Matrix Product States and DMRG

okay so we talked about let's see we talked about Lang shows and then that was the end of the diagonalization part a then we talked about product states which were a state that's not entangled and how it's actually a little bit hard to tell if something is a product state and in tanked an entangled state is just something that's not a product state but we also want to have some measurement of how entangled something is okay the other thing that we talked about is that we usually think about the entanglement with some dividing line in the system and we maybe don't worry about the entanglement within each part of the system we worry about the entanglement across the system okay so you might have two physical objects separated in space and we we wouldn't worry about the lots of entanglement that was inside the object we'd wonder if if those two things were entangled okay and so then we can define a product state with that dividing line okay but then to figure out if something is entangled you need to use the singular value decomposition so the singular value decomposition so that was we haven't used it yet but I was just defining first what the singular value decomposition is it's just a simple matrix factorization that turns a mate any matrix into a product of matrices a unitary tons of diagonal times a unitary okay and I the unit the diagonal has positive elements along the diagonal that are called the singular values and that's sort of a key part of figuring out entanglement okay so how do you turn that into something about a quantum system okay so in you know typically chapter 1 or 2 of a quantum information book you might see the Schmidt decomposition and it really is just sort of applying the singular value decomposition to a quantum system ok so we have a quantum system which is written in terms of the two parts with the dividing line the left part and the right part okay and the structure of the quantum mechanical wave function is that a general wave function will be able to mix different parts of the two sides and so there's this Sai L R that sums you sum over all states and the left on the right and you could have all of these terms coming into the system and if it's got all of these terms coming into the system it might be very entangled okay so sy LR is just the wave function written in this basis but it's got two indices so we can pretend it's it's a matrix usually in quantum mechanics matrices are operators this is still the wave function we're just going to look at it as a matrix so we do an SPD on this sy L R and we get a you d tilde V the same ones we had before I've done that tilde 1 which means that U and V are both unitary ok so let's see what this gives us ok so the normalization of the wave function says that the sum over L + r sy L R squared is 1 that turns out to be cuz it looks like a matrix it turns out to look like the trace a side agar sy and so we'll use that in a little bit okay but now let's take this SVD factorization and use the U and V to define some new states okay so the U and the V are unitary transformations so if I apply them to the states in this fashion it gives me another if the first set is orthonormal that gives me a new set of states that's also orthonormal so I start with the left state cell I apply this unitary transformation so I sum over all the ELLs have the new states I that are the diagonal elements of D and I get this left state labeled by I that's a special kind of state okay and I do the same thing for the right side I take the V Tilda and I sum over all the right states and I change that basis to to this I basis and so then I plug I so then I take sigh and so I haven't used these two lines yet I just insert into the expression Versailles the singular value decomposition with the matrix matrix multiplies expand it out in index notation okay then I rearranged this sum to put the V next to the R and I do that some on R and I find that that I just get this IR piece okay so the R and the V turns into this IR and I do the same thing with the U and the L and that turns into the il piece and I'm left with a an expression for the wave function and it's different from the original expression because it only has one index that gets summed over so it's got a sum of one index I it's got the detailed I I is the coefficient and then this is these are the basis States so these are special basis states that allows the wave function to be written in diagonal form okay so that's a huge improvement over a totally off diagonal form now it's diagonal and this is the Schmidt decomposition and that D tilde III well those are just the singular values which we usually call lambda and so here's our wave function so this works for any wave function it depends on how you cut your system into you know sometimes there's one natural cut to separate systems or sometimes we just have one system and we make imaginary cuts as long as you sort of separate the hilbert space into two parts like one set of spins and there those has bins that's fine okay so then this special basis within those two parts makes it diagonal and it's got simple coefficients the coefficients are real and they're non-negative okay so that's the Schmidt decomposition so the Schmidt decomposition sort of immediately reveals to you whether the system is entangled okay so let's go back we have the normalization okay so we take this form for the normalization and we plug in the SPD and put it plug it in for both use the unitary conditions use the cyclic properties of the trace and the U and the V both disappear we're left with two diagonal matrices which are the same squared and so it just gives you and you trace it and so it turns into just the sum on I of lambda squared equals one okay so whenever you see something in physics where you sum it up and you get one usually think of it as a probability so this lambda I squared is the probability of a particular state it's this state where it's in the Schmitt State pi L on the left and the Schmitt State IR on the right and it's got this diagonal representation okay so suppose suppose sy happens to be a product state then it's already in its schmidt decomposed form it's in a Schmitt decomposed form where lambda 1 is equal to 1 and all the rest of the lambdas are equal to 0 no so then it just looks it just looks like the previous expression for the wave function except we we just read off that Phi is IL and C is IR okay so product wave function translates to SVD where you get only one nonzero piece and that tells you if it's entangled or not so you just do the SPD on this matrix form of the wave function and you immediately see if it's entangled but you also get a lot usually things are usually things are entangled and so then we can try to measure the amount of entanglement by looking at these probabilities you know so the closer it is to all the probability being in one guy that's less entangled it is totally spread out among lots of different states that's very entangled okay so Van Orman came up with the von neumann entanglement entropy and it's just plugging these probabilities it's just project applying these probabilities into the standard statistical mechanics information theory formula for the entropy okay so here it is oh I have a good story about joinin so if it's interesting to read the Wikipedia article on the Noi Minh because it you know they make a pretty good case that he was like the smartest guy of the 20th century and around here Enrico Fermi is one of the particularly famous great physicists and one of our emeritus faculty at Irvine was a postdoc with Enrico Fermi and so he told me this story about andreat and somebody talking to Enrico Fermi about von Norman so there was another postdoc of Enrico Fermi and he said to fair me you know I hear von Neumann is like way smarter than anybody else in the world you know is that really true and Fermi said well yes you know you know how much smarter I am than you well that's how much smarter he is than me I don't know if there's anybody around who still knew and Rico Fermi and would say that well that sounds like him but this guy you know our our faculty member he's retired he's 80 or so but he was okay so it's a von neumann define this and there are other types of entropy that have slightly different formulas that are useful also but this measures the entanglement so let's look at the case