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A short proof of stability of topological order under local perturbations 0 Sergey Bravyi∗, and Matthew B. Hastings†, 1 0 January 25, 2010 2 n a J 5 2 Abstract ] Recently, the stability of certain topological phases of matter under weak perturbations was h proven. Here, we present a short, alternate proof of the same result. We consider models of p - topological quantum order for which the unperturbed Hamiltonian H0 can be written as a sum h of local pairwise commuting projectors on a D-dimensional lattice. We consider a perturbed t a Hamiltonian H = H +V involving a generic perturbation V that can be written as a sum of m 0 short-range bounded-norm interactions. We prove that if the strength of V is below a constant [ thresholdvaluethenH haswell-definedspectralbandsoriginatingfromthelow-lyingeigenvalues 1 of H . These bands are separated from the rest of the spectrum and from each other by a 0 v constant gap. The width of the band originating from the smallest eigenvalue of H decays 3 0 6 faster than any power of the lattice size. 3 4 1. 1 Introduction 0 0 1 Quantum spin Hamiltonians exhibiting topological quantum order (TQO) have a remarkable : propertythattheirgroundstatedegeneracycannotbeliftedbygenericlocalperturbations[1,2]. v i In addition, the spectral gap above the ground state does not close in a presence of such per- X turbations. This property is in sharp contrast with the behavior of classical spin Hamiltonians r a such as the 2D Ising model for which the ground state degeneracy is unstable in a presence of the external magnetic field. Recently the authors of [3] presented a rigorous proof of the gap stability for a large class of Hamiltonians describing TQO. The proof was direct, but very long. This paper presents a short technical note giving an alternate proof of the same result. We feel that both proofs are worth having, since they are complementary. In particular, the present proof is much shorter, but less direct. Since we consider the same models and prove the same result, many of the definitions in this paper are identical. ∗IBM Watson Research Center, Yorktown Heights NY 10594 (USA); [email protected] †Microsoft Research Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106 (USA); ma- [email protected] 1 The interest in proving stability of a spectral gap under perturbations has increased due to recent progress in mathematical physics, where a combination of Lieb-Robinson bounds [4, 5, 6] and the method of quasi-adiabatic continuation [7, 8] with appropriately chosen filter functionsnow providespowerfultechniques forstudyingthepropertiesofgappedlocalquantum Hamiltonians. Perhaps surprisingly, we will use similar techniques to prove the existence of a gap. In contrast, to the best of our knowledge, all previous works addressing the gap stability problem relied on perturbative methods such as the cluster expansion for the thermal Gibbs state [9, 10, 11], Kirkwood-Thomas expansion [12, 13], or the coupled cluster method [14, 15]. Our techniques are particularly well suited to handle topologically ordered ground states, although they can also be applied to topological trivial states such as ground states of classical Hamiltonians (in the classical case stability is possible only for non-degenerate ground states). Further, as shown in [3, 7], the gap stability implies that many of the topological properties of the unperturbed system carry over to the perturbed system. We consider a system composed of finite-dimensional quantum particles (qudits) occupying sites of a D-dimensional lattice Λ of linear size L. Suppose the unperturbed Hamiltonian H 0 can be written as a sum of geometrically local