Invariants of ground state phases in one dimension Sven Bachmann Mathematisches Institut Ludwig-Maximilians-Universität München Joint work with Yoshiko Ogata and Bruno Nachtergaele Warwick Symposium on Statistical Mechanics: Many-Body Quantum Systems Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 1 / 20
What is a quantum phase transition? A simple answer: A phase transition at zero temperature A slightly more precise answer: Consider: ⊲ A smooth family of Hamiltonians H ( s ) , s ∈ [0 , 1] ⊲ The associated family of ground states Ω i ( s ) ⊲ A quantum phase transition occurs at singularities of s �→ Ω i ( s c ) In this talk: ⊲ Quantum spin systems ⊲ Hamiltonians H Λ ( s ) are continuously differentiable ⊲ Spectral gap above the ground state energy γ Λ ( s ) such that � > 0 ( s � = s c ) γ Λ ( s ) ≥ γ ( s ) ∼ C | s − s c | µ ( s → s c ) QPT Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 2 / 20
Local vs topological order Ordered phases ∼ non-unique ground state ⊲ The usual picture: Local order parameter distinguishes between possible ground states Example: Local magnetization in the quantum Ising model ⊲ ‘Topological order’: Local disorder, for any local A , � P Λ AP Λ − C A · 1 � ≤ C | Λ | − α , C A ∈ C , P Λ : The spectral projection associated to the ground state energy The ground state space depends on the topology of the lattice Example: Ground state degeneracy in Kitaev’s 2d model Basic question: What is a ground state phase? Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 3 / 20
Automorphic equivalence � H Λ ( s ) = Φ X ( s ) , s ∈ [0 , 1] X ⊂ Λ with s �→ Φ X ( s ) of class C 1 , and uniform spectral gap: γ := Λ ⊂ Γ ,s ∈ [0 , 1] γ Λ ( s ) > 0 inf Define S Γ ( t ) : ground state space on Γ at s = t . Then there exists an automorphism α t 1 ,t 2 of A Γ such that Γ S Γ ( t 2 ) = S Γ ( t 1 ) ◦ α t 1 ,t 2 Γ α t 1 ,t 2 is local: satisfies a Lieb-Robinson bound Γ Now: Invariants of the equivalence classes? Classification of phases? Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 4 / 20
Finitely correlated states A special class of states on a spin chain A Z with local algebra A ⊲ A finite dimensional C*-algebra B ⊲ A completely positive map E : A ⊗ B → B ⊲ Two positive elements e ∈ B and ρ ∈ B ∗ such that E (1 ⊗ e ) = e, ρ ◦ E (1 ⊗ b ) = ρ ( b ) Notation: E ( A ⊗ b ) = E A ( b ) . Finitely correlated state: ω ( A n ⊗ · · · ⊗ A m ) := ρ ( e ) − 1 ρ ( E A n ◦ · · · ◦ E A m ( e )) Exponential decay of correlations if σ ( E 1 ) \ { 1 } ⊂ { z ∈ C : | z | < 1 } ω ( A ⊗ 1 ⊗ l ⊗ B ) = ρ ( e ) − 1 ρ � E A ◦ ( E 1 ) l ◦ E B ( e ) � Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 5 / 20
Finitely correlated states ⊲ ‘Finite correlation’: The set of functionals on A N defined by ω X ( A ) = ω ( X ⊗ A ) , with X ∈ A Z \ N , generates a finite dimensional linear space. ⊲ Purely generated FCS: Consider B = M k and E ( A ⊗ b ) = V ∗ ( A ⊗ b ) V for V : C k → C n ⊗ C k . ⊲ In a basis { e µ } of C n : V χ = � n µ =1 e µ ⊗ v ∗ µ χ with v i ∈ M k i.e. n � � e µ , Ae ν � v µ bv ∗ E ( A ⊗ b ) = (MPS) ν µ,ν =1 Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 6 / 20
Example: the AKLT model ⊲ Affleck-Kennedy-Lieb-Tasaki, 1987 ⊲ SU(2) -invariant, antiferromagnetic spin- 1 chain ⊲ Nearest-neighbor interaction b − 1 b − 1 � 1 2 ( S x · S x +1 ) + 1 6 ( S x · S x +1 ) 2 + 1 � P (2) � � H [ a,b ] = = x,x +1 3 x = a x = a where P (2) x,x +1 is the projection on the spin- 2 space of D 1 ⊗ D 1 ⊲ Uniform spectral gap γ of H [ a,b ] , γ > 0 . 137194 ⊲ Ground state is finitely correlated: B = M 2 and ( D 1 ⊗ D 1 / 2 ) V = V D 1 / 2 Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 7 / 20
Hamiltonians Let V = ( v 1 , . . . , v n ) ∈ B n,k ( p, q ) and ω V be such that ⊲ v i ∈ M k ⊲ spectral radius of E V 1 is 1 , and it is a non-degenerate eigenvalue ⊲ σ ( E V 1 ) \ { 1 } ⊂ { z ∈ C : | z | < 1 } trivial peripheral spectrum ⊲ there are projections p, q such that pe V p and qρ V q are invertible Then there is a canonical Hamiltonian H V ,p,q such that ⊲ positive, finite range interaction ⊲ uniform spectral gap above the ground state energy ⊲ ground state spaces: S [1 , ∞ ) ∼ S ( −∞ , 0] ∼ S Z = { ω V } , = M ∗ = M ∗ dim( p ) dim( q ) Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 8 / 20
