Non Perturbative Renormalization Group and Bose-Einstein Condensation
These lectures are centered around a specific problem, the effect of weak repulsive interactions on the transition temperature of a Bose gas. This problem provides indeed a beautiful illustration of many of the techniques which have been discussed at this school on effective theories and renormalization group. Effective theories are used first in order to obtain a simple hamiltonian describing the atomic interactions: because the typical atomic interaction potentials are short range, and the systems that we consider are dilute, these potentials can be replaced by a contact interaction whose strength is determined by the -wave scattering length. Effective theories are used next in order to obtain a simple formula for the shift in : one exploits there the fact that near the physics is dominated by low momentum modes whose dynamics is most economically described in terms of classical fields; the ingredients needed to calculate the shift of can be obtained from this classical field theory. Finally the renormalization group is used both to obtain a qualitative understanding, and also as a non perturbative tool to evaluate quantitatively the shift in .
In the first lecture, I recall some known aspects of Bose-Einstein condensation of the ideal gas. Then I turn to an elementary discussion of the interaction effects, and introduce an effective theory with a contact interaction tuned to reproduce the scattering length of the atom-atom interaction. I show that at the mean field level, weak repulsive interactions produce no shift in . Finally, I briefly explain why approaching the transition from the low temperature phase is delicate and may lead to erroneous conclusions.
In the second lecture I establish the general formula for the shift of :
where is the s-wave scattering length for the atom-atom interaction, and the transition temperature of the ideal gas at density . This formula holds in leading order in the parameter which measures the diluteness of the system. Getting formula (1) involves a number of steps. First I explain why perturbation theory cannot be used to calculate , however small is. Then I show that the problem with the perturbative expansion is localized in a particular subset of Feynman diagrams that are conveniently resummed by an effective theory of a classical 3-dimensional field. The outcome of the analysis is the formula (1) where is given by the following integral
where the proportionality coefficient, not written here, is a known numerical factor, and is the self-energy of the classical field, whose calculation requires non perturbative techniques.
In the last lecture, I use the non perturbative renormalization group (NPRG) in order to estimate . This requires the knowledge of the 2-point function of the effective 3-dimensional field theory for all momenta, and in particular in the cross-over between the critical region of low momenta and perturbative region of high momenta. This cross-over region is where the dominant contribution to the integral (2) comes from. In order to obtain an accurate determination of , it has been necessary to develop new techniques to solve the NPRG equations. Describing those techniques in detail would take us too far. I shall only present in this lecture the material that can help the student not familiar with the NPRG to understand how it works, and how it can be used. I shall do so by discussing several simple cases that at the same time provide indications on the approximation scheme that has been developed in order to calculate . I shall end by reporting and discussing the results obtained with the NPRG, and compare them to those obtained using other non perturbative techniques.
Recent discussion of Bose-Einstein condensation can be found in Pitaevskii03 (); Pethick02 (); RMP (); Leggett:2001 (); Andersen:2003qj (). The equation (1) for the shift of is derived and discussed in the series of papers 3/2club (); BigN (); bigbec (); NEW (); Fuchs04 (). Much of the material of the last lecture is borrowed from the papers Blaizot:2004qa (); Blaizot:2005wd (); Blaizot:2006vr (); Blaizot:2005xy (); Blaizot:2006ak ().
2 LECTURE 1. Bose-Einstein condensation
2.1 Bose-Einstein condensation for the non interacting gas
The discussion of Bose-Einstein condensation of the ideal Bose gas in the grand canonical ensemble is standard. We consider a homogeneous system of identical spinless bosons of mass , at temperature . The occupation factor of a single particle state with momentum is
where is the chemical potential. The number density of non-condensed particles is given by
For small density, the chemical potential is negative and large in absolute value, . The gas is then described by Boltzmann statistics:
where is the thermal wavelength:
Unless specified otherwise, we use units such that . Boltzmann’s statistics applies as long as , that is, as long as the thermal wavelength is small compared to the interparticle distance. As one lowers the temperature, keeping the density fixed, the chemical potential increases, and so does the thermal wavelength. Eventually, as , the density of non condensed particles reaches a maximum
where is the Riemann zeta-function and the thermal wavelength. As one keeps lowering the temperature, particles start to accumulate in the lowest energy single particle state. This is the onset of Bose-Eisntein condensation, which takes place then when
At this point the thermal wavelength has become comparable to the interparticle spacing. Eq. (8) defines the critical line in the plane (see Fig. 1). In particular, the critical temperature is given by
It is a function only of the mass of the atom, and of the density.
