An Introduction to Quantum Spin Systems 1 Notes for MA5020 (John von Neumann Guest Lectures) Bruno Nachtergaele and Robert Sims

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1 An Introduction to Quantum Spin Systems 1 Notes for MA5020 (John von Neumann Guest Lectures) Bruno Nachtergaele and Robert Sims Contents 1. Introduction 2 2. Quantum Spin Systems Spins and Qudits Observables States Dirac notation Finite Quantum Spin Systems 6 3. Appendix: C -algebras C -algebras Spectral theory in a C -algebra Positive elements Representations States General Framework for Finite and Infinite Quantum Spin Systems The Dynamics of Finite Systems Infinite Systems Lieb-Robinson Bounds Existence of the Dynamics Ground States and Equilibrium States Ground States Thermal Equilibrium, the Free Energy, and the Variational Principle for Gibbs States The Kubo-Martin-Schwinger condition The Energy-Entropy Balance inequalities Infinite Systems and the GNS representation The GNS Construction Ground States and Equilibrium States for Infinite Systems Symmetry, Excitation Spectrum, and Correlations The Goldstone Theorem The Exponential Clustering Theorem Lie Groups and Lie Algebras Representations of Lie Groups and Lie Algebras Irreducible Representations of SU(2) Tensor products of representations Four Examples Example 1: the isotropic Heisenberg model Example 2: the XXZ model Example 3: the AKLT model Example 4: the Toric Code model Frustration free models The AKLT chain Frustration free spin chains with a unique Matrix Product ground state Some properties of translation invariant matrix product states The commutation property Copyright c 2016 Bruno Nachtergaele and Robert Sims. 1

2 The martingale method and proof of non-vanishing spectral gap for quantum chains with MPS ground states. 82 References 85

3 Frustration free spin chains with a unique Matrix Product ground state. We begin our discussion of general frustration free models in one dimension by stating a result about the structure of the ground states of frustration free spin chains. This will serve as motivation to analyzing in some detail the spin chains with unique MPS ground states and also set the stage for studying some interesting generalizations, such as models with multiple ground states, examples of frustration free chains for which the product structure is not expressed in terms of matrices but in terms of operators on an infinite-dimensional Hilbert space, and frustration free models in more than one dimension. We start by considering spin systems on Z + = {1, 2, }, i.e., a half-infinite chain, with a d- dimensional Hilbert space H x = C d, at each site x Z +. We assume that 0 h B(C d C d ) is a frustration free nearest neighbor interaction. By this we mean that the finite-volume Hamiltonians L 1 H [1,L] = h x,x+1, x=1 where h x,x+1 = h A [x,x+1], have a non-trivial kernel for all L 2. Let G = ker h C d C d. Then the frustration free property is equivalent to (11.20) L 1 x=1 } C d {{ C } d x 1 G C d C d }{{} L 1 x {0}, for all L 2. The set of states η on A Z + such that η(h x,x+1 ) = 0, for x = 1,..., L 1, has weak- limit points. Any such limiting state ω satisfies ω(h x,x+1 ) = 0 for all x Z +. We will call such states zero-energy states and, since they are defined as limits of finite-volume ground states, they are ground states in the sense of Definition 6.2. The set of zero-energy states is a face in the set of all states, meaning that any pure states appearing in the decomposition of a zero-energy state are also zero-energy states. This shows that the set of pure zero-energy states on A Z + for a nearest neighbor interaction h 0 such that G = ker h satisfies (11.20), is non-empty. We have the following theorem about the structure of such states. Theorem 11.3 ( [22]). Let 0 h B(C d C d ) such that G = ker h satisfies (11.20), and suppose ω is a pure state on A Z + such that ω(h x,x+1 ) = 0, for all x 1. Then, given an orthonormal basis { i i = 1,..., d} of C d, there exists a Hilbert space K, a unit vector Ω K, and a set of operators V 1,, V d B(K) for which: i) K is the closed linear span of {V i1 V in Ω i 1, i n {1,, d} and n 0} ii) d α=1 V α V α = 1l. iii) For all n 1, i 1,..., i n, j 1,..., j n {1,..., d}, we have (11.21) ω ( i 1,, i n j 1,, j n ) = V in V ı1 Ω, V jn V j1 Ω iv) For every ψ = d i,j=1 ψ i,j i, j G, we have (11.22) d ψ i,j V j V i = 0. i,j=1 v) Define the operator Ê : B(K) B(K) by d (11.23) Ê(X) = Vi XV i, i=1 for all X B(K).

