Topological order of a two-dimensional toric code

Transcription

1 University of Ljubljana Faculty of Mathematics and Physics Seminar I a, 1st year, 2nd cycle Topological order of a two-dimensional toric code Author: Lenart Zadnik Advisor: Doc. Dr. Marko Žnidarič June 9, 2014

2 Abstract In this seminar, a two-dimensional toric code is presented along with some of its more important properties, such as topologically protected degeneracy and nature of its excitations. These properties are not of interest solely due to their topological nature, but also due to their possible applications, for example in quantum computation. Firstly, a special spin system will be introduced along with a systematic algebraic construction of its ground state, the toric code. Then the ground state s robust nature will be thoroughly explained. From here a short description of its excitations will follow.

3 CONTENTS Topological order of a 2D toric code Contents 1 Introduction 2 2 The square lattice model 2 3 Ground state - the toric code 4 4 Excitations and anyons 8 5 Conclusion 9 1

4 2 THE SQUARE LATTICE MODEL Topological order of a 2D toric code 1 Introduction Different phases of matter can be distinguished with regard to their internal structure, i.e. order. Until 1980-s, it was commonly thought (Landau theory), that these orders could be characterized solely by their symmetry properties, e.g., continuous translational symmetry of a gas vs. discrete one of a crystal. However, in 1982, Horst Störmer and Daniel Tsui found a new quantum state of a two-dimensional electron gas, by cooling it down to very low temperatures and subjecting it to strong magnetic fields.[1] It was named the fractional quantum Hall state and soon it turned out, that it could not be described using only its symmetry properties. A new approach was needed instead of until then commonly used Landau theory, for the fractional quantum Hall state exhibited a new kind of internal structure. Physicists soon discovered, that this new order of matter could be described by means of topological properties of a system s phase space, i.e. properties common to phase space itself as well as its homeomorphic 1 deformations, and so it was named the topological order. The introduction of a global order parameter (as opposed to the local one in Landau theory) was necessary, in order to successfully describe systems exhibiting this new kind of order. Certain physical systems manifesting such topologically induced order could be applied in quantum computation because of their robust nature and manipulation related properties. We shall, in this seminar, analytically discuss one such model, the two-dimensional toric code and show some of its topology - related properties, without explicitly mentioning the corresponding global order parameters. This analytically solvable problem is hard to implement experimentally due to the strange nature of interaction included in it, the four-point spin interaction. However, more natural models that are easier to implement exist and turn out to be, at least in certain regimes, isomorphic to here discussed model, therefore one can use same physical methods to analyse them although their structure is much more complicated. We will show, that the toric code corresponds to a two qubit system and can therefore be used to store information. Even more: this information is protected from localized perturbations, due to the topological order exhibited by the system s ground state. Creation of excitations however, allows one to manipulate the stored information. 2 The square lattice model Let us define a spin system on a square N N two-dimensional lattice, by assigning a qubit state to each of the links connecting the neighbouring lattice points. Such a system is shown in Figure 2. Furthermore, let us apply periodic boundary conditions on this system, so that the problem can be thought of as a problem of 2N 2 qubit states on a 2-torus. One can for example imagine bending the lattice into a hollow cylinder, by joining two of its opposite edges and then bending the cylinder into a torus by joining its openings, corresponding to the remaining two edges of a lattice. Note that since this correspondence of a square lattice with periodic boundary conditions to a torus is homeomorphic, topological properties are retained. Procedure 1 Continuous transformations with continuous inverse. 2

5 2 THE SQUARE LATTICE MODEL Topological order of a 2D toric code is shown schematically in Figure 1. Figure 1: Folding of a plane into a cylinder and then into a torus, as a consequence of periodic boundary conditions. These are imposed on edges represented with blue and red paths. v p Figure 2: Qubit - linked lattice. Links with dots representing qubits are shown, as well as star in blue and plaquette (boundary) in red. We shall refer to intercalated points (centres of squares, labeled by p in Fig. 2) as pseudolattice points, as opposed to lattice points, labeled by v. The Hamiltonian which will be considered is H = v A v p B p, (1.1) where v and p run through all the lattice and pseudo-lattice points respectively and A v -s and B p -s are defined as follows: A v = i star(v) σ x i, B p = j boun(p) σ z j. (1.2) 3