where one of the lambdas is one and the rest are zero well that that gives you entropy of zero you know the factor in front kills outweighs the log and the most entangled is if all the lambdas are equal and then it looks like something proportional to the number of spins in the system you know the the probabilities of all these states will be like if they're all equal to be like 1 over 2 the end now kill off the log and so you get in that case you get something that would be an entropy that's extensive scales with the size of the system so the ordinary stat mech entropy isn't quite like this the the ordinary Stadnyk entropy we sort of think of as a bigger system than the system we're looking at there's a heat bath and we're really thinking about the entanglement between the heat bath and the system of interest but this formula just has the system of interest and so the regular and we're usually talking about the ground state you don't have to but you're usually talking about the ground state so this system the ordinary entropy would be zero because if it's in the ground state it's not entangled with the heat bath okay but we're looking at something different we're saying the heat bath is itself it's just the right-hand side of it versus the left so we cut it in two into just one one system and then this defines what this type of entanglement is it's sort of a more general thing okay so let's look at an example of measuring the entanglement so here's a - spin 1/2 sand so the natural dividing line is of course one spin and the other and I made up this wave function that has just two pieces of it it's normalized the formula only works if the wave function is normalized okay so so then this could be one of these exercises let's take this wave function and the find isn't is fun knowing and entanglement entropy so the first thing you have to do is write it in matrix form remembering that it's not an operator it's this funny left/right thing you know on the left and right sides could be of different sizes so it could be a rectangular matrix here it happens to be square nothing that makes it hermitian or anything so here's the two probability here's up-up is 1 over root 2 that's this element and up down there's this other element okay and you you can do the SVD so of course you can call it numerically but we just have to find some way of writing it in the SVD form ok and so you can do that by in this case by this sort of trick here I've I've taken the wave function itself it's top two elements are root 2 and minus root 2 and then it's got zeros so I've got this part of the wave function here but I I rat it added the other piece that would make this unitary but then I killed this piece off by multiplying by this matrix okay and then I the the rest of the trick is to put in a you which is just a diagonal matrix so that's equal to that and now it's an SPD form okay so this is sort of a it's sort of like two by two you can always figure out by hand how to do it and this is sort of a little tricky way to do it or you can do it more systematically also you can form a density matrix which I'll mention later and diagonalize it that's more systematic okay so so here it is and it's a product state right it's got one singular value and the other one is zero okay and well what's what is what are the two product States well the you can see that lambda you can see that the left spin is definitely up so that's the left guy is just the Upstate and the right hand Smith State is up - down with this factor 1 over root 2 ok so then here are two singular values 1 and 0 but it's a product state so you you do this formula for s and you get 0 and it's on the tangle and there's the for the product form right now ok let's make sure this is the last thing ok any any questions on that switch over to the next set of slides yes singular value decomposition okay Oh boring so okay sorry there's two ways to think about the question so first of all let's suppose that we had a hundred spins okay you don't do the entanglement with each spin you don't divide it into a hundred pieces and do it that way you have to choose a dividing line and all that means is you have to tell me which spins are on the left and then the rest are on the right okay then you can write the matrix in the sai LR form the indices L will go over a huge number of possibilities all the possibilities of that set of spins so will are so you have this huge singular value decomposition to do in most cases if you do a big system you know it won't be practical it'll be sort of the same level of practicality as doing the exact diagonalization okay but that's what in principle what you'd have to do D MRG gives you a shortcut to it you get that you get the entropy as part of the algorithm okay so sort of close cousin of the Schmidt decomposition are reduced density matrices and that's where that's part of the name for D MRG density matrix renormalization group so here's density matrices or more properly reduced density matrices so you have the same split between the left and the right sides oh by the way if you haven't if you haven't seen much of density matrices before fineman's lectures on statistical mechanics has a wonderful chapter two introduction to them talks about examples from polarized light gives a nice little picture of why you should think about quantum mechanics this way okay but so here I'm just going to connect it to the Schmidt decomposition so I write the wave function in the same form okay and then let's imagine that I want to look at an operator an operator oh that only lifts lives on the left hand side if I take the expectation value of that operator within the states I I get these various pieces but the are doesn't connect to the oh so it goes to the R Prime and gives you a chronic or Delta okay and so you're left with a simpler piece involving the left there is a still a sum on our left over so you get rid of the sum on our by defining this reduced density matrix Rho that has two indices for the left an L Prime and an L and it sums over all the are species call it tracing out the bar then with this row there's a simple expression for the expectation value sigh it only involves row and the operator and it's just trace of Rho times oh so and this works we didn't use any properties of oh and doing this derivation so it works for any operator any operator that you want to look at on the left hand side to figure out what's going on the left hand side you can find out what it is by tracing it with row so row has all the information about the left hand side for any operator that when it's the only thing that row doesn't have is how it's connected how its entangled to the right hand side but encapsulates everything on the left hand side okay so that's a that's a nice thing that's a density matrix it's a you know got some other things that about it that I don't have time to talk about like states that are mixed but so Rho is hermitian and so you can diagonalize it so here's a little exercise an analytic exercise that suppose we diagonalize row and I can I can do row by tracing out the right and I get row left or I can trace out the left and go row right and it's sort of just a transpose it that the row is is not a transpose but you just transpose sigh and it switches between them so they're closely related so the exercise did this show that the eigen values of Rho L are the same as for roaaar and they're just lambda i squared from the fit decomposition okay so these are essentially the same thing it's just a different way of writing these singular values which we interpreted as probabilities just become the eigenvalues of the density matrix and the interpretation is the same the land I Squared's gives you the probability of the left state being in this IL state okay so in DMRC programs this actually translates to you can write the program diagonalizing a density matrix or you can write the program doing an SPD and you know typical program might have the option for doing either one they each have some advantages but it's really a tiny technical difference it's really just the same thing okay so what do we know now we know how to calculate the entanglement entropy of on norman entanglement entropy and we can think about doing it for a spin chain the same spin chains that you were working to diagonalize okay so so what would we do so we think of a Heisenberg chain with open boundary