4 Then, for any X B(K), Ê(X) = X if and only if X = λ1l for some λ C. In order to make the connection with the structure of the ground states we found for the AKLT model in the previous section, consider K = C 2, and define the maps E A : B(K) B(K), for A M d, by d E A (X) = i A j Vi XV i = V (A X)V, for all X B(K), i,j=1 where V : K C d K is the isometry defined by Then, V ϕ = d i V i ϕ. (11.24) ω(a 1 A n ) = Tr Ω Ω E A1 E An (1l). i=1 Since the space K for the AKLT model is finite-dimensional, the operators V i and and the transfer operator Ê are represented by matrices. Properties very similar to the ones we proved for the AKLT model in Section 11.1, follow from the product structure whenever dim K < without further assumptions. This will be shown in the next section. We will also prove a general result about the spectral gap of models with matrix product ground state Some properties of translation invariant matrix product states. In this section we derive some general properties of states of a form very similar to (11.24) under the additional assumption that k = dim K <. This is the case of Matrix Product States (MPS). The operators V i are now k k matrices. Under the assumptions of Theorem 11.3 the transfer matrix Ê has a simple eigenvalue 1. Its transpose, ÊT then also has a simple eigenvalue 1 and the Perron-Frobenius theory for completely positive maps (see Appendix??) then implies that there is a unique density matrix ρ M k such that ÊT (ρ) = ρ. Regarding ρ as a positive linear functional on M k, we can express this by the relation (11.25) ρ(ê(b)) = ρ(b), B M k. By replacing Ω Ω by ρ in (11.24), we get expectations ω(a 1 A n ) that are translation invariant. Indeed, using Ê(1l) = 1l and (11.25) it is straighforward to verify that (11.26) ω(a 1 A n ) = ω(1l A 1 A n ) = ω(a 1 A n 1l). The positivity and normalization also follow directly from the definition. Therefore, the finite chain expressions (11.27) ω(a 1 A n ) = TrρE A1 E An (1l). define a unique translation invariant state on A Z. In addition to the assumption that 1 is a simple eigenvalue of Ê, we will also assume in this section that all other eigenvalues of Ê have modulus strictly less than one. This situation is referred as a transfer matrix with trivial peripheral spectrum or, equivalently, that Ê is primitive. (see [71]). Eigenvalues of absolute value 1 other than a simple eigenvalue 1, in general correspond to states that are not pure but have a non-trivial decomposition into pure states. If ρ is not faithful and has support projection P, then one can construct the same translationinvariant pure state by replacing V i with P v i P, i = 1,..., d (see [21]). In the sequel we will assume that P = 1l, i.e., that ρ > 0, or work with the modified v i if, originally, P 1l. In particular, the smallest eigenvalue of ρ is then strictly positive. 77

