Symmetry breaking in quantum 1D jellium Sabine Jansen - PowerPoint PPT Presentation

Symmetry breaking in quantum 1D jellium Sabine Jansen Ruhr-Universit¨ at Bochum joint work with Paul Jung (University of Alabama at Birmingham) Warwick University, March 2014

Context Setting : quantum statistical mechanics. Charged fermions move on a line, homogeneous neutralizing background. Wigner ’34 : in order to understand the effect of electronic interactions in solids, crude approximation: periodic charge distribution (atoms) ≈ homogeneous positive charge distribution. Keep electronic interactions. Jellium, one-component plasma. Possible scenario: at low density, electrons minimize repulsive Coulomb energy by forming a periodic lattice. Wigner crystal. Dimension one: Wigner crystallization proven for the classical jellium at all densities ( Kunz ’74, Brascamp-Lieb ’75, Aizenman-Martin ’80 ), for the quantum and classical jellium at low densities Brascamp-Lieb ’75 . This talk : Wigner crystallization for quantum 1D jellium at all densities. Proof combines arguments of cited works, notably Kunz’s transfer matrix approach.

Outline 1. Setting 2. Main result ◮ existence of the thermodynamic limit of all correlation functions ◮ translational symmetry breaking at all β, ρ > 0 3. Proof ideas ◮ path integrals ◮ transfer matrix, Perron-Frobenius

Electrostatic energy and Hamiltonian ◮ N particles of charge − 1, positions x 1 , . . . , x N ∈ [ a , b ] ⊂ R ◮ one-dimensional Coulomb potential V ( x − y ) = −| x − y | ◮ neutralizing background of homogeneous charge density ρ = N / ( b − a ) ◮ total potential energy � b N � � U ( x 1 , . . . , x N ) := − | x j − x k | + ρ | x j − x | d x a 1 ≤ j ≤ k ≤ N j =1 � b � b − ρ 2 | x − x ′ | d x d x ′ . 2 a a ◮ H N Hilbert space for N fermions = antisymmetric functions in L 2 ([ a , b ] N ). ◮ Hamilton operator N ∂ 2 H N := − 1 � + U ( x 1 , . . . , x N ) . ∂ x 2 2 j j =1 Dirichlet boundary conditions at x = a and x = b .

Free energy and reduced density matrices ◮ β > 0 inverse temperature ◮ Thermodynamic limit N N → ∞ , a → −∞ , b → + ∞ , b − a → ρ. ◮ Canonical partition function Z N ( β ) := Tr exp( − β H N ) = 1 � [ a , b ] N exp( − β H N )( x , x ) d x 1 . . . d x N , N ! exp( − β H N )( x ; y ) integral kernel of exp( − β H N ). ◮ Free energy f ( β, ρ ) = − lim 1 β N log Z N ( β ) . ◮ n -particle reduced density matrices � ρ N [ a , b ] N − n exp( − β H N )( x , x ′ ; y , x ′ ) d x ′ n ( x 1 , . . . , x n ; y 1 , . . . , y n ) ∝ proportionality constant fixed by � [ a , b ] n ρ N n ( x ; x ) d x 1 · · · d x n = N ( N − 1) · · · ( N − n + 1)

Results Theorem (Free energy) �� β log(1 − e − β √ 2 ρ ) 1 ρ 2 + 1 − 1 � f ( β, ρ ) = 12 ρ + β log z 0 ( β, ρ ) . z 0 ( β, ρ ) principal eigenvalue of a transfer operator. Free energy of independent harmonic oscillators + a correction term. Theorem (Symmetry breaking) (i) In the thermodynamic limit along a , b ∈ ρ − 1 Z , all reduced density matrices have uniquely defined limits ρ n ( x 1 , . . . , x n ; y 1 , . . . , y n ) = lim ρ N n ( x 1 , . . . , x n ; y 1 , . . . , y n ) . The convergence is uniform on compact subsets of R n × R n , and ρ N n and ρ n are continuous functions of x and y . (ii) The limit is periodic with respect to shifts by λ = ρ − 1 , ρ n ( x 1 − λ, . . . ; . . . , y n − λ ) = ρ n ( x 1 , . . . ; . . . , y n ) for all n ∈ N and x , y ∈ R n . For every θ / ∈ λ Z there is some n ∈ N and some x ∈ R n such that ρ n ( x − θ ; x − θ ) � = ρ n ( x ; x ) : λ is the smallest period.

Periodicity of the one-particle density Limit state on fermionic observable algebra has smallest period λ = ρ − 1 . Question : periodicity visible at the level of the one-particle density? Brascamp, Lieb ’75 : one-particle density is ∞ − ( x − k λ ) 2 � � � ρ 1 ( x ; x ) = F ( x − k λ ) exp 2 σ 2 k = −∞ F even, log-concave function, 2 σ 2 = [ √ 2 ρ tanh( β � ρ/ 2)] − 1 . At low density ( λ = ρ − 1 ≫ σ ), one-particle density has smallest period λ = ρ − 1 . At high density, we do not know whether this is true. Note A state can have a non-trivial period but constant one-particle density. Example Ψ N = · · · ∧ 1 [ − 1 , 0) ∧ 1 [0 , 1) ∧ · · · ∧ 1 [ n , n +1) ∧ · · · One-particle density � n 1 [ n , n +1) ( x ) ≡ 1, periodicity visible only at the level of two-point correlation functions.

