Renormalization flow and universality for ultracold fermionic atoms

S. Diehl,1 H. Gies,2 J. M. Pawlowski,2 and C. Wetterich2

1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria2Institut für Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg, Germany

�Received 27 March 2007; published 30 November 2007�

A functional renormalization-group study for the Bose-Einstein condensate �BEC�-BCS crossover for ultra-cold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality forbroad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentra-tion akF, besides their dependence on the temperature in units of the Fermi energy. In particular, we computethe universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate towhich level of accuracy universality holds for gases of Li and K atoms.

DOI: 10.1103/PhysRevA.76.053627 PACS number�s�: 03.75.Ss, 05.30.Fk

I. INTRODUCTION

The quantitatively precise understanding of the crossoverfrom a Bose-Einstein condensate �BEC� to BCS superfluidity�1� in gases of ultracold fermionic atoms �2,3� is a theoreticalchallenge. If it can be met, the comparison with future ex-perimental precision results could set a new milestone for theunderstanding of the transition from known microscopiclaws to macroscopic observations at length scales severalorders of magnitude larger than the characteristic atomic ormolecular length scales. Furthermore, these systems couldbecome a major testing ground for theoretical methods deal-ing with the fluctuation problem in complex many-body sys-tems in a context where no small couplings are available.The theoretical progress could go far beyond the understand-ing of critical exponents, amplitudes, and the equation ofstate near a second-order phase transition. Then, a wholerange in temperature and coupling constants could becomeaccessible to precise calculations and experimental tests �4�.

The qualitative features of the BEC-BCS crossoverthrough a Feshbach resonance can already be well repro-duced by extensions of mean-field theory which account forthe contribution to the density from diatom or molecule col-lective or bound states �5�. Furthermore, in the limit of anarrow Feshbach resonance the crossover problem can besolved exactly �6�, with the possibility of a perturbative ex-pansion for a small Feshbach or Yukawa coupling. A system-atic expansion beyond the case of small Yukawa couplingshas been performed as a 1 /N expansion �7�. The crossoverregime is also accessible to �-expansion techniques �8�. Atzero temperature �9� as well as at finite temperature �10�,numerical results based on various Monte Carlo methods areavailable at the crossover. A unified picture of the wholephase diagram has arisen from functional field-theoreticaltechniques, in particular from self-consistent or t-matrix ap-proaches �11�, Dyson-Schwinger equations �6,12�, 2-particle-irreducible �2PI� methods �13�, and the functional renormal-ization group �RG� �14�.

Recently, it has been advocated �6� that a large universal-ity holds for broad Feshbach resonances: all purely “macro-scopic” quantities can be expressed in terms of only twoparameters, namely the concentration c=akF and the tem-

perature in units of the Fermi energy, T=T /�F. Here, kF and�F are defined by the density of atoms, n=kF

3 / �3�2�, �F

=kF2 / �2M� and a is the scattering length. Universality means

that the thermodynamic quantities and the correlation func-tions can be computed independently of the particular real-izations of microscopic physics, as for example, the Fesh-bach resonances in 6Li or 40K. A special point in this largeuniversality region is the location of the resonance, c→�,where the scaling argument by Ho �15� applies.

We emphasize that the universal description in terms oftwo parameters holds even in situations where the under-standing of the microscopic physics may necessitate severalother parameters. It is thus much more than the simple ob-servation that the appropriate model involves only two pa-rameters. The fact that two effective parameters are sufficientis related to the existence of a fixed point in the renormal-ization group flow; for a given T, this fixed point has onlyone relevant direction, namely the concentration c. Then, thequestion of accuracy of a two-parameter description is re-lated to the rate of how fast this fixed point is approached bythe flow towards the infrared. It depends on the scales in-volved and the physical questions under investigation. As instatistical physics, the deviations from universality can berelated to the scaling dimensions of operators evaluated atinfrared fixed points. We will address these questions quan-titatively for Feshbach resonances in 6Li or 40K.

Even for broad Feshbach resonances not all quantities ad-mit a universal description. An example is given by the num-ber density of microscopic molecules which depends on fur-ther parameters �5,12,16�.

The quantitative study of this universality has been per-formed through the solution of Schwinger-Dyson equations,including the molecule fluctuations �6,12,17�. Already in thiswork it has been argued that the key for a proper understand-ing of universality lies in the renormalization flow and its�partial� fixed points. A similar point of view has been advo-cated by Nikolic and Sachdev �7�. A functional renormaliza-tion group study evaluating the renormalization flow for thewhole phase diagram has been put forward in �14�. In thepresent paper, we perform a systematic study of the univer-sality aspects of the renormalization flow for various cou-plings thereby extending the results of �14� concerning uni-versality. We relate the universality for broad Feshbachresonances to a nonperturbative fixed point which is infraredstable except for one relevant parameter corresponding to theconcentration c.

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Our approach is based on approximate solutions of exactfunctional renormalization group equations �18�, for reviewssee �19–21�, and for applications to nonrelativistic fermionssee �26�. These are derived by varying an effective “infrared”cutoff associated to a momentum scale k �in units of kF�. Thesolution to the fluctuation problem corresponds then to thelimit k→0. In this way, a continuous interpolation from themicroscopic Hamiltonian or classical action to the macro-scopic observables described by the effective action isachieved. In this paper, we will use a very simple cutoffassociated to an additional negative chemical potential. Thisworks well on the BEC side of the crossover and in thevacuum limit of vanishing density and temperature. Since thecrucial characteristics of the fixed points relevant for univer-sality can be found in the vacuum limit, this will be sufficientfor our purpose. The investigation of the whole phase dia-gram in �14� has been performed with a different cutoff moreamiable to this global task, see �21,22�.

The simple form of the cutoff allows for analytic solutionsof the flow equations; these can be used for a direct check ofthe more general arguments, resulting from the investigationof fixed points and their stability. As a concrete example, wecompute the universal ratio of the molecular scatteringlength to the atomic scattering length in vacuum. This ratio isreduced as compared to its mean-field value by the presenceof fluctuations of collective bosonic degrees of freedom. Wealso present quantitative estimates to what precision univer-sality is realized for ultracold gases of Li and K.

In Sec. II, we introduce the functional-integral formula-tion of our model for the Feshbach resonance. It containsexplicit bosonic fields for diatom states �23–25�. In the limit

of a divergent Feshbach or Yukawa coupling h�, this modelis equivalent to a purely fermionic formulation with a point-like interaction. In Sec. III, we briefly recapitulate the frame-work of the exact flow equation for the effective averageaction, and specify our cutoff and truncations. The initialvalues of the flow at the microscopic scale are given in Sec.IV, while explicit formulas for the flow of couplings and theeffective potential are computed in Secs. V and VI.

In Sec. VII, we specialize to T=0 and solve the flowequations for various couplings explicitly within our trunca-tion. Section VIII generalizes these results and presents ageneral discussion of the fixed points in the flow of the res-

caled dimensionless parameters h�2 , m�

2 , and ��, correspond-ing to the Yukawa coupling, the energy difference betweenopen- and closed-channel states and an additional pointlikefour-fermion vertex which accounts for a “background scat-tering length.” In particular, the fixed point describing broadFeshbach resonances has only one relevant parameter c,whereas the fixed point characteristic for narrow Feshbach

resonances has two relevant parameters c and h�2 . Section IX

supplements a discussion of the density which is needed fora practical contact to experiment, and Sec. X relates the ini-tial values of the flow �the classical action� to observablequantities, such as the molecular binding energy or the scat-tering length depending on a magnetic field B.

In Sec. XI, we discuss the fixed-point behavior for anadditional parameter, namely the effective four-boson vertex��. This fixed point is responsible for the universal ratio

between the scattering length for molecules and for atoms.We finally present a general discussion of universality forultracold fermionic atoms in Sec. XII, and we estimate thedeviations from the universal broad-resonance limit for Liand K in Sec. XIII. Section XIV contains our conclusions.

II. FUNCTIONAL INTEGRAL

We start from the partition function in presence of sources

��dx=�d3x�d��,

ZB�j�� =� D�D� exp�− S��,�� +� dx�j�*�x���x�

+ j��x��*�x��� , �1�

with

S =� dx�†��� − � − �� +��� + ���

2��†��2

+ �*�� −�

2+ � + � − 2��

− �h� + h����*�1�2 − ��1*�2

*�� .

Here, we have rescaled the fields and couplings, together

with the space and time coordinates x� = kx�, �= �k2 /2M��,with M being the mass of the atoms. Our units are �=c=kB=1. We use the Matsubara formalism with Euclideantime � on a torus with a circumference given by the inverse

temperature T−1. The thermodynamic variables are T

=2MT / k2 and =2M / k2, with denoting the effectivechemical potential.

