spin 2 state in terms of the combined s12 squared operator i will argue that i can write as an operator identity that p2 hat is equal to 1 over 24 times s12 squared into s12 squared minus 2. why on earth would this be the case first of all consider this factor s12 squared from the table i know that if i apply s12 squared to the spin zero state that then i get an eigenvalue of zero so the fact that this projector p2 contains this factor s12 squared means that if i were to operate with p2 on the spin zero state then it would give me this zero eigenvalue and actually that's independent of whatever's going on in this second factor here so this factor here annihilates the s12 equals zero states likewise the factor s12 squared minus two will annihilate the spin 1 states that's because the eigenvalue of the s12 squared operator when acting on a spin one state gives me two so s one two squared minus two will give me zero if i'm acting upon the spin one state so the two factors that i see in here actually annihilate both the spin 0 and spin 1 states all i have left is the spin 2 state so this is therefore a projector onto the spin 2 state it will only give me the spin 2 state back again it will give me 0 if i have a spin one state or a spin zero state okay so this factor of 1 over 24 just gives me the normalization that's because if i actually act with the p2 operator on our spin 2 state then this first term will give me an eigenvalue of 6 and this second term will give me an eigenvalue of 6 minus 2 equals 4. so 6 times 4 is 24 and i need this leading coefficient 1 over 24 so that the eigenvalue of p2 comes out correctly as 1 when i act on it with the s12 equals 2 state okay so that's a little bit convoluted but hopefully you understand the logic i can then express this p2 operator in terms of our individual spin operators for site number one and two in the following way and that's because the s12 operator is simply the sum of s1 plus s2 so s12 squared involves s1 plus s2 dotted into itself if i expand this out i'll get twice the cross terms 2 s1.s2 which is this term and then i will get s1 squared plus s2 squared but here we're talking about spin 1 particles so s1 squared will give me s into s plus one equals two i have that for both of the particles two plus two is equal to four so expanding this all out i obtain the following expression the projector onto the spin two state of a neighboring pair of sites can be given as one third plus one half of s one dot s two plus one sixth of s one dot s two all squared and of course this is very closely related to what we had in our original hamiltonian we can now write the aklt hamiltonian simply as a sum of projectors onto the spin two space of a pair of nearest neighbor sites i and i plus one and then summing over all sites on our one-dimensional chain affleck kennedy lieb and takassi who came up with this hamiltonian basically started from this projector and then wrote the hamiltonian out in terms of these spin operators in this lecture i introduced it the other way around i talked about the hamiltonian first and then i showed that it could be written in terms of these projectors but in general we can consider constructing hamiltonians from these kind of projectors and exploring what kind of physics they have and this is precisely what is at the heart of these aklt constructions so you can see there's a whole family of these you could imagine also a model where we had a projector onto the spin one states or a hamiltonian with some arbitrary spin and other kinds of projectors entering in here the specific example is the one that i wanted to talk about here with this aklt chain with spin 1 particles and the hamiltonian consisting of projectors onto the combined spin two states of neighboring pairs so so far we've understood something about how to rewrite the hamiltonian as a sum of projectors the question now is what is the ground state of such a system we will obtain the lowest energy if all neighboring pairs of spin ones are in their combined spin zero or singlet configuration this will be the ground state because then the hamiltonian will annihilate all of those states this is a projector onto the spin two space of neighboring spins if all neighboring spins are in a spin zero configuration this projector will annihilate it and give us zero energy contribution for that state you might wonder why we can't arrange for our neighboring pairs to be in a spin 1 configuration they would also be annihilated by this spin 2 projector but as it turns out it's actually impossible to have every neighboring pair on the ladders in a spin one configuration whereas we can achieve every pair being in a spin zero configuration let me show you now how that works the question is how do we guarantee that all of these neighboring pairs of spin one sites end up in a combined s equals zero configuration so here i'm depicting our one dimensional chain of spin-one particles now instead of considering each of these particles as a spin one object let's consider it as two spin half particles locked up in the triplet s equals one we can then represent our spin one chain in the following form where each of these units with this blue square box around two black dots represents a spin triplet s equals one configuration this is just a cartoon shorthand notation for the following three degenerate states of spin half particles up up up down plus down up and down down these are the three components of the s equals one triplet in particular each of the dots here represents a spin half particle whereas in the original model each of these dots represents a spin 1 particle we showed in a previous lecture that a spin 1 particle is mathematically identical to two spin half particles locked together in this spin triplet configuration so therefore we have an alternative representation of our spin one chain in terms of pairs of these spin half particles each of which are being locked into this spin triplet configuration how does this help us let's now introduce again our spin singlet cartoon i imagine that if i have two spin half particles in this red bubble here this represents the spin single at s equals zero configuration which looks like one over root two up down minus down up the interesting thing is that i can re-partition my chain originally i had this dimerized form with these neighboring spin half particles in this representation locked up into the spin triplet but now i can imagine drawing these red bubbles to indicate a spin singlet comprising one of the spin half particles from each of the neighboring blue boxes each of these blue boxes of course represents the spin one configuration of two spin halves we take one of the spin halves from each of neighboring boxes and form from them the spin singlet state you might be wondering if this is possible but in fact it's certainly possible because if we take two neighboring spin1 objects indicated by these blue boxes we can form a spin zero configuration of those two things and that's exactly what these red bubbles are supposed to be indicating it's just that we're conceptualizing this as breaking the spin one down into two spin half particles the spin zero states are then just a redimerization of the spin half particles on the chain the state with all of