6 3 GROUND STATE - THE TORIC CODE Topological order of a 2D toric code 2 1 v 1 p Figure 3: star(v) = {1, 2, 3, 4} Figure 4: boun(p) = {1, 2, 3, 4} Here we have defined a star(v) as a set of all links, originating at lattice point v, and boundary(p) = boun(p) as a set of links, surrounding pseudo-lattice point p. Both are shown in Fig. 2. σ α i with α {x, z} represent the standard Pauli spin operators. 2 Due to the periodic boundary conditions, operators A v and B p must satisfy: A v = p v B p = id, (1.3) where products run through all lattice (pseudo-lattice) points. This is a consequence of each link and thus each Pauli operator being accounted for twice in the product (1.3). It is a well known algebraic property of spin systems, that Pauli operators of the same spatial component commute, while those of different components anticommute on the same link and commute on different links. From these relations it follows, that all the operators A v, B p commute for all possible lattice and pseudo-lattice points. The commutation relations [A v, A v ] = [B p, B p ] = 0 are obvious, but commutation of boundary (B p ) and star (A v ) operators needs some explaining. We can see that such operators can have either 0 or 2 links in common. Therefore we have either commutation or 2 anticommutations, the latter ones again giving a commutation. This can be seen from [σ x i σ x j, σ z i σ z j ] = σ x i σ x j σ z i σ z j σ z i σ z j σ x i σ x j, where i and j are shared links. Since A v, B p and H commute, there is a common set of eigenstates of all the operators of these types. 3 Ground state - the toric code Since A v and B p are defined as products of Pauli spin operators, it turns out they can only have eigenvalues of ±1. It can be seen from (1.1), that a ground state ζ must be an eigenstate of all the star and boundary operators, belonging to the eigenvalue Therefore we define a subspace L = { ζ A v ζ = ζ, B p ζ = ζ, v, p}, which we shall henceforth refer to as a ground state manifold, or shortly GSM. It represents the toric code. From (1.1) it follows, that energy of arbitrary element of GSM is E 0 = 4N 2. Although we will discuss excitations in next section, we should on this point mention, that the first excited state can only be achieved 2 For example we have B p = σ z 1σ z 2σ z 3σ z 4, where 1, 2, 3, 4 are links shown in red in Fig The ground state is the one with lowest energy. 4

7 3 GROUND STATE - THE TORIC CODE Topological order of a 2D toric code (due to the constraint (1.3)), by changing eigenvalues of arbitrary pair of either A v -s or B p -s from 1 to 1. Thus we achieve a minimal energy change, δe min = 4. We shall now show, in a non-formal way, two important properties of a ground state manifold, that imply the usefulness of a toric code. These two properties are: 1. The ground state manifold is four-times degenerate. 2. The degeneracy of a ground state manifold is topologically protected. In order to show the first point, we need to define string operators Z C = j C σ z j, (2.1) X C = j C σ x j. (2.2) Here j C means that j runs through all links included in path C between lattice points, while j C means that j runs through all links crossed by path C, between quasi lattice points. Paths are shown in Figure 5. C C Figure 5: String operators on paths between lattice and quasi lattice points. Let the paths C and C be closed loops. If there exist a homotopy of a loop into a point, we say it is contractible, i.e. it can be shrunk into a point without leaving the surface of a torus. Otherwise we refer to a loop as a non-contractible. For a contractible loop, one can write Z C = B p (2.3) X C = p inside C v inside C A v, (2.4) 5

