Decay of correlations in 2d quantum systems Costanza Benassi University of Warwick Quantissima in the Serenissima II, 25th August 2017 Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 1 / 18
The results presented are part of a joint work with J. Fr¨ ohlich and D.Ueltschi (Ann. Henri Poincar´ e 18, 2831–2847, (2017)). Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 2 / 18
Phase transitions and symmetries Symmetry: group of transformations that leaves the system unaltered. Phase transition due to symmetry breaking: if T < T c the system favours an ordered state. Continuous symmetries (e.g. U ( 1 )) VS Discrete symmetries (e.g. Z 2 ) Figure: T < T c Figure: T > T c Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 3 / 18
What about dimensions? Mermin Wagner Theorem: no spontaneous breaking of a continuous symmetry can happen at d ≤ 2 if T > 0. N.B. This statement does not apply to discrete symmetries (e.g. Z 2 symmetry in the ferromagnetic Ising model). Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 4 / 18
Phase transitions and correlation functions Study of the behaviour of the relevant correlation functions to study the absence or presence of symmetry breaking. Expected decay rate in d = 2: power law . Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 5 / 18
A general setting Our aim is to find a general setting for the relevant correlation functions to decay algebraically, with a focus on quantum models on a 2d lattice . We will be interested in systems with a U(1) symmetry . We will show how this setting is fulfilled by a great variety of well studied quantum systems. N. D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two- dimensional isotropic Heisenberg models . Phys. Rev. Lett. 17, 1133- 1136 (1966). O. A. McBryan, T. Spencer, On the decay of correlations in SO(n)-symmetric ferromagnets . Commun. Math. Phys. 53, 299-302 (1977). T. Koma, H. Tasaki, Decay of Superconducting and Magnetic Correlations in One- and Two-Dimensional Hubbard Models . Phys. Rev. Lett. 68, 3248 (1992). J. Fr¨ ohlich, D. Ueltschi, Some properties of correlations of quantum lattice systems in thermal equilibrium . Math. Phys. 56, 053302 (2015). Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 6 / 18
b b b b b b b b b b b b b Some notation Lattice: (Λ , E ) with graph distance. γ = max x ∈ Λ max ℓ ∈ N 1 ℓ |{ y ∈ Λ | d ( x , y ) = ℓ } . Λ Hilbert space H Λ - finite ! (e.g. for quantum spin systems H Λ = ⊗ x ∈ Λ C 2 s +1 ). Linear operators B ( H Λ ). Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 7 / 18
Some assumptions Local algebras: {B A } A ⊂ Λ . Λ A Interaction: { Φ A } A ⊂ Λ , Φ A ∈ B A . Hamiltonian: H Λ = � A ⊂ Λ Φ A . K -norm of interaction { Φ A } A ⊂ Λ : � Φ A � ∞ ( | A | − 1) 2 (diam( A ) + 1) 2+2 K ( | A |− 1) . � � Φ � K = sup y ∈ Λ A ⊂ Λ s.t. y ∈ A � a � = Tr a e − β H Λ Tr e − β H Λ Gibbs state. Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 8 / 18
Some assumptions U(1) symmetry: { S x } x ∈ Λ such that � � � Φ A , S x = 0 ∀ A ⊂ Λ . x ∈ A Correlator O xy ∈ B { x , y } such that [ S x , O xy ] = cO xy . Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 9 / 18
Theorem (C.B., J. Fr¨ ohlich, D. Ueltschi (2017)) Suppose that the constant γ is finite, and that { S x } x ∈ Λ , (Φ A ) A ⊂ Λ , and O xy satisfy the properties above . Then there exist C > 0 and ξ ( β ) > 0 (uniform with respect to Λ and x , y ∈ Λ ) such that |� O xy �| ≤ C ( d ( x , y ) + 1) − ξ ( β ) . Moreover, if there exists a positive constant K such that � Φ � K is bounded uniformly in Λ , then c 2 β →∞ β ξ ( β ) = lim . 8 γ � Φ � 0 Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 10 / 18
What does it mean? For β large enough we have power law decay of correlations with exponent ∝ 1 β : const . |� O xy �| ≤ . const . ( d ( x , y ) + 1) β All the constants are uniform in Λ. Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 11 / 18
Ex. 1 - SU(2) invariant model H Λ = ⊗ x ∈ Λ C 2 s +1 . x = S i ⊗ 1 Λ \ x . S = ( S 1 , S 2 , S 3 ) spin- s operators, with S i � � k � � 2 s S x · � � H Λ = − � k =1 c k ( x , y ) S y . � x , y �∈E Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 12 / 18
Ex. 1 - SU(2) invariant model H Λ = ⊗ x ∈ Λ C 2 s +1 . x = S i ⊗ 1 Λ \ x . S = ( S 1 , S 2 , S 3 ) spin- s operators, with S i � � k � � 2 s S x · � � H Λ = − � k =1 c k ( x , y ) S y . � x , y �∈E Theorem There exist constants C > 0 and ξ ( β ) > 0 , the latter depending on β, γ, s but not on x , y ∈ Λ , such that y �| ≤ C ( d ( x , y ) + 1) − ξ ( β ) . |�S j x S j The exponent ξ ( β ) is proportional to β − 1 for β large enough: β →∞ β ξ ( β ) = (32 s γ 2 ) − 1 . lim Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 12 / 18
Ex. 2- The Hubbard Model H Λ = ⊗ x ∈ Λ span {∅ , ↑ , ↓ , ↑↓} ≃ ⊗ x ∈ Λ C 4 . Hamiltonian (also long range interaction): t xy � � � � c † σ, x c σ, y + c † H Λ = − σ, y c σ, x + V ( { n ↑ , x } x ∈ Λ , { n ↓ , x } x ∈ Λ ) . 2 x , y ∈ Λ σ = ↑ , ↓ Two U(1) symmetries generated by n x = � σ = ↑ , ↓ n σ, x and ∆ x = n ↑ , x − n ↓ , x . Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 13 / 18
Ex. 2 - The Hubbard model Theorem Suppose that t xy = t ( d ( x , y ) + 1) − α with α > 4 . Then there exist C > 0 , ξ ( β ) > 0 (the latter depending on β , γ , α , t, but not on x , y ∈ Λ ) such that |� c † ↑ , x c ↓ , x c † ↓ , y c ↑ , y �| |� c † ↑ , x c † ≤ C ( d ( x , y ) + 1) − ξ ( β ) ↓ , x c ↑ , y c ↓ , y �| |� c † σ, x c σ, y �| where σ ∈ {↑ , ↓} in the last line. Furthermore, r − α +3 � − 1 � 64 γ 2 | t | � β →∞ β ξ ( β ) = lim . r ≥ 1 Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 14 / 18
Wide range of applicability! Other examples: XXZ model, tJ model, Random loop model, . . . Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 15 / 18
Ideas of the proof The proof uses the complex rotation method: “rotate” the correlator and z ∈ Λ e θ z S z . The “angles” { θ z } z ∈ Λ the Hamiltonian with the operator R = � are chosen to encode the expected power law decay. We can then estimate � O xy � by Trotter’s formula and H¨ older inequality for matrices. Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 16 / 18
Conclusions Power-law bound for the decay of correlations in a wide class of U(1)-symmetric systems. The proof relies on simple ingredients – most of all the complex rotation method. Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 17 / 18
Thank you! Costanza Benassi (University of Warwick) Decay of correlations in 2d quantum systems 25th August 2017 18 / 18
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