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Lieb–Liniger model

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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 [de] 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.[2]

Definition

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Given bosons moving in one-dimension on the -axis defined from with periodic boundary conditions, a state of the N-body system must be described by a many-body wave function . The Hamiltonian, of this model is introduced as

where is the Dirac delta function. The constant denotes the strength of the interaction, represents a repulsive interaction and an attractive interaction.[3] The hard core limit is known as the Tonks–Girardeau gas.[3]

For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., for all and satisfies for all .

The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say and are equal; this condition is that as , the derivative satisfies

.

Solution

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Fig. 1: The ground state energy (per particle) as a function of the interaction strength per density , from.[1]

The time-independent Schrödinger equation , is solved by explicit construction of . Since is symmetric it is completely determined by its values in the simplex , defined by the condition that .

The solution can be written in the form of a Bethe ansatz as[2]

,

with wave vectors , where the sum is over all permutations, , of the integers , and maps to . The coefficients , as well as the 's are determined by the condition , and this leads to a total energy

,

with the amplitudes given by

[4]

These equations determine in terms of the 's. These lead to equations:[2]

where are integers when is odd and, when is even, they take values . For the ground state the 's satisfy

Thermodynamic limit

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References

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  1. ^ a b 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. ^ a b c Lieb, Elliott (2008). "Lieb-Liniger model of a Bose Gas". Scholarpedia. 3 (12): 8712. doi:10.4249/scholarpedia.8712. ISSN 1941-6016.
  3. ^ a b Eckle, Hans-Peter (29 July 2019). Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz. Oxford University Press. ISBN 978-0-19-166804-3.
  4. ^ Dorlas, Teunis C. (1993). "Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model". Communications in Mathematical Physics. 154 (2): 347–376. Bibcode:1993CMaPh.154..347D. doi:10.1007/BF02097001. S2CID 122730941.