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| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. This is a service where people write articles using a specific keyword you have given them. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. <br><br> |
| {{technical|date=November 2010}}
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| The '''Lieb–Liniger model''' describes a gas of particles moving in one dimension and satisfying [[Bose–Einstein statistics]].
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| ==Introduction==
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| A model of a gas of particles moving in one dimension and satisfying
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| [[Bose–Einstein statistics]] was introduced in 1963 <ref name="ll63">Elliott H. Lieb and Werner Liniger, ''Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State'', Physical
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| Review 130: 1605–1616, 1963</ref><ref name="l63">Elliott H. Lieb, ''Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum'', Physical Review 130:1616–1624,1963</ref> in order to study whether the available approximate theories
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| of such gases, specifically Bogolubov's theory, would conform to the
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| actual properties of the model gas. The model is based on a well
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| defined
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| Schrödinger Hamiltonian for particles interacting with each other
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| via a two-body potential, and all the eigenfunctions and eigenvalues of
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| this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional [[Bose gas]] with delta interaction. It also can be considered as quantum [[Non-linear Schroedinger equation]].
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| The ground state as well as the low-lying excited states were computed
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| and found to be in agreement with Bogolubov's theory when the potential
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| is small, except for the fact that there are actually two types of
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| elementary excitations instead of one, as predicted by Bogolubov's and
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| other theories.
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| The model seemed to be only of academic interest until, with the
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| sophisticated experimental techniques developed in the first decade of
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| the 21<math>^{\rm st}</math> century, it became possible to produce this
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| kind of gas using real atoms as particles.
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| ==Definition and solution of the Model==
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| There are <math>N</math> particles with coordinates <math> x </math> on the line <math>[0,L]</math>, with periodic boundary conditions. Thus, an allowed wave function <math>\psi(x_1, x_2, \dots, x_j, \dots,x_N)</math> is symmetric, i.e., <math>\psi(\dots, x_i,\dots, x_j, \dots) = \psi(\dots, x_j,\dots, x_i, \dots) </math> for all <math>i \neq j</math> and <math>\psi</math> satisfies <math>\psi( \dots, x_j=0, \dots ) =\psi(\dots, x_j=L,\dots )</math> for all <math>j</math>. The Hamiltonian, in appropriate units, is
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| : <math> H = -\sum\nolimits_{j=1}^N \partial^2/\partial x_j^2 +2c
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| \sum\nolimits_{1\leq i< j\leq N} \delta(x_i-x_j)\ , </math>
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| where <math>\delta </math> is the [[Dirac delta function]], i.e., the interaction is a contact interaction. The constant <math>c\geq 0</math> denotes its strength. The delta function gives rise to a boundary condition when two coordinates, say <math>x_1 </math> and <math>x_2</math> are equal; this condition is that as <math>x_2 \searrow x_1</math>, the derivative satisfies <math>(\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)</math>. The hard core limit <math>c=\infty</math> is known as the [[Tonks–Girardeau gas]].<ref>{{cite journal |first=Marvin |last=Girardeau |title=Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension |journal=Journal of Mathematical Physics |volume=1 |issue=6 |pages=516–523 |year=1960 |doi=10.1063/1.1703687 |bibcode = 1960JMP.....1..516G }}</ref>