For , the chemical potential stays equal to zero and the single particle state is macroscopically occupied, with density given by
The occurrence of condensation relies on the existence of a maximum of the integral (4) fixing the number of non condensed atoms. This depends on the number of spatial dimensions. In 2 dimensions, the integral diverges logarithmically as :
In this case, there is no limit to the number of thermal particles, hence no condensation in the lowest energy mode.
The Bose-Einstein condensation of the non-interacting gas has some pathological features. In particular the compressibility
diverges when . Also the fluctutation of the number of particles in the condensate,
is of the same order of magnitude as the average value . Such features can be related to the large degeneracy of states in Fock space that appears as .
Consider indeed the ground state in Fock space of the hamiltonian of the non interacting Bose gas,
If , then the ground state is the vacuum, with no particle present in the system: adding a particle costs a positive “free-energy” . When , there appears a huge degeneracy in Fock space: all the states with an arbitrary number of particles in the single particle state are degenerate. If , then there is no minimum, the more particle one adds in the state , the more free energy one gains ( for each added particle). Note that here the chemical potential does not fully plays its role of controlling the density: either it is negative, and the density is zero, or it is positive and the density is infinite; if it vanishes the density is arbitrary.
To relate the large degeneracy to the large fluctuations (13), one may use a simple maximum entropy technique to determine the most likely state with particles in the degenerate space. This calculation parallels the corresponding calculation done in the grand canonical ensemble at finite temperature. Maximizing
with the constraints
For large ,
which leads indeed to the large fluctuations (13).
Of course, some of the pathologies of the ideal Bose gas in the grand canonical ensemble could be eliminated by working in the canonical ensemble, where the particle number is fixed (see e.g. Ziff77 ()). However, we shall keep the discussion within the grand canonical ensemble, as it is closer to familiar field theoretical techniques. This allows us in particular to treat Bose-Eisntein condensation (of the interacting gas) as a symmetry breaking phenomenon.
The large degeneracy of states in Fock space implies that infinitesimal interactions could have a large effect, and indeed they do: as soon as weak repulsive interactions are present the large fluctuations are damped, and the compressibility becomes finite .
2.2 Interactions in the dilute gas
As we shall discuss later in this lecture, the dominant effect of the interactions in the dilute gas can be accounted for by an effective contact interaction whose strength is proportional to the -wave scattering length :
In the mean field (Hartree-Fock) approximation, which is also the leading order in , the effect of the interaction is a simple shift of the single particle energies:
where the factor 2 comes from the exchange term.
It is easy to see that this produces no shift in : because the shift of the single particle energy is constant, independent of the momentum, eq. (4) which gives the number density of non condensed particles, can be written
Bose-Einstein condensation now occurs when , instead of in the non interacting case, but clearly the critical line is identical to that given by eq. (8).
Of course, because the Hartree-Fock self-energy in eq. (20) depends on the density, the relation between the chemical potential at the transition and the critical density is more complicated than in the non-interacting case. The transition now takes place at a finite value of the chemical potential, and some of the pathologies of the ideal gas disappear. In particular the compressibility remains finite at (fluctuations remain important though):
The presence of the interactions allows us to treat the phenomenon of Bose Einstein condensation as a symmetry breaking phenomenon. Let us return first to the ground state, and calculate the free energy
where is the volume. is the expectation value of the hamiltonian (14) to which is added the interaction (19), , in a coherent state containing an average number of particles in the state , . In contrast to what happened for the ideal gas, it is now possible to minimize w.r.t. in order to find the optimum ground state for a given . One gets
Thus the ground state has now a finite number of particles, and the fluctuations are normal, proportional to the square root of the mean value. The density in the ground state is , and the pressure is , so that also at zero temperature the compressibility is finite, . Note however that the coherent state is a state with a non definite number of particles: the symmetry related to particle number conservation is spontaneously broken. We shall return later to the similar picture at finite temperature, and come back to this issue of symmetry breaking.
