Free fermions and -determinantal processes Fabio Deelan Cunden - PowerPoint PPT Presentation

Free fermions and α -determinantal processes Fabio Deelan Cunden (University College Dublin) based on J. Phys. A: Math. Theor. 52 , 165202 (2019) joint work with Satya N. Majumdar (Paris) and Neil O’Connell (Dublin) Statistical Mechanics Seminar - University of Warwick, Jan 23, 2020 1 / 23

Determinantal point processes A point process or ‘random point configuration’ on R can be described in terms of its correlation functions (when they exist) P ( exactly one particle in ( x i , x i + δ ) , for all i = 1 , . . . , n ) ̺ n ( x 1 , . . . , x n ) = lim δ n δ → 0 2 / 23

Determinantal point processes A point process or ‘random point configuration’ on R can be described in terms of its correlation functions (when they exist) P ( exactly one particle in ( x i , x i + δ ) , for all i = 1 , . . . , n ) ̺ n ( x 1 , . . . , x n ) = lim δ n δ → 0 Determinantal / fermionic processes (Macchi 70’s) are point processes with ̺ n ( x 1 , . . . , x n ) = 1 ≤ i , j ≤ n K ( x i , x j ) , det n ∈ N . The function K ( x , y ) is called correlation kernel . Macchi-Soshnikov criterion: If K is self-adjoint, locally trace class, with 0 ≤ K ≤ 1 , then the random point configuration exists and is unique (and exhibits repulsion). Examples: • Free fermions (Pauli exclusion principle); • Eigenvalues of random matrices (Dyson, Mehta, Gaudin - level repulsion); • Non-intersecting paths. 2 / 23

α -determinantal point processes Let A be a n × n matrix.and α a scalar. The α -determinant of A (Vere-Jones, 1988) is � det A = sgn( σ ) A 1 σ ( 1 ) A 2 σ ( 2 ) · · · A n σ ( n ) σ ∈ S n where m ( σ ) = # disjoint cycles in the permutation σ . We simply replace the signature sgn( σ ) = ( − 1 ) n − m ( σ ) by α n − m ( σ ) in the definition of the ordinary determinant. It is clear that det − 1 A = det A , det 1 A = per A , det 0 A = A 11 A 22 · · · A nn . 3 / 23

α -determinantal point processes Let A be a n × n matrix.and α a scalar. The α -determinant of A (Vere-Jones, 1988) is � ( − 1 ) n − m ( σ ) A 1 σ ( 1 ) A 2 σ ( 2 ) · · · A n σ ( n ) det A = σ ∈ S n where m ( σ ) = # disjoint cycles in the permutation σ . We simply replace the signature sgn( σ ) = ( − 1 ) n − m ( σ ) by α n − m ( σ ) in the definition of the ordinary determinant. It is clear that det − 1 A = det A , det 1 A = per A , det 0 A = A 11 A 22 · · · A nn . 3 / 23

α -determinantal point processes Let A be a n × n matrix and α a scalar. The α -determinant of A (Vere-Jones, 1988) is � α n − m ( σ ) A 1 σ ( 1 ) A 2 σ ( 2 ) · · · A n σ ( n ) det α A = σ ∈ S n where m ( σ ) = # disjoint cycles in the permutation σ . We simply replace the signature sgn( σ ) = ( − 1 ) n − m ( σ ) by α n − m ( σ ) in the definition of the ordinary determinant. It is clear that det − 1 A = det A , det 1 A = per A , det 0 A = A 11 A 22 · · · A nn . An α -determinantal point process (Shirai and Takahashi, 2003) with kernel K is defined, when it exists, as the point process with n -point correlation functions ̺ n ( x 1 , . . . , x n ) = det α 1 ≤ i , j ≤ n K ( x i , x j ) . α = − 1 : determinantal process; α = 1 : permanental process; α = 0 : Poisson process (with intensity K ( x , x ) ). 3 / 23