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| Schrödinger's time independent equation, <math>H\psi = E\psi</math> is solved by explicit construction of <math>\psi</math>. Since <math>\psi </math> is symmetric it is completely determined by its values in the simplex <math>\mathcal{R} </math>, defined by the condition that <math>0\leq x_1\leq x_2 \leq \dots, \leq x_N \leq L</math>. In this region one looks for a <math>\psi </math> 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 <math>k_1< k_2 < \cdots <k_N</math>, to be determined,
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| : <math> \psi(x_1, \dots, x_N) = \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{P
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| j} x_j\right) </math>
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| where the sum is over all <math>N !</math> permutations, <math>P</math>, of the integers <math>1,2, \dots, N</math>, and <math>P</math> maps <math>1,2,\dots,N</math> to <math> P1,P2,\dots,PN</math>. The coefficients <math>a(P)</math>, as well as the <math>k</math>'s are determined by the condition <math>H\psi =E\psi</math>, and this leads to
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| : <math> E= \sum\nolimits_{j=1}^N\, k_j^2 </math>
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| : <math> a(P) = \prod\nolimits_{1\leq i<j \leq N}
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| \left(1+\frac{ic}{k_{Pi} -k_{Pj}}\right) \ . </math>
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| [[Teunis C. Dorlas|Dorlas]] (1993) proved that all eigenfunctions of <math>H</math> are of this form.<ref>{{cite journal |first=Teunis C. |last=Dorlas |title=Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model |journal=Communications in Mathematical Physics |volume=154 |issue=2 |pages=347–376 |year=1993 |doi=10.1007/BF02097001 |bibcode = 1993CMaPh.154..347D }}</ref>
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| These equations determine <math>\psi</math> in terms of the <math>k</math>'s, which, in turn, are determined by the periodic boundary conditions. These lead to <math>N</math> equations: | |
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| : <math> L\, k_j= 2\pi I_j\ -2 \sum\nolimits_{i=1}^N \arctan
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| \left(\frac{k_j-k_i}{c} \right) \qquad \qquad \text{for } j=1, \,
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| \dots,\, N \ , </math>
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| where <math>I_1 < I_2<\cdots < I_N</math> are integers when <math>N</math> is odd and, when <math>N</math> is even, they take values <math>\pm \frac12, \pm \frac32, \dots</math> . For the ground state the <math>I</math>'s satisfy
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| : <math> I_{j+1} -I_j = 1, \quad {\rm for} \ 1\leq j <N \qquad
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| \text{and } I_1=-I_N. \, </math>
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| The first kind of elementary excitation consists in choosing <math>I_1,\dots, I_{N-1} </math> as before, but increasing <math>I_N</math> by an amount <math>n>0</math> (or decreasing <math>I_1</math> by <math>n</math>). The momentum of this state is <math>p= 2\pi n /L</math> (or <math>-2\pi n /L</math>).
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| For the second kind, choose some <math>0< n \leq N/2</math> and increase <math>I_i\to I_i+1</math> for all <math>i\geq n</math>. The momentum of this state is <math>p= \pi - 2\pi n/L</math>. Similarly, there is a state with <math>p= -\pi +2\pi n/L</math>. The momentum of this type of excitation is limited to <math>|p| \leq \pi.</math>
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| These excitations can be combined and repeated many times. Thus, they are bosonic-like. If we denote the ground state (= lowest) energy by <math>E_0</math> and the energies of the states mentioned above by <math>E_{1,2}(p)</math> then <math>\epsilon_{1}(p) = E_{1}(p)-E_0</math> and <math>\epsilon_{2}(p) = E_{2}(p)-E_0</math> are the excitation energies of the two modes. Later the model was also solved by [[Quantum inverse scattering method]]. This help to calculate correlation functions and to find conservation laws. The Hamiltonian of the model has a family of commuting operators: conservation laws. An explicit expression for higher conservation laws can be found in Davies and Korepin (1989).<ref>http://insti.physics.sunysb.edu/~korepin/davis.pdf</ref>
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| ==Thermodynamic limit==
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| [[File:lieb-liniger-1.gif|thumb|450px|right|F1| Fig. 1: The ground state
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| energy, from.<ref name="ll63"/> See text.]]
