Quantum Lattice Dynamics and Applications to Quantum Information and Computation 1

Transcription

1 1 ESI Vienna, July 21-31, 2008 Quantum Lattice Dynamics and Applications to Quantum Information and Computation 1 Bruno Nachtergaele (UC Davis) based on joint work with Eman Hamza, Spyridon Michalakis, Yoshiko Ogata, Hillel Raz, Benjamin Schlein, and Robert Sims 1 Based on work supported by the U.S. National Science Foundation under grants # DMS , DMS , and DMS , by the Austrian Science Fund (FWF) under Grant No. Y330, a Sofja Kovalevskaja Award of the Humboldt Foundation, and an ESI Junior Fellowship.

2 2 Lecture Plan 1. The basic setup. Lieb-Robinson bounds for the support of time-evolved observables: (i) lattice systems bounded interactions, (ii) (an)harmomic lattice systems. Some comments on applications. 2. The Exponential Clustering Theorem: spectral gap and correlation length of ground states. 3. Structure of gaped ground states. Area Law for the local entropy of a ground with a spectral gap (one dimension). 4. Local perturbations perturb locally: A Lieb-Schultz-Mattis Theorem. Topological phases. My good intentions: (i) I will explain the main ideas and arguments (avoiding the technical details), (ii) I will write lecture notes later.

3 3 Setup finite collection of quantum systems (spins, qudits, (an)harmonic oscillators, atoms, quantum dots,... ) labeled by x Λ. Each system has a Hilbert space H x. The Hilbert space describing the total system is H Λ = x Λ H x. Each system has a dynamics described by a s.a. Hamiltonian H x densely defined on H x. The algebra of observables of the composite system is A Λ = B(H x ) = B(H Λ ). x Λ If X Λ, we have A X A Λ, by identifying A A X with A 1l Λ\X A Λ.

4 4 Interactions We will primarily consider bounded interactions modeled by map Φ from the set of subsets of Λ to A Λ such that Φ(X ) A X, and Φ(X ) = Φ(X ), for all X Λ. The full Hamiltonian is H = H x + Φ(X ). x Λ X Λ E.g., Λ Z d, H x = C 2 ; the Heisenberg Hamiltonian: H = BSx 3 + J xy S x S y x Λ x y =1 The Heisenberg dynamics, {τ t } t R, defined by τ t (A) = e ith Ae ith, A A Λ. For systems of oscillators we will also consider the standard harmonic interaction, which is unbounded.

5 5 Ground states, excitations, dynamics H Λ : finite system, includes boundary conditions described by additional terms in the Hamiltonian. Ground state: eigenvector with eigenvalue E 0 = inf spec H Λ. Spectral gap above the ground state: if dim H Λ <, and H Λ has eigenvalues E 0 < E 1 < E 2 <..., we define γ = E 1 E 0 > 0. In general γ = sup{δ 0 spec H Λ (E 0, E 0 + δ) = } 0. With some care, this definition can also be applied in the thermodynamic limit.

6 6 Excitations: it is often useful to think of excitations as being generated from a ground state by the action of an observable (even in finite systems). ψ = AΩ, H Λ Ω = E 0 Ω Often one studies the dynamics of ψ, bypassing a full spectral analysis of the Hamiltonian: ψ t = τ t (A)Ω. In these lectures we will focus on model-independent properties, valid for general classes of Hamiltonians that share a few essential properties. But models are of course very important.

7 7 Models, types of ground states behavior (1) Standard Heisenberg model: H Λ = x Λ BS 3 x + x,y Λ, x y =1 J xy S x S y - Large B: spins align with the field, no correlations, γ c > 0. - B = 0, J xy = J < 0, ferromagnetically ordered g.s., γ C/L 2 for Λ = [1, L] d Z d. - B = 0, J xy = J > 0, Λ = [1, 2L] d Z d : unique g.s.. In the limit L, antiferromagnetically ordered g.s s, and γ C/L (proved for d 3, and if spin S 1, also for d = 2, Dyson-Lieb-Simon 1978, and others).

8 8 - B = 0, J xy = J > 0, Λ = Z: no long-range order. If spin is halfintegral, γ = 0, power law decay of correlations. If spin is integer, γ > 0, exponential decay of correlations (Haldane s Conjecture). Excitations: - islands of overturned spins aka droplets (Spitzer, Starr) - spin waves - domain walls (Contucci, Koma, Michoel, Spitzer)

9 9 (2) Fancy Heisenberg models: E.g., Λ Z 2, and H Λ = J 1 S x S y + J 2 S x S y x,y Λ, x y =1 x,y Λ, x y = 2 May have fancy ground states and fancy excitations. - Dimerized ground states: dimers in a pattern. - Resonating Valence Bond ground states: dimers without a fixed pattern. - Spin Liquid: fully disordered. - Topological Phases, e.g., the Toric Code model on a surface of genus g has 4 g ground states that are locally identical yet globally distinguishable by a topological index.

