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 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
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