conditions spin one half out to size n okay we diagonalize that we have to decide where to split it well if you split it towards the end it'll give you a small entanglement so we usually split it down the middle to get the biggest entanglement to see what's really going on okay so we divide it in two and so I'll only think about chains with an even number of sites okay so here's the recipe so we divide in two so we first we find the ground state with the exact diagonalization we rewrite that first ground state eigen vector splitting the the spins that are on the left from the spins on the right and then we lump all the spins on the left into one index and the ones on the right into to the right index and treat it like a matrix so you rewrite it in terms of the left and right basis States so it looks like a matrix sort of absorb all the individual spins and single index on each side then we do this SVD just call SVD look at the singular values square them and add them up with the log thing and you get this entanglement entropy okay so this is a program that you know a lot of you are almost ready to do if you like almost had time to finish that that general and diagonalization and so here's what you get okay and so here's a more advanced exercise for for the weeks ahead if you have time you can try to verify this table with Julia okay but so here's that all the even sizes out to 14 and with DRG we can just keep going on this table to any size you want pretty much but here it is with just a simple diagonalization so here's the entanglement entropy for size two the exact answer is log 2 so this point six nine is log 2 okay it's also the maximal possible entanglement entropy so if you if you make all of the Schmidt States equally probable that's the most entanglement you can have that's got a simple formula of n over 2 log 2 so that's the most you can have and so then we can ask the question okay how big is the entanglement entropy compared to the biggest it could be the system is getting bigger this is the state is getting more and more complicated maybe it gets more and more entanglement no it doesn't the entanglement is hardly changing here okay so the maximum is growing with n and so here it's up to 5 but we're still around 0.76 versus starting out at point 6 9 it's actually growing as the log of the system size very slowly we that's one of the things that we've learned in the last 10 or 15 years exactly how this works goes at the log there's also an alternation which gives us a clue about what the system is doing it's like why should they're all even why should it alternate big small big small big small big okay well you can understand that in terms of an RVP rvb picture of the ground state so what the so if you have two spins the ground state is in this singlet state and it's got a nice low energy and on bigger systems it it it makes a very complicated state but it it's got a lot in common with just putting down these singlets so here's I've made a fat line for each singlet and here's two spins and it's the exact singlet if I have N equals four it looks a little bit like a singlet on the left and a singlet on the right okay now the singlet has log two of entanglement in it but if the singlet is sitting entirely in the left it doesn't matter it's entangled within the left we only care about the connections so this N equals four should have a very low entanglement because it looks like two singlets and that that is what we find we find this 0.32 this is not zero so this is just sort of a cartoon like picture yes it would always be log 2 because this is it would always be exactly log 2 the reason is is that it's a it's a it's a symmetric wave function between up and down and so the if you cut it not in sorry if you cut it so it only has one site which is all you can do for N equals four then the one site can is it has two equally probable States up and down that's all that matters so the entanglement entropy is always governed by the smaller system and so in this case for even numbers of sites cutting it just for one side is always log two okay and then if you move the cut steadily to the center there's a particular form that it takes and there's an analytic prediction for this and the large n limit and but it's known about this yes so RV thanks our VP is resonating valence bond state so this singlet here is called a valence bond and just other language and what one way to represent this type of state is it's a superposition of valence bonds in all different places and that's called a resonating valence bond state and this idea dates back to Linus Pauling in the context of chemical rings that the quantum mechanical state can look like that but it really came into physics with Phil Anderson proposing it as a possible ground state for frustrated spin systems and so forth proposed it for the triangular lattice it's now a good description of the CagA may spin liquid lattice which I'll show some results for okay so in this I'm not actually letting these resonate because in 1d they really can't move around with open boundary conditions are kind of stuck from the end so you can only write one pattern well these bonds it pairs up the sites so for N equals six one of the singlets one of the bonds is in the middle and so you're cutting it in two so you get that extra entanglement of cutting that guy in two and so six is big and so it depends on whether you're a multiple of four or not whether you cut the guy in the middle and that gives you the this alternation you can have further neighbors singlets but the near neighbor singlets are directly in the Hamiltonian so you know that we if we put in a jape rhyme it would try to make next neighbor singlets we didn't put that in so it's trying to lower the energy and so it tries to make every neighbor pair a singlet but it can't once you make it a singlet with one guy it can't talk to anybody else so it has to do some sort of combination of fluctuating guys that's mostly near neighbor but sort of fluctuating around but it does have beyond nearest neighbor and you know these numbers aren't going down to zero makes it go and as it goes farther away from the ends the the effects of the ends that pin it in one particular location dies off and the even odd alternation gets smaller and smaller okay so the now let's talk about why this entanglement is so small compared to what it could be there's something called the area law and laws and physics usually are something that's usually true mostly true theories are can be always true laws or like you know there's Hookes law for Springs that's like just a little Taylor series so this is the area law is is better than Hookes law but it's you know you have to be very careful about how you define the system to make it true and there's been some wonderful work proving the area law in certain certain circumstances particularly by Hastings but what it says so one of the things that it's that C log Corrections are not going to be considered here they're sometimes present but what this statement is is that if you have a the ground state of a system with this division of a plus B but it's really all connected then the entropy is proportional to the area of the boundary and that's why it's the area law and it's it's area makes you think of you know three dimensions with a cut makes it an area but so the name isn't quite it's a little misleading but in 1d you divide the system in two and you say what's the what's the number of sites on the edge and it's really only one no matter how long you make the system so this the area law would say that s is a constant because the cut only has one site on it if we go to two dimensions s would be proportional to the length of the line here and so it'd be proportional to ly for an LX by ly system wouldn't depend on LX and 3d you I didn't draw it but you have some 3d thing and you cut it in the middle and there's an area and the entropy should be proportional to that okay and so I can use the RPB picture to give a pictorial justification for that which is here in 2d and the key observation is that these singlets that are on the entirely in the left are entirely in the right don't matter so I draw some sort of singlet pattern here I did it sort of organized but it could be more random and then I say okay now I do a cut and I say how many of those singlets are gonna cut be cut by the line here I'd line them up so that all