Lieb–Liniger model

In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott H. Lieb and Werner Liniger[de] who introduced the model in 1963. The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interacting Bose gas. It can be seen as one model in the theory of generalized hydrodynamics.

Definition

Given N {\displaystyle N} bosons moving in one-dimension on the x {\displaystyle x}-axis defined from [ 0 , L ] {\displaystyle [0,L]} with periodic boundary conditions, a state of the N-body system must be described by a many-body wave function ψ ( x 1 , x 2 , … , x j , … , x N ) {\displaystyle \psi (x_{1},x_{2},\dots ,x_{j},\dots ,x_{N})}. The Hamiltonian, of this model is introduced as

H = − ∑ i = 1 N ∂ 2 ∂ x i 2 + 2 c ∑ i = 1 N ∑ j > i N δ ( x i − x j ) , {\displaystyle H=-\sum _{i=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}+2c\sum _{i=1}^{N}\sum _{j>i}^{N}\delta (x_{i}-x_{j})\ ,}

where δ {\displaystyle \delta } is the Dirac delta function. The constant c {\displaystyle c} denotes the strength of the interaction, c > 0 {\displaystyle c>0} represents a repulsive interaction and c < 0 {\displaystyle c<0} an attractive interaction. The hard core limit c → ∞ {\displaystyle c\to \infty } is known as the Tonks–Girardeau gas.

For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., ψ ( … , x i , … , x j , … ) = ψ ( … , x j , … , x i , … ) {\displaystyle \psi (\dots ,x_{i},\dots ,x_{j},\dots )=\psi (\dots ,x_{j},\dots ,x_{i},\dots )} for all i ≠ j {\displaystyle i\neq j} and ψ {\displaystyle \psi } satisfies ψ ( … , x j = 0 , … ) = ψ ( … , x j = L , … ) {\displaystyle \psi (\dots ,x_{j}=0,\dots )=\psi (\dots ,x_{j}=L,\dots )} for all j {\displaystyle j}.

The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} are equal. The condition is that as x 2 {\displaystyle x_{2}} approaches x 1 {\displaystyle x_{1}} from above (x 2 ↘ x 1 {\displaystyle x_{2}\searrow x_{1}}), the derivative satisfies

( ∂ ∂ x 2 − ∂ ∂ x 1 ) ψ ( x 1 , x 2 ) | x 2 = x 1 + = c ψ ( x 1 = x 2 ) {\displaystyle \left.\left({\frac {\partial }{\partial x_{2}}}-{\frac {\partial }{\partial x_{1}}}\right)\psi (x_{1},x_{2})\right|_{x_{2}=x_{1}+}=c\psi (x_{1}=x_{2})}.

Solution

The time-independent Schrödinger equation H ψ = E ψ {\displaystyle H\psi =E\psi }, is solved by explicit construction of ψ {\displaystyle \psi }. Since ψ {\displaystyle \psi } is symmetric it is completely determined by its values in the simplex R {\displaystyle {\mathcal {R}}}, defined by the condition that 0 ≤ x 1 ≤ x 2 ≤ … , ≤ x N ≤ L {\displaystyle 0\leq x_{1}\leq x_{2}\leq \dots ,\leq x_{N}\leq L}.

The solution can be written in the form of a Bethe ansatz as

ψ ( x 1 , … , x N ) = ∑ P a ( P ) exp ⁡ ( i ∑ j = 1 N k P j x j ) {\displaystyle \psi (x_{1},\dots ,x_{N})=\sum _{P}a(P)\exp \left(i\sum _{j=1}^{N}k_{Pj}x_{j}\right)},

with wave vectors 0 ≤ k 1 ≤ k 2 ≤ … , ≤ k N {\displaystyle 0\leq k_{1}\leq k_{2}\leq \dots ,\leq k_{N}}, where the sum is over all N ! {\displaystyle N!} permutations, P {\displaystyle P}, of the integers 1 , 2 , … , N {\displaystyle 1,2,\dots ,N}, and P {\displaystyle P} maps 1 , 2 , … , N {\displaystyle 1,2,\dots ,N} to P 1 , P 2 , … , P N {\displaystyle P_{1},P_{2},\dots ,P_{N}}. The coefficients a ( P ) {\displaystyle a(P)}, as well as the k {\displaystyle k}'s are determined by the condition H ψ = E ψ {\displaystyle H\psi =E\psi }, and this leads to a total energy

E = ∑ j = 1 N k j 2 {\displaystyle E=\sum _{j=1}^{N}\,k_{j}^{2}},

with the amplitudes given by

a ( P ) = ∏ 1 ≤ i < j ≤ N ( 1 + i c k P i − k P j ) . {\displaystyle a(P)=\prod _{1\leq i<j\leq N}\left(1+{\frac {ic}{k_{Pi}-k_{Pj}}}\right)\,.}

These equations determine ψ {\displaystyle \psi } in terms of the k {\displaystyle k}'s. These lead to N {\displaystyle N} equations:

L k j = 2 π I j − 2 ∑ i = 1 N arctan ⁡ ( k j − k i c ) for j = 1 , … , N , {\displaystyle L\,k_{j}=2\pi I_{j}\ -2\sum _{i=1}^{N}\arctan \left({\frac {k_{j}-k_{i}}{c}}\right)\qquad \qquad {\text{for }}j=1,\,\dots ,\,N\ ,}

where I 1 < I 2 < ⋯ < I N {\displaystyle I_{1}<I_{2}<\cdots <I_{N}} are integers when N {\displaystyle N} is odd and, when N {\displaystyle N} is even, they take values ± 1 2 , ± 3 2 , … {\displaystyle \pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots } . For the ground state the I {\displaystyle I}'s satisfy

I j + 1 − I j = 1 , f o r 1 ≤ j < N and I 1 = − I N . {\displaystyle I_{j+1}-I_{j}=1,\quad {\rm {for}}\ 1\leq j<N\qquad {\text{and }}I_{1}=-I_{N}.}

Thermodynamic limit