pairwise commuting projectors, H = Q , 0 A X A⊆Λ such that its ground subspace P is annihilated by every projector, Q P = 0. We impose two A extra conditions on H and the ground subspace P which are responsible for the topological 0 order. In [3], it was shown that neither of these conditions by itself is sufficient for the gap stability. Let us first state these conditions informally (see Section 2 for formal definitions): TQO-1: The ground subspace P is a quantum code with a macroscopic distance, TQO-2: Local ground subspaces are consistent with the global one. We consider a perturbation V that can be written as a sum of local interactions V = V , r,A X X r≥1 A∈S(r) where S(r) is a set of cubes of linear size r and V is an operator acting on sites of A. We r,A assume that the magnitude of interactions decays exponentially for large r, max kV k ≤ Je−µr, r,A A∈S(r) where J,µ > 0 are some constants independent of L. Our main result is the following theorem. Theorem 1. There exist constants J ,c > 0 depending only on µ and the spatial dimension 0 1 D such that for all J ≤ J the spectrum of H +V is contained (up to an overall energy shift) 0 0 in the union of intervals I , where k runs over the spectrum of H and I is the closed k≥0 k 0 k interval S I = [k(1−c J)−δ,k(1+c J)+δ], k 1 1 for some δ bounded by J times a quantity decaying faster than any power of L. 2 Using the quasi-adiabatic continuation operators in [23], this superpolynomial bound on δ can be improved to a bound by an exponential of a polynomial of L, where the polynomial may be taken to be arbitrarily close to linear. Thestability proofpresentedin[3]involved twomainsteps: (i)provingstability foraspecial classofblock-diagonalperturbationsV composedoflocalinteractionsV preservingtheground r,A subspace of H , and (ii) reducing generic local perturbations to block-diagonal perturbations. 0 The most technical part of the proof was step (ii) which required complicated convergence analysis for Hamiltonian flow equations. In the present paper we show how to simplify step (ii) significantly using an exact quasi- adiabatic continuation technique [7, 8]. In Section 3 we define two types of block-diagonal perturbations: the ones that preserve the ground subspace of H locally ([V ,P] = 0) and 0 r,A globally ([V,P] = 0). We prove stability under locally block-diagonal perturbations in Section 5 which mostly follows [3] and uses the technique of relatively bounded operators. The reduction fromglobally block-diagonal to locally block-diagonal perturbationsis intwo steps; in Lemma7 we exploit the idea of [16, 17] to write a Hamiltonian of a gapped system as a sum of terms such that the ground states are (approximate) eigenvectors of each term separately, and then we show that such terms can be written as locally block-diagonal perturbations in Section 4. Finally, we usean exact quasi-adiabatic continuation technique presented in Section 6 to reduce generic local perturbations to globally block-diagonal perturbations in Section 7. In contrast to [3] we use certain self-consistent assumptions on the spectral gap of the perturbed Hamiltonian H = H +sV. In Section 7 we prove that if the minimal gap ∆ s 0 min along the path 0 ≤ s≤ 1 is bounded by some constant (say 1/2) then, in fact, ∆ is bounded min by a larger constant (say 3/4). We use continuity arguments to translate this result into an unconditional constant bound on ∆ , see Section 7. The continuity arguments we use rely on min the fact that the spectrum of H +sV is a continuous function of s for any finite sized system; 0 however, our theorem gives bounds that are independent of system size. 2 Hamiltonians describing TQO Tosimplifynotation weshallrestrictourselves