The Density Matrix for the Ground State of 1-d Impenetrable Bosons in a Harmonic Trap Institute of Fundamental Sciences Massey University New Zealand 29 August 2017 A. A. Kapaev Memorial Workshop Michigan Center for Applied Interdisciplinary Mathematics August 25-29, 2017
The Plan 1-d Impenetrable Bosons
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation Homogeneous and Periodic Case
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation Homogeneous and Periodic Case Recurrence Relations
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation Homogeneous and Periodic Case Recurrence Relations Orthogonal Polynomial System & Semi-Classical OPS Class
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation Homogeneous and Periodic Case Recurrence Relations Orthogonal Polynomial System & Semi-Classical OPS Class Lax pairs
The Plan 1-d Impenetrable Bosons Density Matrix and Bose Condensation Homogeneous and Periodic Case Recurrence Relations Orthogonal Polynomial System & Semi-Classical OPS Class Lax pairs L(1, 1, 3; 2) member of degenerations from 2-variable Garnier system
1-d Impenetrable Bosons Hamiltonian N bosons x j (, ) N 2 H = x 2 i=1 i N + x 2 j + 2c δ(x i x j ), c > 0 j=1 1 i<j N Ground State eigenfunction Ψ 0 (x 1,..., x N ) satisfies subject to boundary conditions 1. Ψ 0 xi =x j +0 = Ψ 0 xi =x j 0 ( Ψ0 2. Ψ ) 0 x i x j xi =x j +0 HΨ 0 = E 0 Ψ 0 ( Ψ0 Ψ ) 0 = 2c Ψ x i x j 0 xi =x j ±0 xi =x j 0 Bethe Ansatz solution without harmonic trap [Lieb+Liniger 1963] Impenetrable Bose wavefunction c [Girardeau 1960]. The wave function for impenetrable bosons must vanish at coincident coordinates, Ψ 0 (x 1,..., x i,..., x j,..., x N ) = 0 for x i = x j, (i j), This means that for any fixed ordering of the particles x 1 < x 2 <... < x N there is no distinction between impenetrable bosons and free fermions.
Ground State Wavefunction Schrödinger Operator N 2 H R = x 2 j=1 j N + x 2 j j=1 Normalised single particle eigenstates {φ k (x)} k=0,1,... φ k (x) = 2 k c k e x2 /2 H k (x), (c k ) 2 = π 1/2 2 k k! H k (x) denotes the Hermite polynomial of degree k. Ground state wave function Ψ 0 as a Slater determinant of distinct single-particle states 1 Ψ 0 (x 1,..., x N ) = (N!) 1/2 det[φ k(x j )] j=1,...,n k=0,...,n 1 where the factor of (N!) 1/2 is included so that dx 1 dx N Ψ 0 (x 1,..., x N ) 2 = 1.
Density Matrix Density Matrix is defined as ρ N (x; y) N dx 2... dx N Ψ 0 (x, x 2,..., x N )Ψ 0 (y, x 2,..., x N ), R R so that dx ρ N (x; x) = N R Using this one must solve the eigenvalue problem ρ N (x; y)ψ k (y) dy = λ k ψ k (x), k Z 0. R Note λ 0 > λ 1 >... > λ N 1 > 0, N 1 λ j = N j=0
GUE Averages Recall ρ N (x; y) = N dx 2... dx N Ψ 0 (x, x 2,..., x N )Ψ 0 (y, x 2,..., x N ) R R Vandermonde determinant formula with p j (x) a monic polynomial of degree j, det[p j 1 (x k )] j,k=1,...,n = det[x j 1 ] k j,k=1,...,n = (x k x j ) 1 j<k N Product formula for Ψ 0 Ψ 0 (x 1,..., x N ) = 1 C N N j=1 e x2 j /2 1 j<k N N 1 x k x j, (C N ) 2 = N! (c l ) 2. This coincides precisely with the multivariate probability distribution function (p.d.f.) for a particular class of random matrices l=0 Ψ 0 2 = E GUEN namely, GUE N, the Gaussian unitary ensemble of N N complex Hermitian matrices, Density matrix ρ N+1 (x; y) can be written as an average over this eigenvalue p.d.f ρ N+1 (x; y) = 1 N c 2 e x2 /2 y 2 /2 E GUEN x x l y x l N l=1
