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HOT SCALAR THEORY IN LARGE N :BOSE-EINSTEIN CONDENSATION ∗PETER ARNOLD repor
ting on work done in collaboration with BORIS TOM ´A ˇSIK Department of Physics,University of Virginia,P.O.Box 400714,Charlottesville,VA 22904-4714,USA I review the Bo-Einstein condensation pha transition of dilute gas of cold atoms,for particle theorists acquainted with methods of field theory at finite tem-perature.I then discuss how the dependence of the pha transition temperature on the interaction strength can be computed in the large N approximation.1Pha Transitions in Hot Scalar Theories The standard example from particle physics of a scalar theory is the Higgs ctor of electroweak theory,which has a pha transition (or a crossover)at a temperature of order the weak scale:say,a few hundred GeV or so.Let’s focus on pure scalar theory by imagining tting the gauge coupling constant g w to zero.At finite temperature,the Higgs picks up a thermal contribution #λT 2to its effective mass,and the effective potential becomes roughly of the form V (φ)∼m 2eff(T )φ2+λφ4with m 2eff(T )=−M 2+#λT 2.At sufficiently high temperature,the #λT 2turns the potential from being concave down at the origin to concave up,and so restores the symmetry that is spontaneously broken at zero temperature.Standard techniques for analyzing the pha transition between the hot,symmetry-restored pha and the cold,symmetry-broken pha are as follows.(i)Work in Euclidean time (for studying non-dynamical questions).The Euclidean time direction becomes periodic at finite temperature,with period
β=1/T .(ii)Near the cond-order pha transition (T →T c )of the purely scalar model,the correlation length becomes infinite.Large distance physics becomes important,and there are large,non-perturbative,large-wavelength fluctuations.(iii)At large distances (E k
∼k ≪T ),the compact Euclidean time direction decouples,and one can reduce the original Euclidean theory to
粉笔网页版a purely3-dimensional effective theory of the zero-Euclidean-frequency modes:
S eff= d3x |∇φ|2+m2eff|φ|2+λeff|φ|4+··· .(1)
(iv)Figure out what to do with the3-dimensional theory(put it on a lattice, or whatever).
2Today’s Talk:Bo-Einstein Condensation
The purpo of today’s talk is to show that the exact same techniques par-ticle physicists u to study relativistically hot scalar theories can also be ud to study the Bo-Einstein condensation pha transition of dilute gas of(for example)Rubidium atoms at T∼0.1µK.There’s an identical three-dimensional effective theory to study the non-perturbative long-distance physics(E k≪T)near the critical temperature:
S eff= d3x |∇φ|2+r|φ|2+u|φ|4+··· ,(2)警示线
where I’ve switched to typical condend-matter names(r and u)for the coefficients.For a non-relativistic problem,E k∼k2/(2m),and the long-distance condition E k≪T for the validity of this effective theory becomes √网络游戏有哪些
塞翁失马歇后语
k≪
ψ+(chemical potential term).(4)
2m∇2−V(x)
意气风发的意思whereψis a complex bosonicfield and I’ve written the path integral,for the moment,in real time rather than Euclidean time.Why is this the path inte-
gral?Note that the equation of motion,obtained by varying with respect to ψ∗,is just the Schr¨o dinger equation i∂t+∇2/2m−V(x) ψ=0.The above path integral therefore describes the cond quantization of the Schr¨o dinger
equation:it describes arbitrary numbers of particles,just like the standard path integral for QED descri
bes arbitrary numbers of photons.In canonical quantization language,thefieldψreprents an operator
ˆψ(x,t)= nˆa nψn(x)e−iωn t,(5)
北京必去十大景点
where theψn(x)are eigenstates of the Schr¨o dinger equation,theωn are the corresponding eigen-energies,and theˆa n are the corresponding annihilation operators for particles in that mode.If we specialize to the ca where there is no external potential[V(x)=0],then this becomes
ˆψ(x,t)→ kˆa k e i k·x−iωk t,(6)
which looks just like the quantization offield in terms of plane waves that you’re ud to from relativistic quantumfield theory.
Recall that in single-particle QM,ψ∗ψgives you the probability density. In cond-quantized QM,the analogous statement is thatˆψ∗ˆψgives you the number density,so
ˆN= xˆψ∗ˆψ= nˆa†nˆa n.(7)
To describe a system of particles with a given number density n,it’s convenient to u the grand-can
onical enmble and introduce a chemical potential term µN in the Hamiltonian or Lagrangian.So,ourfinal Lagrangian for a free non-relativistic Bo gas is
L= d3xψ∗ i∂t+1
2 xyψ∗ψ(x)U(x−y)ψ∗ψ(y).(9)
If U is localized,then,in the low-energy limit(wavelength≫a),we can replace it by an effectiveδ-function:
4πa
U(x−y)→
ψ−2πa
2m∇2+µ−V(x)
ψ0+2πa
2m∇2−µeff+V(x)
2mT:
β0dτL E→β d3x |∇φ|2+r|ψ|2+u|φ|4 ,(13)
where r=−2mµeffand u=4πamT.Note that the diluteness condition a≪n1/3implies that the coupling u should be considered as“small,”since
it is proportional to a.So,we have here an O(2)field theory in3-dimensions (sinceφis complex)which is weakly-coupled at short distances.b
6A Goal:Calculate T c as a Function of n
An interesting thing to try with this effective theory is to calculate the correc-tion,due to interactions,to the ideal gas result for the Bo-Einstein conden-sation temperature T c.Actually,it turns out to be technially slightly easier to calculate the shift∆n(T)in the critical density due to interactions,at fixed temperature,rather than the shift∆T(n)in the critical temperature, atfixed density.The two are easily related.Then recall that the density n= ψ∗ψ ∼ φ∗φ .If one were to do perturbation theory,the sort of dia-grams one would form
where the cross reprent the operatorφφ.But perturbation theory breaks down at the transition;so what to do?
One possible technique,implemented by Baym,Blaizot,and Zinn-Justin,1 is to try the large N approximation for solving the3-dimensional theory, tting N=2at the end.At leading order in large N,the graphs which contribute to φ∗φ → φ· φ are
Baym et al.find T c=T0(1+2.33an)plus higher orders in1/N and in an1/3.Boris Tom´aˇs ik and I2have analyzed the next-order corrections in1/N andfind that they change the coefficient2.33by only-26%for N=2.This correction is surprisingly small and suggests that large N might not be too bad for T c!
Acknowledgments
This work was supported by the U.S.DOE,grant DE-FG02-97ER41027.
2mT where it breaks down,where one wants to match it to the original theory.
References
1.G.Baym,J.-P.Blaizot,and J.Zinn-Justin,Europhys.Lett.49,150
(2000).
2.P.Arnold and B.Tom´aˇs ik,cond-mat/9912306,to appear in Phys.Rev.
A.