8 3 GROUND STATE - THE TORIC CODE Topological order of a 2D toric code due to contributing factors from links inside the loop canceling out (each Pauli operator from inside the loop is accounted for twice). 4 But in the ground state, all A v -s and B p -s have eigenvalue 1 and therefore (2.3) and (2.4) imply Z C = X C = id, on the GSM. The only nontrivial closed operator strings are those non-contractible. There are four: X 1, Z 1, X 2, Z 2, where paths 1 (horizontal) and 2 (vertical) are shown in Figure 6. They share either 0 or 2 links with operators A v and B p and thus commute with them. It can be shown that in the ground state one can translate these four operators arbitrarily, without changing their eigenvalues. Figure 6: Non-trivial strings: labeled as 1 (red) and 2 (blue). Due to the translational invariance of non-contractible loops, we have [X 1, Z 1 ] = [X 2, Z 2 ] = 0 (paralel strings can be translated so that they possess no common links). On the other hand, we have {X 1, Z 2 } = 0 (perpendicular paths share one link, that contributes anticommutation). Hence there exist a bijective correspondence X 1 σ x 2, X 2 σ x 1, Z 1 σ z 1, Z 2 σ z 2 (1, 2 have nothing to do with links between lattice points). But this means that non-contractible loop operators span algebra of Pauli matrices on a tensor product of two newly introduced qubit spaces, H 1 and H 2, which is four-dimensional. Thus we have shown, that L corresponds to a four-dimensional space H 1 H 2 and therefore diml = 4, in other words: ground state is 4-fold degenerate. 5 In order to show the second point (the topological protection), we have to consider a local perturbation operator e.g., an additive term in Hamiltonian, given by Ω = σj α σ β k σγ l... (2.5) Note that since our model is essentially a spin system, one can assume that arbitrary perturbation can be written as a sum of terms of the form (2.5). Here links j, k, l are localized (close together in comparison with lattice width). Because non-contractible loops can be translated arbitrarily, this perturbation commutes with X 1, X 2, Z 1, Z 2. Mathematically adept reader might note that, by restriction and projection onto a GSM 6, this operator becomes a map 4 Here inside the loop means laying inside a disk, bounded by a loop. 5 The toric code is referred to as a two- dimensional since it is a result of a two-dimensional model. 6 Operators X 1, X 2, Z 1, Z 2 must also be projected and restricted onto a GSM. These projections and restrictions however, are just the Pauli operators on two qubit subspaces. 6

9 3 GROUND STATE - THE TORIC CODE Topological order of a 2D toric code P (Ω L ) : L L, which is Pauli group - equivariant (here P is a projector onto a GSM, L a restriction onto a GSM and ({±I, ±ii, ±iσ x,y,z }, ) the Pauli group). Thus by Schur s lemma it is a homothety, i.e. a scalar multiple of identity. This means that one cannot move, by means of this perturbation, from one state in L (GSM) to another, without leaving this manifold, i.e. one cannot manipulate ground states without exciting them. If we label the ground states with i, j, this can be written as i P (Ω L ) j = λδ ij, (2.6) If this is used in perturbation theory, it results in corrections to the energy, that fall exponentially with N, the width of the lattice. Therefore this corrections to the ground state energy vanish in thermodynamic limit and degeneracy of GSM is thus topologically protected. Even more: if perturbation is small, compared to the energy gap between the ground state and the first excited state, this also implies the protection of ground state itself. Such a system could prove very useful in quantum computation. Should one somehow store information into this manifold, this information would be well protected from the effects of local perturbations. Altohugh this particular toric code model cannot be applied to a real situation as of today, due to the interaction of four qubit states which is very rare, quasiparticle statistics corresponding to the excitations (which are discussed in the next section) of this system has, in certain experiments, already been observed.[2] There also exists a similar model, the Kitaev s honeycomb model (Figure 7), where excitations obey different statistics, that might have also been experimentally observed.[2] The Hamiltonian of this model involves a nearest-neighbour two-spin interaction, with spin states applied to vortices of a planar honeycomb lattice. Due to the two-point interaction, this model is more likely to be implemented. Its analysis is much more complicated and will be avoided here, but let us nontheless state, that Kitaev s honeycomb model provides us with only a partial topological protection in parameter space and is, in certain regime, isomorphic to our model. Figure 7: Kitaev s honeycomb model. Spin operators are labelled with σ α, α {x, y, z}. Hamilton s function includes products of nearest-neighbour Pauli spin operators. Qubits are assigned to vortices. This is schematically shown in this figure.[3] 7