Density Matrix and Bose Condensation Criteria for Bose-Einstein condensation (BEC) are - Onsager-Penrose 1956 criteria for BEC λ 0 (N) N = e O(1) BEC o(1) no BEC as N Off Diagonal Long Range Order criteria of Yang 1962 ρ (x) x C > 0 NB: already in thermodynamic limit N
Thermodynamic limit of Homogeneous System [Jimbo Miwa Mori Sato 1980] Density Matrix ρ (x; 0) is now identified with a τ-function of the fifth Painlevé system ρ (x; 0) = ρ 0 exp ( πρ 0 x σ V (t) dt ), 0 t where ρ 0 denotes the bulk density. σ V satisfies the non-linear equation subject to the x 0 boundary condition (xσ V )2 + 4(xσ V σ V 1) ( xσ V σ V + (σ V )2) = 0 σ V (x) x 0 x2 3 + x3 3π + O(x4 ). [Lenard 1972] Eigenvalues of the density matrix are Fourier coefficients L λ k = ρ N (x; 0)e 2πikx/L dx. 0 The N asymptotic expansion ( ) 1/2 π ρ N (x; 0) ρ 0 A, A = G4 (3/2) N sin(πρ 0 x/n) 2π where G(x) denotes the Barnes G-function, valid for x/n fixed. Therefore a specific c computable from the above. λ 0 c N
Finite N Periodic System [FFGW 2003] Finite system of N + 1 particles on a ring of circumference L, x [0, L]. Wavefunction 1 Ψ 0 (x 1,..., x N ) = L N/2 (N!) 1/2 2 sin π(x k x j ) L 1 j<k N Density Matrix ρ N+1 (x; 0) ρ N+1 (x; 0) = 1 N L E ( πx U(N) 2 sin L θ ) l ( ) θl 2 sin 2 2 l=1 Density Matrix can be identified with a member of a τ-function sequence in the sixth Painlevé system with Okamoto parameters b = ( 1 2 N, 1 + 1 2 N, 1 2 N, 1 1 2 N). Thermodynamic limit made by substituting x = e 2it/N and taking N with N/L fixed.
Determinants The general Heine identity N! det [ Determinant formula with matrix elements ] dx w(x)x j+k 2 = R j,k=1,...,n dx 1 R dx N R N w(x l ) l=1 ρ N+1 (x; y) = 1 c 2 e x2 /2 y 2 /2 det[a j,k (x; y)] j,k=1,...,n N a j,k (x; y) = 2 j k+2 c j 1 c k 1 To simplify further, we note 1 j<k N dt x t y t H j 1 (t)h k 1 (t)e t2 (x t)(y t), t [x, y] x t y t = (x t)(y t), t [x, y] x j x k 2
Orthogonal Polynomial System (OPS) Orthogonal polynomial system {p n (t)} defined by the orthogonality relations n=0 dt w(t)p n (t)t m 0 0 m < n = R h n m = n A consequence of the orthogonality relation is the three term recurrence relation a n+1 p n+1 (t) = (t b n )p n (t) a n p n 1 (t), n 1 Central objects in our probabilistic model are the Hankel determinants n := det[µ j+k 2 ] j,k=1,...,n, n 1, 0 := 1, in terms of the moments of the weight {µ n } n=0,1,..., µ n := dt w(t)t n R Alternatively n = 1 n! dt 1... R dt n R result in orthogonal polynomial theory n w(t l ) l=1 1 j<k n a 2 n = n+1 n 1 2, n 1, n (t k t j ) 2, n 1
Spectral Derivative/1st Lax pair [Bauldry 1990, Bonan and Clark 1990] Orthogonal polynomials satisfy a system of coupled first order linear differential equations with respect to t ( d/dt) Wp n = (Ω n V)p n a n Θ n p n 1, n 1, with the coefficient functions V(t), W(t), Θ n (t), Ω n (t). Compatibility Condition Coefficient functions satisfy the recurrence relations Ω n+1 + Ω n = (t b n )Θ n, n 0, (Ω n+1 Ω n )(t b n ) = W + a 2 n+1 Θ n+1 a 2 nθ n 1, n 0