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| To discuss a gas we take a limit <math>N</math> and <math>L</math> to
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| infinity with the density <math>\rho =N/L</math> fixed. The ground state
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| energy per particle <math>e = E_0/N</math>, and the
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| <math>\epsilon_{1,2}(p)</math> all have limits as <math>N\to
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| \infty</math>. While there are two parameters, <math>\rho</math> and
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| <math>c</math>, simple length scaling <math>x\to \rho x</math> shows
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| that there is really only one, namely <math>\gamma =c/\rho</math>. | |
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| To evaluate <math>E_0</math> we assume that the ''N'' <math>k</math>'s lie between numbers <math>K</math> and −''K'', to be determined, and with a density <math>L\, f(k)</math>. This <math>f</math> is found to satisfy the equation (in the interval <math>-K \leq k \leq K</math>)
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| : <math> 2c\int\nolimits_{-K}^K \frac{f(p)}{c^2 +(p-k)^2} dp = 2\pi f(k)
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| -1 \quad {\rm and} \quad \int\nolimits_{-K}^K f(p) dp = \rho \ , </math>
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| which has a unique positive solution. An excitation distorts this density <math>f</math> and similar integral equations determine these distortions. The ground state energy per particle is given by
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| : <math> e = \frac{1}{\rho}\int\nolimits_{-K}^K k^2 f(k) dk .</math>
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| Figure 1 shows how <math>e</math> depends on <math>\gamma</math> and also shows Bogolubov's approximation to
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| <math>e</math>. The latter is asymptotically exact to second order in <math>\gamma</math>, namely, <math>e\approx \gamma
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| -4\gamma^{3/2}/\pi</math>. At <math>\gamma =\infty</math>, <math>e =
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| \pi^2/3</math>.
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| [[File:lieb-liniger-2.gif|thumb|200px|right|F2| Fig. 2: The energies of the two types of excitations, from.<ref name="l63"/> See text.]]
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| Figure 2 shows the two excitation energies
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| <math>\epsilon_1(p)</math> and <math>\epsilon_2 (p)</math> for a small value of <math>\gamma = 0.787</math>. The two curves are similar to these for all values of <math>\gamma >0</math>, but the Bogolubov approximation (dashed) becomes worse as <math>\gamma </math> increases. | |
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| [[Correlation function (statistical mechanics)]] of the model was evaluated in the book entitled of ''Quantum Inverse Scattering Method and Correlation Functions'' by
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| V.E. Korepin, N.M. Bogoliubov and A.G. Izergin (Cambridge University Press, 1993).
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| <ref>V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, ''Quantum Inverse Scattering Method and Correlation Functions'', Cambridge University Press, 1993</ref>
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| [[Quantum entanglement]] of the model was calculated in Physical Review Letters, vol 92, issue 9, electronic identifier 096402, 5 March 2004 .
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| ==From three to one dimension.==
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| 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<ref>{{cite journal |first=Elliott H. |last=Lieb |first2=Robert |last2=Seiringer |first3=Jakob |last3=Yngvason |title=One-dimensional Bosons in Three-dimensional Traps |journal=[[Physical Review Letters]] |volume=91 |issue=15 |pages=150401 |year=2003 |doi=10.1103/PhysRevLett.91.150401 |arxiv=cond-mat/0304071 |bibcode = 2003PhRvL..91o0401L }}</ref> and for excited states.<ref>{{cite journal |first=Robert |last=Seiringer |first2=Jun |last2=Yin |title=The Lieb–Liniger Model as a Limit of Dilute Bosons in Three Dimensions |journal=Communications in Mathematical Physics |volume=284 |issue=2 |pages=459–479 |year=2008 |doi=10.1007/s00220-008-0521-6 |arxiv=0709.4022 |bibcode = 2008CMaPh.284..459S }}</ref> 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 <math>e</math>.
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| ==References==
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| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| ==External links==
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| * See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.[http://www.scholarpedia.org/article/Lieb-Liniger_model_of_a_Bose_Gas]
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| * See also V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, ''Quantum Inverse Scattering Method and Correlation Functions'', Cambridge University Press, 1993.[http://books.google.com/books?id=kaZ0pKIHhxAC&dq=quantum+inverse+scattering+method&printsec=frontcover&source=bl&ots=4AaoICh4Q3&sig=8RoPelCcAOLTG1vY21DDLCulNGY&hl=en&ei=I6zuScSBE5a-M9q1pOwP&sa=X&oi=book_result&ct=result&resnum=8#v=onepage&q&f=false]
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| {{DEFAULTSORT:Lieb-Liniger model}}
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| [[Category:Statistical mechanics]]
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