5 78 For any faithful state ρ, i.e., a density matrix with trivial kernel, we can define a non-degenerate inner product on M k by (11.28) A, B ρ = TrρA B for all A, B M k, and let ρ denote the corresponding norm. Let ρ min = min spec(ρ). Since ρ is finite-dimensional, we have ρ min > 0. It follows that the norm ρ is equivalent to the Hilbert-Schmidt norm on M k, A 2 = TrA A, i.e. [ ] ρ (11.29) A 2 2 = Tr[A A] Tr A A = 1 A 2 ρ for any A M k. ρ min ρ min As in the case of the AKLT model, the trivial peripheral spectrum of C > 0 and λ (0, 1) such that (11.30) a(n) := Ên 1l ρ Cλ n. Ê implies that there exists Since Ê(1l) = 1l, we have (11.31) Ên+1 1l ρ = Ê (Ên 1l ρ ) Ên 1l ρ and therefore a(n) is monotone decreasing in n. The following maps Γ n : M k H [1,n] generalize the maps ψ (n) ( ) of the AKLT model: (11.32) Γ n (B) = i 1,...,i n Tr[Bv in v i1 ] i 1,..., i n, B M k. These are the Matrix Product Ground states for finite chains of length n. We will now study the vectors Γ n (B) and the linear subspace spanned by them in some detail. We begin with a lemma that estmates the inner product of two such vectors. Lemma For any B, C M k, (11.33) Γ n (B), Γ n (C) B, C ρ ka(n) B 2 C 2 k ρ min a(n) B ρ C ρ Proof. Recall (11.32). Γ n (B), Γ n (C) = (11.34) Tr[Bv in v i1 ] Tr[Cv in v i1 ] i 1,,i n = Tr[vi 1 vi n B ]Tr[Cv in v i1 ] i 1,,i n = α vi 1 vi n B α β Cv in v i1 jβ α,β i 1,,i n = α,β α Ên (B α β C) β = α,β α 1l ρ (B α β C) β + α,β α (Ên 1l ρ ) (B α β C) β, where we have used the indices α and β to denote summation over the orthonormal basis of C k. Now the first term above is clearly (11.35) α 1l β Tr[ρ(B α β C)] = Tr[ρB C] = B, C ρ α,β

6 79 and the remainder can then be estimated by (11.36) Ên 1l ρ B α C β a(n)k B 2 C 2 a(n)k B ρ C ρ ρ min α,β where we have used both (11.29) and (11.42). It is clear that in the special case of C = B, the above lemma yields: (11.37) Γ n (B) 2 B 2 ρ k a(n) B 2 ρ ρ min and if B ρ 0, then (11.38) Γ n (B) 2 B 2 ρ 1 k ρ min a(n) Thus, if kρ 1 min a(n) < 1, Γ n is injective. Let us set b(n) = kρ 1 min a(n). Let n 0 be the smallest integer such that Γ n is injective for all n n 0. We will refer to n 0 as the injectivity length. As the ρ inner product is non-degenerate, a simple consequence of this bound is that Γ n is eventually injective. In fact, the following corollary is immediate. Corollary For any B M k, the bound (11.39) B ρ 1 b(n) Γn (B) B ρ 1 + b(n) holds for n sufficiently large. Here ρ min b(n) = ka(n) and n large means b(n) < 1. Proof of Corollary The bound (11.40) Γ n (B) 2 B 2 ρ b(n) B 2 ρ follows immediately from (??). If B = 0, there is nothing to prove. Otherwise, this bound can be re-written as (11.41) b(n) Γ n(b) 2 B 2 1 b(n) ρ from which the above claim readily follows. This result shows that Γ n is injective for sufficiently large n, i.e. that there exists n 0 such that for all n n 0, we have dim ranγ n = k 2. One can also show that, if Ê is primitive, regardless of the values of λ and ρ min, Γ n is injective for n k 4 [71]. Let α 1,..., α k be an orthonormal basis of C k. Using Cauchy-Schwarz, we then have the following pair of inequalities, which we will use to prove the next lemma: k (11.42) B α j k B α j 2 = k B 2 = k k B 2 B ρ ρ min j=1 j for any B M k. Now consider three consecutive intervals (organized left-middle-right) with lengths l, m, and r respectively. We wish to estimate the inner product of vectors ϕ G l+m (C d ) r and ψ (C d ) l G m+r. If m n 0, the maps Γ m+r and Γ l+m are injective. Therefore there exist unique matrices B ϕ (k 1,..., k r ), B ψ (i 1,..., i l ) M k, such that (11.43) ϕ = k 1,...,k r Γ l+m (B ϕ (k 1,..., k r )) k 1,..., k r

