Lieb–Liniger model explained

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. t is named after Elliott H. Lieb and Werner Liniger who introduced the model in 1963.^[1] The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interaction Bose gas.

Definition

Given

N bosons moving in one-dimension on the x -axis defined from [ 0 ,L] with periodic boundary conditions, a state of the N-body system must be described by a many-body wave function \psi(x _{1,} x _{2,} ... , x _{j,} ... ,x _{N)} . The Hamiltonian, of this model is introduced asH = - \sum ^{N} _{i =1} (\partial ^{2})/(\partial x ^{2} _{i}) + 2 c \sum ^{N} _{j>i} \delta(x _{i-x} _{j) ,}

where

\delta is the Dirac delta function. The constant c denotes the strength of the interaction, c> 0 represents a repulsive interaction and c< 0 an attractive interaction.^[2] The hard core limit c\to inf ty 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.,

\psi( ... , x _{i,...,} x _{j,} ... ) = \psi( ... , x _{j,...,} x _{i,} ... ) for all i ≠ j and \psi satisfies \psi( ... , x _{j=0,} ... ) = \psi( ... , x _{j=L,...} ) for all j .

The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say

x _{1} and x _{2} are equal; this condition is that as x _{2} \searrow x _{1} , the derivative satisfies \left.\left( (\partial)/(\partial x _{2}) - (\partial)/(\partial x _{1}) \right) \psi (x _{1,} x _{2)\right|} ^{x _{2=x} _{1+}} = c \psi (x _{1=x} _{2)} .

Solution

H\psi = E\psi , is solved by explicit construction of \psi . Since \psi is symmetric it is completely determined by its values in the simplex l{R} , defined by the condition that 0 \leq x _{1} \leq x _{2} \leq ... , \leq x _{N} \leq L .

The solution can be written in the form of a Bethe ansatz as^[3]

\psi(x _{1,} ... , x _{N)} = \sum _{P} a(P)\exp \left( i \sum ^{N} _{j =1} k _{P} x _{j\right)} ,

with wave vectors

0 \leq k _{1} \leq k _{2} \leq ... , \leq k _{N} , where the sum is over all N ! permutations, P , of the integers 1 , 2 , ... , N , and P maps 1 , 2 , ... ,N to P _{1,P} _{2,...,P} _{N} . The coefficients a(P) , as well as the k 's are determined by the condition H\psi = E\psi , and this leads to a total energyE = \sum ^{N} _{j =1} k ^{2} _{j} ,with the amplitudes given bya(P) = \prod _{1\leq} \left( 1+ (ic)/(k _{Pi} - k _{Pj}) \right) . ^[4]

These equations determine

\psi in terms of the k 's. These lead to N equations:L k _{j=} 2 \pi I _{j} -2 \sum ^{N} _{i =1} \arctan \left( (k _{j-k} _{i})/(c) \right) for j =1 , ... , N ,

where

I _{1} < I _{2} < … < I _{N} are integers when N is odd and, when N is even, they take values \pm (12 ,)/(\pm) (32 ,)/(...) . For the ground state the I 's satisfyI _{j+1} - I _{j} = 1 , {\rm for} 1 \leq j <N and I _{1=-I} _{N.}

Notes and References

1. Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Physical Review 130: 1605–1616, 1963
2. Book: Eckle, Hans-Peter . Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz . 2019-07-29 . Oxford University Press . 978-0-19-166804-3 . en.
3. Lieb . Elliott . 2008 . Lieb-Liniger model of a Bose Gas . Scholarpedia . en . 3 . 12 . 8712 . 10.4249/scholarpedia.8712 . free . 1941-6016.
4. Teunis C. . Dorlas . Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model . Communications in Mathematical Physics . 154 . 2 . 347–376 . 1993 . 10.1007/BF02097001 . 1993CMaPh.154..347D . 122730941 .