Low-temperature sprectrum of correlation lengths of the XXZ chain in - PowerPoint PPT Presentation

Low-temperature sprectrum of correlation lengths of the XXZ chain in the massive antiferromagnetic regime Frank G¨ ohmann Bergische Universit¨ at Wuppertal Fachgruppe Physik Firenze 21.5.2015

Correlation functions of 1d lattice models 1d lattice models Quantum many-body systems: Defined by a Hamiltonian H ( L ) depending on the systems size L Simple prototypical class: ‘spin models’ on a 1d lattice. These are ‘fully regularized’: Discrete space, lattice spacing a 1 Finite number of lattice sites L 2 Finite local Hilbert space ∼ = C d 3 � C d � ⊗ L 1. and 2. imply that the space of states is finite dimensional ∼ = QFTs (relativistic and non-relativistic) as certain scaling limits involving a → 0, L → ∞ Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 2 / 28

Correlation functions of 1d lattice models 1d lattice models Quantum many-body systems: Defined by a Hamiltonian H ( L ) depending on the systems size L Simple prototypical class: ‘spin models’ on a 1d lattice. These are ‘fully regularized’: Discrete space, lattice spacing a 1 Finite number of lattice sites L 2 Finite local Hilbert space ∼ = C d 3 � C d � ⊗ L 1. and 2. imply that the space of states is finite dimensional ∼ = QFTs (relativistic and non-relativistic) as certain scaling limits involving a → 0, L → ∞ Main example here the integrable XXZ Hamiltonian � �� L / 2 L / 2 � − h j + σ y j − 1 σ y ∑ σ x j − 1 σ x σ z j − 1 σ z ∑ σ z H ( L ) = J j +∆ j − 1 j 2 j = − L / 2 + 1 j = − L / 2 + 1 ∆ = ( q + q − 1 ) / 2 = ch ( γ ) , L even, σ α j , α = x , y , z , Pauli matrices acting on � C d � ⊗ L factor j in Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 2 / 28

Correlation functions of 1d lattice models What needs to be calculated? Quantum mechanics H ( L ) | n � = E n | n � spectrum and eigenstates 1 E 0 ∼ L for L → ∞ (thermodynamic limit), e 0 = lim L → ∞ E 0 / L ground state energy 2 per lattice site Dispersion relation of elementary excitations ε ( p ) for L → ∞ 3 Typically known for integrable systems Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 3 / 28

Correlation functions of 1d lattice models What needs to be calculated? Quantum mechanics H ( L ) | n � = E n | n � spectrum and eigenstates 1 E 0 ∼ L for L → ∞ (thermodynamic limit), e 0 = lim L → ∞ E 0 / L ground state energy 2 per lattice site Dispersion relation of elementary excitations ε ( p ) for L → ∞ 3 Typically known for integrable systems Statistical mechanics and thermodynamics � e − H ( L ) / T � Partition function of the canonical ensemble Z = Tr 1 , T temperature Z ∼ e − fL / T , f = − lim L → ∞ T ln ( Z ) / L free enery per lattice site 2 Not systematically known for integrable systems Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 3 / 28

Correlation functions of 1d lattice models What needs to be calculated? Quantum mechanics H ( L ) | n � = E n | n � spectrum and eigenstates 1 E 0 ∼ L for L → ∞ (thermodynamic limit), e 0 = lim L → ∞ E 0 / L ground state energy 2 per lattice site Dispersion relation of elementary excitations ε ( p ) for L → ∞ 3 Typically known for integrable systems Statistical mechanics and thermodynamics � e − H ( L ) / T � Partition function of the canonical ensemble Z = Tr 1 , T temperature Z ∼ e − fL / T , f = − lim L → ∞ T ln ( Z ) / L free enery per lattice site 2 Not systematically known for integrable systems Statistical mechanics and static correlation functions Static correlation functions of local operators X , Y 1 � � e − H ( L ) / T X 1 Y m � X 1 Y m � = lim L → ∞ Tr / Z have been studied by means of the (reduced) density matrix � e − H ( L ) / T � D m ( T ) = lim / Z L → ∞ Tr − L / 2 + 1 ,..., 0 , m + 1 ,..., L / 2 � C d � ⊗ m well defined for every m ∈ N ∈ End Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 3 / 28

Correlation functions of 1d lattice models What needs to be calculated? Static correlation functions of local operators continued � � � , where V n ∼ = C d and X trivial Infinite chain formalism: X ∈ End 2 n ∈ Z V n β outside chain segment [ 1 ,ℓ ] : [ X , e j α ] = 0 for j ∈ { 1 ,...,ℓ } , local. Maximal such ℓ is called the length of X Local operators span a vector space W 3 � � � ∀ m ≥ ℓ Local operators have a natural restriction X [ 1 , m ] to End n ∈{ 1 ,..., m } V n 4 5 This allows us to properly define the expectation value of a local operator X on the infinite chain � X � = Tr 1 ,...,ℓ { D ℓ ( T ) X [ 1 ,ℓ ] } where ℓ is the length of X Construction (in generalized form) nicely compatible with the integrable structure 6 of XXZ [B OOS , J IMBO , M IWA , S MIRNOV , T AKEYAMA 2006-09] Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 4 / 28

