1 Kyoto, July 29, 2013 Gapped Ground State Phases of Quantum Lattice Systems 1 Bruno Nachtergaele (UC Davis) based on joint work with Sven Bachmann, Jutho Haegeman, Eman Hamza, Spirydon Michalakis, Tobias Osborne, Norbert Schuch, Robert Sims, Frank Verstraete, and Amanda Young. 1 Work supported by the National Science Foundation (DMS-1009502).
2 Outline ◮ What is a gapped ground state phase? ◮ Automorphic equivalence ◮ Example: the AKLT model ◮ Symmety protected phases ◮ Particle-like elementary excitations ◮ Concluding remarks: locality and its implications
3 What is a quantum ground state phase? By phase, here we mean a set of models with qualitatively similar behavior. E.g., a g.s. ψ 0 of one model could evolve to a g.s. ψ 1 of another model in the same phase by some physically acceptable dynamics and in finite time. For finite systems such a dynamics is provided by a quasi-local unitary U Λ . When we take the thermodynamic limit Λ ↑ Γ U ∗ lim Λ AU Λ = α ( A ) , A ∈ A Λ 0 , this dynamics converges to an automorphism of the algebra of observables. To make this more precise, we need some notation.
4 Ground states of quantum spin models By quantum spin system we mean quantum systems of the following type: ◮ (finite) collection of quantum systems labeled by x ∈ Λ, each with a finite-dimensional Hilbert space of states H x . E.g., a spin of magnitude S = 1 / 2 , 1 , 3 / 2 , . . . would have H x = C 2 , C 3 , C 4 , . . . . ◮ The Hilbert space describing the total system is the tensor product � H Λ = H x . x ∈ Λ with a tensor product basis |{ α x }� = � x ∈ Λ | α x �
5 ◮ 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 ⊗ 1 l Λ \ X ∈ A Λ . Then �·� � A = A Λ Λ A common choice for Λ’s are finite subsets of a graph Γ (often called the ‘lattice’). E.g., if Γ = Z ν , we may consider Λ of the form [1 , L ] ν or [ − N , N ] ν .
6 Interactions, Dynamics, Ground States The Hamiltonian H Λ = H ∗ Λ ∈ A Λ is defined in terms of an interaction Φ: for any finite set X , Φ( X ) = Φ( X ) ∗ ∈ A X , and � H Λ = Φ( X ) X ⊂ Λ For finite-range interactions, Φ( X ) = 0 if diam X ≥ R . Heisenberg Dynamics: A ( t ) = τ Λ t ( A ) is defined by τ Λ t ( A ) = e itH Λ Ae − itH Λ For finite systems, ground states are simply eigenvectors of H Λ belonging to its smallest eigenvalue (sometimes several ‘small eigenvalues’).
7 The quasi-locality property is expressed as follows: there exists a rapidly decreasing function F ( d ), such that for observables A supported in a set X ⊂ Γ, there exists A d ∈ A X d such that � α ( A ) − A d � ≤ � A � F ( d ) where X d ⊂ Γ is all sites of distance ≤ d to X . α is the time evolution for a given unit of time. For a short-range real dynamics we would have something of the form � τ t ( A ) − A d � ≤ � A � F ( d − v | t | ) where v is often referred to as the Lieb-Robinson velocity.
8 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 then Y ∩ supp A = ∅ . So, one can find the support of A by looking which B it commutes with. A more general statement is true: if the commutators are uniformly small for B ∈ A Y , then A is close to A Λ \ Y .
9 Lemma Let A ∈ A Λ , ǫ ≥ 0 , and Y ⊂ Λ be such that � [ A , B ] � ≤ ǫ � B � , for all B ∈ A Y (1) then there exists A ′ ∈ A Λ \ Y such that � A ′ ⊗ 1 l − A � ≤ ǫ ⇒ we can investigate supp τ Λ t ( A ) by estimating [ τ Λ t ( A ) , B ] for B ∈ A Y . This is what Lieb-Robinson bounds are all about.
10 Lieb-Robinson bounds Theorem ( Lieb-Robinson 1972, Hastings-Koma 2006, N-Sims 2006, N-Ogata-Sims 2006) Let F : [0 , ∞ ) → (0 , ∞ ) be a suitable non-increasing function such that the interaction Φ satisfies F ( d ( x , y )) − 1 � � Φ � F = sup � Φ( X ) � < ∞ x � = y X ∋ x , y Then, ∃ constants C and v (depending only on F , � Φ � F , and the lattice dimension, s. t. ∀ A ∈ A X and B ∈ A Y , � ≤ C � A � � B � min( | X | , | Y | ) e v | t | F ( d ( X , Y )) . � � [ τ Λ � t ( A ) , B ] where d ( X , Y ) is the distance between X and Y .
11 Suppose Φ 0 and Φ 1 are two interactions for two models on lattices Γ. Each has its set S i , i = 0 , 1, of ground states in the thermodynamic limit. I.e., for ω ∈ S i , there exists � ψ Λ n g.s. of H Λ n = Φ i ( X ) , X ⊂ Λ n for a sequence of Λ n ∈ Γ such that ω ( A ) = lim n →∞ � ψ Λ n , A ψ Λ n � .
