Lieb-Liniger model of a Bose Gas
| Elliott H. Lieb (2008), Scholarpedia, 3(12):8712. | doi:10.4249/scholarpedia.8712 | revision #91428 [link to/cite this article] |
The Lieb-Liniger model describes a gas of particles moving in one-dimension and satisfying Bose-Einstein statistics.
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Introduction
A model of a gas of particles moving in one-dimension and satisfying Bose-Einstein statistics was introduced in 1963 (Lieb-Liniger 1963, Lieb 1963) in order to study whether the available approximate theories of such gases, specifically Bogolubov's theory, would conform to the actual properties of the model gas. The model is based on a well defined Schrödinger Hamiltonian for particles interacting with each other via a two-body potential, and all the eigenfunctions and eigenvalues of this Hamiltonian can, in principle, be calculated exactly.
The ground state as well as the low-lying excited states were computed and found to be in agreement with Bogolubov's theory when the potential is small, except for the fact that there are actually two types of elementary excitations instead of one, as predicted by Bogolubov's and other theories.
The model seemed to be only of academic interest until, with the sophisticated experimental techniques developed in the first decade of the 21\(^{\rm st}\) century, it became possible to produce this kind of gas using real atoms as particles.
Definition and Solution of the Model
There are \(N\) particles with coordinates \(x\) on the line \([0,L]\ ,\) with periodic boundary conditions. Thus, an allowed wave function \(\psi(x_1, x_2, \dots, x_j, \dots,x_N)\) is symmetric, i.e., \(\psi(\dots, x_i,\dots, x_j, \dots) = \psi(\dots, x_j,\dots, x_i, \dots) \) for all \(i \neq j\) and \(\psi\) satisfies \(\psi( \dots, x_j=0, \dots ) =\psi(\dots, x_j=L,\dots )\) for all \(j\ .\) The Hamiltonian, in appropriate units, is
\( H = -\sum\nolimits_{j=1}^N \partial^2/\partial x_j^2 +2c \sum\nolimits_{1\leq i< j\leq N} \delta(x_i-x_j)\ , \)
where \(\delta \) is the Dirac delta function, i.e., the interaction is a contact interaction. The constant \(c\geq 0\) denotes its strength. The delta function 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 \((\frac{\partial}{\partial x_2} - \frac{\partial}{\partial x_1} ) \psi (x_1, x_2)|_{x_2=x_1+}= c \psi (x_1=x_2)\ .\) The hard core limit \(c=\infty\) is known as the Tonks-Girardeau gas (Girardeau 1960).
Schrödinger's time independent equation, \(H\psi = E\psi\) is solved by explicit construction of \(\psi\ .\) Since \(\psi \) is symmetric it is completely determined by its values in the simplex \(\mathcal{R} \ ,\) defined by the condition that \(0\leq x_1\leq x_2 \leq \dots, \leq x_N \leq L\ .\) In this region one looks for a \(\psi \) of the form considered by H.A. Bethe in 1931 in the context of magnetic spin systems -- the Bethe Ansatz. That is, for certain real numbers \(k_1< k_2 < \cdots <k_N\ ,\) to be determined,
\( \psi(x_1, \dots, x_N) = \sum_P a(P)\exp (\ i \sum_{j=1}^N k_{P j} x_j)\ , \)
where the sum is over all \(N !\) permutations, \(P\ ,\) of the integers \(1,2, \dots, N\ ,\) and \(P\) maps \(1,2,\dots,N\) to \( P1,P2,\dots,PN\ .\) The coefficients \(a(P)\ ,\) as well as the \(k\)'s are determined by the condition \(H\psi =E\psi\ ,\) and this leads to
\( E= \sum\nolimits_{j=1}^N\, k_j^2 \)
\( a(P) = \prod\nolimits_{1\leq i<j \leq N} \left(1+\frac{ic}{k_{Pi} -k_{Pj}}\right) \ . \)
T.C. Dorlas (Dorlas, 1993) proved that all eigenfunctions of \(H\) are of this form.