Correlation functions of 1d lattice models What needs to be calculated? Static correlation functions of local operators continued � � � , where V n ∼ = C d and X trivial Infinite chain formalism: X ∈ End 2 n ∈ Z V n β outside chain segment [ 1 ,ℓ ] : [ X , e j α ] = 0 for j ∈ { 1 ,...,ℓ } , local. Maximal such ℓ is called the length of X Local operators span a vector space W 3 � � � ∀ m ≥ ℓ Local operators have a natural restriction X [ 1 , m ] to End n ∈{ 1 ,..., m } V n 4 5 This allows us to properly define the expectation value of a local operator X on the infinite chain � X � = Tr 1 ,...,ℓ { D ℓ ( T ) X [ 1 ,ℓ ] } where ℓ is the length of X Construction (in generalized form) nicely compatible with the integrable structure 6 of XXZ [B OOS , J IMBO , M IWA , S MIRNOV , T AKEYAMA 2006-09] Dynamical (= time-dependent) correlation functions � X 1 ( t ) X m + 1 � 1 Most correlation functions encountered in experiments are of this type 2 By definition the time dependence is X 1 ( t ) = e i H ( L ) t X 1 e − i H ( L ) t Problem: X 1 ( t ) is not a local operator, does not fit into the formalism based on reduced density matrix Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 4 / 28

Correlation functions of 1d lattice models What needs to be calculated? Dynamical correlation functions continued Alternative: Spectral (or Lehmann) representation. Keep L finite. Then 3 e − E k / T e − E k / T � X 1 ( t ) Y m + 1 � = ∑ � k | X 1 ( t ) Y m + 1 | k � = ∑ � k | X 1 ( t ) | ℓ �� ℓ | Y m + 1 | k � Z Z k ,ℓ k � � e − E k / T = ∑ e − i ( E ℓ − E k ) t − ( p ℓ − p k ) m � k | X 1 | ℓ �� ℓ | Y 1 | k � Z k ,ℓ � � e − i ( E ℓ − E 0 ) t − ( p ℓ − p 0 ) m T → 0 ∑ − → � 0 | X 1 | ℓ �� ℓ | Y 1 | 0 � ℓ Integrable case: eigenstates of H ( L ) are eigenstates of the transfer matrix 4 Matrix elements of the form � ℓ | Y 1 | 0 � are often called form factors. They can be 5 calculated by ‘integrable methods’ So far form factors are the only way to access dynamical correlation functions of 6 integrable systems Summation is a problem 7 Often interest is in asymptotic analysis e.g. m → ∞ , t → ∞ 8 Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 5 / 28

Correlation functions of 1d lattice models Example of a factor series XXZ for ∆ > 1, 0 < h < h ℓ from D UGAVE , FG, K OZLOWSKI , S UZUKI 2015 based on Bethe Ansatz for finite L , then L → ∞ m + 1 ( t ) � = ( q 2 ; q 2 ) 4 ∞ � σ z 1 σ z ( − 1 ) m ( − q 2 ; q 2 ) 4 ∞ � �� � π / 2 1 ( − 1 ) ι m d n h ν � � n h � ( 2 π ) n h F ( 2 ) { ν a } n h ε ( 0 ) ( ν a ) t − 2 π p ( ν a ) m ι = 0 ∑ ∑ ∑ + exp i ι 1 n h ! − π / 2 a = 1 n h ∈ 2 N Previous work: J IMBO , M IWA 95, L ASHKEVICH 03 where � � F ( 2 ) 2 n χ n χ � � d n χ ψ { ν h a } ; { ψ a } 1 � F ( 2 ) ι { ν a } n h 1 1 = ( 2 π i ) n χ · � � � ι 1 n χ n χ 2 n χ n χ ! � { ψ b } Γ ε ( { ν a } ) ψ a 1 ; { ν h c } ∏ a = 1 Y 0 1 and � � ψ ∈ C n χ � � � � n χ 2 n χ � Y 0 � { ψ b } Γ ε ( { ν a } ) = ψ a 1 ; { ν h c } = ε , a = 1 ,..., n χ 1 with ε > 0 small enough Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 6 / 28

Correlation functions of 1d lattice models Example large-distance asymptotics for equal times This (combined with a result of L ASHKEVICH 03 ) allows one to obtain an explicit formula for the next-to-leading term in the asymptotics of the static longitudinal two-point function � � = ( q 2 ; q 2 ) 4 ∞ σ z 1 σ z ( − 1 ) m m + 1 ( − q 2 ; q 2 ) 4 ∞ � �� + A · k ( q 2 ) m ( − 1 ) m − th 2 ( γ / 2 ) ( q ; q 2 ) 4 � ∞ 1 + O ( m − 1 ) m 2 ( − q ; q 2 ) 4 ∞ where k ( q 2 ) = ϑ 2 2 ( 0 | q 2 ) ( − q ; q 2 ) 4 ( q 4 ; q 4 , q 4 ) 8 1 ∞ ∞ 3 ( 0 | q 2 ) , A = ϑ 2 π sh 2 ( γ / 2 ) ( q 2 ; q 2 ) 2 ( q 2 ; q 4 , q 4 ) 8 ∞ ∞ generalizing the result of the correlation length of J OHNSON , K RINSKY AND M C C OY 73 Time dependent case can be analyzed in a similar way Frank G¨ ohmann (BUW – FG Physik) Correlation lengths of massive XXZ 21.5.2015 7 / 28