10 10 The support of observables and small commutators Recall that we can identify A A X with a unique A A Y, for all Y that contain X. The smallest set X such that A A X, is called the support of A, denoted by supp A. I.e., A A X iff supp A X. Even if interactions are between nearest neighbors only, for generic A A Λ, supp τ t (A) = Λ for all t 0, Therefore nothing can be described locally, but this contradicts common experience.

11 11 For X, Y Λ, s.t., X Y =, A A X, B A Y, AB BA = [A, B] = 0: observables with disjoint supports commute. Conversely, if A A Λ satisfies [A, B] = 0, for all B A Y (1) then Y supp A =. A more general statement is true: if the commutators in (1) are uniformly small, than A is close to A Λ\Y.

12 12 Lemma Let A A Λ, ɛ 0, and Y Λ be such that [A, B] ɛ B, for all B A Y (2) then there exists A A Λ\Y such that A 1l A cɛ with c = 1 if dim H Y <, and c = 2 in general. we can investigate supp τ t (A) by estimating [A, B] for B A Y. Lieb-Robinson bounds do that for [τ t (A), B].

13 13 Let d be a metric on Λ. Often, Λ is a graph and d is the graph distance: d(x, y) is the length of the shortest path from x to y over edges in the graph. For X, Y Λ, we define d(x, Y ) = min{d(x, y) x X, y Y }. The diameter, diam(x ), of a subset X Λ is diam(x ) = max{d(x, y) x, y X }.

14 14 Lieb-Robinson bound : C, v, a > 0 s.t. [ τ t (A), B ] C A B e a(d v t ) where t R, A A X, B A Y, for finite X, Y V, v t and d = d(x, Y ). The first such estimates x were proved by Lieb & Robinson (1972). X Y y d = d(x, Y )

15 15 Lieb-Robinson bounds, what do they mean? Lieb, E.H. and Robinson, D.W.: The Finite Group Velocity of Quantum Spin Systems, Commun. Math. Phys. 28, (1972). For a specific model, the connection with the group velocity was made mathematically explicit by Ch. Radin (1973), but this interpretation follows in general from the following observation. Up to small corrections, the diameter of the support of τ t (A) does not grow faster than linearly in t. More precisely, by the Lemma, one has the following result: There exist C > 0, such that for all δ > 0, there exists A δ t, supported in a ball of radius v t + δ such that A δ t τ t (A) C A e aδ.

16 16 Lieb-Robinson bounds, why are they useful? - Philosophically, they show that non-relativistic theories defined by local Hamiltonians, such as the ones used in condensed matter physics, have a dynamics that does not violate locality. They provide a finite bound for the velocity of propagation of anything in quantum lattice systems with local interactions. - They can be used for practical applications. M. Hastings (2004) was the first to realize this and since then Lieb-Robinson bounds have been used to prove several new results about the ground states of quantum lattice systems: Hastings, N-Sims, Hastings-Koma, Bravyi, Verstraete, Eisert, Osborne, and others. Quasi-locality of the dynamics + other conditions imply quasi-locality properties of ground states.

17 17 General Assumption on the Interactions: We will assume that there is a non-increasing function F : [0, ) (0, ), with the following properties: (i) F := sup x V y V F (d(x, y)) <, (ii) there is a constant C > 0 such that for pairs x, y V, F (d(x, z))f (d(z, y)) CF (d(x, y)) z V If V = Z ν, we can take F (r) = 1 (1+r) ν+ɛ, for any ɛ > 0. Suppose we have V with a function F as above, and some a 0, such that Φ a := sup x,y V e ad(x,y) F (d(x, y)) Φ(X ) < X x,y

18 18 Lieb-Robinson bounds (bounded interactions) Theorem (Lieb-Robinson 1972, Hastings 04, N-Sims 06, H-Koma 06, N-S-Ogata 06, Eisert-Osborne 06, N-S 2007) Assume Φ a < for some a > 0. For local observables A A X and B A Y, one has the bound [τ t (A), B] 2 A B F min( Φ X, Φ Y )e a(d(x,y ) 2a 1 Φ ac t ) C where Φ X is given by Φ X = {x X Z with x Z, Z X c, and Φ(Z) 0 }. This means that the Lieb-Robinson velocity, v, is bounded by a 1 2 Φ a C.

19 19 The improvements obtained by the authors listed in the theorem have mostly to do with the dependence of the prefactor on the dimensions of the local Hilbert spaces and the size of the supports. For interactions that decay only as a power law, a similar bound with power law decay can be derived in the same way. If one assumes X Y =, one can replace e a(d(x,y ) 2a 1 Φ ac t ) by e ad(x,y ) (e 2 Φ ac t ) 1), which vanishes for t = 0. In general, for interactions of finite range R, the commutator behaves as t k, for small t, with k d(x, Y )/R.