The model parameters are the following:�1� The detuning of the magnetic field B−B0 �with

=2 B for 6Li and =1.57 B for 40K�,

� =2M

k2 �B − B0� , �2�

�2� The Feshbach or Yukawa coupling h� which can beextracted from the properties of the quantum-mechanicaltwo-atom system as the molecular binding energy or thescattering cross section.

�3� A pointlike interaction for the fermionic atoms param-

eterized by ��. The shifts �, ��, and h� are counter termsthat are removed by the ultraviolet renormalization. Detailsof the rescaling and the formulation can be found in �12,14�.

The scale k �which sets the units� is arbitrary. �A typical

value is k=1 eV.� Under a rescaling of k→ k / all quantitiesscale according to their canonical dimension, i.e.,

x = kx → x/ , � = �k2/�2M��� → �/ 2,

T → 2T, → 2, � → 2�, h� → 1/2h�,

�� → ��/ , � → 3/2�, � → 3/2� . �3�

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We observe that the canonical dimension of time is minustwo and therefore the nonrelativistic Lagrangian has dimen-sion five and not four, as for a relativistic quantum fieldtheory. The total atom density n defines the Fermi momen-tum kF,

n =kF

3

3�2 . �4�

If one associates k with kF all quantities are expressed inunits of the Fermi momentum �12�.

From the partition function the effective action � is ob-tained by the usual Legendre transform, �= ��,

���� = − ln Z†j����‡ +� dx�j�*� + j��*� . �5�

The order parameter �0 for superfluidity corresponds to theminimum of � for j�=0 and obeys the field equation

��

��

�0

= 0. �6�

The effective action generates the 1PI Green’s functions suchthat the propagators and transition amplitudes can be directlyrelated to the functional derivatives of �. The constructionabove is easily extended to an effective action that is also afunctional of fermionic fields by introducing the appropriatefermionic sources, see below.

III. EXACT FUNCTIONAL FLOW EQUATION ANDTRUNCATION

The variation of � with the change of an effective infraredcutoff k is given by an exact renormalization-group equation�18–21�. For the present theory, this approach has beenimplemented in �14� within a study of the phase diagram; seealso �26�. For RG studies of the purely bosonic system, see�27�.

For the purpose of the present work, we extend the trun-cation used in �14� as well as using a particular version of theflow equation where the cutoff acts like a shift in the respec-tive chemical potentials for � and �, as also done in �28�. Tothat end, we introduce an infrared-regularized partition func-tion Zk by adding a cutoff term to the action in Eq. �1�,

S��,�� → S��,�� + �Sk��,�� , �7�

where the cutoff term �Sk is chosen as

�Sk��,�� =� dx�Rk�F��†� + Rk

�B��*�� , �8�

with

Rk�F� = Z��k�k2, Rk

�B� = 2Z��k�k2,

�Rk�F�

�k2 = Q�Z�,�Rk

�B�

�k2 = 2Q�Z�,

Q�,� = 1 − ��,�/2, ��,� = − k�

�kln Z�,�. �9�

The k-dependent wave function renormalization Z�,� will bedetermined below, and we note that k is measured in units of

the fixed scale k. Introducing the effective average action �kin terms of a modified Legendre transform,

�k��,�� = − ln Zk�j�, j�� − �Sk��,��

+� dx�j�*� + j��* + j�

†� − �†j�� , �10�

the exact functional renormalization group equation �flowequation� can straightforwardly be derived,

��k

�k2 =� dx�Z�Q� �†��c + 2Z�Q� �*��c�

= Tr�− Z�Q���k�2� + Rk��*�

−1+ 2Z�Q���k

�2� + Rk��*�−1 � .

�11�

Here, we have expressed the propagators of the fermionicand the bosonic fields by the corresponding components ofthe inverse of the matrix of second functional derivatives of�k. The trace Tr contains an integration over x or a corre-sponding momentum integration as well as a trace over allinternal indices. Both �k and �k

�2� are functionals of arbitrary

fields � and � which are kept fixed for the k derivative in Eq.�11�.

The flow equation �11� is a nonlinear functional differen-tial equation and we can only hope to find approximate so-lutions by suitable truncations of the most general form of�k. In this work, we exploit the ansatz

�k =� dxZ��†��� − A�� − �� + Z��*��� − A����

+ u��,� − h���*�1�2 − ��1*�2

*� +��

2��†��2�

=� dx�†��� − A�� − �� + �*��� − A���� + u��,�

− h���*�1�2 − ��1*�2

*� +��

2��†��2� . �12�

In the second equation, we have used renormalised fields

� = Z�1/2�, � = Z�

1/2� , �13�

and renormalized couplings

h� = h�Z�−1Z�

−1/2, �� = ��Z�−2. �14�

The ansatz �12� generalizes that of �14� by the four-fermioninteraction with coupling ��. This term is particularly impor-tant in the limit of broad Feshbach resonances as will bediscussed later.

The global symmetry of U�1� phase rotations implies thatthe effective potential u can only depend on �=�*�. If theground state corresponds to a homogeneous �0 the pressureand density can be computed from the properties of the ef-fective potential

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� �u

���

�0

= 0, u0�� = u��0,� ,

p = −k5

2Mu0, n = − k3�u0

�. �15�

At this point, some comments concerning the wave functionrenormalizations are in order:

�i� Analytic continuation to “real time” and a Fouriertransform to �real� frequency space results in ��→−�. Wedefine Z�,� by the coefficient of the term linear in � in the

full inverse propagator P�,� given by the terms quadratic in

� or � in �k. More precisely, we choose in Fourier space

Z� = − � � P�

���

q�=0,�=0. �16�

�ii� With this definition of Z� and Z�, the renormalizedfields � ,� have a unit residuum for the pole in the propaga-tor for q� →0 if the “on shell value” of � vanishes for q� =0.

�iii� We use the same Z�,� in the definition of the cutoff�9� as in Eq. �12�.

�iv� The fields � and � describe “microscopic” or “bare”atoms and molecules while the renormalized field � de-scribes dressed molecules. The wave function renormaliza-tion Z� accounts for a description of dressed molecules as amixing of microscopic molecules and diatom states �5,6,14�.

IV. INITIAL CONDITIONS

Inserting the truncation �12� into the flow equation �11�,and taking appropriate functional derivatives, leads to acoupled set of differential equations for the couplings Z�, A�,Z�, A�, h�, �� as well as the effective potential u���. Thetask is to follow the flow of these couplings as the cutoffscale k is changed. The initial values for large k are taken as

Z� = Z� = 1, A� = 12 , A� = 1,

u = m�2� ,

m�2 = � + � − 2 . �17�

Here � is related to the detuning of the magnetic field

� =2M

k2 �B − B0� �18�

and we have to fix � �i.e., the renormalized counterpart of� in Eq. �1�� such that the Feshbach resonance in vacuumoccurs for �=0. The initial values of �� and h� are chosensuch that the molecular binding energy and scattering of twoatoms in vacuum is correctly described.

A realistic model contains an effective ultraviolet cutoff �which is given roughly by the inverse of the range of the van

der Waals force. For k�� / k we can take the limit �→�since all momentum integrals are ultraviolet finite. All ultra-violet “divergencies” are already absorbed in the computa-

tion of �� �12,14�. For 6Li or 40K and k=1 eV one has � / kof order 102 to 103.

V. RUNNING COUPLINGS AND ANOMALOUSDIMENSIONS

The flow equations for the various couplings and the den-sities derived from Eq. �11� have a simple interpretation asrenormalization group improved one-loop equations �18�with full propagators and dressed vertices, but are exact. Therenormalization constants Z�, Z� are related to the depen-dence of the unrenormalized inverse propagators �for � and

�� on the �Minkowski� frequency, i.e., the coefficients of theterms linear in ��=−� /��� in ��2�. The second derivative ofEq. �11� yields exact flow equations for the inverse propaga-tor �18–21� and therefore for Z�, Z�. We define

��,� = ��,�/2k2 = − � ln Z�,�/�k2 �19�

and obtain in our approximation

�� = −h�

2Q�

2

�

�k2 I��F� + ��

�B�. �20�

Here the fermion loop integral

I��F� =

1

8T2� d3q

�2��3���−3�tanh �� − �� cosh−2 ��� �21�

involves

�� =1

2T��A�q2 + k2 − �2 + h�

2� ,

� =1

2T�A�q2 + k2 − � . �22�

In our truncation, the boson fluctuations contribute to ��

only in the superfluid phase ����B���=0�=0� �14�.

Crucial quantities for the investigation of this problem arethe running of the Yukawa coupling h�,

�

�k2h�2 = ��� + 2���h�

2 + 2h�2��Q�J�

�F�, �23�

with

J��F� =

1

16T2� d3q

�2��3���−5��3�2 − ��

2�tanh ��

+ �� cosh−2 ���− 3�2 + ��2 + 2�����

2 − �2�tanh ���� ,

�24�

and the pointlike four-fermion interaction

�

�k2�� = 2���� + ��2Q��I�

�F� + I��� ,

I��= −

1

4T2� d3q

�2��3���−1 tanh �� cosh−2 ��. �25�

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For both quantities, we neglect the contribution from themolecule fluctuations. This is a valid approximation for theparameter ranges for which we give quantitative results. Inother ranges, the molecule fluctuations give more dominantcontributions, see �14� and the discussion below. For �=0,we note J�

�F�= I��F�.