10 4 EXCITATIONS AND ANYONS Topological order of a 2D toric code 4 Excitations and anyons It has been shown that operators on non-contractible closed loops are the only non-trivial operators with their action restricted solely on the GSM. If, however, one removes this restriction and therefore allows excitations to happen, it is no longer so. If we relabel string-operators (2.1), (2.2) as S z (C) = j C σ z j (3.1) S x (C ) = j C σ x j, (3.2) where strings C and C are not closed and are shown in Figure 5, we can see that as (3.1) anticommutes with A v -s in endpoints of a path C and similarly (3.2) with B p -s in endpoints of a path C, these two strings create a minimal excitation from the ground state. That the excitation is minimal, i.e. δe = 4, follows from the fact that each endpoint of a string contributes an energy change of ɛ = 2 to a system. The endpoints of this strings can be thought of as quasiparticles. We refer to them as abelian anyons. Their nature is a manifestation of Pauli commutation relations on a two-dimensional toric code. We will show that they obey neither the fermionic nor bosonic statistics. Hence they are named anyons. Figure 8 shows operator strings S z (R) connecting lattice points and S x (G), S x (B) connecting quasi lattice points. The operator S x (B) is equal to an identity when acting on a ground state manifold, due to being on a contractible loop, as we have shown in the preceding section. R B G Figure 8: String operators corresponding to excitations. Now let us excite ground state ζ first with S x (G) and S z (R), then again with S x (B), and define corresponding excited states as in = S z (R)S x (G) ζ, (3.3) f = S x (B) in. (3.4) 8

11 5 CONCLUSION Topological order of a 2D toric code (3.4) can be rewritten as follows: f = S x (B) in = S x (B)S z (R)S x (G) ζ. (3.5) Since paths B and G have no links in common they commute, while B and R anticommute due to sharing one link. So we arrive at f = S z (R)S x (G)S x (B) ζ = S z (R)S x (G) ζ = in, (3.6) where we have used S x (B) ζ = ζ, along with commutation relations of operator strings in equation (3.5). This process has an interesting interpretation. If endpoints are regarded as quasiparticles, then operation S x (B) corresponds to winding the quasiparticle corresponding to the endpoint of the path G, around the quasiparticle at the endpoint of the path R. By means of such a process we get a phase factor of 1 = exp(iπ). Since this winding process is topologically equivalent to the process of interchanging the involved quasiparticles twice, interchange of these anyons contributes a phase factor i = exp(i π ). Since interchange of abelian 2 anyons manifests in a phase factor i, these quasiparticles are neither fermions nor bosons, where phase factor would be ±1. As we have mentioned in the conclusion of the preceding section, the abelian anyon statistics might have been observed experimentally, as well as the statistics of their counterparts, the non-abelian anyons, which show up in the analysis of Kitaev s honeycomb model. 5 Conclusion We have analysed a special qubit system, defined on a two-dimensional square lattice with periodic boundary conditions and saw, that the ground state of such a system is robust under local perturbations, its degeneracy remaining unaltered in thermodynamic limit. The ground state was 4-fold degenerate and corresponded to a two-qubit system. If experimentally implemented, such a system could provide us with means of a strong preservation of information, i.e. quantum memory. Due to the four-point interaction (the Hamiltonian includes products of four spin operators), this Hamiltonian has unnatural form and thus such a model is hard, if not impossible, to realize. The Kitaev s honeycomb model which turns out to have similar physics, but is nontheless more analytically complicated, could be easier to implement experimentally [4], but would only guarantee partial protection of information. We have seen, that quasiparticles corresponding to the excitations of the toric code, the so-called abelian anyons, obey a special statistics that is neither fermionic nor bosonic. Certain experiments (see [2]) hint the existence of both types of anyons (abelian and non-abelian, the latter ones showing up in Kitaev s honeycomb model), although there has been no final confirmation as of today. For conclusion we should also mention without proof and explanation, that due to the high tolerance of operator strings to small local translations, the toric code could also be used as a fault-tolerant computation method with unitary transformations and measurements performed through manipulation of abelian anyons.[5] 9

12 LITERATURE AND REFERENCES Topological order of a 2D toric code Literature and references [1] Xiao-Gang, W. An introduction of topological orders. Available at wen/toparts3.pdf [2] Sanghun An, Jiang, P., Choi, H., Kang, W., Simon, S.H., Pfeiffer, L. N., West, K. W., Baldwin, K. W. Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect. arxiv: v1 [cond-mat.mes-hall] [3] Burnell, F. J., Nayak, C. (2011). Slave fermions in the Kitaev honeycomb model, 10. Available at (April 1, 2014) [4] Leggett, A. J. (2013). Lecture 26, The Kitaev models, PHYS598PTD. University of Illinois, Department of physics. [5] Kitaev, A., Laumann, C. (2008). Lectures, Topological phases and quantum computation. arxiv: [cond-mat.mes-hall] [6] Kitaev, A. (2003). Fault-tolerant quantum computation by anyons. Ann., Phys., 303, arxiv:quant-ph/