Energy as a sum of squares Observation : when particles are labelled from left to right a ≤ x 1 ≤ · · · ≤ x N ≤ b , energy is a sum of squares N ( x j − m j ) 2 + N m j = a + ( j − 1 � U ( x 1 , . . . , x N ) = ρ 12 ρ, 2) λ. j =1 Baxter ’63 . Elementary computation: x j − a + b � � � 2 � − ( x k − x j ) + ρ 2 j < k j x j − a + b � 2 , � � � � = ( k − 1) x k − ( N − j + 1) x j + ρ 2 k j j then complete the squares. Remark: Boltzmann weight: a Gaussian times a characteristic function (of a convex set). Starting point for Brascamp, Lieb ’75 .

Transfer matrix for the classical jellium Partition function for the classical system: � b � b N � ( x j − m j ) 2 � � Z N ( β ) ∝ d x 1 · · · d x N exp − βρ � x 1 ≤ · · · ≤ x N � . 1 a a j =1 Three easy steps: 1. change variables y j = x j − m j 2. define Gaussian measure µ ( d y ) = exp( − βρ y 2 ) d y 3. write indicator that particles are ordered as product of pair terms N N � � � x 1 ≤ · · · ≤ x N � = 1 ( y j − 1 ≤ y j + λ ) = K ( y j − 1 , y j ) 1 j =2 j =2 Remember m j − m j − 1 = λ = ρ − 1 . Partition function becomes � Z N ( β ) ∝ R N µ ( d y 1 ) · · · µ ( d y N ) F ( y 1 ) K ( y 1 , y 2 ) · · · K ( y N − 1 , y N ) G ( y N ) . Functions F ( y 1 ) = 1 ( y 1 + m 1 ≥ a ) and G ( y N ) = 1 ( y N + m N ≤ b ) encode boundary conditions. Representation used in Kunz’s proof.

Path integrals I Work in L 2 (Weyl chamber) instead of antisymmetric wave functions. W N ( a , b ) = { x | a ≤ x 1 ≤ · · · ≤ x N ≤ b } , Fermionic Hilbert space is isomorphic to L 2 ( W N ( a , b )). Hamiltonian becomes ∂ 2 − 1 + N � + ρ ( x j − m j ) 2 � � H N = 12 ρ. 2 ∂ x 2 j 1 ≤ j ≤ N Fermi statistics ⇒ Dirichlet boundary conditions at x j = x j +1 . Apply Feynman-Kac formula in Weyl chamber. Path space E = { γ : [0 , β ] → R | γ continuous } µ xy = Brownian bridge measure on E (not normalized). Non-colliding paths N ( a , b ) := { ( γ 1 , . . . , γ N ) ∈ E N | ∀ t ∈ [0 , β ] : a < γ 1 ( t ) < · · · < γ N ( t ) < b } . W β Feynman-Kac formula: � β � e − ρ � N 0 ( γ j ( t ) − m j ) 2 d t 1 W β � e − β H N ( x ; y ) ∝ µ x 1 y 1 ⊗ · · · ⊗ µ x N y N N ( a , b ) ( γ ) j =1

Path integrals II � � β � e − ρ � N 0 ( γ j ( t ) − m j ) 2 d t 1 W β � Z N ( β ) ∝ µ x 1 x 1 ⊗· · ·⊗ µ x N x N j =1 N ( a , b ) ( γ ) d x 1 · · · d x N . W N ( a , b ) Probability measure on non-colliding paths W β N ( a , b ) ⊂ E N . Gaussian measure conditioned on non-collision. Particle positions recovered as path starting points x j = γ j (0).

Transfer matrix for the quantum jellium Step 1 : change variables η j ( t ) = γ j ( t ) − m j . Step 2 : Define Gaussian measure ν on 1-particle path space � β � 1 � � � � η ( t ) 2 d t ν ( d η ) f ( η ) = µ xx ( d η ) exp − ρ f ( γ ) . d x c ( β, ρ ) E R E 0 Step 3 : Transfer operator in L 2 ( E , ν ) encoding non-collision: � � � ( K f )( η ) = K ( η, ξ ) f ( ξ ) ν ( d ξ ) , K ( η 1 , η 2 ) = 1 ∀ t : η 1 ( t ) < η 2 ( t ) + λ . E Partition function Z N ( β ) ∝ � F , K N − 1 G � , suitable F , G ∈ L 2 ( E , ν ). Operator K is compact (Hilbert-Schmidt), irreducible ⇒ || K || = largest eigenvalue z 0 ( β, ρ ) > 0 (Krein-Rutman / Perron-Frobenius). Asymptotics of the partition function ↔ principal eigenvalue z 0 ( β, ρ ) of K . Infinite volume measure on E Z : Shift-invariant, ergodic. Theorems on free energy, existence and uniqueness of the limits of correlation functions follow.

Symmetry breaking I ◮ It is enough to look at “diagonal” correlation functions ρ ( x ; x ) / expectations of multiplication operators. Instead of dealing with full quantum state, look at probability measure P on point configurations ω = { x j | j ∈ Z } . Shifted configuration is τ θ ω = { x j + θ | j ∈ Z } . ◮ Correlation functions are factorial moment densities of P � ρ n ( x ; x ) d x 1 · · · d x n = E � N I ( N I − 1) · · · ( N I − n + 1) � , I ×···× I N I = # ω ∩ I = # { j | x j ∈ I } number of particles in interval I . Correlation functions determine measure P uniquely (moment problem). ◮ If measure P and shifted measure P ◦ τ θ are mutually singular, then there must be some correlation function ρ n and some x 1 , . . . , x n such that ρ n ( x 1 − θ, . . . ; . . . , x n − θ ) � = ρ n ( x 1 , . . . ; . . . , x n ). We prove P ◦ τ θ ⊥ P whenever θ / ∈ λ Z .