these spin half particles on neighboring sites now locking up into spin singlets is then illustrated in this way affleck kennedy liebentasaki argued that this must be the exact ground state of this hamiltonian and as you can see this is a kind of valence bond solid we have these singlet states these valence bonds which span the whole chain it's just that we've formulated our spin ones in terms of these pairs of spin halves but in this language um the states that i've drawn here is indeed a valence bond solid also as argued on the previous slide this must be the lowest energy state the ground state of the system the hamiltonian consists of a projector onto the spin two configuration of neighboring sites and as you can see here all of the neighboring sites are in a spin zero configuration that's what these red bubbles are telling us therefore the hamiltonian which projects onto the spin 2 configuration of nearest sights will give us an eigenvalue of zero for this state that is actually the lowest possible energy in this system and therefore this is the ground state by locking up each spin half pair into a singlet we guarantee that neighboring spin-one pairs cannot combine into a spin two state therefore this valence bond solid state is a zero energy eigenstate of the hamiltonian and is in fact the ground state this idea of separating out our original spin-1 particles into two spin-half particles in a triplet configuration might seem rather artificial and just like a mathematical trick however i want to convince you that this is a real physical thing and actually this relates to an important topic in modern quantum condensed metaphysics called fractionalization and it has to do with topological defects that appear in this system at the boundaries consider now not an infinitely long chain of spin-one particles but a finite chain here i'm imagining that i have a boundary where the chain is cut on both ends according to the aklt construction the lowest energy state the ground state of this system will be the one in which we dimerise these effective spin half particles in these neighboring blue boxes into effective spin at singlet configurations as denoted by these red bubbles however you can see that at the ends of the chain we will have a single unpaired effective spin half particle because the ground state of our many particle spin system has this valence bond solid kind of structure that implies that the end of our chain we must have these effective spin half particles and that they're basically free what i mean by free is that they're local moments that are not coupled to anything else in the system they're zero energy states and they're pinned and perfectly localized to the boundary of this system this really is a physical effect these free uncoupled effective spin half particles at the boundary of the system can really be directly measured for example if you did a numerical calculation of this system and measured the spin susceptibility at the end of the chain you would see that these are free local moments these are topological because they depend on the coupling in the rest of the system to get rid of these free spin half particles at the boundary you'd have to do something drastic to the whole system to destroy this valence bond solid ground state there is some information in the system that is stored non-locally here in order to obtain these localized boundary states to get rid of these states we'd actually have to go through a phase transition and change the ground state of the system radically this is an example of fractionalization because we started off with single spin 1 particles and we end up with excitations in the system that are actually behaving like spin half particles again this illustrates the point that in many body quantum condensed metaphysics the laws that are governing our system are not necessarily the microscopic laws that we started out with as anderson said moore really is different these considerations relate to a whole field of study in modern quantum connects metaphysics relating to topological quantum matter finally i'll comment that the aklt chain is obviously very special and peculiar system which has an exact valence bond solid ground state however it is representative of a wider class of system that shows these effects of topological defects and fractionalization the aklt system is nice because it has an exact solution and we can understand the nature of these topological defects in fact earlier haldane actually showed that the regular spin-1 heisenberg chain has the same fractionalization and topological spin-half edge states even though the ground state of that system is not exactly the valence bond solid state this is called the huldan spin gap problem and the concepts of fractionalization and topological protection of localized surface states in the system has been incredibly important and influential in physics and is today a huge area of modern research with proposed applications to quantum computation and there'll be more on that later haldane won the 2016 nobel prize in physics for his research in this area and it really started with this seminal result so in summary in this lecture we have considered spin systems with antiferromagnetic interactions in the first part we saw how this could give rise to magnetic frustration which manifests as a ground state degeneracy in these systems the classical niall state with a pure staggered magnetization is not the exact eigenstate of these systems however one way of understanding the eigenstates of these systems is in terms of so-called resonating valence bonds where we consider singular states of spins on neighboring sites but these singlet states are fluctuating between different sites in some systems for example the heisenberg model on the triangular lattice we have the phenomenon of a quantum spin liquid this is a system with a macroscopic ground state degeneracy and a high level of frustration that suppresses the magnetic ordering in those systems even down to very low temperatures we understand the ground states of those systems in terms of resonating valence bonds in the second part of the lecture we considered a class of exactly sulfur models which can be regarded as valence bond solids these are product states of these valence bond singlets a certain class of hamiltonian will give these valence bond solid states as exact ground states in these systems we see that if we cut the chain and consider the boundary we often see topological defects which can exhibit the phenomenon of fractionalization of course there's much more to say about this fast topic but in this lecture course this is as far as we'll go in the coming lectures we're going to change gear and discuss feminic systems where the electrons can really move around on the lattice we can discuss the dynamics in terms of the propagation of the electrons through the many body system as well as just the spin dynamics we will see that in a limit of the hubbard model describing those fermionic systems we obtain the heisenberg model that we've been discussing in these lectures and this will bring us full circle

Channel: Dr Mitchell's physics channel

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Length: 64min 50sec (3890 seconds)

Published: Tue Feb 23 2021