of them and the line were it could be half of them or something and you can see that the the ones that are cut are is proportional to the area of the line for the area the area of the boundary and so that's where the area law comes from it's just sort of using the singlet picture saying the interior of this guy doesn't talk to the interior of that guy in a typical ground state okay but there's a there's this sort of gives you the picture that okay this kind of makes sense in this RvB type of language it doesn't give you a sense that it should always be true and of course there are log corrections to this but this is a sort of you know detailed research area that people have thought a lot about there's a lot to know about this but each of these singlets that were cut would contribute their log to to the 2s and it would just add that up okay so the area law is the thing that makes d MRG work it says the area law tells us that so the biggest biggest entropy would be the volume of volume law and if the volume law applied d MRG wouldn't work at all but the entropy is a lot smaller and that allows us to throw away states okay and so that's the next section how do we throw away States if the entanglement is small okay so truncating low-probability States so the first thing is that if the von neumann entanglement entropy is small that means that there has to be a bunch of state with nearly zero probability so here's a little schematic plot of the lambda squared versus the index I and I've made it so that it's just getting really close to zero beyond a few of these so this would be a low entanglement state and I'm imagining that these guys over here are really small like to the minus six or ten to minus ten and we're willing to make that sort of error so we throw them away okay so here's the Schmidt decomposition of our state and it goes all the way up to the full size of the system to the N over two and we say no I'm going to throw away all the guys to the right of my pointer and just keep them up there say there's m of them little m and so i'm gonna keep those and this is my approximation and m can be much less than M over two okay and this is just like that approximation of a matrix directly from a singular value decomposition where you only keep a certain other rows and columns exactly the same okay so then that basic truncation which cuts down this the it compresses the system exponentially you know two of the n over two goes down to something like a hundred or a thousand where n is maybe a hundred thousand per thousand it's also to the N is huge so this is the basic idea of DMR G so it uses the density matrix or the Schmidt decomposition finds the lambdas and uses that to truncate down the system okay but but this is what I told you so far isn't enough because in order to get the Schmidt decomposition we need the wave function but we're trying to get the wave function so it's like a chicken and the egg if you have the wave function you could compress it down to something small but you don't have it so how you compress it it's I like call this a chicken and the egg problem which came first the chicken or the egg anybody know the answer eggs dinosaurs head eggs and how did it out it how did the how did evolution solve the chicken and the egg problem it did it iteratively improved till we got a chicken egg a chicken and an egg it started with primitive features with crude eggs and it gradually evolved everything together in the self consistent way till it got to a chicken and an egg and that's what we do in DMR G it's the exact same thing we start off with something that's like wait how can you have one without the other we don't have the wave function so how can we get the Schmitt decomposition so what you do is you build it up together and self consistently so the the term our G comes from the historical background of DMR G so Ken Wilson had a numerical our G method that was used for the condo impurity and worked great for other impurity problems and so that numerical our G looked much more like a traditional RG but it was a numerical implementation and sodium RG fixes that by introducing the density matrix as a fix okay and back then I didn't know about quantum information quantum information you know was just starting and so there was sort of a duplication of ideas in DMR G and ideas of quantum information at the same time and then at some point they came together we we learned about each other and so now I'm describing you something that is not the way I could have described it twenty years ago I'm using the ideas of entanglement and quantum information because that's the way we think about it now but 20 years ago I would have said oh this this is an RG method and it's altered by the usual RG method to make it work by this density matrix and so it doesn't look quite as much like an RG method so because has drifted away from traditional RG methods you know the name is sort of old you still think about it that way and RG is still a part of this whole field but now we think of the best RG of this type is a different tensor network called the Mira that's a that's a bit long story okay yes oh well it needs a name that's catchy and Dia Margie worked and so you should still use that but now you know you have to think about what Nate if you find something you need to get a good name and some people are not very good at it and you know has to have a good sound and you know this worked so you know and it was the way it was the way the way I figured it out so but let's see matrix product States is the heart of DM RG and then the other part of the ERG is the way you iteratively optimize it and so you'd have to have a name that included like iterative optimization of matrix product states the other line shows methods something you know something horrible like that okay so so the solution here is involves doing the Smith decomposition not just on the middle but everywhere you can so you don't just lump the sites into left and right you give them an order like there are one-dimensional chain 1 through m and then you draw a dividing line between 1 & 2 between 2 & 3 3 & 4 etc and you do you think about doing the Schmidt decompositions on every dividing line each dividing line gives you a big compression of the wavefunction so you do them all and in the end you're left with something that is very small it's really highly compressed so if you just do it in the middle the compression you get is like the square root of the size you sort of cut the system in half but if you do it on every side it just completely kills off the exponential and gives you something much smaller so that's the first step okay and and so you know we have to work up a little bit to this and then the second step if you want to optimize so this gives us when we cut it at every possible place this gives us something called the matrix product state which I'll explain and then we optimize the matrix product state to minimize the energy and you always so it's like working with the wave function but always in compressed form and you it's like you have a zipped up file and you can access a little bit of it and unzip it and look at it and then zip it back up and move over and unzip the rest but if you unzip it all at once it'll be exponentially large so you can't do that so that's sort of the way it works okay so I'll get to that but first I have to talk about diagrams so the algebra of matrix product states gets a little bit messy and it gets even worse if you do generalizations for higher dimensions and so one of the things that some of the people who came from quantum information and started working on this you know they were used to quantum circuit diagrams and so they started doing circuit diagrams for DM RG and it made everything much easier and so we use these all the time ok so so how do these diagrams work ok so here's a wave function for 4 spins and the diagram for that is a box which represents the wave function and the external legs that come out are just each one has a spin attached to it and the total size of this thing each one has a factor of 2 so it's 2 to the 4 to the 4 numbers in this box and so this just is the representation for sigh of s1 through s4 ok so suppose we had something that only had 2 legs coming out so every external leg like this and there's some daughter box in the center this is a tensor with two indices well that's a matrix but we're