Invariants of gapped phases Theorem. Consider I ∈ B n,k i ( p i , q i ) and F ∈ B n,k f ( p f , q f ) and the canonically associated Hamiltonians H I ,p i ,q i , H F ,p f ,q f . There is a continuous path H ( s ) , s ∈ [0 , 1] such that 1. H (0) = H I ,p i ,q i and H (1) = H F ,p f ,q f 2. H ( s ) are uniformly gapped 3. There is a unique ground state on Z if and only if dim( p i ) = dim( p f ) and dim( q i ) = dim( q f ) . � � In words: The pair dim( p ) , dim( q ) is the invariant of the gapped phase with a unique state on Z . Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 9 / 20
Corollary & Comments Corollary. Each gapped phase contains a model with a pure product state in the thermodynamic limit Remarks: ⊲ The theorem emphasizes the role of edge states in the non-trivial classification of gapped phases in d = 1 ⊲ No bulk-edge correspondence ⊲ No symmetry requirements ⊲ Conjecture: The theorem extends to arbitrary gapped models with a unique ground state in the thermodynamic limit ⊲ The interaction length is constant and the smallest such l is l ≤ ( k 2 − n + 1) k 2 ⊲ The case of the AKLT model: belongs to the phase (2 , 2) Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 10 / 20
About the proof Key: E V ω V H V V = ( v 1 , . . . , v n ) − → − → − → Gap ( E V Gap ( H V ) 1 ) − → and i.e. Construct a gapped path of Hamiltonians by constructing a path V ( s ) with the right properties But: V �→ H V not always continuous! The theorem reduces to a statement about the pathwise connectedness of a certain subspace of ( M k ) × n Note: n � v µ bv ∗ E V 1 ( b ) = µ µ =1 the matrices v µ are the Kraus operators for the CP map E V 1 . Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 11 / 20
Primitive maps One way to enforce the spectral gap condition: Perron-Frobenius theory ⊲ Irreducible positive map = ⇒ 1. Spectral radius r is a non-degenerate eigenvalue 2. Corresponding eigenvector e > 0 3. Eigenvalues λ with | λ | = r are r e 2 π i α/β , α ∈ Z /β Z ⊲ A primitive map is an irreducible map with β = 1 Lemma. A CP map with Kraus operators { v 1 , . . . , v n } is primitive iff there exists m ∈ N such that span { v µ 1 · · · v µ m : µ i ∈ { 1 , . . . , n }} = M k Note: m fixed! Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 12 / 20
Primitive maps How to construct paths of primitive maps? Consider k � � � Y n,k := V : v 1 = λ α | e α � � e α | , and � v 2 e α , e β � � = 0 α =1 with the choice ( λ 1 , . . . , λ k ) ∈ Ω := { λ i � = 0 , λ i � = λ j , λ i /λ j � = λ k /λ l } Then, | e α � � e β | ∈ span { v µ 1 · · · v µ m : µ i ∈ { 1 , 2 }} for m ≥ 2 k ( k − 1) + 3 . Problem reduced to the pathwise connectedness of Ω ⊂ C k Use transversality theorem Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 13 / 20
Backbone of proof 1. Embed I , F into a common matrix algebra M k 2. Construct V ( s ) , s ∈ [0 , 1] such that ⊲ V (0) = I , V (1) = F ⊲ V ( s ) ∈ Y n,k for s ∈ (0 , 1) At the edges s ∈ { 0 , 1 } : perturb the Jordan blocks of v 1 3. If dim( p i ) = dim( p f ) , then p f = u ∗ p i u and interpolate in SU( k ) If dim( q i ) = dim( q f ) , then q f = w ∗ q i w and interpolate in SU( k ) Result: continuous V ( s ) , p ( s ) , q ( s ) generating a continuous H ( s ) := H V ( s ) ,p ( s ) ,q ( s ) with uniform spectral gap Note: If dim( p i ) � = dim( p f ) then dim( S i, [0 , ∞ ) ) � = dim( S f, [0 , ∞ ) ) : There is no automorphism, different phases Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 14 / 20
Local symmetries Next question: What if H ( s ) all share a symmetry? Automorphic equivalence and local symmetries: ⊲ Lie group G , and π g the action of G on A Γ ⊲ G is a local symmetry of the interaction if π g (Φ X ( s )) = Φ X ( s ) for all g ∈ G , X ⊂ Γ and s ∈ [0 , 1] Then: ◦ π g = π g ◦ α t 1 ,t 2 α t 1 ,t 2 Γ Γ i.e. α t 1 ,t 2 is compatible with the symmetries Γ Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 15 / 20
Edge representations Let now Π Γ ( s ) be the subrepresentation of G on S Γ ( s ) Proposition. Assume H ( s ) , s ∈ [0 , 1] is a smooth path of gapped Hamiltonians with G -invariant interactions. Then the representations Π Γ ( t 1 ) and Π Γ ( t 2 ) are equivalent for all t 1 , t 2 ∈ [0 , 1] . Follows from � � � � ( α t 1 ,t 2 ∗ α t 1 ,t 2 ◦ π g ( A ) Π Γ ( t 2 ) ( ω ) ( A ) = ω Γ Γ � π g ◦ α t 1 ,t 2 � = ( α t 1 ,t 2 ∗ = ω ( A ) (Π Γ ( t 1 )( ω )) ( A ) Γ Γ The representations Π Γ are invariants of symmetric gapped phases Now: concrete observables? Sven Bachmann (LMU) Invariants of ground state phases Warwick 2014 16 / 20
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