VI. FLOW OF THE EFFECTIVE POTENTIAL

Keeping now � �and not �� fixed, the flow of the effectivepotential obeys

�

�k2u��� = �F��� + �B��� + ���u���� , �26�

with

�F��� = − Q�� d3q

�2��3 �

��

tanh �� − 1� ,

�B��� = Q�� d3q

�2��3� + �2

��

coth �� − 1� . �27�

Here, we define

�� =1

2T��A�q2 + 2k2 + u���A�q2 + 2k2 + u� + 2�u���1/2,

� =A�q2 + 2k2 + u�

2T,

�2 =�u�

2T, �3 =

�2u�

2T, �28�

and the primes denote derivatives with respect to �. Thecontribution of the boson fluctuations is computed in a basiswhere the complex field � is written in terms of two realfields with inverse bosonic propagator

P� = A�q2 + 2k2 + u� + 2�u� − 2�nT

2�nT A�q2 + 2k2 + u�� . �29�

Then, Eq. �11� evaluated for constant � becomes

�B��� = TQ� tr�n� d3q

�2��3P�−1 − 1

1

A�q2 + 2k2� , �30�

in accordance with �29�. The subtraction of a �-independentpart renders the momentum integral ultraviolet finite. Forextensive numerical studies one would rather rely on cutoffchoices as in �14� which do not require any ultraviolet sub-traction, and minimize the numerical costs. However, this isof no importance for the quantities computed in this paper—

the dependence of A� on and T is negligible, and simplic-ity of the approach is the more important property.

In the symmetric phase �SYM�, one has �0=0, u��0�=m�

2 ,u��0�=��, whereas in the presence of spontaneoussymmetry breaking �SSB� we use �0�0, u���0�=0, u���0�

=��. Equation �26� yields in the symmetric phase

�

�k2m�2 = h�

2Q�I��F��� = 0� − Q�I�

�B��� = 0� + ��m�2 . �31�

Here, we define the boson loop integral

I��B� =

��

2T� d3q

�2��3���2� + �2�u� + ��u��� + �2

��2 sinh2 ��

+ ��2��2 − �� − ��3�u�coth ��

��3 � , �32�

such that

I��B��� = 0� =

��

T� d3q

�2��3 sinh−2 � . �33�

The flow starts with a positive m�2 for large k and is there-

fore in the symmetric regime. As k decreases, m�2 decreases

according to Eq. �31�, with �� being positive. At high tem-perature, m�

2 stays positive for all k and the system is in thesymmetric phase and ungapped for all momenta. At tempera-tures below a pseudocritical temperature Tp, m�

2 hits zero atsome critical kc. For k�kc, the flow first continues in thebroken-symmetry regime with nonzero �0. Whereas fermi-onic fluctuations tend to increase �0, molecule fluctuationscause its depletion. For temperatures in between the criticaltemperature and the pseudocritical temperature, Tc�T�Tp,molecule fluctuations eventually win out over the fermionicfluctuations and �0 vanishes again at smaller k; here, thesystem is in the symmetric phase. Identifying the k depen-dence of the fermionic two-point function with the momen-tum dependence, the nonzero value of �0 for finite k can beassociated with a pseudogap �30�. Below the critical tem-perature Tc, �0 stays finite for k→0, corresponding to thesuperfluid phase with a truly gapped spectrum.

Molecule fluctuations are particularly important duringthose stages of the flow where �0 is nonzero, i.e., for k�kcand below the pseudocritical temperature Tp, see �14�. In thiswork, we focus on the properties of the flow in the symmet-ric regime, where �0=0 for all values of k under consider-ation. We still include molecule fluctuations for all quantitiesthat are associated with the effective potential u���, in par-ticular the molecule density and the molecule-molecule scat-tering length, see Secs. IX and XI below. But molecule fluc-tuations are neglected for the other running couplings whichare dominated by fermion fluctuations in the parameterranges considered here.

VII. SOLVING THE FLOW EQUATION FOR ZEROTEMPERATURE

Let us first concentrate on T=0 and k2− �0. We define

K2 = k2 − �34�

and employ � /�k2=� /�K2. Then the flow equation for utakes the explicit form

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�

�K2u��� = ���u� −1

2�2�0

�

dxx2�Q� A�x2 + K2

��A�x2 + K2�2 + h�2�

− 1� − Q� A�x2 + 2k2 + u� + �u���A�x2 + 2k2 + u���A�x2 + 2k2 + u� + 2�u��

− 1�� .

�35�

For �=0, T=0, �0, one also finds

I��F� =

1

16�A�

−3/2K−1, I��= 0, �36�

and, in our approximation, the contributions of the bosonloops to �� ,�h�

2 /�k2 and ��� /�k2 vanish for m�2 �0.

At T=0 and �0, we can use Z�=A�=1, Q�=1. This isan exact result �7� as long as all propagators in the relevantdiagrams have simple poles in the imaginary q0 plane. With

�� =h�

2

64�K−3, �37�

we find the coupled system of flow equations for the sym-metric phase

�h�2

�K=

h�4

32�K−2 +

h�2��

4�,

���

�K=

��2

8�. �38�

The solution of the last equation,

��−1 = ��,in

−1 +1

8��Kin − K� = ��,0

−1 −K

8�, �39�

renders �� almost independent of K if K�Kin+8� /��,in.Here, we assume implicitly that ��,in is not too much nega-tive such that �� remains finite in the whole k range of in-terest. For positive ��, the self-consistency of the flow re-quires an upper bound ��,0 / �8���1 /Kin.

In contrast, for ��=0 the solution for h�2 ,

h�−2 = h�,in

−2 +1

32��K−1 − Kin

−1� , �40�

is dominated by small K.For ���0, the flow of h� is modified, however, without

changing the characteristic behavior for K→0. Indeed, in thelimit K→0 the term ��� becomes subdominant for the evo-lution of h�

2 . The flow for the ratio h�2 /K reaches a fixed point

32�. We can explicitly solve the system �38� by consecutive

integrations. One obtains for h�2 =Z�h�

2 ,

h�2 = h�,in

2 exp�−1

4��

K

Kin

dx���x��= h�,in

2 �1 − c0Kin�2�1 − c0K�−2 = h�,02 �1 − c0K�−2.

�41�

This yields for the wave function renormalization

Z� = 1 +1

32��

Kin−1

K−1

d�x−1�h�2�x−1�

= 1 +h�,0

2

32� 1

K1 −

c0

1/K − c0�

+ 2c0 ln 1

K− c0� − �K → Kin�� , �42�

with

c0 =��,0

8�,

1

K� c0. �43�

For K→0, the ratio h�2 / h�,in

2 obviously reaches a constantwhich depends on ��,in ,Kin, whereas Z� diverges

�h�,02 / �32�K�, as is consistent with the fixed-point behavior

h�2 �32�. Neglecting c0 �as always appropriate for small

enough K� and assuming a broad resonance h�,02 �32�, we

arrive at the simple relation

Z� �h�,0

2

32�K. �44�

In this limit, the factor Q� appearing in the flow equations isgiven by

Q� = 1 −k2

2�k2 − ��45�

and approaches 1 /2 for k2�−. We note that in this rangethe bosonic cutoff function Rk

�B�=2Z�k2 is effectively linearin k and not quadratic.

We finally investigate the flow equation for m�2 in the

symmetric phase. As we will see below, the boson loops �thelast contribution in Eq. �35�� vanish in our truncation. Thisyields

�m�2

�K=

h�2

8�+

h�2m�

2

32�K−2, �46�

or

�m�2

�K=

�

�K�Z�m�

2� =h�

2

8�. �47�

The solution reads

DIEHL et al. PHYSICAL REVIEW A 76, 053627 �2007�

053627-6

m�2 =

2M �B − B0�

k2− 2 + �

−h�,in

2 �1 − c0Kin�2

8�c0 1

1 − c0Kin

−1

1 − c0K� . �48�

The counterterm � is determined from the condition m�2�B

=B0 , =0,k=0�=0, such that �up to minor corrections�

m�2 =

2M �B − B0�

k2+

h�,02

8�

K

1 − c0K− 2 . �49�

In particular, for k=0 and �0 one has K=�−, and Eq.�49� yields the expression for m�

2 in the presence of all fluc-tuations

m�2 =

2M �B − B0�

k2+

h�,02 �−

8� − ��,0�−

− 2 . �50�

At this point, we have the explicit solution of the flow equa-tion for T=0. In the present simple truncation, we expect thatthe description of the system is qualitatively correct. Higherquantitative precision will require a more extended trunca-tion; however, this is not the main emphasis of the presentwork which rather concentrates on the structural propertiesrelated to the fixed points.