2.3 Atoms in a trap
Although our main discussion concerns homogeneous systems, it is instructive to contrast the situation in homogeneous systems to what happens in a trap. We shall consider here a spherical harmonic trap, corresponding to the following external potential
Consider first the non interacting gas. We assume the validity of a semiclassical approximation allowing us to express the particle density as the following phase space integral:
where . This requires that the temperature is large compared to the level spacing, , a condition well satisfied near the transition if , the number of particles in the trap, is large enough. The number density in the trap can be written as
where is the function (4). Thus in a wide trap for which the semiclassical approximation is valid, the particles experience the same conditions as in a uniform system with a local effective chemical potential . In particular, eq. (27) shows that the density at the center of the trap is related to the chemical potential by the same relation as in the homogeneous gas. It follows that is determined by the same condition as for the homogeneous gas, that is, where is the density at the center of the trap. An explicit calculation gives
with . Note that while the condensation condition is “universal” when expressed in terms of the density at the center of the trap, the dependence of on depends on the form of the confining potential. In the present case, the dependence can be obtained from the following heuristic argument. For a temperature the particle density is approximately given by the classical formula
so that the thermal particles occupy a cloud of radius , where is the characteristic radius of the harmonic trap and is the thermal wavelength. The average density in the thermal cloud is . At the transition, the interparticle distance is of the order of the thermal wavelength, so that, , or , from which the relation follows.
Note that the confining potential makes condensation easier than in the uniform case. This is related to the fact that the density of single particle states in a trap decreases more rapidly with decreasing energy than in a uniform system: it goes as in a trap and as in a uniform system, where is the number of spatial dimensions. It follows in particular that, in a trap, condensation can occur in , in contrast to the homogeneous case; the heuristic argument presented above yields then . Note however that the effects of the interactions in a 2-dimensional trap are subtle (for a recent discussion see PNAS ()).
The leading effect of repulsive interactions in a trap is to push the particles away from the center of the trap, thereby decreasing the central density. This effect is analogous to that produced by an increase of the temperature. One expects therefore the interactions to lead to a decrease of the transition temperature. To estimate this, we note that, in the presence of interactions, the density is still given by eq. (30) after substituting A simple calculation gives then the density (for not too large, and to leading order in ) in terms of the density at the center of the trap Fuchs04 ():
This result suggests that the effect of the interaction can also be viewed as a modification of the oscillator frequency, . This is enough to estimate the shift in :
where, in the last step, we have used the fact that at the transition. A more explicit calculation yields the proportionality coefficient Giorgini96 ().
It is important to keep in mind that this effect of mean field interactions on is very different from the one that leads to eq. (1) (indeed the sign of the effect is different). If we were comparing systems at fixed central density rather than at fixed particle number, there would be no shift of (see the discussion in Fuchs04 ()).
2.4 The two-body problem
We now come back to the construction of the effective interaction that can be used in many-body calculations of the dilute gas. We shall in particular recall how effective field theory can be used to relate the effective interaction to the low energy scattering data. More on the use of effective theories can be found in the lecture by T. Schäfer in this volume Schafer:2006yf (). A pedagogical introduction is given in Ref. Kaplan:2005es ().
Consider two atoms of mass interacting with the central two-body potential , with the relative coordinate and . The relative wave function satisfies the Schrödinger equation
The scattering wave function is given by ()
where is the initial relative momentum (with ), while is the final relative momentum. The scattering is elastic and because the potential is rotationally invariant, the scattering amplitude is a function only of the scattering angle in the center of mass frame, and of the energy . For short-range interactions, the interaction takes place predominantly in the -wave, and the scattering amplitude becomes of a function of the energy only. It has then the following low momentum expansion:
where is the scattering length, the effective range, and the neglected terms in the denominator involve higher powers of . In the very low momentum limit, when , one can ignore the effective range. Then the scattering amplitude depends on a single parameter, .