tothespatialdimensionD = 2. Ageneralization to an arbitrary D is straightforward. Let Λ = Z ×Z be a two-dimensional square lattice of L L linear size L with periodic boundary conditions. We assume that every site u ∈ Λ is occupied by a finite-dimensional quantum particle (qudit) such that the Hilbert space describing Λ is a tensor product H = H , dimH = O(1). (1) u u O u∈Λ Let S(r)bea set of all squareblocks A⊆ Λ of size r×r, wherer is a positive integer. Note that S(r) contains L2 translations of some elementary square of size r×r for all r < L, S(L) = Λ, and S(r) = ∅ for r > L. We shall assume that the unperturbed Hamiltonian H involves only 0 2×2 interactions (otherwise consider a coarse-grained lattice): H = Q . (2) 0 A X A∈S(2) 3 Here Q is an interaction that has supporton a square A. We also assume that the interactions A Q are pairwise commuting projectors, A Q2 = Q , Q Q = Q Q for all A,B ∈ S(2). (3) A A A B B A Well known examples of Hamiltonians composed of commuting projectors are Levin-Wen models [18] and quantum double models [2]. We assume that H has zero ground state energy, i.e., the ground states of H are zero- 0 0 eigenvectors of every projector Q . Let P and Q be the projectors onto the ground subspace A and the excited subspace of H , that is, 0 P = (I −Q ) and Q = I −P. (4) A Y A∈S(2) For any square B ∈ S(r), r ≥ 2 define local versions of P and Q as P = (I −Q ) and Q = I −P . (5) B A B B Y A∈S(2) A⊆B Note that P and Q have support on B. B B We shall need two extra properties of H related to TQO defined in [3]. We shall assume 0 that there exists an integer L∗ ≥ La for some constant a > 0 such that one has the following properties: 1. TQO-1: Let A ∈ S(r) be any square of size r ≤ L∗. Let O be any operator acting on A A. Then PO P = cP A for some complex number c. 2. TQO-2: Let A ∈ S(r) be any square of size r ≤ L∗ and let B ∈ S(r+2) be the square that contains A and all nearest neighbors of A. Let O be any operator acting on A such A that O P = 0. Then A O P = 0. A B The property TQO-1 is often taken as a definition of TQO, see Ref. [1, 2, 19, 20]. One can also show that L∗ ≥ d/4 where d is the distance of P considered as a codespace of a quantum error correcting code, i.e., the smallest integer m such that erasure of any subset of m particles can be corrected, see Ref. [21] for details. We will need later a corollary of TQO-1 and TQO-2. We use b (A) to denote the square l containing square A as well as all first, second,...,l-th neighbors of square A. That is, it is a “ball” of distance l around A. Corollary 1. Let O be any operator supported on a square A. Let C = b (A) and suppose A 2 that C has size bounded by L∗. Then P O P = cP , (6) C A C C 4 for some constant c. Thus, all states ψ with P ψ = ψ have the same reduced density matrix on A, and in C particular the reduced density matrix of ψ on square A is equal to the reduced density matrix of any ground state on A. Finally, we have the useful equality: kO Pk= kO P k. (7) A A C Proof. Let B = b (A). Note that P O P commutes with the Hamiltonian, and hence com- 1 B A B muteswithP. So,P (P O P )P = (P O P )P. SobyconditionTQO-1,(P O P )P = cP B A B B A B B A B for some c. Define O′ = O − c. Then, (P O′ P )P = 0. So, by condition TQO-2, A A B A B (P O′ P )P = 0. Note that P O′ P commutes with P . So, P (P O′ P )P = 0. Hence, B A B C B A B C C B A B C P O′ P = 0. So, P O P = cP . C A C C A C C Now consider the second claim. Let ψ be any state such that P ψ = ψ. Let ρ (ψ) be the C A reduced density matrix of ψ on A. Then, tr(ρ O ) = tr(P |ψihψ|P O ) = c(O )tr(|ψihψ|) = A A C C A A c(O ), where the constant c(O ) depends only on O and not on ψ. Thus, the reduced density A A A matrix is the