not going to use any of the mathematical properties of tensors and coordinate systems we'll just call something with lots of indices a tensor and so here's a matrix a vector has one index coming out here's a three leg tensor which we'll use a lot of ok so those are some of our basic units okay so let's use this to write down a diagram for this simple matrix multiply so I've got a matrix a in a matrix B I say a times B is C okay so first I go to the summation convention say C IJ equals aik bkj so K is the internal link that gets summed over okay so here's the diagram for this so see the tensor just has these two external legs I and J and then a times B has this and there's a leg that's internal and it's just like Fineman diagrams for QED the internal legs get summed over or you know integrated over in diagrams in in the field theory but summed over here and so that's that's a matrix multiplying so the rule is you contract over all the interior indices and all the external ones are determining the answer that you have yeah yeah yeah or the you know there's there's different let's see I'm not sure the all the right terminology but there's something that just gives you you know the number of indices that's the number of legs yes right well I I that will fak that's like eating one cookie or one potato chip you can't just have one you can't just have five minutes on spooky action at a distance I love to lecture about that in my quantum mechanics class but we spend a you know week or so on it and so we can talk about it one-on-one but I don't want to try to get into that especially like right now okay so so that's that's the simple form of these diagrams but suppose you you say well what does the Smith decomposition look like okay so you take one matrix and you expand it out so it's like the reverse of this there's a diagonal matrix in here which can sort of be absorbed onto the left side or the right side so you can put a little dot in here or not it doesn't do very much but this is the Schmidt decomposition but if this index I has Israel goes over very small range this is a big improvement so suppose that L goes from one to a million and R goes from one to a million the whole matrix is 12 is is 10 to the 12 elements but if I do this Schmidt decomposition and if I goes from 1 to 10 I've reduced it down to a 10 million plus a ten million matrix so 10 to the 7 instead of the 10 to 12 so this is a big compression and so that's what that looks like so in general we want to take these complicated tensors and cut them into and insert to the spdm and assume that there's a small dimension that goes in okay okay so the matrix product state is when you do this SVD at every possible step so suppose we started with four sites and here's the wave function for four sites and I first step I cut it in two and I have this extra I to index the two just goes to this that's where I cut it and so this index is small so I've actually cut the size of the problem by taking its square root essentially okay and then I take each side on the line I take the left side and I cut that into with another singular value decomposition and I cut the right-hand side with another singular value decomposition and I've got something that's totally spread out and there's only a few legs on each tensor so this has four tensors and the edge ones only have two indices and the middle ones have three you know the middle the middle ones are like a line and then one coming down and that's a matrix product state and we usually just sort of quickly write it like a comb like this but every time there's a junction here that means there's a tensor living in that place and it's a three ranked three tensor except on the edges so that's a matrix product state so the storage so so this is an approximation to a wave function and it's an approximation that works great if there's low enough entanglement if there's high entanglement it totally fails but if there's low entanglement it'll work great so this storage if it works great you know you go from to the N down to about n tensors and each of them are M Squared they look like matrices here's what each one looks like the indices I will say our M so it's M by M plus another index that only goes over up and down so 2m squared so we have I left out the two but an M squared times two and so that's an exponential compression yeah so there's a truncation that came so you decide how much probability you're going to throw away maybe you throw away everything less than 10 the minus 8 you know that's a typical number for D MRG so a weight 10 demise 8 so then you're at your answer is only good to pay digits well that's fine is it what is it yes very well very well behaved we understand a lot about it you have to talk about we'd be a little bit more specific but it generally it's very well behaved when you do that you know there's details about how you do the SVD which I don't want to get into you know you sort of have to it's like sort of a putting in a gauge to the pieces as you go along to make it especially well behaved but it's basically just keeping everything orthonormal on the sides before you do a singular value decomposition but as long as you don't do that it works very well some of the other tensor Network methods should work really well in principle but they they have they're less controlled numerically than the MRG so they have some trouble that makes them a little bit difficult to work with one more detail about doing the SVD you know so when I did the first cut and I did an SVD I wanted to think of that as a matrix that I cut into but the left side had two indices so I wanted to treat the two indices as one and so that's I see this little three leg thing I call a combiner it's some call sometimes called something with fusion there's there's different names for it but it it puts two indices together and gives you a one index coming out but there's no loss there's no truncation it's like if you have two and two going in it just real Able's it as four one two four so it's like here's here's an index that you know if these two guys go into that it's like just relabeling them like this so it looks like one index and then it makes it into you call your SVD with this form so it just looks like a matrix so you combine all of the indices on the left into one and then all of them on the right into one and do an ordinary SVD so then DRG is an MPs wavefunction optimized by an iterative sweeping lanczos plus Schmitt decomposition algorithm so it's got all of these got these pieces that we were working on and but it keeps it in this compressed form and sweeps back and forth until it converges directly in the compressed form the matrix product form okay and I put up some references here so the I worked the original PR L is pretty short I worked really hard on the P Arby and so people were able to really do the algorithm based on the long PRB but more recently Lully show walk has written two nice reviews one was in Reb Maude fizz in 2005 but then subsequent to that the matrix product language really took off and so he has this second one that is a good place to start in annals of physics that just titled something like DMR G in the in the era of matrix product states Frank Pullman is coming in a week or so and he's also going to be talking about DMR G but some more advanced topics that I won't get to so he's an expert in DMR G and so is Bela Bauer he'll come maybe in the last week and so they'll be both be here later and I have to get to another conference tomorrow morning so you won't see me after tonight okay so the basic steps of DMR G okay so we have this matrix product state and we focus on two tensors to improve it works better to have to to improve rather than try to improve one at a time because they they talk to each other and it helps makes them go faster so two of them together look like this for index guy okay and alpha goes over this interior index but it's really going over a Schmitt State for a cut right there and beta goes over the commit states on the right and so these are both size m and then you have two sizes of two so it's about four M Squared object that you work with okay you do line shows on this guy so this guy is not in its ground state if you do line shows you can make it have lower energy you can replace the wave function here so you treat this little tensor thing as a wave function it's a wave function in a