The scattering amplitude can be expressed as the matrtix element between plane wave states of the T-matrix:
and the T-matrix itself can be calculated in terms of the Green function
and with real for the retarded Green’s function . The following formal relations are useful
Note that both and are analytic functions of , with poles on the negative real axis corresponding to the energies of the bound states, and a cut on the real positive axis.
Let us now turn to the many-body problem. Assuming that the atoms interact only through the two body potential , we can write the interaction hamiltonian as
We demand that the local effective theory
reproduce the same scattering data in the two particle channel at low momentum as the original potential . We know that in the long wavelength limit the scattering amplitude depends only on the scattering length , so we expect to be related to .
To establish this relation we calculate the scattering amplitude for the effective theory. For a contact interaction is given by
To calculate , which is divergent, we introduce a cut-off on the momentum integral
It is then convenient to define a “renormalized” strength by
where in the last relation is the scattering length. This relation between and is obtained by comparing obtained from eq. (44) (
Remark. One may improve the description by including the effective range correction. This is done by adding to the hamiltonian a term of the form Braaten:2000eh ()
and adjusting so as to reproduce the scattering amplitude of the original two body problem, at the precision of the effective range. At tree level in the effective theory, the calculation of the scattering amplitude yields
where in the second line is the magnitude of the relative momentum. Note that in order to make the identification with the two-body problem discussed above, we have to pay attention that the two-body problem traditionally treats the two particles as distinguishable, whereas the present calculation involves matrix element of the -matrix between symmetric two particle states. Thus the tree level calculation of the -matrix for the hamiltonian (41) yields which differs by a factor 2 with the conventional definition of the scattering length. Staying with the usual convention, we therefore write
from which the identification of follows
2.5 One-loop calculation
Having at our disposal an effective many-body hamiltonian, we may now perform detailed calculations of the effect of the interactions. The one loop calculation that we present here gives us the opportunity to come back to the issue of symmetry breaking, illustrates the use of the delta potential and points to the difficulty of approaching the phase transition from below.
The grand canonical partition function can be written as a path integral:
and is the inverse temperature. The complex field to be integrated over is a periodic function of the imaginary time , with period . The action is invariant under a symmetry:
It is convenient to add an external source linearly coupled to the bosonic field (thereby breaking the symmetry):
The loop expansion is an expansion around the field configurations which make the action stationnary. These field configurations, which we denote by are determined by the equation
There are two solutions. For , or . The solution corresponds to the vacuum state and is the stable solution when . For (and ) the second solution corresponds to broken symmetry and BE condensation; in that case the solution is a maximum of the free energy.
We now expand the field around the solution which corresponds to a minimum of the (classical) free energy:
with having only Fourier components, and keep in terms which are at most quadratic in the fluctuations. That is, we write , with the classical action
and the one-loop correction
The gaussian integral is standard and, after a Legendre transform to eliminate , it yields the following expression of the thermodynamic potential:
where is the Bogoliubov quasiparticle energy:
Note that in this calculation we have used the relation (46) in order to replace the bare coupling constant by the renormalized one. This is the origin of the fourth term in the r.h.s. of eq. (2.5) which eliminates the divergence in the sum over the zero point energies. Note that this replacement assumes that and are perturbatively related, which can only be possible if the cut-off in (46) is not too large, i.e., if , or .
This calculation was first made in the context of Bose-Einstein condensation by Toyoda Toyoda (), and led him to the erroneous conclusion that . It is essentially a mean field calculation, the mean field being entirely due to the condensed particles. This approximation, equivalent to the lowest order Bogoliubov theory (see e.g. FW ()), describes correctly the ground state at zero temperature and its elementary excitations. However, its extension near the critical temperature meets several difficulties; in particular it predicts a first order phase transition BG (), a point apparently overlooked in Ref. Toyoda (). In fact, above , , and the present one-loop calculation yields the free energy of an ideal gas, with no shift in the critical temperature.