VIII. FLOW EQUATIONS AND FIXED POINTS

The explicit solution of the preceding section is indeedvery useful for verifying and explicitly demonstrating thegeneral fixed-point properties. The following features of theflow equations hold in a much wider context of differentmicroscopic actions and different cutoffs. The overall patternof the flow is governed by the existence of fixed points.Some of these fixed points may correspond to particularlysimple situations, being less relevant for the physics underdiscussion. By contrast, the stability or instabilities of smalldeviations from the various fixed points are much more im-portant, as they determine the topology of the flow in thespace of coupling constants, as demonstrated in Fig. 1. If thesystem is in the vicinity of any of the different fixed pointsthe number of effective couplings needed for a description ofthe macrophysics �beyond T� corresponds to the number ofrelevant directions.

A particularity of our system concerns a certain redun-dancy in the description: a pointlike interaction can be de-scribed either by the four-fermion coupling �� or by a lim-

iting behavior of the scalar exchange. Indeed, for h�,in2 →�

and fixed h�,in2 / m�,in

2 the molecule exchange interaction be-comes effectively pointlike. Therefore, we may define an ef-fective pointlike coupling

��,eff = �� −h�

2

m�2 = �� −

h�2

m�2 , �51�

which describes the interaction in the zero-momentum limit.

Combining the flow equations �38� and �46� for �� , h�2 , m�

2 ,one obtains

���,eff

�K=

��,eff2

8�. �52�

This is the same flow equation as for �� �38�.Next, we may consider the renormalized couplings in

units of kk instead of k. This will reveal the relevant fixedpoints for the flow more clearly. We define, according to thescaling dimensions of Eq. �3�,

�� = ��k, ��,eff = ��,effk ,

h� =h�

�k, m�

2 = m�2 /k2,

t = ln k/kin, �53�

and obtain the dimensionless flow equations

�t�� = �� +��

2y

8�, �t��,eff = ��,eff +

��,eff2 y

8�,

�th�2 = − h�

2 +h�

4y3

32�+

h�2 ��y

4�,

�tm�2 = − 2m�

2 +h�

2y

8�+

h�2m�

2y3

32�, �54�

with

y = k2

k2 − �1/2

. �55�

Let us first consider =0, or, more generally, k2�−,such that y=1. In this case, we observe two fixed points for

��, either ��=0 or ��=−8�. The corresponding fixed points

for h�2 �0 are

0 1 2 3

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

(A)(C)

(B)

(D)h2

ϕ

32π

λψ8π

FIG. 1. Location of fixed points �A�–�D� projected onto the

plane which is spanned by the couplings �� / �8�� and h�2 / �32��.

The arrows characterize the flow of the couplings towards theinfrared.

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�A�: �� = 0, h�2 = 32�, m�

2 = 4,

�B�: �� = − 8�, h�2 = 96�, m�

2 = − 12. �56�

�We will discuss later the fixed points with h�2 =0, m�

2 =0.�The fixed point �A� is infrared stable for �� and h�

2—bothcouplings run towards their fixed-point values as k is low-ered, as is visualized in Fig. 1. In contrast, m�

2 is infraredunstable—the detuning B−B0 corresponds to a relevant per-

turbation of the fixed point. With h�2 at the fixed-point value,

m�2 deviates from its fixed point with an anomalous dimen-

sion

m�2 = 4 + min

2 kin

k. �57�

Comparing this with the explicit solution �49� for ��=0,

=0, h�= h�,0, namely

m�2 =

h�2

8�+

2M �B − B0�

Z�k2k2�58�

with Eq. �42�,

Z� �h�,0

2

32�k, �59�

we can identify

m�2 =

64�M �B − B0�

k2h�,02 kin

. �60�

We conclude that broad Feshbach resonances �large enoughYukawa couplings� can be characterized by the fixed point

�A�, with h� and �� being irrelevant and m�2 �B−B0 the

relevant coupling.

For the second fixed point �B�, �� becomes a relevant

parameter. For h�2 , the fixed point �56� with h�

2 =96� remainsinfrared attractive, but the fixed point for m�

2 occurs for nega-tive m�

2 where our computation in the symmetric phase is nolonger valid.

Finally, we turn to the fixed points with h�2 =0. In this

case, h�2 is always a relevant coupling and increases as k is

lowered. The fixed point

�C�: �� = 0, h�2 = 0, m�

2 = 0 �61�

is infrared attractive for ��. For this case of a narrow Fesh-bach resonance, the crossover problem can be solved exactly�6�, and perturbation theory around the exact solution be-

comes valid for small h�2 , ��. The coupling m�

2 is a secondrelevant parameter around this fixed point. As one increases

h�,02 , a crossover to the fixed point �A� occurs �6�. The

“narrow-resonance fixed point” �C� describes the limit of acombined model with free fermions and free bosons, sharingthe same chemical potential. We emphasize that, for small

h�, the equivalent purely fermionic description typically hasa nonlocal interaction.

For the fourth fixed point of our system,

�D�: �� = − 8�, h�2 = 0, m�

2 = 0, �62�

all three parameters are relevant. This point correspondsagain to strong attractive interactions between the fermionicatoms. The flow away from this fixed point for nonvanishing

h�2 / m�

2 can be characterized by the flow of the ratio �y=1�

�t h�2

m�2 � = 1 +

��

4�� h�

2

m�2 −

1

8� h�

2

m�2 �2

, �63�

which does not have a fixed point for positive h�2 , m�

2 . For-mally, the flow runs for negative m�

2 towards the fixed point

�B� with h�2 / m�

2 =−8�. Actually, it may be possible to con-sider �� as a redundant parameter using partial bosonizationand rebosonization during the flow �31�, see also �21�. Then,

the fixed points with ��=−8� may again be associated withbroad Feshbach resonances. We show the four fixed points A,

B, C, D and the infrared flow of the couplings h�2 and �� in

Fig. 1.We should emphasis that the inclusion of the omitted con-

tributions from boson fluctuations to the flow of �� couldresult in corrections �y=1�,

�t�� = �� +��

2

8�+

c1

32�h�

4 +c2

32���h�

2 , �64�

such that the flow of �� remains no longer independent of h�2 .

This will change the precise location of fixed points �A� and�B�. We expect that the qualitative characteristics of the fixedpoint �A� remain unchanged, whereas the fate of the fixedpoint �B� is less clear. We may consider the flow of the ratio

�� / h�2 ,

�t ��

h�2 � = 2 ��

h�2 � +

h�2

32�c1 + �c2 − 1���

h�2

− 4 ��

h�2 �2� .

�65�

For any possible fixed point, one has

h�2 = 32�1 −

��

4�� ⇒

h�2

32�= 1 + 8

��

h�2 �−1

, �66�

and therefore obtains the fixed-point condition for Q

= �� / h�2 as

c1 + �c2 + 1�Q + 12Q2 = 0. �67�

In general, this quadratic equation has two distinct solutions,where the larger value of Q is infrared stable and corre-sponds to �A�, while the smaller value of Q is unstable andcorresponds to �B�.

We finally include the effect of a nonzero negative chemi-cal potential . As soon as k2 becomes smaller than −, theparameter y �55� rapidly goes to zero. Then, only the firstterm in the flow equations �54� matters. In this range of k, the

couplings ��, h�2 , m�

2 remain constant. A negative acts asan additional infrared cutoff, such that the flow is effectively

DIEHL et al. PHYSICAL REVIEW A 76, 053627 �2007�

053627-8

stopped for k2�−. This yields a simple rough picture: the

couplings ��, h�2 , m�

2 flow according to Eq. �54� for k2�− until the flow stops when k2�−.

The fixed points are relevant not only for T=0. We dem-onstrate the influence of the fixed point �A� on the flow of

the Yukawa coupling for T=0.5 and c=1 in Figs. 2 and 3. InFig. 2, we observe that the renormalized Yukawa coupling h�

at small k becomes almost independent of its initial value

h�,02 if the latter is large. Figure 3 demonstrates the influence

of the background scattering length abg for a fixed value of a,

again for T=0.5 and c=1.