7 80 and similarly, (11.44) ψ = It will be convenient to define (11.45) C ϕ = and (11.46) D ψ = i 1,...,i l i 1,..., i l Γ m+r (B ψ (i 1,..., i l )). k 1,...,k r B ϕ (k 1,..., k r )ρv k 1 v k r ρ 1 i 1,...,i l v i 1 v i l B ψ (i 1,..., v l ) It will be useful to use the following notations: i = (i 1,..., i l ), j = (j 1,..., j m ), k = (k 1,..., k r ), where each individual index takes value in {1,..., k}. We also define v i = v il v i1, v i = v i 1 v i l, etc. Lemma Suppose that m is large enough so that b(m) < 1. (Thus m n 0.) Let l 0 and r 0. For every ϕ G l+m (C d ) r and ψ (C d ) l G m+r, we have the estimate (11.47) ϕ, ψ C ϕ, D ψ ρ b(m) 1 b(m) ϕ ψ. Proof. ϕ, ψ = i,j,k Tr[v i v j B ϕ(k) ]Tr[B ψ (i)v k v j ] (11.48) = Γ m (v i B ϕ (k)), Γ m (B ψ (i)v k ) Now we apply Lemma See that ϕ, ψ vi B ϕ(k), B ψ (i)v k ρ (11.49) by an application of Cauchy-Schwarz. One now sees that v i B ϕ (k) 2 ρ = (11.50) where we have used that (11.51) = k By (11.38), it is clear that (11.52) B ϕ (k) 2 ρ k b(m) v i B ϕ (k) ρ B ψ (i)v k ρ b(m) v i B ϕ (k) 2 ρ B ψ (i)v k 2 ρ Tr [ρb ϕ (k) v i v ib ϕ (k)] Tr [ρb ϕ (k) B ϕ (k)] = k vi v i = 1l i 1 1 b(l + m) B ϕ (k) 2 ρ Γ m+l (B ϕ (k)) 2 k

8 81 and by the orthonormality of the basis k, we have (11.53) Γ m+l (B ϕ (k)) 2 = ϕ 2 The other factor is seen to be: B ψ (i)v k 2 ρ = Tr [ρvk B ψ(i) B ψ (i)v k ] = Tr [(Êt ) r ] (ρ)bψ (i) B ψ (i) i k = i B ψ (i) 2 ρ (11.54) 1 1 b(m + r) where we used that Êt (ρ) = ρ. Thus the right-hand-side of (11.49) is bounded by Γ m+r (B ψ (i)) 2 = 1 1 (11.55) b(m) ϕ ψ 1 b(l + m) 1 b(m + r) i ψ 2 1 b(m + r) b(m) 1 b(m) ϕ ψ where we have used the monotonicity property that follows from (11.31). Note also that v i B ϕ (k), B ψ (i)v k ρ = Tr [ ρb ϕ (k) vi B ψ(i)ρρ 1 ] v k [ ( ) ( = Tr ρ ρ 1 v k ρb ϕ (k) i (11.56) as claimed. k = C ϕ, D ψ ρ v i B ψ(i) The commutation property. Consider a frustration free quantum spin chain with MPS ground states such as, e.g., the AKLT chain. To start, we assume that Ê is primitive with a unique density matrix ρ that is an eigenvector with eigenvalue 1 of ÊT, and WLOG we can assume that ker ρ = {0}, the smallest eigenvalue of ρ, ρ min, is non-zero. Let G n M n d denote the orthogonal projections onto G n. Proposition 11.7 (Commutation Property). For all m 1, l 0, r 0, we have (11.57) (G l+m 1l r )(1l l G m+r ) G l+m+r ɛm, where (11.58) ɛ m = b(m) (1 b(m)) 2. Proof. Since G l+m+r projects onto a subspace of the ranges of both G l+m 1l r and 1l l G m+r, we have the identity (G l+m 1l r )(1l l G m+r ) G l+m+r = (G l+m 1l r G l+m+r )(1l l G m+r G l+m+r ). From this it is easy to see that to prove (11.57) is equivalent to showing that for all ϕ G l+m (C) r, ψ C l G m+r, such that G l+m+r ϕ = G l+m+r ψ = 0, the have the following bound (11.59) ϕ, ψ ɛ m ϕ ψ. )]