α -determinantal point processes An α -DPP with kernel K is defined, when it exists, as the point process with n -point correlation functions ̺ n ( x 1 , . . . , x n ) = det α 1 ≤ i , j ≤ n K ( x i , x j ) . Existence criterion when α < 0 and K is self-adjoint and locally trace class: 4 / 23

α -determinantal point processes An α -DPP with kernel K is defined, when it exists, as the point process with n -point correlation functions ̺ n ( x 1 , . . . , x n ) = det α 1 ≤ i , j ≤ n K ( x i , x j ) . Existence criterion when α < 0 and K is self-adjoint and locally trace class: • The point process exists (and is unique) iff − 1 α ∈ N and 0 ≤ − α K ≤ 1 . 4 / 23

α -determinantal point processes An α -DPP with kernel K is defined, when it exists, as the point process with n -point correlation functions ̺ n ( x 1 , . . . , x n ) = det α 1 ≤ i , j ≤ n K ( x i , x j ) . Existence criterion when α < 0 and K is self-adjoint and locally trace class: • The point process exists (and is unique) iff − 1 α ∈ N and 0 ≤ − α K ≤ 1 . • The α -DPP is a superposition (union) of − 1 α i.i.d. DPPs with kernel − α K . 4 / 23

Free fermions in a harmonic potential N spin-polarized noninteracting fermions in a one-dimensional harmonic potential. Suppose that the system is in a stationary state (an eigenstate of H ). 5 / 23

Free fermions in a harmonic potential N spin-polarized noninteracting fermions in a one-dimensional harmonic potential. Suppose that the system is in a stationary state (an eigenstate of H ). Qestions: 1. Find the particle density of the system. 2. Find the pair correlation function. 3. Find the n -point correlation function. 4. Study the particle statistics asymptotics as N → ∞ . Fact: the number statistics of free fermions forms a determinantal point process. 5 / 23

Qantum harmonic oscillator Single particle Schr¨ odinger equation: − ∂ 2 ∂ x 2 + x 2 � � ψ ( x ) = E ψ ( x ) , x ∈ R . 4 ψ k ( x ) = h k ( x ) e − x 2 / 4 , Solutions: E k = k + 1 / 2 , k ∈ Z + 6 / 23

Qantum harmonic oscillator Single particle Schr¨ odinger equation: − ∂ 2 ∂ x 2 + x 2 � � ψ ( x ) = E ψ ( x ) , x ∈ R . 4 ψ k ( x ) = h k ( x ) e − x 2 / 4 , Solutions: E k = k + 1 / 2 , k ∈ Z + Non-interacting fermions: antisymmetric superpositions of the ψ k ’s 1 √ Ψ k 1 ,..., k N ( x 1 , . . . , x N ) = 1 ≤ i , j ≤ N ψ k i ( x j ) , det with 0 ≤ k 1 < k 2 < · · · < k N . N ! They are normalised eigenfunctions of the Hamiltonian � − ∂ 2 + x 2 � � i H = with eigenvalues E = k 1 + · · · + k N + N / 2 ∂ x 2 4 i i in the subspace of completely antisymmetric states Ψ( x σ ( 1 ) , . . . , x σ ( N ) ) = sgn( σ )Ψ k 1 ,..., k N ( x 1 , . . . , x N ) . 6 / 23

Free fermions in a harmonic potential Denote by J = { k 1 , . . . , k N } ⊂ Z + the occupied levels. | Ψ J ( x 1 , . . . , x N ) | 2 = 1 1 ≤ i , j ≤ N ψ k i ( x j ) = 1 1 ≤ i , j ≤ N ψ k i ( x j ) det det 1 ≤ i , j ≤ N K J ( x i , x j ) , det N ! N ! � K J ( x , y ) = ψ k ( x ) ψ k ( y ) : integral kernel of projection onto span { ψ k ( x ): k ∈ J } k ∈ J 7 / 23