same for all vectors such that P ψ =ψ. C Finally, let ψ, Ψ satisfy kO P k = |O ψ| and kO Pk = |O Ψ |. Then, we have that 0 A C A A A 0 P ψ = ψandPΨ = Ψ ,whichimpliestr(|O |2|ψihψ|) = tr(|O |2|Ψ ihΨ |),sincetheirreduced C 0 0 A A 0 0 density matrices agree on A. Hence, kO Pk= kO P k, as claimed. A A C 3 Local Hamiltonians and the Lieb-Robinson bounds Throughout this paper we restrict ourselves to Hamiltonians with a sufficiently fast spatial decay of interactions. Any such Hamiltonian V will be specified using a decomposition into local interactions, V = V , (8) r,A X X r≥1 A∈S(r) † whereV = V is an operator acting non-trivially only on a squareA. We shall often identify r,A r,A a Hamiltonian and the corresponding decomposition unless it may lead to confusions. Let us define several important classes of Hamiltonians. Definition 1. AHamiltonian V has support near asite u∈ Λifall squaresinthe decomposition Eq. (8) contain u. Definition 2. A Hamiltonian V is globally block-diagonal iff it preserves the ground subspace P, that is, [V,P] = 0. A Hamiltonian V is locally block-diagonal iff all terms V in the r,A decomposition Eq. (8) preserve the ground subspace, that is, [V ,P] = 0 for all r,A. r,A Definition 3. A Hamiltonian V has strength J if there exists a function f : Z → [0,1] + decaying faster than any power such that kV k ≤ Jf(r) for all r ≥ 1 for all A∈ S(r). r,A 5 Our main technical tool will be the following corollary of the Lieb-Robinson bound for systems with interactions decaying slower than exponential [24, 20]. Lemma 1. Let H be a (time-dependent) Hamiltonian with strength O(1). Let V be a Hamil- tonian with strength J. Let U(t) be the unitary evolution for time t, |t| ≤ 1, generated by H. Define V˜ = U(t)VU(t)†. Then V˜ has strength O(J). If V has support near a site u then V˜ also has support near u. Wewillalsoneedananalogue ofLemma1foraninfiniteevolution time. Itwillbeapplicable only to evolution under Hamiltonians H having a finite Lieb-Robinson velocity [4, 5, 6, 24], for example, Hamiltonians with exponentially decaying interactions. Definition 4. A Hamiltonian V has strength J and decay rate µ iff kV k≤ J exp(−µr) for all r ≥ 1 for all A∈ S(r). r,A Lemma 2. Let H be a (time-dependent) Hamiltonian with strength O(1) and decay rate µ > 0. Let V be a Hamiltonian with strength J. Let U(t) be the unitary evolution for time t generated by H. Finally, let g(t) be any function decaying faster than any power for large |t|. Define ∞ V˜ = dtg(t)U(t)VU(t)†. Z −∞ Then V˜ has strength O(J). If V has support near a site u then V˜ also has support near u. The superpolynomially decaying function of r bounding the norm of V˜ depends on the r,A superpolynomially decaying function of r bounding the norm of V as well as on µ and the r,A function g(t). (Remark: In fact, one can show that there exist g(t) that have the properties neededlater andthatdecay as anexponentialofapoweroft, makingitpossiblealsotoconsider Hamiltonians H that do not have a decay constant µ > 0 but that instead have sufficiently fast stretched exponential decay. We omit this for simplicity.) 4 Reduction from global to local block-diagonality In this section we prove that a certain class of globally block-diagonal perturbations can be reduced to locally block-diagonal perturbations with a small error by simply rewriting the Hamiltonian in a different form. Lemma 3. Consider a Hamiltonian H = H + X , (9) 0 u X u∈Λ where H obeys TQO-1,2 and X is a perturbation with strength J that has support near u. 0 u Suppose that [X ,P] = 0 for all u. u 6 Then we can rewrite H = H +V′+∆, (10) 0 where V′ is a locally block-diagonal perturbation with strength O(J), and k∆k decays faster than any power of