reduced basis you can treat it as just an ordinary wave function you can lower its energy with this exact diagonalization step with length shows okay so it turns into a sigh prime and it makes it look like a just one for index tensor instead of a product of two guys it's sort of erases the the product nature that it had but it lowers the energy sigh prime has a lower energy than sigh then you do a shim it decomposition to split it back up and so it turns this guy into that and you put this back into the NPS because it now it look just looks like two links again but it's it's better links okay then you shift over by one site and you repeat okay so you work your way from site 1 2 to 3 you go all the way the end then you reverse and go back all the time you're reducing the energy but keeping it in them you're only briefly for two sites out of the matrix product form then you go right back and then you then you keep going back and forth each back and forth is called a sweep and you repeat it however many times you have time for or until it converges or until you run out of computer time by the number of times you might go back and forth for a for a spin chain might be only three or four - even if converges really fast for a bigger challenging system you might do dozens of times but usually it's not so many so it's a it's a nicely converging procedure okay so what I want to switch to now is this is the end of the sort of what I would do is blackboard lecture if we had a lot of time but I'm gonna switch over to one of my standard you know talk that I gave in June in Brazil and now hopefully with this background you're sort of ready to understand you know DMR G as you would hear it in a you know regular talk and there'll be a number of interesting new things along the way you know it'll go a little bit faster now but you'll sort of see a lot more of what we can do okay so I'm sort of jumping forward skipping some historical stuff and starting with this slide that I love to show which is just talking about entanglement entropy and it comes from a 12-sided exact ionization just like you could do now or if you work a little bit harder you can do it in a day or two okay so here's I took that eight sidechain diagonalized it took a 12 sidechain diagonalized it I got every not just the ground state but I got every energy level and I did the Schmidt decomposition on each of those states okay so the Schmidt decomposition isn't a property of the ground state you can put any state into the Schmidt decomposition and calculate its entanglement energy entropy and we were just seeing how low the entanglement entropy is for the ground state but suppose you look at the other states so the first thing you see of course is that boy if I draw all these levels it just turns black the energy spacing between levels becomes exponentially small you know and if you really can't distinguish these states up here they're kind of weird pathological States but here's the entanglement entropy that you get for all of these states on the 12 site system and it's plotted versus energy so the ground state has the red circle on it and you can see how that has a low entanglement entropy and what does it look like it looks like it's below it's a little bit below point five so it's it's small but then this n over 2 log 2 we saw on the table that's the maximum possible that's up here and so the the ones in the middle the black area here those go up pretty close to the maximum you know there's a little bit that keeps from going to the maximum things like conservation of angular momentum they can't quite get up to the maximum but basically they get huge entanglement now that this system is Heisenberg chains if you if you ask you know well is this a strongly correlated system or not you know people would say oh yes it's as strongly correlated there's no way that you could think about treating it with just a simple mean field theory but in terms of entanglement it's very unentangled but it's not not near zero so this is sort of a strongly correlated ground state over here the maximum energy has zero entanglement can anybody tell me why what those to start states are with zero entanglement ferromagnetic now why aren't they degenerate I put in a little random field I didn't tell you about to split degeneracies so just a tiny little field just to make all the circles separate random field ok so that otherwise these should be exactly the same energy up and down ok but then all the guys in the middle are are huge ok so this is a fundamental difference DMR G does not work for states in the middle in an ordinary system the entanglement is not small it gets big there's nothing that works to get one state here now in fact in there is an exception to that which is if you have strong disorder you can have a many-body localized state you might hear about this later on I'm not sure but in this this school but there are certain types of states that everybody has low entanglement and that's because they have this little disorder field I just put a tiny thing you've got a huge thing on and it'll make everything low entangled and then there's a couple of papers that just have come out in the last week or so telling you how to do DMR G in that case even for high energy states and one of the papers has Frank Pullman who will be here in a week or so so you can hear about that's a little bit advanced but he might mention it ok ok so it's a little bit different to think about ok the states up here those are at high temperature well you don't have to isolate an individual eigenstate to do high temperature you just have to do combinations of them in the right way so we do have a number of ways of doing finite temperature that are quite nice okay I'm gonna show you just a few slides just so you see what you've just been hearing and it you know in sort of the usual way of talking about it there's a few extra details so why is the entanglement of ground state small because I give you KB the picture of the RvB but there's also a quantum information idea called monogamy of entanglement which basically says that if you really entangle two things together here neither one can be entangled with anybody else monogamy means you just stay married to one partner so you can't have polygamous marriages in quantum entanglement it doesn't work and so this is a sort of well-developed theory and that helps explain part of the area law it's like once this guy is bound to that guy you can't talk to anybody over here okay and the rest we talked a little bit about okay so here's the Schmidt decomposition the way I usually explain it in a brief fashion you know cutting the system into entanglement entropy just gone through that exploiting the low entanglement in the 1d case cutting the system into sort of inserting complete sets of or incomplete sets of schmitt states at each point to get the matrix product state okay and then this one has a little animation thing super crude it just shows how you move back and forth improving the wavefunction on two sites okay little more carefully drawn a tree oh so there's one thing that I should have mentioned the other way of writing instead of just jumping to the diagrams here's the other way of writing a matrix product state it so this a 1 a 2 a n you think of as a matrix with an extra label on it which is just like the Pauli spin matrices they're matrices that have an XYZ label on them so here's an extra label on them that is in the square brackets and it's just the value of the spin so you can think of it as like a pair of matrices and so the rule for getting the the wave function called the value for a particular s 1 s 2 s n you know if we have the rule for a function is that ok you give me particular values of s 1 s 2 etc say all up then I can tell you the number okay so the way that rule works is you give me these numbers the square brackets will pick one particular matrix and you multiply them all together and the first and last or a vector and it collapses to one number but what that number it is depends on the S is because there's really two guys at each place and you had to pick which one and that's what gives you the that's how the compression works it just has these matrices but it can give you every possible value and expanding it out okay okay so how well does this work so this is a two thousand site spin 1/2 Heisenberg chain didn't have any extra disorder now