9 82 The inner product can be estimated by applying Lemma 11.6 and using the extra information we have about the vectors ϕ and ψ. Since ϕ G l+m (C) r, there exists A M k such that B ϕ (k) = Av k, which allows us to determine C ϕ : (11.60) C ϕ = k Now, G l+m+r ψ = 0 implies Av k ρv k ρ 1 = A(ÊT ) r (ρ)ρ 1 = A. (11.61) Γ l+m+r (A), ψ = 0, for all A M k. In combination with Lemma 11.6 this gives (11.62) A, D ψ ρ b(m) 1 b(m) ψ A ρ, and hence (11.63) D ψ ρ b(m) 1 b(m) ψ. By a similar reasoning, we find (11.64) C ϕ ρ b(m) 1 b(m) ϕ. Combining this information with Lemma 11.6 proves the proposition. Proposition 11.7 is referred to as the Commutation Property of the the ground states projections because it directly implies a bound on the commutator: [G l+m 1l r, 1l l G m+r ] (G l+m 1l r )(1l l G m+r ) G l+m+r + G l+m+r (1l l G m+r )(G l+m 1l r ) 2ɛ m. This property is closely related to the Factorization Property proved in [36] which, in turn, can be seen to underly the Area Law for the entanglement entropy [33] The martingale method and proof of non-vanishing spectral gap for quantum chains with MPS ground states. The Commutation Property of Proposition 11.7 in combination with the frustration freeness of the spin chains with MPS ground states we are considering in this chapter, can be seen to imply a uniform lower bound for the spectral gap of these models. See, e.g., [21] or [65]. The martingale method is a simple argument that relates the quantity ɛ m to a lower bound for the spectral gap in an efficient way [49] and is based on an idea that has been very useful in the context of classical interacting particle models [45] and other many-body systems [12, 13]. To present the martingale method in a transparant way, it is useful to identify the crucial properties that make it work in the context of quantum spin systems. The ground state space of a finite chain is the intersection of the ground state spaces of shorter chains. We will describe this as a sequence of decreasing subspaces H N G 1 G 2 G N, which are the null spaces of a corresponding increasing sequence of non-negative Hamiltonians. For n = 1,..., N, H n B(H N ), G n = ker H n, H 1 H 2 H N. Note however, that although the sequence H n, n = 1,..., N, will typically be associated with the system on an increasing sequence of finite volumes, the H n do not have to coincide with the finite-volume Hamiltonians in terms of which the model is defined. In general, we will have constants c, C > 0 such that (11.65) ch N H ΛN CH N, where H ΛN is the finite-volume Hamiltonian of which we want the estimate the spectral gap.

10 83 Define (11.66) h n = { H 1 if n = 1 H n H n 1 if n = 2,..., N and let G n and g n B(H) N be the orthogonal projection onto ker H n and ker h n, respectively. Of course, (11.66) is equivalent to assuming that H n = n k=1 h k, and H n will be increasing if we require h k 0. Furthermore, define 1l G 1 if n = 0 (11.67) E n = G n G n+1 if 1 n N 1. if n = N G N It is easily verified that the E n are mutually orthogonal orthogonal projections forming a resolution of the identity: N n=0 E n = 1l and E n E m = δ n,m E n. Assumptions for the Martingale Method: (i) There is a constant γ > 0 such that h n γ(1l g n ), n = 1... N. (ii) There are integers l 0, r 1, such that E k g n = g n E k, if k [n l, n + r]. (iii) There exists ɛ [0, l + r], such that E n g n+1 E n ɛ 2 E n, n = 1,..., N 1. Theorem In the setup described immediately here above, assume the Assumption (i) (iii) hold. Let ψ H N such that G N ψ = 0. Then Proof. By assumption E N ψ = G N ψ = 0. Hence (11.68) ψ 2 = Using this, for all n = 0,..., N 1, we have (11.69) ψ, H N ψ γ 2 (1 ɛ 1 + l + r) 2 ψ 2. N 1 n=0 E n ψ 2. E n ψ 2 = ψ, (1l g n+1 )E n ψ + ψ, g n+1 E n ψ ( N 1 ) = ψ, (1l g n+1 )E n ψ + ψ, E m g n+1 E n ψ. By Assumption (ii) and the mutual orthogonality of the projections E m, the summation in the last term can be reduced and we obtain ( n+r ) (11.70) E n 2 ψ, (1l g n+1 )E n ψ + ψ, E k g n+1 E n ψ. After applying the inequality m=0 k=n l (11.71) ϕ 1, ϕ 2 1 2c ϕ c 2 ϕ 2 2, ϕ 1, ϕ 2 H, c > 0, two each of the two terms in (11.70), we find E n ψ 2 1 ψ, (1l g n+1 )ψ + c 1 2c 1 2 ψ, E nψ + 1 2c 2 ψ, E n g n+1 E n ψ + c 2 2 ψ, ( n+r k=n l E k ) 2 ψ.