L∗. Proof. Consider some fixed site u and let X ≡ X . By assumption, X has a decomposition u X = X(r), where X(r) = X and the norm kX(r)k decays faster than r≥1 A∈S(r),A∋u r,A any pPower of r. By adding consPtants to the different terms X and using TQO-1 we can r,A achieve PX P = 0 for all r ≤ L∗. Define ∆ = PX = PXP = PX(r)P. Note that r,A r>L∗ k∆k decays faster than any power of L∗. Define also X′ = X −∆Psuch that X′P = 0. We can assume that X′ has strength O(J) and its support is near u if we treat ∆ as a single interaction on a square of size L. To simplify notation we set X = X′ in the rest of the proof. Choosing any l ≤ L∗ and applying Eq. (7) to O = l X(r) we arrive at r=1 P l l k X(r)P k = k X(r)Pk ≤ kXPk+k X(r)Pk ≤ Jf(l) (11) bl+2(u) X X X r=1 r=1 r>l for some function f decaying faster than any power of l. Consider a given site u. Let B be the union of all squares of size r that contain u, such r that X(r) has support on B . We have B ⊂ B ⊂ ... ⊂ B = Λ for some integer M. Define r 1 2 M an orthogonal unity decomposition I = M+1E by m=1 m P E = Q , 1 B1 E = Q P for 2 ≤ m ≤ M, m Bm Bm−1 E = P = P. (12) M+1 BM Taking into account that E X = XE = 0 we arrive at M+1 M+1 X = E X(q)E = Y(j)+ Z(q), (13) p r X X X 1≤p,q,r≤M j≥1 q≥1 where max(p,r)−2 Y(j) = E  X(q)E . (14) p r X X 1≤p,r≤M q=1   p+r=j and Z(q)= E X(q)E . (15) p r X 1≤p,r≤q+1 All operators Y(j) and Z(j) are hermitian, annihilate P, and act only on sites in the square B . The norm of Z(q) decays faster than any power of q because of the decay of the norm of j+1 X(q). 7 We claim that the norm of Y(j) decays faster than any power of j. Indeed, Y(j) is a sum over j −1 different terms corresponding to different choices of p,r. We show that the norm of each such term p,r decays fast enough. Assume without loss of generality that p ≥ r. Then, max(p,r)−2 max(p,r)−2 kE X(q) E k ≤ kE X(q) k, which decays faster than any power of p q=1 r p q=1 (cid:16) (cid:17) (cid:16) (cid:17) p by EPq. (11). Since p ≥ j/2, it decaPys faster than any power of j. Thus, we have decomposed X as a sum of terms which annihilate P, supported on squares of increasing radius about site u, with norm decreasing faster than any power of the size of the square. This completes the proof. 5 Stability under locally block-diagonal perturbation Inthissection,wedefinetheconceptofrelativelyboundedperturbations(seeChapterIVof[22]) and show that the spectral gaps separating low-lying eigenvalues of H are stable against such 0 perturbations. We then demonstrate that locally block-diagonal perturbations satisfy the rela- tiveboundnessconditionwhichprovesthegapstability forlocally block-diagonal perturbations. Note that our condition of relative boundness is different from the one used in [10]. In particular, our condition provides an elementary proof of the gap stability without the need to use cluster expansions as was done in [10]. Let H and W be any Hamiltonians acting on some Hilbert space H. We shall say that W 0 is relatively bounded by H iff there exist 0≤ b <1 such that 0 kWψk ≤ bkH ψk for all |ψi ∈ H. (16) 0 The following lemma asserts that a relatively bounded perturbation can change eigenvalues of H at most by a factor 1±b. 0 Lemma 4. Suppose W is relatively bounded by H . Then the spectrum of H +W is contained 0 0 in the union of intervals [λ (1−b),λ (1+b)] where λ runs over the spectrum of H . 0 0 0 0 Proof. Indeed, suppose (H +W)|ψi = λ|ψi, that is, 0 (H −λI)|ψi = −W |ψi. (17) 0 The relative boundness then implies k(H −λI)ψk ≤ bkH ψk, that is, 0 0 hψ|(H −λI)2|ψi ≤ b2hψ|H2|ψi. (18) 0 0 LetH = λ P bethespectraldecomposition of H . Herethesumrunsover thespectrum 0 λ0 0 λ0 0 of H andPP is a projector onto the eigenspace with an eigenvalue λ . Define a probability 0 λ0 0 distribution p(λ )= hψ|P |ψi. Substituting it into Eq. (18) one gets 0 λ0 (λ −λ)2p(λ ) ≤ b2λ2p(λ ). (19) 0 0 0 0 X X λ0 λ0 Therefore there exists at least one eigenvalue λ such that 0 (λ −λ)2 ≤ b2λ2. (20) 0 0 This is equivalent to λ (1−b)≤ λ ≤ λ (1+b). 