for the Heisenberg chains there's an exact solution due to Hans bethe back in 1931 or so and it also works for finite chains or infinite chains but here you use it for finding change you have to write a little program to evaluate the answer but you can know the exact energy and a few excited states so the left hand picture is showing the energy that's an e it's the total energy of the system is a thousand sites so it's a big number and the M is how many schmitt states I was keeping and there was sweeping back and forth and the index I is where you are in the sweep now it's two thousand sites this only goes to the center and then it uses reflection symmetry to replace the right hand side with the left and it sort of didn't turn around so a little bit of efficiency okay so you do M equals 10 then you do you sort of get that converged then you increased M equals 15 the energy goes down change it to M equals 20 and you the way you typically do this is you slowly increase the size of the matrices which is the same as the numbers unit states you slowly increase them until you get whatever conversions you want and so then the Stars here are the exact ground state and you know a whole bunch of sweeps have just sort of sat there at the scale then the right-hand side is showing the deviation in the total energy from the exact beta on thoughts and answer and so as you increase the size of the matrices its falling this exponential scale and it's the total energy it's not per site so this is a really accurate wave function and it's got some curvature here but it sort of straightens out once you're below the first excited state this is a gapless system a critical system and so this would be a system that is hard to do because gapless systems have long-range correlations and so they're considered hard but because it's not strictly infinite it's only 2,000 sites there's a very tiny finite size energy gap and as soon as you iterate so that your energy is below that first date it starts looking like a gap system and then DM RG tends to converge exponentially well for gap systems and so it sort of goes to a straight line and you know of course we could push this to more accuracy how big an M can we do want to you know on a desktop you know maybe M equals 5,000 something like that so you can make this essentially exact to all digits that you might have on the double precision so for 1d systems DRG sort of is that you know get you what you want very accurately sort of the best method around we've learned how to get other properties that are of interest in experiment so finite temperature spectral functions dynamics out of equilibrium dynamics things with disorder some things are harder than a lot harder than this but we know how to do a pretty good job on lots of different things okay so here's my usual introduction to diagrams for matrix product states you've seen that here's here's one where if you connect them up it actually gives you a trace and then they're all matrices matrix products state matrix product States as variational states I didn't talk much about calculating observables okay so you have this comb state this matrix row state then you want to calculate a property the way you measure something is there's two wave functions there's a sigh on the right and the sigh on the left and you have to do a a contraction so the way it looks in the diagrams is you put the first the guy on the right on the bottom and the guy on the left on the top and you contract them together and contracting is summing over all the values of the spins so that just links up all of the lines but and if you just did all all of them linked up with nothing in between it would just be the normalization and you'd get one but if you want to measure something you insert a spin operator at that particular place if you want to measure a product of two things you insert two of them and then when you contract over everything it gives you a number and that's just that particular property so with this you can calculate correlation functions and lots of other things it doesn't matter how many pieces you put in here but it has to be sort of you know in this simple way it should just be one term okay so there's often diagrams that look sort of like this with things contracted top and bottom and then other things in between there's another let me see if I've got that in a slide I guess I don't have it so let me just mention it you can also put the Hamiltonian in between here and you can write the Hamiltonian as something that looks like a matrix product state is called a matrix product operator and you'll see that a little bit in this afternoon in the I tensor the the I tensor library because that's the usual way the Hamiltonian is handled with I tensor so a matrix product operator let me just draw it in the diagram so here's the usual comb for a matrix product state the basic unit looks like that and this is an M PS and a matrix product operator and MPO looks like this okay so this is like s1 s2 we end it s 1 s 1 prime s 2 s 2 prime ok so this this general if I if I make this into a box this can represent any operator on those spins okay but it turns out you can compress the usual Hamiltonian operators to make them look like this they have the same sort of structure of tensors as up here they typically have pretty small bond dimensions pretty small m like 5 and so you can write the whole Hamiltonian in this form and the the fact that it looks like this guy means it just fits into the algorithm really easily so you can write an algorithm that just assumes that you've figured out the Hamiltonian like this and then you do all the all the programming with this totally arbitrary and then you only have the Hamiltonian the system you're doing coming into one part of the calculation everything else kind of looks the same ok so M POS are one of the big advances the last 10 years or so in terms of organizing DMR G programs and making them easier okay okay I want a skip matrix product basis that connects a little bit to the old RG way of thinking about things okay and this is what I talked about comparing with the old way of thinking about RG in the new way so I want to skip this also okay so let me talk a little bit about doing DM r g4 TV systems okay so this will just be sort of an overview show a few things of interest okay so the algorithm that I presented was all for a one-dimensional chain and it really used that because it kept splitting up the system on each bond okay so suppose we wanted to do DMR G for 2d systems well if the 2d system is infinite you're out of luck but if if if you want to do a strip and then you can do wider strips and see what the trend is has an approximation to an infinite 2d you can do this sort of scheme this is sometimes called a snake but you can take here's this 2d grid it's only five wide here but I make it into 1d just by connecting the sites up with the blue lines okay and it then looks like a 1d system but it has longer range bonds so for instance the first site is connected very strongly to site number six on the path okay that's that's something that you don't usually get with a to 1d system okay all of these connections take you sort of deep into the other system and it messes up the area law the 1d area law it's saying oh yeah I got something inside the system here talking to a guy inside the other system now it doesn't go infinitely far across it sort of only goes five across so it messes it up a little bit but you have to think about this as a 2d system to find out how well DMR G is gonna work and so you do a cut like this and say okay how big is the area well the number of sites on the boundary is five so the entanglement should be bigger by a factor of five than in 1d well a factor of five doesn't sound too bad run it five times as long except it's an exponential it's a fifth power so it actually gets much harder so the the the number of states that you have to keep em will go as some exponential of the width of the system so at first people some people thought that you wouldn't be able to do DMR G at all for 2d systems but it turns out that this exponential is you know effectively the the a coefficient is fairly small so you can do a reasonably big system without without running out of numerical power you they become big calculations but if you keep say up to M equals 10,000 then you can do a system of you know with 10 or 14 