20 20 Lieb-Robinson bounds for Anharmonic Lattices Λ Z ν, H Λ = x Λ L2 (R), p x = i d dq x, and H Λ = x Λ p 2 x +ω 2 q 2 x +V (q x ) + x y =1 λ(q x q y ) 2 +Φ(q x q y ) We proved Lieb-Robinson bounds under the following conditions on V and Φ: (i) If λ = 0, it is sufficient that V is such that px 2 + ω 2 qx 2 + V (q x ) is self-adjoint, and Φ L (R). (ii) If λ > 0, V and Φ have to be C 1, L 1, and such that k V (k) L 1 (R).

21 21 Theorem (N-Raz-Schlein-Sims, 2008) Case (i): Assume λ = 0, V such that px 2 + ω 2 qx 2 + V (q x ) is self-adjoint, and Φ L (R). For local observables A A X and B A Y, we have bound [τ t (A), B] 2 A B min[ Φ X, Φ Y ]e (ad(x,y ) 2ν+2 Φ e a t ). (3) for all a > 0. Here Φ X is given by Φ X = {x X y Λ \ X, such that x y = 1}.

22 22 For f : Λ C, define the Weyl operator W (f ) by W (f ) = e i P x Λ (qx Refx +px Imfx ). Clearly, W (f ) is a unitary operator in A Λ. Theorem (N-Raz-Schlein-Sims, 2008) Case (ii): Let λ 0, and assume V is C 1, L 1, and such that k V (k) L 1 (R). Then, for all f, g with supp f X and supp g Y, we have [τ t (W (f )), W (g)] C f g min( X, Y )e for all a > 0, and where v = 6 ω 2 + 4νλ + 3C 4a k V (k) 1. a(d(x,y ) v t )

23 23 Sketch of the Proof (bounded interactions only) Consider the function f : R A defined by f (t) := [τ t (A), B]. (4) Differentiate to see that f satisfies the following diff. eqn f (t) = i [f (t), τ t (H X )] i [τ t (A), [τ t (H X ), B]], (5) with the notation H Y = Z Λ: Z Y Φ(Z), (6) for any subset Y V. The first term in (5) above is norm-preserving, and therefore we have [τ t (A), B] [A, B] + 2 A t 0 [τ s (H X ), B] ds (7)

24 24 Define the quantity then (7) implies that [τ t (A), B] C B (X, t) := sup, (8) A A X A C B (X, t) C B (X, 0) + 2 Z V : Z X Clearly, one has that Φ(Z) t 0 C B (Z, s)ds. (9) C B (Z, 0) 2 B δ Y (Z), (10) where δ Y (Z) = 0 if Z Y = and δ Y (Z) = 1 otherwise. Using this fact, iterate (9) and find that (2 t ) n C B (X, t) 2 B a n, (11) n! n=0

25 25 where the coefficients are given by a n = Z1 Λ: Z 2 Λ: Z 1 X Z 2 Z 1 Zn Λ: Z n Z n 1 n Φ(Z i ) δ Y (Z n ). i=1 Using the properties of the function F a and the norm Φ a one can estimate a n : a 1 x Φ X y Φ Y Z Λ: x,y Z Φ(Z) x X Φ a F a (d(x, y)). y Y where we defined F a (r) = e ar F (r). Note that if F satisfies (i) and (ii) then so thus F a (r), for all a 0, with F a F, and C a C.

26 26 Next, a 2 x Φ X y Φ Y z Λ Z 1 Λ: Φ a Φ 2 a Φ(Z 1 ) x,z Z 1 x Φ X y Φ Y Φ 2 a C a x Φ X y Φ Y z Λ x Φ X y Φ Y Φ(Z 2 ) Z 2 Λ: z,y Z 2 F a (d(z, y)) Φ(Z 1 ) z Λ Z 1 Λ: x,z Z 1 F a (d(x, z)) F a (d(z, y)) F a (d(x, y)), Similarly a n Φ n ac n 1 a x Φ X y Φ Y F a (d(x, y)), and this completes the proof.

27 27 Comments: The existence of local approximations for τ t (A), for local A, by observables whose support has a radius v t has many applications. In addition to a few we will discuss, there are others such as speeding up and analyzing practical numerical algorithms for calculating ground states, equilibrium states, form factors, of quantum lattice models. Open problems: prove Lieb-Robinson bound for more general unbounded Hamiltonians, e.g., the anharmonic lattice with a quartic potential.

28 28 Existence of the dynamics in the thermodynamic limit It is well-known that one can use the Lieb-Robinson bound to establish the existence of the dynamics for infinite lattice systems. Let Λ n be an increasing exhausting sequence of finite subsets of an infinite system, V, with interaction Φ. The essential observation is the following bound: for n > m τt Λn (A) τt Λm (A) x Λ n\λ m X x t 0 [Φ(X ), τt Λm (A)] ds.