12 If the two models are in the same phase, we have a suitably local automorphism α such that S 1 = S 0 ◦ α This means that for any state ω 1 ∈ S 1 , there exists a state ω 0 ∈ S 0 , such that the expectation value of any observable A in ω 1 can be obtained by computing the expectation of α ( A ) in ω 0 : ω 1 ( A ) = ω 0 ( α ( A )) . The quasi-local character of α guarantees that the support of α ( A ) need not be much larger than the support of A in order to have this identity with small error. Where do such quasi-local automorphisms α come form?
13 Fix some lattice of interest, Γ and a sequence Λ n ↑ Γ. Let Φ s , 0 , ≤ s ≤ 1, be a differentiable family of short-range interactions for a quantum spin system on Γ. Assume that for some a , M > 0, the interactions Φ s satisfy e ad ( x , y ) � sup � Φ s ( X ) � + | X |� ∂ s Φ s ( X ) � ≤ M . x , y ∈ Γ X ⊂ Γ x , y ∈ X E.g, Φ s = Φ 0 + s Ψ with both Φ 0 and Ψ finite-range and uniformly bounded. Let Λ n ⊂ Γ, Λ n → Γ, be a sequence of finite volumes, satisfying suitable regularity conditions and suppose that the spectral gap above the ground state (or a low-energy interval) of � H Λ n ( s ) = Φ s ( X ) X ⊂ Λ n is uniformly bounded below by γ > 0.
14 Theorem (Bachmann, Michalakis, N, Sims (2012)) Under the assumptions of above, there exist a co-cycle of automorphisms α s , t of the algebra of observables such that S ( s ) = S (0) ◦ α s , 0 , for s ∈ [0 , 1] . The automorphisms α s , t can be constructed as the thermodynamic limit of the s-dependent “time” evolution for an interaction Ω( X , s ) , which decays almost exponentially. Concretely, the action of the quasi-local transformations α s = α s , 0 on observables is given by n →∞ V ∗ α s ( A ) = lim n ( s ) AV n ( s ) where V n ( s ) solves a Schr¨ odinger equation: d ds V n ( s ) = iD n ( s ) V n ( s ) , V n (0) = 1 l , with D n ( s ) = � X ⊂ Λ n Ω( X , s ) .
15 The α t , s satisfy a Lieb-Robinson bound of the form � [ α t , s ( A ) , B ] � ≤ � A �� B � min( | X | , | Y | ) e C | t − s | F ( d ( X , Y )) , where A ∈ A X , B ∈ A Y , d ( X , Y ) is the distance between X and Y . F ( d ) can be chosen of the form d − b (log d )2 . F ( d ) = Ce with b ∼ γ/ v , where γ and v are bounds for the gap and the Lieb-Robinson velocity of the interactions Φ s , i.e., b ∼ a γ M − 1 .
16 The AKLT model (Affleck-Kennedy-Lieb-Tasaki, 1987) Antiferromagnetic spin-1 chain: [1 , L ] ⊂ Z , H x = C 3 , L L � 1 l + 1 2 S x · S x +1 + 1 � � � P (2) 6( S x · S x +1 ) 2 H [1 , L ] = 3 1 = x , x +1 x =1 x =1 The ground state space of H [1 , L ] is 4-dimensional for all L ≥ 2. In the limit of the infinite chain, the ground state is unique, has a finite correlation length, and there is a non-vanishing gap in the spectrum above the ground state (Haldane phase). Theorem (Bachmann-N, CMP 2013, to appear) There exists a curve of uniformly gapped Hamiltonians with nearest neighbor interaction s �→ Φ s such that Φ 0 is the AKLT interaction and Φ 1 defines a model with a unique ground state of the infinite chain that is a product state.
17 J 2 Sutherland SU(3) AKLT J 1 ferro Haldane dimer Bethe Ansatz Potts SU(3) H = � x J 1 S x · S x +1 + J 2 ( S x · S x +1 ) 2
18 Symmetry protected phases in 1 dimension For a given system with λ -dependent G -symmetric interactions, we would like to find criteria to recognize that the system at λ 0 is in a different gapped phase than at λ 1 , meaning that the gap above the ground state necessarily closes for at least one intermediate value of λ . This is the same problem as before but restricted to a class of models with a given symmetry group (and representation) G . Our goal is to find invariants, i.e., computable and, in principle, observable quantities that can be different at λ 0 and λ 1 , only if the model is in a different ground state phase.
19 The case G = SU (2) and the Excess Spin Models to keep in mind: antiferromagnetic chains in the Haldane phase and generalizations. Unique ground state with a spectral gap and an unbroken continuous symmetry. Let S i x , i = 1 , 2 , 3, x ∈ Z , denote the i th component of the spin at site x . Claim: one can define + ∞ � S i x , x =1 as s.a. operators on the GNS space of the ground state and they generate a representation of SU (2) that is characteristic of the gapped ground state phase. We can prove the existence of these excess spin operators for two classes of models (Bachmann-N, arXiv:1307.0716): 1) models with a random loop representation; 2) models with a matrix product ground state (MPS).
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