Free fermions in a harmonic potential Denote by J = { k 1 , . . . , k N } ⊂ Z + the occupied levels. | Ψ J ( x 1 , . . . , x N ) | 2 = 1 1 ≤ i , j ≤ N ψ k i ( x j ) = 1 1 ≤ i , j ≤ N ψ k i ( x j ) det det 1 ≤ i , j ≤ N K J ( x i , x j ) , det N ! N ! � K J ( x , y ) = ψ k ( x ) ψ k ( y ) : integral kernel of projection onto span { ψ k ( x ): k ∈ J } k ∈ J Ψ J defines a determinantal point process on R with correlation kernel K J ( x , y ) : N ! ˆ | Ψ J ( x 1 , . . . , x N ) | 2 d x n + 1 · · · d x N ̺ n ( x 1 , . . . , x n ) = ( N − n )! N ! ˆ 1 = 1 ≤ i , j ≤ N K J ( x i , x j ) d x n + 1 · · · d x N det ( N − n )! N ! = 1 ≤ i , j ≤ n K J ( x i , x j ) det (by Mehta-Gaudin integration lemma) 7 / 23

Free fermions in a harmonic potential Particle density: ̺ 1 ( x ) = K J ( x , x ) � K J ( x 1 , x 1 ) � K J ( x 1 , x 2 ) Pair correlation fcn: ̺ 2 ( x 1 , x 2 ) = det K J ( x 2 , x 1 ) K J ( x 2 , x 2 ) = K J ( x 1 , x 1 ) K J ( x 2 , x 2 ) − | K J ( x 1 , x 2 ) | 2 . . . ̺ n ( x 1 , . . . , x n ) = 1 ≤ i , j ≤ n K J ( x i , x j ) det . . . � with K J ( x , y ) = ψ k ( x ) ψ k ( y ) , J = { k 1 , . . . , k N } . k ∈ J 8 / 23

Free fermions in a harmonic potential Particle density: ̺ 1 ( x ) = K J ( x , x ) � K J ( x 1 , x 1 ) � K J ( x 1 , x 2 ) Pair correlation fcn: ̺ 2 ( x 1 , x 2 ) = det K J ( x 2 , x 1 ) K J ( x 2 , x 2 ) = K J ( x 1 , x 1 ) K J ( x 2 , x 2 ) − | K J ( x 1 , x 2 ) | 2 . . . ̺ n ( x 1 , . . . , x n ) = 1 ≤ i , j ≤ n K J ( x i , x j ) det . . . � with K J ( x , y ) = ψ k ( x ) ψ k ( y ) , J = { k 1 , . . . , k N } . k ∈ J Large- N asymptotics of ̺ n ( x 1 , . . . , x n ) ? Need to understand the asymptotics of the correlation kernel K J ( x , y ) . 8 / 23

Ground state: J = { 0 , 1 , . . . , N − 1 } 9 / 23

Ground state and the GUE eigenvalue process Suppose that J = [ 0 . . N ) : 1 √ Ψ GS ( x 1 , . . . , x N ) = 1 ≤ i , j ≤ N ψ i − 1 ( x j ) . det N ! This is the ground state. The corresponding projection kernel N − 1 � K J ( x , y ) = ψ k ( x ) ψ k ( y ) k = 0 is the kernel of the Gaussian Unitary Ensemble of random matrix theory. 10 / 23

Ground state and the GUE eigenvalue process Suppose that J = [ 0 . . N ) : 1 √ Ψ GS ( x 1 , . . . , x N ) = 1 ≤ i , j ≤ N ψ i − 1 ( x j ) . det N ! This is the ground state. The corresponding projection kernel N − 1 � K J ( x , y ) = ψ k ( x ) ψ k ( y ) k = 0 is the kernel of the Gaussian Unitary Ensemble of random matrix theory. Let X be a N × N random complex Hermitian matrix with independent standard Gaussian entries. Then, the eigenvalues of X have joint probability density p ( x 1 , . . . , x N ) = | Ψ GS ( x 1 , . . . , x N ) | 2 . (from the Weyl denominator formula. Mehta, Gaudin, Dyson, ...) 10 / 23