Semi-Classical OPS Class [Maroni,... 1985 -> ] Semi-Classical weights defined by 1 d 2V(t) w(t) = w(t) dt W(t) with V(t) and W(t) irreducible polynomials in t. Deformed Hermite Orthogonal Polynomials with weight w(t; x, y) = t x t y e t2, t R, = [1 2χ (x,y) (t)](t x)(t y)e t2 where 1 if t I χ I (t) = 0 otherwise
Spectral Data [Magnus 1995,...] Numerator and denominator polynomials W(t) = (t x)(t y), 2V(t) = 2t 3 + 2(x + y)t 2 + 2t(1 xy) x y N.B. we have finite singularities at t = x, y. Coefficient functions are and Θ n (t) = 2t 2 + 2(x + y b n )t Ω n (t) = t 3 + (x + y)t 2 + (n + 1 xy 2a 2 n)t + 2n + 1 + 2(x + y)b n + 2(1 xy) 2[a 2 n+1 + a2 n + b 2 n], n 0 n 1 + b i + (x + y)(2a 2 n n 1 2 ) 2a2 n(b n + b n 1 ), n 1 i=0
Recurrence Relations: Discrete Garnier Equations Proposition Density matrix is given by (N + 1)! ρ N+1 (x; y) = C 2 e x2 /2 y 2 /2 N, N+1 and N+1 = a 2 2 N N, N 1 subject to 0 = 1, 1 = µ 0. Coupled recurrences for the polynomial coefficients are n b i + (n + 1 xy)b n b 3 n + (x + y)[b 2 n n 1] i=0 + a 2 n+1 [x + y 2b n b n+1 ] a 2 n[x + y 2b n b n 1 ] = 0 and a 2 n+1 [2n + 5 2xy + 2(x + y)(b n+1 + b n ) 2(a 2 n+2 + a2 n+1 + b2 n+1 + b2 n + b n+1 b n )] a 2 n[2n + 1 2xy + 2(x + y)(b n + b n 1 ) 2(a 2 n + a 2 n 1 + b2 n + b 2 n 1 + b nb n 1 )] + (b n x)(b n y) = 0
Initial Data and Moments The initial data for b n and a 2 n are given by where for x < y b 1 = 0, b 0 = µ 1 µ 0, a 2 0 = 0, a2 1 = µ 0µ 2 µ 2 1 µ 2 0 µ 0 = π( 1 2 + xy)[1 + erf (x) erf (y)] + ye x2 xe y2 µ 1 = 1 2 π(x + y)[1 + erf (x) erf (y)] e x 2 + e y2 µ 2 = π( 3 4 + 1 2 xy)[1 + erf (x) erf (y)] + (y 1 2 x)e x2 (x 1 2 y)e y2
Classical Solutions to L(1, 1, 3; 2) System Deformed Hermite Orthogonal Polynomials with weight w(t; x, y) = (t x) µ (t y) ν e t2, t Γ where Γ {, x, y, }. Numerator and denominator polynomials W(t) = (t x)(t y), 2V(t) = 2t(t x)(t y) + ν(t x) + µ(t y) Coefficient functions are and Θ n (t) = 2(t x)(t y) 2b n (t x y) + θ n, n 0 Ω n (t) = t(t x)(t y) (2a 2 n n)(t x y) + 1 2 ν(t x) + 1 2 µ(t y) + ω n, n 1
Discrete Garnier Equations Proposition Coupled first order recurrences n n + 1 for system {a 2 n, b n, θ n, ω n } θ n + 2b n x = [ω n + (2a 2 n n)x][ω n + (2a 2 n n)x ν(x y)] a 2 n(θ n 1 + 2b n 1 x) θ n + 2b n y = [ω n + (2a 2 n n)y][ω n + (2a 2 n n)y + µ(x y)] a 2 n(θ n 1 + 2b n 1 y) a 2 n+1 + a2 n = b 2 n + 1 2 (2n + 1 + µ + ν θ n) ω n+1 + ω n = (x + y)θ n + b n (2xy θ n ) µx νy
2 2 matrix system Spectral Derivatives Y n (t; x, y) = ( pn ) ɛ n /w p n 1 ɛ n 1 /w where the associated functions are ɛ n (s; x, y) = dt w(t) p n(t) Γ s t, Proposition s Γ Assume Reµ > 0, Reν > 0 and x y. The spectral derivatives satisfy { t Y n = 2t ( ) ( ) 0 0 0 1 + 2a 0 1 n + A x 1 0 t x + A } y Y n AY n t y where the residue matrices are ω n + (2a 2 n n)y x y a n(θ n + 2b n y) x y A x (t; x, y; µ, ν) a n (θ n 1 + 2b n 1 y) x y ω n + (2a 2 n n)y x y µ, (1) and A y = A x x y,µ ν. Properties: TrA x = µ, det A x = 0