11 84 To estimate the firs term of the RHS, we use Assumption (i). The second term can be combined with the LHS. For the third term we sue Assumption (iii). In the fourth term we can use the mutual orthogonality of the E m. This gives (11.72) (1 c 1 2 ɛ2 ) E n ψ 2 c 2 2c 2 2 n+r k=n l E k ψ 2 1 2c 1 γ ψ, h n+1ψ. We now sum over n = 0,..., N 1, use the fact the {E n n = 0,..., N} is a resolution of the identity, and recall that E n ψ = G N ψ = 0. The result is [ (11.73) ψh N ψ 2c 1 γ 1 c 1 2 ɛ2 c ] 2(1 + l + r) ψ 2. 2c 2 2 We can maximize the constant in the RHS by choosing c 1 = 1 ɛ 1 + l + r and c 2 = ɛ/ 1 + l + r. This yields the inequality stated in the theorem. Let us now apply this theorem to the spin chains with a unique pure MPS ground state, such as the AKLT chain. For simplicity of the notation, assume that the model is defined in terms of a frustration-free nearest neighbor interaction 0 h M d M d. Then, let m 1 and define (11.74) h n = Then, we have H n = n k=1 h k (n+1)m 1 x=(n 1)m+1 H [1,(n+1)m] H n 2H [1,(n+1)m] h x,x+1. E n = G (n+1)m 1l (N n)m G (n+2)m 1l (N n 1)m g n = 1l (n 1)m G 2m 1l (N n)m. Assumption (i) is satisfied with γ given by the spectral gap of H [(n 1)m+1,(n+1)m]. Since m is fixed, and the model is translation invariant this gap is positive and independent of n. Assumption (ii) is satisfied with l = 0, r = 1. For the spin chains with a unique pure MPS ground states (iii) follows from the Commutation Property (Proposition 11.7), To see this, it suffices to express the quantities E n and g n+1 in terms of the ground state projection operators: (11.75) g n+1 E n = (1l (n 1)m G 2m )(G (n+1)m 1l m G (n+2)m ) ɛ m where ɛ m is the quantity estimate in Proposition orthogonal projections G and E, one has (11.76) GE ɛ EGE ɛ 2 E. It them suffices to note that, for two Other choices for the sequence H n are possible and may provide more precise information in some cases. E.g., if one is interested in estimating the spectral gap for finite systems with periodic boundary conditions, it is useful to let H N 1 be comparable to the Hamiltonian of the chain with open boundary conditions and H N the Hamiltonian for the system with the same Hilbert space but with the additional term(s) that corresponds to considering periodic boundary conditions. Further refinements of the method exist. See, e.g., [61]. The method can also be applied to some quantum spin models in higher dimensions as long as one can find sequences of finite volumes and associate Hamiltonians such that the Assumptions (i)-(iii) are satisfied. See, e.g., [7, 9] for a few example in two and more dimensions.

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