IX. DENSITY AND CONDENSATE FRACTION

Let us next compute the density

n =n

k3= −

�u0

�, �68�

where the derivative will be taken at fixed � and k. This ispossible since for the minimum du0 /d=�u /���0

+ ��u /���

����0 /��=�u /���0. Before doing a more detailed calcula-

tion, we approximate the potential by

u��,� = u�� + m�2��� +

1

2���2, �69�

with �� independent of . The density then receives contri-butions from unbound atoms and molecules �nF+2nM� andatoms in the condensate �nC� for �0�0,

nF + 2nM = −� u

�, nC = −

�m�2

��0. �70�

For an estimate of the condensate fraction �C= nC / n, we use�up to small corrections involving Kin�

�m�2

�= − 2Z�1 + Oc0K ln

1

K�� , �71�

where we have used Eqs. �44� and �50�. For small K, theterm −2Z� dominates and we conclude that the condensatefraction is given by the renormalized order parameter �0,

15 12.5 10 7.5 5 2.5 0

200

400

600

800

1000

k2

h2ϕ

(b)

10000 8000 6000 4000 2000 00

500

1000

1500

2000

2500

k2

h2ϕ

(a)

FIG. 2. Sensitivity of the renormalized Yukawa coupling h�

to the two-body Yukawa coupling h�,0 at fixed ��,0: �a� uv and

�b� ir flow of the Yukawa coupling. We use k=kF=1 eV, abgk

= ��,0 / �8��=0.38 as appropriate for 6Li and T=0.5, c−1=1. Thedifferent curves correspond to the two-body value of the Yukawa

coupling h�,02 =3.72� �103 ,104 ,105� �from top to bottom�, where

the last value corresponds to 6Li while the first one is comparable tothat of 40K. Universality with respect to the value of the two-bodyYukawa coupling is very strong even for a comparatively large

�fixed� ��.

15 12.5 10 7.5 5 2.5 0

200

400

600

800

1000

k2

h2ϕ

(b)

10000 8000 6000 4000 2000 00

2500

5000

7500

10000

12500

15000

k2

h2ϕ

(a)

FIG. 3. Sensitivity of the renormalized Yukawa coupling h� to

the two-body background scattering length �� for fixed two-body

Yukawa coupling h�,0=3.72�105, k=kF=1 eV, as appropriate for6Li: �a� uv and �b� ir flow of the Yukawa coupling. We plot curves

for abgk= �� / �8��=−0.38� �10−3 ,10−2 ,10−1 ,1� �the last value cor-

responds to 6Li� and T=0.5, c−1=1. The uppermost line in �b� hasthe largest abg and starts in �a� as the lowest line.

RENORMALIZATION FLOW AND UNIVERSALITY FOR… PHYSICAL REVIEW A 76, 053627 �2007�

053627-9

�C =2�0

n. �72�

The above formula is one viable definition of the condensatefraction. In general, it still needs to be related to the defini-tion of a condensate fraction as measured in a particular ex-periment.

The density receives contributions from atoms and mol-ecules with different momenta. In order to describe each mo-mentum mode accurately, we first compute the change of thedensity with k,

d

dk2 n = −d

dk2

�u0

�= −

�

�k2� �u0

��

�0

− � �2u

�� � �

�0

d�0

dk2 ,

�73�

and then integrate from kin to k=0. The right-hand side willbe dominated by momenta q2�k2. The first term in Eq. �73�is given by the derivative of Eq. �27�. The different con-tributions have an approximate interpretation in terms of un-bound atoms, molecules, and the condensate density

dnF

dk2 = −�

��F��0�, 2

dnM

dk2 = −�

��B��0� ,

dnC

dk2 = − � �2u

�� � �

�0

d�0

dk2 + ���0� . �74�

For example, for �0=0 one rediscovers the integral over theusual Fermi distribution �Q�=1�

dnF

dk2 =�

�k2�F�0� =�

�k2 � d3q

�2��2

2

exp��A�q2 + k2 − �/T� + 1.

�75�

In Eq. �75� we have traded the derivative for a k2 deriva-tive, since � depends only on the combination k2− . As longas A� depends only on K2=k2− , the k2 derivative also actson A�, and Eq. �75� can be integrated,

nF =� d3q

�2��3

2

exp��A�q2 − �/T� + 1+ nF�kin� − �F�� = 0,kin� .

�76�

We may evaluate the initial value nF�kin� in perturbationtheory, finding both quantities being exponentially sup-pressed, nF�kin�=�F��=0,kin��0. In our approximation,�A�=1� nF is simply the density of a free gas of fermionicatoms as long as �0=0. In the presence of a condensate ��0

�0�, we typically find a flow where �0�k�=0 for k�kc, suchthat the flow of nF�k� remains unchanged in this range. Onthe other hand, the contribution from modes with q2�kc

2 willbe suppressed by the presence of a gap �=h�

��0 in thepropagator. We emphasize that the derivative � /�k2 in Eq.�75� does not act on �0, since the derivative in Eq. �74� istaken at fixed �. Therefore the flow for nF has to be inte-grated numerically if �0�0.

For the computation of the molecule density nM, we need��B /�. Let us concentrate on �0=0 and neglect the de-pendence of Q� and A�, such that

��B

�= 2Q�� d3q

�2��3

�

��exp��A�q2 + 2k2 + m�

2�/T� − 1�−1

= Q�

�m�2

�

�

�k2 � d3q

�2��3

��exp��A�q2 + 2k2 + m�2�/T� − 1�−1. �77�

We next use Eq. �71�,

dm�2

d� − 2 − m�

2 �

�ln Z�. �78�

The second term can be approximated for small K by

m�2 �

�ln Z� �

m�2

2�k2 − �. �79�

If we neglect this term and set Q�=1 we find the intuitiveformula

nM =� d3q

�2��3

1

exp��A�q2 + m�2�/T� − 1

. �80�

This is the expression for a free boson gas. What is crucial,however, is the appearance of the inverse propagator for therenormalised bosonic field � instead of the microscopic or“bare” molecule field �. The propagators for the renormal-ized and bare fields are related by a relative factor Z�, suchthat we find for the density of bare molecules

nM,bare = Z�−1nM. �81�

As a consequence, one may have a substantial molecule frac-tion 2nM / n even if the density of bare molecules is tiny.These features reproduce the results from a solution of theSchwinger-Dyson equations �6� where a bare molecule den-sity in accordance with the experimental result of Partridgeet al. �32� has been found. It is one of the important advan-tages of our functional renormalization group approach thatit accounts for the distinction between renormalized and baremolecules in a very direct and straightforward manner �14�.Within a Hamiltonian formulation, this issue is related to themixing between open-channel and closed-channel atoms, andits correct treatment is crucial for a quantitatively reliabledescription of the crossover. We emphasize that the renor-malized molecule field plays an important role even on theBCS side of the crossover. The composite bosons are crucialeffective degrees freedom, even though the microscopictheory can be very well approximated by a pointlike interac-tion between fermionic atoms without any reference to mol-ecules. On the BCS side, the renormalized field � describesCooper pairs. Nevertheless, one never needs these physicalinterpretations explicitly, since the density only involves the dependence of the potential at its minimum, which is awell-defined quantity.

DIEHL et al. PHYSICAL REVIEW A 76, 053627 �2007�

053627-10

We finally turn to the condensate density. If we can ap-proximate the dependence of �u− u� by a term −2�, assuggested by Eq. �78�, we infer

dnC

dk2 = 2d�0

dk2 + 2���0 = 2Z�

d

dk2

�0

Z�

. �82�

�This approximation needs to hold only in the range of kwhere �0 differs from zero.� The dominant flow of nC typi-cally arises from a region where the k dependence of Z� isalready subleading, resulting in nC�2�0, as found above. Inpractice, all these various approximations of our analyticaldiscussion need not be made, since it is sufficient to followthe flow of n�k� numerically, starting from an initial valuen�kin��0 and extracting the physical density for k=0.

X. VACUUM AND TWO-ATOM SCATTERING

It is one of the advantages of our method that it can accesssimultaneously the many-body physics of a gas in thermalequilibrium and the two-body physics of atom scattering andmolecular binding. Indeed, the two-body physics describesexcitations above the vacuum. In turn, the vacuum and theproperties of its excitations obtain in our formalism simplyby taking the limit of vanishing density and temperature.�More precisely, the limit should be taken such that the ratio

T / n2/3 is large enough that no condensate occurs.� The scat-tering cross section between two atoms can then be directlyinferred from the Yukawa coupling and the propagator of themolecule field in the vacuum.

For T=0, the condition of zero density requires �0 inorder to ensure nF=0, cf. Eq. �76�. On the other hand, weinfer from Eq. �80� that m�

2 �0 is needed for nM=0, whereasa vanishing condensate density requires �0=0. The vacuumis the state that is reached as density and temperature ap-proach zero from above. It therefore corresponds to theboundary of the region where �0,m�

2 �0. There are twobranches of vacuum states �for a more detailed discussion,see �17��. The first has a negative = A�0 and m�

2 =0. Inthis case, the single-atom excitations have a gap −A�0which corresponds to half the binding energy of the stablemolecules,

�M = k2A/M, �M = 2A. �83�

We identify this state with the “molecule phase” where stablemolecules exist in the vacuum. This phase is realized for B�B0, with �M or A being a function of B that vanishes forB→B0. The other branch corresponds to =0, m�

2 �0. This“atom phase” of the vacuum is realized for B�B0, with m�

2 afunction of B vanishing for B→B0. In the atom phase the

“binding energy” �M= k2m�2 / �2M� is positive and the “mol-

ecules” correspond to unstable resonances. We observe acontinuous phase transition between the molecule and atomphase �6,14� for m�

2 =0, =0, corresponding to the locationof the Feshbach resonance at B=B0. This fixes � in theinitial value of m�

2 �18� by the requirement that for �=0 themass term m�

2�� vanishes precisely for =0 if T=0.The vacuum state can be used to fix the parameters of our

model by direct comparison to experimentally measured

binding energies and cross sections. Let us first consider themolecular binding energy �M�B� that can be computed fromEq. �50� by requiring m�

2 =0, i.e.,

�M�B� =A�B�k2

M

= �B − B0� +kh�,0

2

16��M���M�1 −

��,0�M��M�

8�k�−1

.

