Free-Fermion entanglement and Leonard pairs Nicolas CRAMPE Institut Denis-Poisson, Tours based on work done in collaboration with Pierre-Antoine Bernard (CRM) Krystal Guo (CRM) Rafael Nepomechie (U. Miami) Luc Vinet (CRM) Nicolas CRAMPE (IdP) 1 / 21
Introduction • Physical interest: Free-Fermion models on 1D system or graph ⇓ Computation of entanglement entropy = Spectrum of the chopped correlation matrix C • Numerical issue: C is hard to be diagonalized numerically • Surprising fact (V. Eisler and I. Peschel): Tridiagonal matrix T commutes with C and is easy to be diagonalized • Goals: − Identify T as an algebraic Heun operator of Leonard pairs − Classify the models where T exists Nicolas CRAMPE (IdP) 2 / 21
Outline Ground state of free-Fermion Hamiltonian Chopped correlation matrix and entanglement entropy Leonard pairs and algebraic Heun operators Algebraic Heun operator and chopped correlation matrix Example: Uniform chain Further results and concluding remarks Nicolas CRAMPE (IdP) 3 / 21
Ground state of free-Fermion Hamiltonian Open quadratic free-Fermion inhomogeneous Hamiltonian N − 1 N N � J n ( c † n c n + 1 + c † B n c † c † m � ∑ ∑ ∑ H = n + 1 c n ) − n c n = H mn c n , n = 0 n = 0 m , n = 0 J n and B n real parameters, { c † { c † m , c † m , c n } = δ m , n , n } = { c m , c n } = 0 . To diagonalize � H , first diagonalize ( N + 1 ) × ( N + 1 ) matrix B 0 J 0 0 J 0 B 1 J 1 0 0 J 1 B 2 J 2 0 H = | � � H mn | 0 ≤ m , n ≤ N = ... ... ... 0 J N − 2 B N − 1 J N − 1 0 J N − 1 B N Nicolas CRAMPE (IdP) 4 / 21
Ground state of free-Fermion Hamiltonian Two orthonormal basis • Position basis {| 0 � , | 1 � ,..., | N �} � H | n � = J n − 1 | n − 1 �− B n | n � + J n | n + 1 � , • Momentum basis {| ω k �} N � ∑ H | ω k � = ω k | ω k � | ω k � = φ n ( ω k ) | n � with n = 0 We order the N + 1 eigenvalues ω 0 < ω 1 < ··· < ω N φ n ( ω k ) , the eigenfunctions, are related to orthogonal polynomials. Nicolas CRAMPE (IdP) 5 / 21
Ground state of free-Fermion Hamiltonian Having diagonalized � H , we see that Hamiltonian � H can be rewritten as N � c † ∑ H = ω k ˜ k ˜ c k , k = 0 where the annihilation operators ˜ c k are given by N N ∑ ∑ c k = φ n ( ω k ) c n , c n = φ n ( ω k ) ˜ ˜ c k , n = 0 k = 0 c † and creation operators ˜ k obtained by Hermitian conjugation. These obey c † c † c † { ˜ k , ˜ c p } = δ k , p , { ˜ k , ˜ p } = { ˜ c k , ˜ c p } = 0 Nicolas CRAMPE (IdP) 6 / 21
Ground state of free-Fermion Hamiltonian Eigenvectors of � H given by c † c † | Ψ �� = ˜ k r | 0 �� , k 1 ... ˜ with k 1 ,..., k r ∈ { 0 ,..., N } pairwise distinct Vacuum state | 0 �� is annihilated by all the annihilation operators c k | 0 �� = 0 , ˜ k = 0 ,... , N The energy eigenvalues are given by r ∑ E = ω k i i = 1 Nicolas CRAMPE (IdP) 7 / 21
Ground state of free-Fermion Hamiltonian Ground state | Ψ 0 �� is constructed by filling the Fermi sea: c † c † | Ψ 0 �� = ˜ K | 0 �� , 0 ... ˜ where K ∈ { 0 , 1 ,..., N } is the greatest integer below the Fermi momentum, such that ω K < 0 , ω K + 1 > 0 . K can be modified by adding a constant term in the external magnetic field B n . Nicolas CRAMPE (IdP) 8 / 21
Chopped correlation matrix and entanglement entropy The 1- particle correlation matrix � C in the ground state is the ( N + 1 ) × ( N + 1 ) matrix with entries � C mn = �� Ψ 0 | c † m c n | Ψ 0 �� . It is seen K � ∑ C = | ω k �� ω k | , k = 0 i.e. � C is projector onto subspace of C N + 1 spanned by vectors | ω k � with k = 0 ,..., K Nicolas CRAMPE (IdP) 9 / 21
Chopped correlation matrix and entanglement entropy To discuss entanglement, one needs bipartition : sites { 0 , 1 ,...,ℓ } Part 1: sites { ℓ + 1 ,ℓ + 2 ,..., N } Part 2: Entanglement properties in ground state is provided by reduced density matrix ( 2 ℓ + 1 × 2 ℓ + 1 ) ρ 1 = tr 2 | Ψ 0 ���� Ψ 0 | Observation (Peschel, Vidal et al.): ρ 1 is determined by "chopped" correlation matrix C ( ℓ + 1 ) × ( ℓ + 1 ) submatrix of � C : C = | � C mn | 0 ≤ m , n ≤ ℓ Nicolas CRAMPE (IdP) 10 / 21