3 LECTURE 2. The formula for
Let us start by considering again the phase diagram in Fig. 1. We want to determine the change of the critical line due to weak repulsive interactions. For small and positive values of the -wave scattering length , we expect the change illsutrated in Fig. 2, with the two critical lines close to each other. One can then relate, in leading order, the shift of the critical temperature at fixed density, , to that of the critical density at fixed temperature, . Since, for , , we have
It turns out that it is easier to calculate at fixed temperature than at fixed density, and in the following we shall set up the calculation of .
In the previous lecture, we have seen that, under appropriate conditions, the effective hamiltonian is of the form
where is related to the scattering length by (see eq. (46)):
This effective hamiltonian provides an accurate description of phenomena where the dominant degrees of freedom have long wavelength, , and the system is dilute, . Recall also that we shall be working in the vicinity of the transition where , with the thermal wavelength.
3.1 Condensation condition and critical density
In order to exploit standard techniques of quantum field theory and many-body physics (see e.g.FW (); KB (); self-consistent (); BR ()), we shall first relate the particle density to the single particle propagator. The particle density can be written as
are respectively the creation and annihilation operators in the Heisenberg representation and T denotes (imaginary) time ordering (see e.g. BR ()). The single particle propagator is
It is a periodic function of : for , , where is the inverse temperature. Because of its periodicity, it can be represented by a Fourier series
where the ’s are called the Matsubara frequencies:
and we have also taken a Fourier transform with respect to the spatial coordinates. We are making here an abuse of notation: we denote by the same symbol the function and its Fourier transform, with the implicit understanding that the arguments, whether space-time or energy-momentum variables, are enough to specify which function one is considering. The inverse transform is given by
In the absence of interactions, the hamiltonian is of the form
In these formulae, is the volume of the system, and the creation and annihilation operators satisfy . The free single particle propagator can be obtained by a direct calculation. It reads
or, in imaginary time,
Thus, for non interacting particles,
The full propagator is related to the bare propagator by Dyson’s equation
where is the self-energy. In this case eq. (67) yields the following expression for the density
or equivalently the occupation factor in the interacting system
We shall approach the condensation from the high temperature phase. Then the system remains in the normal state all the way down to . The Bose-Einstein condensation occurs when the chemical potential reaches a value such that (see e.g. PPbook ()):
At that point,
By using the general relation (80) between the Green function and the density, one can then write the following formula for the shift in the critical density caused by the interaction:
The first term in this expression is the critical density of the interacting system at temperature , the second term is the critical density of the non interacting system at the same temperature. This formula makes it obvious that vanishes if the self-energy is independent of energy and momentum. This is the case in particular when the interactions are treated at the mean field level, as we already observed. In this case, , and the formula above yields
which vanishes since .
3.2 Breakdown of perturbation theory
Because the interactions are weak, one may imagine calculating by perturbation theory. However the perturbative expansion for a critical theory does not exist for any fixed dimension ; infrared divergences prevent a complete calculation, as we shall recall. If one introduces an infrared cutoff to regulate the momentum integrals, one finds that perturbation theory breaks down when , all terms being then of the same order of magnitude.
The leading order on is given by the diagram in Fig. 3. As we just saw, the contribution of this diagram to is just the mean field value , and the net effect on is zero. We shall then examine the second order contribution, given by the diagram displayed in Fig. 4. We shall see that this diagram is infrared divergent. Next, we shall show, using simple power counting, that such infrared divergences occur in higher orders and signal a breakdown of perturbation theory as one approaches the critical point.
Second order perturbation theory
The second order self-energy diagram is the lowest order diagram that is momentum dependent and can therefore yield corrections to the critical density. It is displayed in Fig. 4.
Its contribution is given by
Anticipating that infrared divergence can occur, we focus on the contribution where , and calculate the difference , which we write from now on simply as :
In this calculation we have used Hartree-Fock propagators, and set
where . The quantity