29 29 Theorem Let a 0, and Φ such that Φ a <. Then, the dynamics {τ t } t R corresponding to Φ exists as a strongly continuous, one-parameter group of automorphisms on A. In particular, Λn lim τt (A) τ t (A) = 0 n for all A A = n A Λ n. The convergence is uniform for t in compact sets and independent of the choice of exhausting sequence {Λ n }.

30 30 The Exponential Clustering Theorem In a relativistic quantum field theory, the speed of light plays the role of an automatic bound for the Lieb-Robinson velocity. This implies decay of correlations in the vacuum of a QFT with a gap and a unique (Ruelle, others). Fredenhagen proved an exponential bound for the decay: e γc 1 x, i.e., ξ c/γ. In a QFT the gap γ is interpreted as the mass of the lightest particle. In condensed matter physics, a gap also implies exponential decay under general conditions (Hastings 2004, N-Sims 2006, Hastings-Koma 2006).

31 31 Theorem (N-Sims 2006, Hastings-Koma 2006) Case (i): bounded interactions with Φ a < for some a > 0. Suppose H has a spectral gap γ > 0 above a unique ground state. Then, there exists µ > 0 and a constant c = c(f, γ) such that for all A A X, B A Y, AB A B c A B min( Φ X, Φ Y )e µd(x,y ). One can take µ = aγ γ + 4 Φ a.

32 32 Theorem (N-Raz-Schlein-Sims, 08) Let H be the anharmonic lattice Hamiltonian with λ 0 satisfying the conditions of Case (ii), and suppose H has a unique ground state and a spectral gap γ > 0 above it. Denote by the expectation in the ground state. Then, for all functions f and g with finite supports X and Y in the lattice, we have the following estimate: W (f )W (g) W (f ) W (g) d(x,y )/ξ C f g min( X, Y )e where ξ = (4av + γ)/(aγ) and, if we assume d(x, Y ) ξ, C is a constant depending only on the dimension ν.

33 33 Idea of the proof Suppose H 0 with unique ground state Ω, HΩ = 0, with a gap γ > 0 above 0. Let A A X and B A Y, d(x, Y ) > 0, a, C, v > 0, such that [τ t (A), B] C A B e a(d(x,y ) v t ). We can assume Ω, AΩ = Ω, BΩ = 0. We want to show that there is a ξ <, independent of X, Y, A, B, s.t. Ω, ABΩ Ce d(x,y )/ξ.

34 34 For z C, Im z 0, define f (z) = Ω, Aτ z (B)Ω = γ e ize d A Ω, P E BΩ. For T > b > 0, and Γ T the upper semicircle of radius T centered at 0: f (ib) = 1 2πi Γ T f (z) z ib dz. Then 1 Ω.ABΩ lim sup b 0,T 2π Γ T f (t) t ib dz.

35 35 Next, introduce a Gaussian cut-off and remember f (t): 1 Ω, ABΩ lim sup b 0,T 2π T T e αt2 Ω, Aτ t(b)ω t ib dt /(4α) +Ce γ2 assuming γ > 2αb. For α(d(x, Y )/v) 2 1, the Lieb-Robinson bounds lets us commute τ t (B) with A in this estimate. Using the spectral representation of τ t, we get 1 T Ω, ABΩ lim sup dt e iet e αt2 b 0,T 2π t ib d B Ω, P E AΩ +err. The t integral can be uniformly bounded by e γ2 /(4α). Optimizing α gives the result. γ T

36 36 Structure of gaped ground states The Exponential Clustering Theorem says that a non-vanishing gap γ implies a finite correlation length ξ. Can one say more about the structure of the ground state? E.g., with the goal of devising better algorithms to compute ground states? Related: use Lieb-Robinson bounds to help compute dynamical properties (Eisert, Hastings, Osborne, Verstraete,... ). The AKLT model and the structure of Valence Bond states give a hint. L 1 HL AKLT = [ S x S x (S x S x+1 ) 2 ] = x=1 L 1 x=1 P (2) x,x+1 acting on (C 3 ) L. P (2) x,x+1 is the projection onto the spin 2 subspaces of two spin 1 s at x and x + 1. E 0 = 0.

37 37 Let φ C 2 C 2 be the singlet state: φ = 1 2 ( ). Ground states of the AKLT model: for α, β =,, define ψ L) αβ = (P(1) P (1) ) α φ φ β Then, HL AKLT ψ (L) α,β = 0, four mutually almost orthogonal ground states. One can show that ker H AKLT = span{ψ (L) αβ α, β =, } and that γ > 0, uniform in L. The thermodynamic limit has a unique ground state with exponential decay of correlations. To describe the special structure of this ground state, suppose we have orthogonalized ψ (L) αβ. The density matrix of the limiting ground state restricted to A [1,L] is then given by 4ρ [1,L] = P [1,L] = ψ (L) αβ ψ (L) αβ α,β

38 38 Then, for l 0, a 1, such that [a l, a + l + 1] [1, L], one can show P [a l,a+l+1] [P [1,a] P [a+1,l] ] P [1,L] Ce l/ξ (12) This property allows one to prove a uniform lower bound for the gap (Fannes-N-Werner 1992, Spitzer-Starr 2002). Idea: the AKLT model may be somewhat special but the states but the property (12) is general for gapped ground states. The existence of a large class VBS models supports this; also a density result (Fannes, N, Werner 1992). VBS models also exist in higher dimensions and their ground state are sometimes called PEPS (Verstraete).