0 0 8 In the rest of this section we proof stability under locally block-diagonal perturbations. Let H be a Hamiltonian satisfying TQO-1,2. 0 Lemma 5. Let V be a locally block-diagonal perturbation with strength J. Then the spectrum of H +V is contained in the union of intervals I , where k runs over the spectrum of H 0 k≥0 k 0 and S I = {λ ∈ R : k(1−b)−δ ≤ λ ≤ k(1+b)+δ} k for some b = O(J) and for some δ decaying faster than any power of L∗. Proof. For any integer r ≥ 1 define W(r)= V . r,A X A∈S(r) By assumptions of the lemma we have kV k ≤ Jf(r) (21) r,A for some function f decaying faster than any power and [V ,P] = 0. Performing an overall r,A energy shift and using TQO-1 we can assume that V P =PV = 0 for all 1 ≤ r ≤ L∗ and for all A ∈S(r) (22) r,A r,A Proposition 1. Suppose 1 ≤ r ≤ L∗. Then W(r) is relatively bounded by H with a constant 0 b(r)= O(Jr2f(r)). Thispropositionimmediately impliesthat W(r)isrelatively boundedbyH witha 1≤r≤L∗ 0 constantb = L∗ b(r)= O(J). Lemma4thenPimpliesthatthespectrumofH + W(r) r=1 0 1≤r≤L∗ iscontained inPtheunionofintervals [k(1−b),k(1+b)]. TreatingtheresidualtermPs W(r) r>L∗ using the standard perturbation theory then leads to the desired result. P Proof of the proposition. Let Λ = B ∪ B ∪ ... ∪ B be a partition of the lattice into 1 2 M contiguous squares of size r such that any 2×2 square is contained in exactly one square B . a If a 2×2 square is on the boundary of two, or more, of the B , we include it in the bottom a left one in the partition. Moreover, if r does not divide L, then squares at the boundary may be truncated to rectangles, to fill in the partition. We refer to B as boxes to distinguish them a from the squares involved in the decomposition of V. For any binary string Y ∈ {0,1}M define a projector M R = [Y Q +(1−Y )P ]. Y a Ba a Ba Y a=1 Clearly the family of projectors R defines an orthogonal decomposition of the Hilbert space, Y that is, R = I. Given a string Y ∈ {0,1}M, we shall say that a box B is occupied iff Y Y a Y = 1. P a 9 We claim that any operator V acting on a square A ∈ S(r) and satisfying Eq. (22) has r,A only a few off-diagonal blocks with respect to this decomposition. Specifically, TQO-2 implies that R V R 6= 0 (23) Y r,A Z only if A has distance O(1) from some occupied box in Y and A has distance O(1) from some occupied box in Z, and the configurations Y,Z differ only at those boxes that overlap with A. Clearly, for any fixed Y such that Y has k occupied boxes the number of pairs (A ∈ S(r),Z) that could satisfy Eq. (23) is at most O(kr2). Denote for simplicity W ≡ W(r), w ≡ Jf(r). For any state |ψi we get hψ|W2|ψi = hψ|R WR WR |ψi Y Z V X Y,Z,V⊆[M] ≤ kR WR k·kR WR k·kR ψk·kR ψk Y Z Z V Y V X Y,Z,V 1 ≤ kR WR k·kR WR k· (hψ|R |ψi+hψ|R |ψi) Y Z Z V Y V 2 X Y,Z,V = kR WR k·kR WR k·hψ|R |ψi Y Z Z V Y X Y,Z,V ≤ O(k2r4w2)hψ|R |ψi =O(w2r4)hψ|G|ψi, (24) Y X X k≥0 Y:|Y|=k where G= k2R . (25) Y X X k≥0 Y:|Y|=k The inequality Eq. (24) follows from the fact that Y and Z differ at at most O(1) boxes and an obvious bound k(k+O(1)) = O(k2). Finally, note that G≤ H2 since any configuration with k 0 occupied boxes must have at least k defects (squares A ∈ S(2) for which Q has eigenvalue 1) A and since creating a defect costs at least a unit of energy. We arrive at hψ|W2|ψi ≤ b2hψ|H2|ψi, b= O(wr2). (26) 0 This completes the proof. 6 Exact quasi-adiabatic continuation We define a continuous family of Hamiltonians, H = H +sV, (27) s 0 10