�84�

In the vicinity of the Feshbach resonance �small ��M�� thebinding energy depends quadratically on B,

�M = −�16��2M 2�B0 − B�2

k2h�,04

. �85�

Using

a0 =��,0

8�k, h�,0

2 =kh�,0

2

4M2 ,

a = −h�,0

2 M

4�� �B − B0� − �M�+ a0, �86�

Eq. �84� reads

�M��M� =1

a. �87�

The scattering length for the fermionic atoms is deter-mined by the total cross section in the zero-momentum limit,=4�a2, such that

a =M

4���,eff =

��,eff

8�k. �88�

More details are described in the Appendix. For the atomphase ��M�0�, one has �6�

��,eff = ��,0 −h�,0

2

m�2� = 0�

= ��,0 + ��,R. �89�

For the molecule phase, the effective interaction depends onthe energy � of the exchanged molecule

��,R�� = 0,p� = 0� = −h�

2�A�2Z��A�A

,

��,R�� = − 2A,p� = 0� = −h�

2�A�4Z��A�A

. �90�

We find for the atom phase

a = a0 −h�,0

2 M

4� �B − B0�= a0 + ares, �91�

which agrees with a �86� for �M=0. Equation �87� thereforerelates the binding energy in the molecule phase to the scat-tering length in the atom phase.

RENORMALIZATION FLOW AND UNIVERSALITY FOR… PHYSICAL REVIEW A 76, 053627 �2007�

053627-11

The value of h�,0 can now be extracted from the resonantpart, ares, whereas ��,0 follows from the B-independent“background scattering length” a0=abg which has been mea-sured as abg

�Li�=−1420aB, abg�K�=174aB �aB the Bohr radius�,

or, expressed in units �=c=kB=1 �aB=2.6817�10−4 eV−1�,abg

�Li�=−0.38 eV−1, abg�K�=4.67�10−2 eV−1. For k=1 eV corre-

sponding to a density n=4.4�1012 cm−3, we find the dimen-sionless expressions for 6Li and 40K,

h�,02�Li� = 3.72 � 105, h�,0

2�K� = 6.1 � 103,

��,0�Li� = − 9.55, ��,0

�K� = 1.17. �92�

From these values, one can compute the initial values��,in ,h�,in

2 . For 6Li, we use the values

kin = 103, h�,in2�Li� = 2.56, ��,in

�Li� = − 2.51 � 10−2, �93�

whereas for 40K we take

kin = �300, h�,in2�K� = 1.67 � 105, ��,in

�K� = 6.14. �94�

For 40K, we observe large values of h�,in2 and ��,in. Similar

values for 6Li are observed in the corresponding K range,

e.g., h�,in2 �K2=300�=6490, ��

�Li��K2=300�=2024. The broadFeshbach resonances indeed describe strongly coupled sys-tems. In our approximation, the solutions of the flow equa-tions for h�

2 ,�� and Z� do not depend on m�2 provided m�

2

remains positive during the flow.

XI. SCATTERING LENGTH FOR MOLECULES

So far, we have only discussed the contributions of themolecule fluctuations to �u /� in order to extract the mol-ecule density nM. In this section, we discuss their influenceon the effective potential for T=0 more systematically. Inparticular, we will extract the scattering length for molecule-molecule scattering in vacuum from ��=�2u /��2 ��0. FromEq. �26�, we infer for the derivative of the potential �primesdenote derivatives with respect to �� for the simple case ofQ�=1 and A�=Z�=Q�=1,

�

�k2u��B� = −1

4�2

��0

�

dxx2

�A�x2 + 2k2 + u� + 2�u�

1�A�x2 + 2k2 + u�

� �u�2

A�x2 + 2k2 + u�−

3�u�2 + 2�2u�u�A�x2 + 2k2 + u� + 2�u�

� .

�95�

This contribution vanishes for �=0, and we find in the sym-metric phase no influence on the running of m�

2 .The bosonic contribution to the running of ��=u��0�,

�

�k2���B� =

��2

2�2�0

�

dxx2

�A�x2 + 2k2 + m�2�2

=1

8�

��2

A�3/2�2k2 + m�

2�1/2 , �96�

combines with the fermionic contribution and the anomalousdimension

�

�k2�� =h�

2��

32��k2 − �−3/2 +

�

�k2���F� +

�

�k2���B�,

�

�k2���F� = −

3h�4

8�2�0

�

dxx2

�x2 + k2 − �4 = −3h�

4

256��k2 − �−5/2.

�97�

Similar to the flow equation for ��, the equation for thegradient coefficient A� decouples from Eqs. �38� and �46� inthe symmetric phase at T=0 and reads

�

�k2A� =h�

2

64��k2 − �−3/2A� −

1

2� . �98�

There are no boson contributions in this regime. The equa-tion is solved by A�=1 /2 for our initial condition A�,in=1 /2 �the initial condition is actually irrelevant, since A�

=1 /2 is an ir attractive fixed point of Eq. �98��.It is interesting to compare the scattering length for mol-

ecules aM= ��M / �2�� with the fermionic scattering length a�88�. For this purpose, we are interested in the flow of theratio Ra=aM /a=2�� /��,eff. For the sake of a simple analyticdiscussion, we take A�=1 /2, ��,0=0 and consider the broad-resonance limit. Let us first consider

R =8���k2 − �

h�2 , �99�

which approaches Ra in the infrared limit k→0, R�k→0�=Ra. With K2=k2− , we find the flow equation

K2 �R

�K2 = R −8K4��

h�4

�h�2

�K2 +8K4

h�2

���

�K2

= R +1

8���K −

3

32�

h�2

K+

2�2��2K4

�h�2�2k2 + m�

2

= R +h�

2

32�K1

2R − 3 +� k2 −

k2 + m�2 /2

R2� .

�100�

Inserting the fixed-point behavior h�2 =32�K yields

K2 �R

�K2 =3

2�R − 2� +� k2 −

k2 + m�2 /2

R2. �101�

If we neglect the contribution from the molecule fluctuations�the last term in Eq. �101�� we find an infrared stable fixed

point at R=2. For this fixed point, one obtains for k→0, → A,

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�� = −h�

2

4A

= −h�

2

2�M

, �102�

to be compared with ��,eff=−h�2 / �2�M�, as appropriate for the

molecule phase. For the atom scattering in the moleculephase, the propagator of the exchanged molecule has to beevaluated for nonzero �=−�M /2, cf. Eq. �90�. We discussthis issue in the Appendix. We conclude that this fixed pointcorresponds to the mean-field result aM=2a, Ra=2, see also�14�.

The bosonic fluctuations will lower the infrared value of

R. Using the fixed point for h�2 and the expression �84� for

the binding energy in Eq. �49�, one finds

k2 +m�

2

2= 3k2 − �M − �− �M

�2k2 − �M. �103�

At the resonance point �M=0, the flow equation for R be-comes

k2 �R

�k2 = − 3 +3

2R +

1�3

R2. �104�

The infrared-stable fixed point now occurs for

R* =�3

2�9

4+

12�3

−3

2� � 1.325. �105�

Precisely on the Feshbach resonance �B=B0�, the asymptoticbehavior obeys the scaling form

�� =h�

2R*

8k2 =4�R*

k. �106�

For B�B0, the mass term m�2 is smaller as compared to

the critical value for B=B0. For a given K the expression�2k2+m�

2 =�2K2+2+m�2 is therefore smaller and the term

�R2 in the flow equation is enhanced. As a consequence, ��

turns out smaller than the critical value �106�.It is instructive to study the flow in the range where

k2�−�M /2,K��−�M /2+k2 /�−2�M and therefore

k2 +m�

2

2� 2k2. �107�

The flow equation for R reads

k2 �

�k2 R = �3

2�R − 2� +

�2k2 − �M

2kR2� k2

k2 − �M/2.