39 39 The Area Law for the entanglement entropy VBS states satisfy an area bound on the entropy of their local restrictions. Let X Λ and ρ X A X is the density matrix describing the restriction of a VBS state to A X, then S(ρ X ) = Trρ x log ρ X C X Conjecture: area law holds for gapped ground states in general of arbitrary quantum spin systems with bounded spins and bounded finite-range interactions. Next: An approximation theorem similar to (12) with Area Law for chains as an application.

40 40 An approximation theorem for gapped ground states We will consider a system of the following type: Let Λ be a finite subset of Z d. At each x Λ, we have a finite-dimensional Hilbert space of dimension n x. Let H V = Φ(x, y), {x,y} Λ, x y =1 with Φ(x, y) J. Suppose H V has a unique ground state and denote by P 0 the corresponding projection, and let γ > 0 be the gap above the ground state energy. For a set A Λ, the boundary of A, denoted by A, is A = {x A there exists y Λ \ A, with x y = 1}. and for l 1 define B(l) = {x Λ d(x, A) < l}.

41 41 l A B Λ

42 42 The following generalizes a result by Hastings (2007): Theorem (Hamza-Michalakis-N-Sims, in prep.) There exists ξ > 0 (given explicitly in terms of d, J, and γ), such that for any sufficiently large m > 0, and any A Λ, there exist two orthogonal projections P A A A, and P Λ\A A Λ\A, and an operator P B A B(m) with P B 1, such that P B (P A P Λ\A ) P 0 C(ξ) A 2 e m/ξ where C(ξ) is an explicit polynomial in ξ.

43 43 Sketch of the proof (several ideas by Hastings & many estimates by Hamza and Sims) (1) The first step is the bring the Hamiltonian in a form similar to the Hamiltonian of the AKLT model. Assume E 0 = 0. We aim at a decomposition, for each sufficiently large l, H Λ = K A + K B(l) + K Λ\A, with the following properties: - supp K X X, for X = A, B(l), Λ \ A. - K X ψ 0 e cl, for each X and for some c > 0 (we only assumed H Λ ψ 0 = 0).

44 44 We start from where H Λ = H I + H B + H E, (13) I = I (l) = {x A for all y A, d(x, y) l} E = E(l) = {x V \ A for all y A, d(x, y) l}. The sets I (l) and E(l) are the interior and exterior of A. B(l) is boundary of thickness 2l: B(l) = {x Λ d(x, A) < l}. Note that Λ is the disjoint union of I, B, and E. Now define H I = X Λ: X I Φ(X ), H B = X Λ: X B Φ(X ), H E = X Λ: X E Φ(X ). For l > 1, there are no repeated terms and (13) holds.

45 45 H X ψ 0 is not necessarily small but we can arrange it so that each term has 0 expectation in ψ 0. The next step: for X {I, B, E} define (H X ) α = for α > 0. We still have α τ t (H X ) e αt2 dt, π H Λ = (H I ) α + (H B ) α + (H E ) α, We can show H X ψ 0 is small for α small, but the support is no longer X. The easiest way to correct this is by redefining them with a suitably restricted dynamics:

46 46 e.g., K (α) α A = e ith A H I e ith A e αt2 dt, π and similarly define K (α) Λ\A using H Λ\A, and K (α) B using H B(2l). A good choice for α is av 2 /(2l), where a and v are the constants appearing in the Lieb-Robinson bound. With this choice all errors are bounded by ɛ(l) C(d, a, v)j 2 A l d 1/2 e l/ξ with ξ = 2 max(a 1, av 2 /γ 2 )

47 47 To summarize: H Λ (K (α) A and for X = A, B, Λ \ A, + K (α) B K (α) X ψ 0 ɛ(l) + K (α) Λ\A ) ɛ(l)

48 48 (2) Now define the projections P A and P Λ\A as the spectral projections of K (α) A and K (α) Λ\A corresponding to the eigenvalues ɛ(l). This gives (1l P A )ψ 0 1 K (α) ψ 0 ɛ(l) ɛ(l) and similary for P Λ\A. Since the projections commute we have the identity 2(1l P A P Λ\A ) = (1l P A )(1l + P Λ\A ) + (1l P Λ\A )(1l + P A ), and we get P 0 P 0 P A P Λ\A ) = P 0 (1l P A P Λ\A ) 2 ɛ(l). A