These equations determine \(\psi\) in terms of the \(k\)'s, which, in turn, are determined by the periodic boundary conditions. These lead to \(N\) equations\[ L\, k_j= 2\pi I_j\ -2 \sum\nolimits_{i=1}^N \arctan \left(\frac{k_j-k_i}{c} \right) \qquad \qquad {\rm for}\ \ j=1, \, \dots,\, N \ , \]
where \(I_1 < I_2<\cdots < I_N\) are integers when \(N\) is odd and, when \(N\) is even, they take values \(\pm \frac12, \pm \frac32, \dots\ .\) For the ground state the \(I\)'s satisfy
\( I_{j+1} -I_j = 1, \quad {\rm for} \ 1\leq j <N \qquad {\rm and } \ \, I_1=-I_N \ . \)
The first kind of elementary excitation consists in choosing \(I_1,\dots, I_{N-1} \) as before, but increasing \(I_N\) by an amount \(n>0\) (or decreasing \(I_1\) by \(n\)). The momentum of this state is \(p= 2\pi n /L\) (or \(-2\pi n /L\)).
For the second kind, choose some \(0< n \leq N/2\) and increase \(I_i\to I_i+1\) for all \(i\geq n\ .\) The momentum of this state is \(p= \pi - 2\pi n/L\ .\) Similarly, there is a state with \(p= -\pi +2\pi n/L\ .\) The momentum of this type of excitation is limited to \(|p| \leq \pi.\)
These excitations can be combined and repeated many times. Thus, they are bosonic-like. If we denote the ground state (= lowest) energy by \(E_0\) and the energies of the states mentioned above by \(E_{1,2}(p)\) then \(\epsilon_{1}(p) = E_{1}(p)-E_0\) and \(\epsilon_{2}(p) = E_{2}(p)-E_0\) are the excitation energies of the two modes.
Thermodynamic Limit
To discuss a gas we take a limit \(N\) and \(L\) to infinity with the density \(\rho =N/L\) fixed. The ground state energy per particle \(e = E_0/N\ ,\) and the \(\epsilon_{1,2}(p)\) all have limits as \(N\to \infty\ .\) While there are two parameters, \(\rho\) and \(c\ ,\) simple length scaling \(x\to \rho x\) shows that there is really only one, namely \(\gamma =c/\rho\ .\)
To evaluate \(E_0\) we assume that the \(N\) \(k\)'s lie between numbers \(K\) and \(-K\ ,\) to be determined, and with a density \(L\, f(k)\ .\) This \(f\) is found to satisfy the equation (in the interval \(-K \leq k \leq K\))
\( 2c\int\nolimits_{-K}^K \frac{f(p)}{c^2 +(p-k)^2} dp = 2\pi f(k) -1 \quad {\rm and} \quad \int\nolimits_{-K}^K f(p) dp = \rho \ , \)
which has a unique positive solution. An excitation distorts this density \(f\) and similar integral equations determine these distortions. The ground state energy per particle is given by
\( e = \frac{1}{\rho}\int\nolimits_{-K}^K k^2 f(k) dk .\) Figure 1 shows how \(e\) depends on \(\gamma\) and also shows Bogolubov's approximation to \(e\ .\) The latter is asymptotically exact to second order in \(\gamma\ ,\) namely, \(e\approx \gamma -4\gamma^{3/2}/\pi\ .\) At \(\gamma =\infty\ ,\) \(e = \pi^2/3\ .\)
Figure 2 shows the two excitation energies \(\epsilon_1(p)\) and \(\epsilon_2 (p)\) for a small value of \(\gamma = 0.787\ .\) The two curves are similar to these for all values of \(\gamma >0\ ,\) but the Bogolubov approximation (dashed) becomes worse as \(\gamma \) increases.
From three to one dimension.
This one-dimensional gas can be made using real, three-dimensional atoms as particles. One can prove, mathematically, from the Schrödinger equation for three-dimensional particles in a long cylindrical container, that the low energy states are described by the one-dimensional Lieb-Liniger model. This was done for the ground state in (Lieb-Seiringer-Yngvason 2003) and for excited states in (Seiringer-Yin 2008). The cylinder does not have to be as narrow as the atomic diameter; it can be much wider if the excitation energy in the direction perpendicular to the axis is large compared to the energy per particle \(e\ .\)
References
- 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
- Elliott H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Physical Review 130:1616-1624,1963.
- Teunis C. Dorlas, Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model, Communications in Mathematical Physics, 154:347-376,1993.
- Marvin Girardeau, Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension, Journal of Mathematical Physics, 1:516-523,1960.
- Elliott H. Lieb, Robert Seiringer and Jakob Yngvason, One-dimensional Bosons in Three-dimensional Traps, Physical Review Letters, 91:#150401--1-4,2003.
- Robert Seiringer and Jun Yin, The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions, Communications in Mathematical Physics, 284:459-479,2008
See also
Bogolubov's theory, Bose-Einstein statistics, Bose condensation: mathematical aspects, Schrödinger Hamiltonian