�108�

For k→0, the second term dominates,

�R

�k=

�2R2

�k2 − �M/2, �109�

and the running stops for k→0, resulting in

R−1�k = 0� � R−1�ktr� + �2 arcsinh �2ktr

�− �M� . �110�

Here, ktr is a typical value for the transition from the scalingbehavior �104� for k2�−�M /2 to the boson dominated run-ning �109� for k2�−�M /2. We may give a rough estimate

using ktr=�−�M /2 and R�ktr�= R*,

R�0� =R*

1 + �2 arcsinh�1�R*

� 0.5. �111�

The value for a numerical solution turns out somewhathigher and we find for k→0

R�k → 0� =aM

a= 0.81. �112�

This is mainly due to the presence of fermion fluctuations

which enhance R.Following the flow of �� in a simple truncation thus pre-

dicts aM /a�0.81, in qualitative agreement with rather re-fined quantum mechanical computations �33�, numericalsimulations �9�, and sophisticated diagrammatic approaches�34�, aM /a�0.6. The resummation obtained from our flowequation is equivalent to the one performed in �35�, yieldingaM /a�0.78±4. Functional RG flows in the present trunca-tion with optimized cutoffs �21,22� lead to aM /a�0.71 �14�.It should be possible to further improve the quantitative ac-curacy by extending the truncation, for example by includingan interaction of the type �†��*� corresponding to atom-molecule scattering. Already at this stage, it is clear that thefinal value of Ra results from an interesting interplay betweenthe fermionic and bosonic fluctuations. In a model describingonly interacting bosonic particles �atoms or molecules�, theflow of �� can be taken from Eq. �96� for m�

2 =0 ,A�= 12 ,

���

�k=

1

2���

2 . �113�

�For an application to bosonic atoms, one should recall thatour normalization of the particles corresponds to atom num-ber 2 and mass 2M.� The bosonic fluctuation effects have thetendency to drive �� towards zero, according to the solutionof Eq. �113� for k→0,

�� =��,in

1 + ��,inkin/2�. �114�

This solution reflects the infrared-stable fixed point for ��

=��k,

����* = 0, �115�

resulting from

k�

�k�� = �� +

��2

2�. �116�

On the other hand, the fermion fluctuations have the op-posite tendency: they generate a nonvanishing �� even for aninitial value ��,in=0 and slow down the decrease of �� as

RENORMALIZATION FLOW AND UNIVERSALITY FOR… PHYSICAL REVIEW A 76, 053627 �2007�

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compared to Eq. �113� in the range of small k. The resultingRa results from a type of balance between the opposite driv-ing forces. If the scaling behavior �104� is approximatelyreached during the flow, the final value of Ra is a universalratio which does not depend on the microscopic details. Thisis a direct consequence of the “loss of microscopic memory”for the fixed point �105�.

XII. UNIVERSALITY

The essential ingredient for the universality of the BEC-BCS crossover for a broad Feshbach resonance is the fixed

point in the renormalization flow for large h�2 �fixed point �A�

in Sec. IX�. This fixed point is infrared stable for all cou-plings except one relevant parameter corresponding to thedetuning B−B0. For T=0, this fixed point always dominatesthe flow at B=B0 as long as effects from a nonzero densitycan be neglected. However, for any kF�0 the flow will fi-nally deviate from the scaling form due to the occurrence ofa condensate �0�0. Typically, this happens once k reachesthe gap for single fermionic atoms, k��=h�

��0. �If we use

k=kF the relevant scale is k=16�c / �3��.� For k��, thecontribution from fermionic fluctuations to the flow becomessuppressed. Also the contribution from the “radial mode” ofthe bosonic fluctuations will be suppressed due to a massliketerm m�

2 �2���0. Only the Goldstone mode corresponding tothe phase of � will have unsuppressed fluctuations.

For nonzero density it is convenient to express all quan-tities in units of the Fermi momentum kF and the Fermi en-

ergy �F, i.e., to set k=kF. As compared to the scaling form atzero density the flow equations are now modified by twoeffects. They concern the influence of �0�0 and the nonzerovalue for = * which corresponds to B=B0. �Typically, thisvalue * is positive, as opposed to *=0 for the zero densitycase.� Nevertheless, all these modifications only concern theflow for small k. If the flow for large k has already ap-proached the fixed point close enough no memory is leftfrom the microscopic details. We can immediately concludethat all physical quantities are universal for B=B0.

The same type of arguments applies for nonzero tempera-

ture. For T�0, an effective infrared cutoff is introduced forthe fermionic fluctuations. Also the contributions from thebosonic fluctuations become modified. Again, this only con-

cerns the flow for k��T. The loss of microscopic memorydue to the attraction of the flow for large k towards the fixedpoint remains effective.

In a similar way, we can consider the flow for B�B0.Away from the exact location of the resonance, the observ-able quantities will now depend on the relevant parameter.The latter can be identified with the concentration c=akF �6�.Still, the deviation from the scaling flow only concerns smallk, namely the range when the first term in Eq. �49� becomesimportant. This happens for

k �16�M �B − B0�

h�,02 kF

2=

1

c. �117�

If the flow for large k has been attracted close enough to thefixed point all physical quantities at nonzero density are uni-

versal functions of two parameters, namely c and T=T /�F.These functions describe all broad resonances.

XIII. DEVIATIONS FROM UNIVERSALITY

The functional flow equations also allow for simple quan-titative estimates for the deviations from universality for agiven atom gas. Typically, these corrections are power sup-pressed ���ir /�uv�p. The microscopic scale �uv roughly de-notes the inverse of the characteristic range of van der Waalsforces and we may associate �uv�1 / �100aA� with aA the“radius” of the atoms. More precisely, �uv corresponds to thescale at which the attraction to the fixed point sets in. Insome cases, this may be substantially below the van derWaals scale, as for the case of 40K, as we will argue below.The relevant infrared scale �ir depends on the density, �ir

=L�c , T�kF. In view of the discussion in the preceding sec-tion, we approximate L by the highest value of k where thedeviation from the scaling form of the flow equations sets in,i.e.,

L = max��c−1�,�T,���,�� . �118�

We note that and �=h���0 depend on c and T.

The power p of the suppression factor reflects the“strength of attraction” of the fixed point. More precisely, wemay consider the flow of various dimensionless couplings �i.If we denote their values at the fixed point by �i*, the flow ofsmall deviations �i=�i−�i*

is governed by the “stabilitymatrix” Aij

k�

�k�i = Aij� j . �119�

The relevant parameter corresponds to the negative eigen-value of A. All other eigenvalues of A are positive, and p isgiven by the smallest positive eigenvalue of A.

Typical couplings �i are ��� , h�2 , m�

2 , R�. For these cou-plings, the stability matrices reads for fixed point �A�:

A =�1, 0, 0, 0

8, 1, 0, 0

0, 14� , − 1, 0

0, e1, e2, e3

� �120�

with e1= �R*−6+2R*2 � / �32��, e2=−R*

2 / �6�3�, e3=3

+4R* /�3. The eigenvalues are �1,1−1,e3� and we concludep=1. This situation is not expected to change if more cou-plings like A�, A� or the �3 term in the potential are included.

We may use our findings for a rough estimate for the

range in c−1 and T for which deviations from universality aresmaller than 1%. From our previous considerations this holdsfor

�c−1� ��uv

100kF, �T �

�uv

300kF. �121�

As a condition for �uv, we require that all couplings are in awider sense in the “vicinity” of the fixed point at the scale

DIEHL et al. PHYSICAL REVIEW A 76, 053627 �2007�

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kuv=�uv /kF. Of course, one necessary condition is �uv��.Also, the flow equations must be meaningful for k�kuv.While this poses no restriction on Li where ��,0�0, the caseof K with positive ��,0 requires kuvc0�1 or kuv�20. If wewould try to extrapolate the flow to even higher k the value

of h�2�k� would diverge as can be seen from Eq. �41�. As a

further condition for the flow to be governed by the universalbehavior of the fixed point we require Z�−1�1. This tells usthat the notion of a diatom state � starts to be independent ofthe detailed microscopic properties. For Li and K, this hap-pens for scales only somewhat below kin.

A reasonable estimate at this stage may be kuv�Li�=800,

kuv�K�=15. One percent agreement with the universal behavior

would then hold for

Li: �c−1� � 8, �T � 3,

K: �c−1� � 0.15, �T � 0.05, �122�

which corresponds to

Li: �akF� � 0.13, T/TF � 9,

K: �akF� � 6.7, T/TF � 0.0025. �123�

Already at this stage, we conclude that the universal behav-

ior for K covers only a much smaller range in c and T ascompared to Li. In particular, experiments with Li are indeedperformed within the universal regime which extends far offresonance and to temperatures well above the quantum de-generate regime T /TF�1, while for K it will be hard to reachthe truly universal domain, since the lowest available tem-peratures range down to T /TF�0.04, and the magnetic fieldtuning is too low to resolve a regime where �akF � �6.7.

At first sight, the limitation to a rather small value kuv�K�

=15 for 40K may seem to be of a technical nature, enforcedby the breakdown of the flow for �� for k�kuv

�K�. However,this technical shortcoming most likely reveals the existenceof some additional physical scale close to kuv

�K�, as, for ex-ample, associated to a further nearby resonance state not ac-counted for in our simple model. Indeed, if our model �with-out modifications and additional effective degrees offreedom� would be valid for k�kuv

�K�, one could infer an

effective upper bound for �� from its flow. Starting at some

kin with an arbitrarily large positive ��,in, the value of ���k�would be renormalized to a finite value, bounded by ���K=0��8� /Kin �cf. Eq. �39��. This would lead to a contradic-tion with the observed value unless Kin is sufficiently low.