49 49 (3) As the final step, we need to replace the P 0 multiplying P A P Λ\A by something supported in B(m) for a suitable m. We start from the observation that for a s.a. operator with a gap, such as H Λ, the ground state projection P 0 can be approximated by P α defined by P α = α e ith V e αt2 dt. π If the gap is γ, and with our choice of α, we have P α P 0 e γ2 /(4α) e l/ξ

50 50 We will modify this formula for P α in two ways: replace e ith Λ by e (α) (α) (α) it(k A +K B +K Λ\A ) (α) (α) it(k A +K e result with an operator supported in B(3l): Λ\A ), then approximate the P B = α e π P B = Tr HΛ\B(3l) PB (α) (α) (α) it(k A +K B +K Λ\A ) (α) (α) it(k A +K e Λ\A ) e αt2 dt Then, P 0 P A P Λ\A P B P A P Λ\A and P B P B can both be estimated to be small.

51 51 Area Law for Gapped 1-Dim l Systems Suppose d = 1, V = [1, L] Z and n x n, for all x [1, L]. For 1 a b L, let ρ [a,b] denote the density matrix of P 0 restricted to A [a,b]. Theorem (Hastings 07) There exist constants C and ξ < depending only on J and γ, such that S(ρ [a,b] ) Cξ log(ξ) log(n)2 ξ log(n) One can take ξ = 12v/γ.

52 52 The approximation theorem is not sufficient to prove the area law by itself, but it is an essential step. Here is how. Consider the (essentially unique) Schmidt decomposition of the pure state ψ 0 H A H Λ\A : r ψ 0 = σ α ψa α ψλ\a α, α=1 with o.n. vectors ψx α H X and σ α > 0. r is called the Schmidt rank of ψ 0. For any density matrix ρ, we have S(ρ) log(rankρ) If ρ A is the restriction of ψ 0, the rank of ρ A is given by the Schmidt rank of ψ 0 with respect to this decompostion. Therefore, we would like to show that the Schmidt rank is bounded by C A.

53 53 The rank of ρ A when P 0 is of the form P B (P A P Λ\A ) can be bounded in a similar way. First, suppose P A and P Λ\A are of rank 1, and P B = R E j A E j Λ\A j=1 for operators E j X A X. Then P 0 is of rank one, projecting on a vector of Schmidt rank R. Note that supp P B B implies R (dim H B A ) 2. By using the spectral decomposition for general P A and P Λ\A, we see that rankρ A R (dim H B A ) 2 rankp A rankp Λ\A.

54 54 For general models, one has to struggle with the fact that it is not clear how to bound rankp A and rankp Λ\A. In one dimension, Hastings succeeded in bypassing this difficulty in a circuitous way. For VBS models, in any dimension, rankp A and rankp Λ\A are easily seen to be bounded by c A. So all VBS models satisfy an area law (even without a gap, as long as they satisfy a non-degeneracy condition). For the AKLT model we obviously have rankp [a,b] 4.

55 55 Lieb-Schultz-Mattis Theorem The Lieb-Schultz-Mattis (LSM) theorem is a classic result (Annals of Physics, 1961) about the spin-1/2 Heisenberg antiferromagnetic chain. It was generalized to other 1D models by Affleck and Lieb (Lett. Math. Phys., 1985). A standard example where the LSM theorem applies is the half-integer spin antiferromagnetic Heisenberg chain: V = [1, L]: L 1 H L = S x S x+1 x=1 It also applies to the XXZ chain in the disordered regime.

56 56 The LSM Theorem states that if the ground state of H L is unique, then the gap to the first exited state is bounded by C/L. A result of Lieb and Mattis (1966), shows that for the particular model H L with L even, this is indeed the case. The generalization by Affleck and Lieb in 1985 includes chains with spins of arbitrary half-integer magnitude. I will discuss the proof of a higher-dimensional version of this result tas given by N-Sims (CMP 2007), which closely follows Hastings (PRB 2004).

57 57 Setup For simplicity, we will only consider systems defined on subsets V Z d, d 1 of the following form: V = [1, L] V L Zd, with V L Zd 1, such that V L CLd 1. We will assume periodic boundary conditions in the first coordinate. The spin systems on each copy of VL same. V L are the can have a mixture of spins of different magnitudes, s x, but we have to assume that the total spin in each copy of V is a half-integer: x VL some nonnegative integer k. s x = k + 1/2, for

58 58 V L V L V L V L x 1 = 1 x 1 = m x 1 = L

59 59 The new LSM Theorem The result holds for a large class of models H V. For this talk it suffices to focus on the simplest examples. E.g., the theorem applies is the half-integer spin antiferromagnetic Heisenberg Hamiltonian on V, with odd VL, and translation invariance in the first coordinate. Theorem (Hastings PRB 2004, N-Sims CMP 2007) If the ground state of H V is non-degenerate, then the gap above it, γ L, satisfies for some constant C and all L. γ L C log L L

60 60 The interactions should be finite-range and real, but are not limited to pair interactions. Only rotation invariance about one axis is used, but the ground state has to be non-degenerate. The spins can have a mixture of magnitudes as long as they are uniformly bounded and the total spin in each copy of V L is half-integral. The lattice structure of V L is not used except for the translation invariance in the horizontal direction.