for these spin systems which is really pretty big and it could be a cylinder so it's connected with periodic boundary conditions that helps reduce the finite size effects and so that can be enough to get really good answers for some 2d systems it depends exactly on how much is going on on one site so the simplest case is if there's spin 1/2 and so there's only 2 possibilities and so the area law coefficient is especially small if you had a Hubbard model there's four states on each site and it's like twice as hard you can only do half this wider system yes sorry I could sew there I'm assuming that the 2d system has nearest neighbor connections and then I put a number you know that's number one go down to five oh so I guess the way this is numbered it goes up to ten so there is a nearest neighbor connection that connects actually sites one and ten and that would be as long as it gets in this arrangement I could have made it go down and then jump back up to the top and go down again and then it would only be five away or six away so it see it's not so far right so a 2d finite strip is the same as a 1d system with long range interactions okay but the entanglement is that of the 2d system it is the same we use the simple 2d system to estimate the entanglement we look at this picture it is equivalent they're always all equivalent to a 1d system with long enough bonds so part of the D MRG algorithm doesn't care that it's 2d you write the same code but then you find that it doesn't converge very well with only ten states or 100 States you have to crank it up to a lot more states to get good accuracy so it sort of knows that it's really two-dimensional well if each dot so so the little guys sticking out here that those are the degrees of freedom of the site and they're not labeled here you know I'm thinking about them as up and down so each little angled thing is up or down if it was Hubbard it would be empty up electron down electron or doubly occupied so there'd be four possibilities okay so there'd be it'd be the same picture but there would be more entanglement because of those extra degrees of freedom it's got spin fluctuations and charge fluctuations or twice as many yeah well I haven't drawn all the bonds the horizontal bonds that are in there yeah yeah that's I guess that's uh that's unclear so it's a 2d system so all those horizontal bonds are really there in the Hamiltonian but they look like long-range bonds in the matrix product state snake okay so I've talked about DMR G since people from quantum information came along particularly and he mentioned some names defray we'd all found sort of the so one of the first key connections between quantum information and and DMR G he sort of reinvented part of DMR G without knowing about it and then Ignacio serac and Frank Estrada were two of the other key people that start in quantum information and now our I have done a lot of major work in this improving DM RG and so to do two dimensions here's this thing that I just showed you but there's if as soon as you start drawing the diagrams you can draw a better tensor network like this one that really directly represents 2d so a basic tensor here has five legs not three as the one coming out which has up or down and then it has north-south east-west the four directions and it doesn't have a snake path it directly represents two so there's a tensor leg that connects every nearest neighbor well that matches what the Hamiltonian has it's a much more natural representation it allows much better compression of the wavefunction okay so the MRG the energy is equivalent to this in 1d you know there's there's three legs in each in in these strips think that entropy starts blowing up with the width and so your M has to blow up in paps the M doesn't have to change very much and so you might have M equals 10 and get a pretty good answer and so yeah it's more difficult because it has more legs so that's still a lot of coefficients but it's a much better compression so this has been a very promising algorithm since you know the early 2000s about ten ten or fifteen years ago when it was proposed however it's much more difficult to work with numerically and in particular you know we talked about estimating the calculation time of an algorithm you know diagonals a dial izing a matrix is M cubed well you know indium RG that M cube is the biggest piece so the calculation time indium RG is M cubed times a few factors involving the size of the system okay in this case the calculation time is much harder to work with this and you get up to M to the twelfth it's it's not exponential and you know exponential is the worst except that depending on the coefficient m to the twelfth is worse than an exponential for quite a while and so this is the key difficulty of working with peps and so it took quite a while to for this to turn into something that's really useful now it is but there's still just a couple of group or well there's mostly politte Corbeau who has the nicest work with this sort of peps and and it's just very difficult calculations there's a number of groups now that do two dimensions with DMR G because it's it's almost like the same sort of coding as in 1d so these two sorts of methods are they're both their cousins you know they're both based on this sort of matrix product state or the generalization the the whole field is tensor networks and so they're close cousins just adapted for different things one of the things that you can do with this is you can actually make the lattice go out to infinity and the way you do that is you just say well let me make every tensor the same and then just assert it goes out to infinity and then the question is well can you work with that and actually evaluate properties and it turns out you can and so it's amazing that you can actually sort of do numerical works not with a finite size effect but it's strictly infinite and a lot that's what Philippe's work does it goes out to infinity and so that's that's a neat thing but you still have this M you know the the biggest M that I've seen Phillipe have is a maybe 14 or so and so it's sort of competitive with the best DM RG work okay so I am going to save this for later and we can take any more questions and otherwise we can break for lunch yes well you you have a different slightly different peps for different lattices it for instance for the Kagame lattice which has a interesting structure it took a little while to figure out that putting a tensor on every site wasn't the best thing to do and so the Kagame lattice has little triangles that touch on corners and they put a tensor in the center of each triangle and then that works better a lot better and so each thing you have to figure out a little bit the bet is still a pretty general purpose method but these subtle little differences that have to be worked out more questions yes so we always use the commuting symmetries so if it's a spin system we keep track of s of Z if it's a fermion system like the Hubbard model we would keep track of two numbers the number of particles and the total s of Z other groups have put in full su 2 symmetry which means that you can work with a smaller m and it's like you're working with a big bigger m by maybe a factor of 10 or something so that's a that's a nice thing to do when you have that symmetry it's also a much more complicated program and you know and many systems don't have the symmetries that would make it work so we haven't our group doesn't do it but other groups are have you know some systems they do better than us because they put in the full su 2 symmetry and there are other other symmetries that you can put in it's difficult to put in the spatial symmetry that you might want to put in because as you step through the lattice you're breaking it up you're killing the symmetry and so spatial symmetries are much harder but the sort of local symmetries are you sort of do as much as you can this convenient okay so we we gather again at 2:00 in the computer labs you

Channel: ICTP Condensed Matter and Statistical Physics

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Keywords: ICTP, Abdus Salam International Centre for Theoretical Physics, Materials Science, Chemical Physics, Physical Chemistry, Energy and Sustainability, High Performance Computing

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Length: 86min 20sec (5180 seconds)

Published: Thu Nov 30 2017