Topological Charge of Elementary Excitations in Lieb-Liniger Model - PowerPoint PPT Presentation

Topological Charge of Elementary Excitations in Lieb-Liniger Model Vladimir Korepin Prepared by: You Quan Chong Department of Physics and Astronomy Stony Brook University

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Outline 1 Lieb-Liniger model 2 Periodic boundary conditions 3 Thermodynamic limit at zero temperature 4 Excitations at zero temperature 5 Fermionization 6 Excitations at finite temperature

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Outline 1 Lieb-Liniger model 2 Periodic boundary conditions 3 Thermodynamic limit at zero temperature 4 Excitations at zero temperature 5 Fermionization 6 Excitations at finite temperature

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Lieb-Liniger model Hamiltonian � � � ∂ x Ψ † ( x ) ∂ x Ψ( x ) + c Ψ † ( x )Ψ † ( x )Ψ( x )Ψ( x ) H = dx Canonical equal-time commutation relations: [Ψ( x, t ) , Ψ † ( y, t )] = δ ( x − y ) [Ψ( x, t ) , Ψ( y, t )] = [Ψ † ( x, t ) , Ψ † ( y, t )] = 0 Heisenberg equation of motion i∂ t Ψ = − ∂ 2 x Ψ + 2 c Ψ † ΨΨ

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Eigenfunctions Eigenfunctions 1 � d N z χ N ( z 1 , . . . , z N | λ 1 , . . . , λ N )Ψ † ( z 1 ) . . . Ψ † ( z N ) | 0 � | ψ N ( λ 1 , . . . , λ N ) � = √ N ! H | ψ N � = E N | ψ N � where χ N is a symmetric function of all z j . Quantum mechanical Hamiltonian N � � − ∂ 2 � � H N = + 2 c δ ( z j − z k ) , c > 0 ∂z 2 j j =1 N ≥ j>k ≥ 1 H N χ N = E N χ N

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations χ N − 1 / 2    [( λ j − λ k ) 2 + c 2 ]  � χ N =  N !  j>k N � � � ( − 1) [ P ] exp � � i z n λ P n [ λ P j − λ P k − icǫ ( z j − z k )] P n =1 j>k   N ( N − 1) � N � = ( − i ) 2   ( − 1) [ P ] exp � � � √ ǫ ( z j − z k ) i z k λ P k N !   N ≥ j>k ≥ 1 P k =1   i   � × exp ǫ ( z j − z k ) θ ( λ P j − λ P k ) 2  N ≥ j>k ≥ 1  � � ic + λ − µ x where θ ( λ − µ ) = i ln ; ǫ ( x ) = ic − λ + µ | x | χ N is antisymmetric in momenta (Pauli principle in momentum space) χ N ( z 1 , . . . , z N | λ 1 , . . . , λ j , . . . , λ k , . . . , λ N ) = − χ N ( z 1 , . . . , z N | λ 1 , . . . , λ k , . . . , λ j , . . . , λ N )

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations χ N Normalization � ∞ N d N z χ ∗ N ( z 1 , . . . , z N | λ 1 , . . . , λ N ) χ N ( z 1 , . . . , z N | µ 1 , . . . , µ N ) = (2 π ) N � δ ( λ j − µ j ) −∞ j =1 where the momenta { λ } and { µ } are ordered: λ 1 < λ 2 < . . . < λ N , µ 1 < µ 2 < . . . < µ N Completeness � ∞ N d N λ χ ∗ N ( z 1 , . . . , z N | λ 1 , . . . , λ N ) χ N ( y 1 , . . . , y N | λ 1 , . . . , λ N ) = (2 π ) N � δ ( z j − y j ) −∞ j =1 where the coordinates { z } and { y } are ordered: z 1 < z 2 < . . . < z N , y 1 < y 2 < . . . < y N Energy, E N = � N j =1 λ 2 j Momentum, P N = � N j =1 λ j

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Outline 1 Lieb-Liniger model 2 Periodic boundary conditions 3 Thermodynamic limit at zero temperature 4 Excitations at zero temperature 5 Fermionization 6 Excitations at finite temperature

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Periodic boundary conditions Traditionally, we have a periodic wave function: χ N ( z 1 , . . . , z j + L, . . . , z N | λ 1 , . . . , λ N ) = χ N ( z 1 , . . . , z j , . . . , z N | λ 1 , . . . , λ N ) Bethe equations N λ j − λ k + ic � exp { iλ j L } = − λ j − λ k − ic , j = 1 , . . . , N (1) k =1 Log form of Bethe equations ψ j = 2 π ˜ j = 1 , . . . , N (2) n j , where ˜ n j are integers, and N � ψ j = Lλ j + ψ ( λ j − λ k ) k =1 k � = j � λ + ic � ψ = i ln ; − 2 π < ψ ( λ ) < 0 , Im λ = 0 λ − ic

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Periodic boundary conditions Using the antisymmetric θ ( λ ) instead of ψ ( λ ), θ ( λ ) = ψ ( λ ) + π ; θ ( λ ) = − θ ( − λ ) � ic + λ � θ ( λ ) = i ln ic − λ The Bethe equations become N � Lλ j + θ ( λ j − λ k ) = 2 πn j (3) k =1 where n j + N − 1 n j = ˜ 2 and they can be integers (N odd) or half-integers (N even)