61 61 Structure of the proof If ψ 0 is the normalized unique ground states, the variational principle gives, for any normalized ψ 1 such that ψ 0, ψ 1 1, the bound 0 < γ ψ 1, (H E 0 )ψ 1 1 ψ 0, ψ 1 2 So, there are three steps to obtain such a bound: (1) construction of the trial state ψ 1 (2) estimate of its energy (3) estimate of its inner product with the ground state. The upper bounds provided by these estimates contain γ L in such a way the assumption that γ L > C(log L)/L, with sufficiently large C, leads to a contradiction.

62 62 The proof is complicated by the fact that the ground state is unknown, but interesting because it required new (mathematical) understanding of the general properties of quantum spin dynamics and ground state correlations. Conceptually, the variational state is the ground state of a modified Hamiltonian, H θ, in which the interactions in one hyperplane have been twisted by an angle θ. The idea of Hastings (2004) was to describe this state as the solution of a differential equation in the variable θ with the original ground state as initial condition. Lieb-Robinson bounds and the Exponential Clustering Theorem play a crucial role in estimating the energy of the variational state and proving its orthogonality to the ground state.

63 63 1: Construction of the trial state Twisted interactions: let (x, y) be a horizontal bond in V : x = (m, v), y = (m + 1, v), for some m [1, L], and v V L. Then, for θ R we define the twisted Heisenberg interaction at (x, y) by h xy (θ) = S x e iθs3 y Sy e iθs3 y Twisted Heisenberg Hamiltonian on V L = [1, L] VL : H θ,θ = h xy (θ xy ) x y =1 where θ if x = (m, v), y = (m + 1, v) for some v VL θ xy = θ if x = (m + L/2, v), y = (m L/2, v) for v VL 0 else

64 64 V L V L V L θ θ L/2 x 1 = 1 m m + 1 x 1 = L

65 65 The three lowest energies of H(θ, 0), as a function of θ, behave differently depending on whether the spins are half-integer (left), or integer (right): 2.9 Spin 1/2 10 Spin Smallest three eigenvalues Smallest three eigenvalues pi/2 pi 3pi/2 2pi Twist angle pi/2 pi 3pi/2 2pi Twist angle

66 66 Hastings idea (PRB 2004): in the half-integer spin case we can obtain the first excited state by applying a quasi-adiabatic evolution to the ground state, where θ is the evolution parameter. It is easy to see that H θ, θ is unitarily equivalent to H = H 0,0. In particular, for the ground state energy E 0 (θ, θ ) we have θ E 0 (θ, θ) = 0. By using this in the derivative of the eigenvalue equation H θ, θ ψ 0 (θ, θ) = E 0 ψ 0 (θ, θ), one obtains 1 ψ 0 (θ) = [ θ H θ, θ ]ψ 0 (θ). H θ, θ E 0

67 67 This can formally be written using the Heisenberg dynamics τ t (A) = e ith Ae ith, for any observable A, as follows: ψ 0 (θ) = 0 τ it ( θ H θ, θ )ψ 0 (θ) dt To make things well-defined we introduce parameters a, T > 0 and define where B a,t (A, H) = T 0 A a (it, H) = 1 2πi dt[a a (it, H) A a (it, H) ] + ds e as2 s it τ s(a)

68 68 Note that θ H θ, θ contains terms localized at bonds (x, y) of the form x = (m, v), y = (m + 1, v) and x = (m + L/2, v), y = (m L/2, v), for some v VL. We denote these two types of terms grouped together by 1 H and 2 H. Then, B a,t ( θ H θ, θ, H) = B a,t ( 1 H, H) B a,t ( 2 H, H) Hastings insight: we can use this equation for ψ 0 (θ, θ) with B a,t ( θ H θ, θ, H) replaced by B a,t ( 1 H, H), and suitably chosen parameters a and T, to obtain an equation for an approximation of ψ 0 (θ, 0).

69 69 Hastings Equation for the variational state is then: θ ψ a,t (θ) = B a,t (θ)ψ a,t (θ), with B a,t (θ) = B a,t ( 1 H, H θ, θ ) and initial condition ψ a,t (0) = ψ 0 (0, 0) Then, ψ 1 = ψ a,t (2π), with a = γ L /L, and T = L/2, is the variational state we will use. Remarks: 1) this is equation is norm preserving. 2) It is easy to derive on O(1) a priori upper bound for γ L.