Chopped correlation matrix and entanglement entropy Introduce the projectors ℓ K | ω k �� ω k | = � ∑ ∑ π 1 = | n �� n | π 2 = and C , n = 0 k = 0 the chopped correlation matrix can be written as C = π 1 π 2 π 1 To calculate entanglement entropies one has to compute the eigenvalues of C Not easy to do numerically because the eigenvalues of that matrix are exponentially close to 0 and 1 Parallel between study of entanglement properties of finite free-Fermion chains and time and band limiting problems will indicate how this can be circumvented Nicolas CRAMPE (IdP) 11 / 21
Leonard pairs and Algebraic Heun operators Definition Leonard pairs ( A , A ∗ ) A and A ∗ are linear transformation of V (dim V < + ∞ ) such that In a basis B of V , A is diagonal and A ∗ is irreducible tridiagonal In a basis B ∗ of V , A ∗ is diagonal and A is irreducible tridiagonal Remarks Leonard pairs have been classified Leonard pairs satisfy Askey–Wilson algebra Bispectral problems and orthogonal polynomials Tridiagonalization The operator W = r 0 + r 1 A + r 2 A ∗ + r 3 { A , A ∗ } + r 4 [ A , A ∗ ] is the more general operator tridiagonal in both bases B and B ∗ . W is called algebraic Heun operator. Nicolas CRAMPE (IdP) 12 / 21
Leonard pairs and Algebraic Heun operators Why “Heun” ? Let us choose A = x ( x − 1 ) d 2 dx 2 +( α + 1 − ( α + β + 2 ) x ) d dx A ∗ = x Then the operator W becomes � γ � d W ∼ d 2 αβ x − q δ ε dx 2 + x + x − 1 + dx + x ( x − 1 )( x − d ) , x − d and is the standard differential Heun operator (Fuchsian 2nd order differential equation with four regular singularities). Nicolas CRAMPE (IdP) 13 / 21
Algebraic Heun operator and chopped correlation matrix Strategy Pick J n and B n in the Hamiltonian so that � H is one element of a Leonard pair ( � H , � X ) Construct an algebraic Heun operator and prove that it commutes with the chopped correlation matrix Recall � � H | ω k � = ω k | ω k � , H | n � = J n − 1 | n − 1 �− B n | n � + J n | n + 1 � By definition of Leonard pairs � � X | ω k � = J k − 1 | ω k − 1 �− B k | ω k � + J k | ω k + 1 � , X | n � = λ n | n � Nicolas CRAMPE (IdP) 14 / 21
Algebraic Heun operator and chopped correlation matrix Introduce algebraic Heun operator: T = { � � X , � H } + µ � X + ν � H In position basis � T | n � = J n − 1 ( λ n − 1 + λ n + ν ) | n − 1 � +( µλ n − 2 B n λ n − ν B n ) | n � + J n ( λ n + λ n + 1 + ν ) | n + 1 � , Recall that π 1 = ∑ ℓ n = 0 | n �� n | . One gets [ � T , π 1 ] = 0 if ν = − ( λ ℓ + λ ℓ + 1 ) Nicolas CRAMPE (IdP) 15 / 21
Algebraic Heun operator and chopped correlation matrix In momentum basis � T | ω k � = J k − 1 ( ω k − 1 + ω k + µ ) | ω k − 1 � +( νω k − 2 B k ω k − µ B k ) | ω k � + J k ( ω k + ω k + 1 + µ ) | ω k + 1 � and K ∑ π 2 = | ω k �� ω k | k = 0 One gets [ � T , π 2 ] = 0 if µ = − ( ω K + ω K + 1 ) Since C = π 1 π 2 π 1 , one gets [ T , C ] = 0 Nicolas CRAMPE (IdP) 16 / 21
Algebraic Heun operator and chopped correlation matrix The tridiagonal matrix d 0 t 0 t 0 d 1 t 1 t 1 d 2 t 2 T = ... ... ... t ℓ − 2 d ℓ − 1 t ℓ − 1 t ℓ − 1 d ℓ with t n = J n ( λ n + λ n + 1 − λ ℓ − λ ℓ + 1 ) d n = − B n ( 2 λ n − λ ℓ − λ ℓ + 1 ) − λ n ( ω K + ω K + 1 ) . commutes with the chopped correlation matrix C Nicolas CRAMPE (IdP) 17 / 21
The homogeneous chain Let us choose J 0 = ... = J N − 1 = − 1 2 , B n = 0 . The associated eigenvalues of the Hamiltonian � H are � π ( k + 1 ) � ω k = − cos , k = 0 , 1 ,..., N . N + 2 The matrix T is then given by with t n = 1 2 [ cos ( θ n )+ cos ( θ n + 1 ) − cos ( θ ℓ ) − cos ( θ ℓ + 1 )] d n = − cos ( θ n )[ cos ( θ K )+ cos ( θ K + 1 )] This readily recovers recent results of Eisler & Peschl. Nicolas CRAMPE (IdP) 18 / 21
Further results and concluding remarks Shown that for chains associated to Leonard pairs (bispectral orthogonal polynomials), algebraic Heun operator readily provides a tridiagonal matrix that commutes with correlation matrix The approach provides such commuting matrices for the many chains corresponding to finite discrete polynomials of Askey scheme N. Crampé, R. Nepomechie, L. Vinet, Entanglement in Fermionic Chains and Bispectrality , Roman Jackiw 80th Birthday Festschrift, arXiv:2001.10576 N. Crampé, R. Nepomechie, L. Vinet, Free-Fermion entanglement and orthogonal polynomials , J. Stat. Mech. (2019) arXiv:1907.00044 Nicolas CRAMPE (IdP) 19 / 21
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