Two-variable Garnier Systems L(1, 1, 1, 1, 1; 2) κ 0, κ 1, κ, θ 1, θ 2 L(1, 1, 1, 2; 2) κ 0, κ 1, κ, θ 2 L(1, 1, 3; 2) κ 0, κ 1, κ L(1, 2, 2; 2) κ 0, κ 1, κ L(1, 4; 2) κ 0, κ L(2, 3; 2) κ 0, κ L(5; 2) α L( 9 2 ; 2)
Deformation Derivatives Proposition Assume Reµ > 0, Reν > 0 and x y. The deformation derivatives with respect to x or y satisfy { 1 x Y 1 n = 2 x y { 1 y Y 1 n = 2 x y ( ) θn 2b n y 0 0 θ n 1 + 2b n 1 y ( ) θn + 2b n x 0 0 θ n 1 2b n 1 x A } x Y n B x Y n t x A } y Y n B y Y n t y
N Asymptotics Special diagonal case: x = y = 0 yields b n = 0 and with a 2 n+1 [ 2n + 5 2a 2 n+2 [ ] n+1] 2a2 = a 2 n 2n + 1 2a 2 n 2a 2 n 1 a 2 0 = 0, a2 1 = 3 2 Solution?
N Asymptotics Special diagonal case: x = y = 0 yields b n = 0 and with a 2 n+1 [ 2n + 5 2a 2 n+2 [ ] n+1] 2a2 = a 2 n 2n + 1 2a 2 n 2a 2 n 1 a 2 0 = 0, a2 1 = 3 2 Solution? a 2 n = n + 1 2 1 2 ( 1)n
N Asymptotics Special diagonal case: x = y = 0 yields b n = 0 and with a 2 n+1 [ 2n + 5 2a 2 n+2 [ ] n+1] 2a2 = a 2 n 2n + 1 2a 2 n 2a 2 n 1 a 2 0 = 0, a2 1 = 3 2 Solution? or for general y = x 0 a 2 n = n + 1 2 1 2 ( 1)n
N Asymptotics Special diagonal case: x = y = 0 yields b n = 0 and with a 2 n+1 [ 2n + 5 2a 2 n+2 [ ] n+1] 2a2 = a 2 n 2n + 1 2a 2 n 2a 2 n 1 Solution? or for general y = x 0 a 2 0 = 0, a2 1 = 3 2 a 2 n = n + 1 2 1 2 ( 1)n [ a 2 n = n (n + 2)H 2 n+1 (n + 1)H ] [ n+2h n n H 2 n 1 (n 1)H ] nh n 2 2 [ ] 2 (n + 1)H 2 n n H n+1 H n 1
Where to from here? Can the density matrix eigenvalues (or just the lowest one) be accessed directly and analytically using these ideas? the theory is easily generalisable to calculate other quantities, higher correlations etc, take the thermodynamic limit, finite temperature extensions may be possible within the structures, the situation for finite interparticle repulsion c < is more challenging, certainly other non-integrable potentials can be handled with a little more effort with the resulting difference and differential systems being non-integrable extensions to 2 dimensions would require a major task at the level of the basic mathematical theory
Selected References M. D. Girardeau. Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys., 1:516-523, 1960. E. H. Lieb and W. Liniger. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. (2), 130:1605-1616, 1963. A. Lenard. Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys., 5(7):930-943, 1964. H. G. Vaidya and C. A. Tracy. One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature. J. Math. Phys., 20(11):2291-2312, 1979. M. Jimbo, T. Miwa, Y. Môri, and M. Sato. Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D, 1(1):80-158, 1980. H. Kimura. The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian structure. Ann. Mat. Pura Appl. 155:(4), 25-74, 1989. A. P. Magnus. Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math., 57(1-2):215-237, 1995. P. J. Forrester, N. E. Frankel, T. M. Garoni, and N. S. Witte. Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable bosons. Comm. Math. Phys., 238(1-2):257-285, 2003. H. Kawamuko. On the Garnier system of half-integer type in two variables. Funkcial. Ekvac., 52(2):181-201, 2009.