70 70 Locality property of the trial state Let Λ m,l V L be the subvolume of the form [m + 2 L/4, m 2 + L/4] VL. This is slightly less than half of the system. Let ρ a,t (θ) and ρ 0 (θ, θ) be the density matrices corresponding to the states ψ a,t (θ) and ψ 0 (θ, θ). Theorem Suppose there exists a constant c > 0 such that Lγ L c and choose a = γ L /L and T = L/2. Then, there exists constants C > 0 and k > 0 such that sup Tr VL \Λ m,l [ρ a,t (θ) ρ 0 (θ, θ)] 1 CL 2d e klγ L. θ [0,2π]

71 71 This theorem states that dropping one of the terms in the B operator in Hastings Equation does not change much the effect of the other term in its own half of the system. The proof of this theorem combines to important facts: (i) ψ a,t (θ) X (θ)ψ 0 (0, 0), for some X (θ) A Λm,L, i.e., an operator localized in Λ m,l. (ii) The spatial correlations in ψ 0 (0, 0) decay exponentially fast with a rate γ L. These two facts are provided by the Lieb-Robinson bound and the Exponential Clustering Theorem, respectively.

72 72 2: Energy Estimate The main idea here is to use the locality property of the trial state and the unitary equivalence of the Hamiltonians H θ, θ to obtain a uniform estimate of the derivative of the function E(θ) = ψ a,t (θ), H θ,0 ψ a,t. Since E(0) = E 0 and E(2π) is the energy of the trial state ψ 1 = ψ a,t (2π), this provides a bound for the difference. Explicitly, we can prove the following result. Theorem Suppose there exists a constant c > 0 such that Lγ L c and choose a = γ L /L and T = L/2. Then, there exists constants C > 0 and k > 0 such that ψ 1, H L ψ 1 E 0 CL 3d 1 e klγ L.

73 73 3: Orthogonality Estimate By construction, the quasi-adiabatic dynamics is norm preserving. So, ψ 1 = ψ 0 = 1. To prove that ψ 1 is sufficiently orthogonal to ψ 0 we need to use that the total spin in each transversal system V L is half-integral. Theorem Suppose there exists a constant c > 0 such that Lγ L c and choose a = γ L /L and T = L/2. Then, there exists constants C > 0 and k > 0 such that ψ a,t (2π), ψ 0 CL 2d e klγ L.

74 74 This result is proved by using the invariance of ψ 0 (θ, θ) under twisted translations. Let T be a unitary implementing the translation symmetry in the horizontal direction (in which we have periodic boundary conditions), chosen such that T ψ 0 = ψ 0. This is possible since ψ 0 is the unique ground state and T HT = 0. Define T θ,θ = TU m (θ)u m+l/2 (θ ) where U n (θ) applies the rotation e iθs3 x x = (n, v), for some v V L. to all sites for the form

75 75 Then, it is easy to see that H θ, θ = W H(0, 0)W, with W = U n (θ) m<n m+l/2 and that W TW = T θ, θ commutes with H θ, θ. Therefore T θ, θ ψ 0 (θ, θ) = ψ 0 (θ, θ).

76 76 The idea is again to use the locality properties to study the solution of Hastings Equation. We can show T θ,0 ψ a,t (θ) = ψ a,t (θ). Note that, due to the half-integral total spin in each VL, U m (2π) = 1l. Therefore T 2π,0 = T and therefore T ψ 1 = ψ1. Since T ψ 0 = ψ 0, this implies that ψ 1 is (nearly) orthogonal to ψ 0.

77 77 Concluding remarks Studying specific models is important but thinking about general mechanisms is useful too. The fun is not over yet! There are enough open problems for everyone: - area law in d > 1 - Haldane s conjecture - general stability of gapped ground states - lower bounds for small gaps; nature of the excitations - long-time dynamics? - topological phases, what do they really look like? - provable convergence of numerical algorithms

78 78 Some References 1. O. Bratteli and D.W. Robinson D.W., Operator Algebras and Quantum Statistical Mechanics. Volume 2., Second Edition. Springer-Verlag, E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. (N.Y.) 16, (1961) 3. E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Comm. Math. Phys. 28, (1972) 4. M.B. Hastings, Lieb-Schultz-Mattis in higher dimensions, Phys. Rev. B 69, (2004) 5. B. Nachtergaele and R. Sims: Lieb-Robinson Bounds and the Exponential Clustering Theorem. Comm. Math. Phys. 265, (2006) 6. M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265, (2006) 7. B. Nachtergaele, Y. Ogata, and R. Sims: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124, no. 1, 1 13 (2006) 8. B. Nachtergaele and R. Sims: A multi-dimensional Lieb-Schultz-Mattis theorem. Comm. Math. Phys. 276, (2007), 9. B. Nachtergaele, H. Raz, B. Schlein, and R. Sims, Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems, Commun. Math. Phys., to appear, arxiv.org: B. Nachtergaele, and R. Sims: Locality Estimates for Quantum Spin Systems, in V. Sidovaricius (Ed), Proceedings of ICMP XV, Rio de Janeiro 2006, to appear, arxiv: E. Hamza, S. Michalakis, B. Nachtergaele, and R. Sims, in preparation.