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Existence of solutions to Bethe equations Theorem: Solutions to the Bethe equations (3) exist and can be uniquely parameterized by { n j } Proof: Yang-Yang action: N N N S = 1 n j λ j + 1 � λ 2 � � j − 2 π θ 1 ( λ j − λ k ) 2 L 2 j =1 j =1 j,k � λ where θ 1 ( λ ) = 0 θ ( µ ) dµ Extremum conditions (minima) for S give the Bethe equations (3) : ∂S = 0 ∂λ j

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Existence of solutions to Bethe equations Consider � N � ∂ 2 S = ∂ψ j � = ψ ′ jl = δ jl L + K ( λ j , λ m ) − K ( λ j , λ l ) ∂λ j ∂λ l ∂λ l m =1 where 2 c K ( λ, µ ) = ψ ′ ( λ − µ ) = θ ′ ( λ − µ ) = c 2 + ( λ − µ ) 2 So, the Yang-Yang action is convex: N N N ∂ 2 S K ( λ j , λ l )( v j − v l ) 2 ≥ L � � Lv 2 � � v 2 v j v l = j + j > 0 , ∂λ j ∂λ l j,l j =1 j>l =1 j =1 which means that solutions to Bethe equations exist and are unique. If the wavefunction is non-zero, then the Bethe equations are non-degenerate: � ∂ψ j � L ∂ 2 S � � � d N z | χ N | 2 = det = det ∂λ j ∂λ l ∂λ l 0

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Outline 1 Lieb-Liniger model 2 Periodic boundary conditions 3 Thermodynamic limit at zero temperature 4 Excitations at zero temperature 5 Fermionization 6 Excitations at finite temperature

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Preliminaries To pass to the thermodynamic limit, we need: λ j ’s are separated by some interval 2 π ( n j − n k ) ≥ | λ j − λ k | ≥ 2 π ( n j − n k ) 2 π ≥ ; j � = k L (1 + 2 D L (1 + 2 D L c ) c ) where density D = N/L N � λ 2 Energy j is minimized under the condition that { λ j } j =1 satisfy the Bethe equations, given that � N − 1 � n j = − + j − 1 , j = 1 , . . . , N 2

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Some remarks at c = + ∞ At c = + ∞ , the Bethe equations become e iLλ j = ( − 1) N +1 or in log form Lλ j = 2 π ˜ ˜ n j where ˜ ˜ n j can be integers (N odd) or half-integers (N even) We get a non-interacting, free fermion model (due to Pauli principle) for c = + ∞ , with N N j = 2 π � λ 2 � ˜ n 2 E = ˜ j L j =1 j =1 N N λ j = 2 π � � ˜ P = ˜ n j L j =1 j =1

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Thermodynamic limit at zero temperature Thermodynamic limit: D = N N → ∞ ; L → ∞ ; L = const. Bethe equation of ground state (lowest energy): � N + 1 N � �� � Lλ j + θ ( λ j − λ k ) = 2 π j − , j = 1 , . . . , N 2 k =1 We define the density of particles in momentum space: 1 ρ ( λ k ) = lim L ( λ k +1 − λ k ) > 0 Integral equation for ρ ( λ ) (Lieb-Liniger equation) � q ρ ( λ ) − 1 K ( λ, µ ) ρ ( µ ) dµ = 1 2 π 2 π − q And � q D = N L = ρ ( λ ) dλ − q

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Energy Let the ground state at T = 0 be | Ω � Microcanonical ensemble: � q � Ω | H | Ω � λ 2 ρ ( λ ) dλ = E L = L � Ω | Ω � − q Grand canonical ensemble: Let H h = H − hQ � Ψ † ( x )Ψ( x ) dx where Q = Energy N E h � ( λ 2 N = j − h ) j =1

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Outline 1 Lieb-Liniger model 2 Periodic boundary conditions 3 Thermodynamic limit at zero temperature 4 Excitations at zero temperature 5 Fermionization 6 Excitations at finite temperature

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations Excitations We start with periodic boundary conditions (3): N � Lλ j + θ ( λ j − λ k ) = 2 πn j k =1 We define the “shift function” F : ( λ j − ˜ λ j ) F ( λ j | λ p , λ h ) ≡ ( λ j +1 − λ j ) . satisfying the following integral equation: � q dν 2 π K ( µ, ν ) F ( ν | λ p , λ h ) = θ ( µ − λ p ) − θ ( µ − λ h ) F ( λ j | λ p , λ h ) − 2 π − q

Lieb-Liniger PBC Thermodynamic limit Excitations Fermionization Excitations One particle + one hole We consider excitations consisting of a particle and a hole Observable energy (particle + hole): ∆ E ( λ p , λ h ) = E excited ( λ p , λ h ) − E gs � [ ε 0 (˜ = ε 0 ( λ p ) − ε 0 ( λ h ) + λ j ) − ε 0 ( λ j )] j � q ε ′ = ε 0 ( λ p ) − ε 0 ( λ h ) − 0 ( µ ) F ( µ | λ p , λ h ) dµ − q Observable momentum (particle + hole) ∆ P ( λ p , λ h ) = P excited ( λ p , λ h ) − P gs � q = λ p − λ h − F ( µ | λ p , λ h ) dµ − q � q = λ p − λ h + [ θ ( λ p − µ ) − θ ( λ h − µ )] ρ ( µ ) dµ − q