The observed value of ��,0 therefore implies the existence ofa scale near kuv

�K� where our simple description needs to beextended. This issue is very similar to the “triviality bound”in the standard model for the electroweak interactions in par-ticle physics.

As further concern, one may ask if h�2�kuv� is already close

enough to the fixed point value 32� and if ���kuv� is closeenough to zero. This is an issue, since we know of the exis-tence of a different fixed point for narrow Feshbach reso-

nances at h�2 = ��=0. At the scale kuv, the flow should not be

in the vicinity of this “narrow-resonance fixed point” any-more, but at least in the crossover region towards the “broad-resonance fixed point.” If we evaluate the couplings at thescales kuv

�Li�=800, kuv�K�=15, we obtain for c−1=0,

Li: h�2�kuv� = 5.00 � 10−3, ���kuv� = − 25.0,

m�2�kuv� = 6.07 � 10−2;

K: h�2�kuv� = 4.53 � 103, ���kuv� = 58.8, m�

2�kuv� = 54.0.

�124�

For K, the relevance of the broad-resonance fixed point

seems plausible, but the small value of h�2�kuv� for Li may

shed doubts. However, one should keep in mind that for astrong enough ��,eff the distribution between �� and −h�

2 /m�2

concerns mainly the dependence of the scattering length onthe magnetic field B and not so much the physics for a givena or c. Indeed, we could absorb �� by partial bosonization�Hubbard-Stratonovich transformation� in favor of a changeof h�

2 and m�2 . Up to terms �q4, which are subleading for the

low-momentum physics, a model with given ��, h�2 , m�

2 isequivalent to a model with ��� =0, but new values h��

2 andm��

2 related to the original parameters by

h��2 = h�

2 − 2��m�2 +

��2m�

4

h�2 ,

m��2 = m�

2 −��m�

4

h�2 . �125�

The value of the new Yukawa coupling for Li, h��2�kuv�

=4.65�102 is much larger and seems acceptably close to thebroad resonance fixed point. Using partial bosonization andthe technique of rebosonization during the flow �31�, it mayactually be possible to treat �� as a redundant parameter,thus enlarging the “space of attraction” of the fixed point�A�. In this context, we note that we should include the con-tribution of the bosonic fluctuations to the running of �� forour truncation. This will shift the precise location of the fixed

points. In a language with rebosonization where �� remainszero, these additional contributions will be shifted into the

flow of m�2 and h�

2 .Our estimate for the deviations from universality concerns

only the overall suppression factor. Detailed proportionalitycoefficients cO for the deviation of a given observable Ofrom the universal strong-resonance limit, O=cO�ir /�uv,depend on the particular observable. The deviations fromuniversality should therefore be checked by explicit solutionsof the flow equation for Li and K.

XIV. CONCLUSIONS

In this paper, we have performed a functional renormal-ization group study for ultracold gases of fermionic atoms.We have concentrated on four parameters: the Yukawa cou-pling of the molecules to atoms h�, the offset between mo-

RENORMALIZATION FLOW AND UNIVERSALITY FOR… PHYSICAL REVIEW A 76, 053627 �2007�

053627-15

lecular and open channel energy levels m�2 which is related to

the detuning, the background atom-atom-interaction ��, andthe molecule-molecule interaction ��. This system exhibits afixed point for the rescaled couplings which is infrared stableexcept for one relevant parameter. This parameter can beassociated with the inverse concentration c−1= �akF�−1.

Whenever for a given system the trajectories of the flowin coupling-constant space approach this fixed point closeenough, the macroscopic physics loses the memory of themicrophysical details except for one parameter, namely, c−1.In consequence, for a certain range in c−1 around 0 and for a

certain range in temperature T all dimensionless macroscopic

quantities can only depend on c and T. Here dimensionlessquantities are typically obtained by multiplication with ap-propriate powers of kF or �F.

The macroscopic quantities include all thermodynamicvariables and, more generally, all quantities that can be ex-pressed in terms of n-point functions for renormalized fieldsat low momenta. In particular, the correlation functions foratoms and “molecules” as well as their interactions depend

only on c and T, independently of the concrete microscopicrealization of a broad Feshbach resonance. Here “molecules”refers to bosonic quasiparticles as collective diatom stateswhich may be quite different from the notion of microscopicor “bare” molecules. In a certain sense this situation has ananalog in the universal critical behavior near a second-orderphase transition. In contrast to this, however, the universal

description includes now a temperature range between T=0and substantially above the critical temperature, and the uni-versal quantities depend on an additional relevant parameterc−1. We may also compare to a quantum phase transition atT=0 which is realized in function of a parameter analogousto c−1. In our case, universality is extended to nonzero tem-perature as well.

The “broad resonance fixed point” is not the only fixedpoint of the system. Another “narrow resonance fixed point”

at h�2 =0, ��=0 allows for an exact solution of the crossover

problem for narrow Feshbach resonances �6,36� and a per-turbative expansion around this solution. The narrow-resonance fixed point has two relevant parameters c−1 and

h�2 . A typical flow away from this fixed point at small enough

c−1 and �� describes a crossover towards the broad-resonance fixed point.

For a given physical system characterized by some micro-scopic Hamiltonian, it is important to determine how close itis to the universal behavior of the broad-resonance limit. Wehave performed here a first estimate for the range in c−1 and

T for which universality holds within 1% accuracy for theexperimentally studied Feshbach resonances in 6Li and 40K.It will be an experimental challenge to verify or falsify thesepredictions of universality.

ACKNOWLEDGMENTS

We thank S. Flörchinger, H.C. Krahl, M. Scherer, and P.Strack for useful discussions. H.G. acknowledges support bythe DFG under Contract No. Gi 328/1-3 �Emmy-Noetherprogram�.

APPENDIX: SCATTERING LENGTH AND CROSSSECTION

In this appendix, we collect useful formulas for the scat-tering of molecules in vacuum. In quantum field theory, thescattering cross section for identical nonrelativistic bosons isgiven by

B =1

2� d�

d

d�. �A1�

The factor 1/2 is a convention, motivated by integration overhalf the spatial angle in order to avoid double counting of theidentical particles. The differential cross section is related tothe scattering amplitude by

d

d�=

MB2 �AB�2

16�2 , �A2�

such that, in case of isotropic �e.g., s-wave� scattering

B =MB

2 �AB�2

8�. �A3�

Next, we relate the above results to nonrelativistic quantummechanics: The scattering length a is a quantity which isdirectly meaningful in nonrelativistic quantum mechanicsonly. It measures the strength of the 1 /r decay of the scat-tered wave function at low energies as a /r. Equivalently, a isdefined by the l=0 phase shift for the scattered wave func-tion. This definition leads to the relation between scatteringlength and cross section �identical bosons�

B = 8�aB2 . �A4�

We can use Eqs. �A3� and �A4� in order to relate the scatter-ing amplitude to the scattering length or rather to define thisrelation,

�AB� =8�aB

MB=

2��

Z�2 . �A5�

The scattering amplitude is given by the effective four-bosonvertex, as obtained by functional derivatives of the effective

action, Z�2AB=��4�=2��. In the limit of a pointlike interac-

tion, �� is a constant. �Note that the notion of scatteringamplitude is used to describe different quantities in quantummechanics and quantum field theory �QFT�.�

For fermions, the definitions are similar,

F =� d�d

d�. �A6�

Now, the integration covers the full space angle, since thefermions can be distinguished by their spin. The differentialcross section is related to the scattering amplitude by

d

d�=

MF2�AF�2

16�2 , �A7�

such that for isotropic scattering

DIEHL et al. PHYSICAL REVIEW A 76, 053627 �2007�

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F =MF

2�AF�2

4�. �A8�

For distinguishable fermions, one has in quantum mechanics

= 4�aF2 , �A9�

such that the scattering amplitude and scattering length arerelated by

�AF� =4�aF

MF= ��,eff��,q�� . �A10�

The scattering amplitude AF is given by the tree-levelgraph for the molecule exchange fermions plus a contribu-

tion from the fermionic 1PI vertex ��. The energy- andmomentum-dependent resonant four-fermion vertex gener-ated by the molecule exchange reads

��,eff��,q�� − ��,0

= −h�

2�A�

P���,q��

= −h�

2�A�

− � +q�2

4M+ m�

2 + ��P���,q�� − �P��0,0���

�A11�

with

�P���,q�� =h�

2�A�M3/2

4��− � − 2A +

q�2

4M. �A12�

With q�1 ,q�2 the momenta of the scattered atoms, one findsfor the momentum and energy of the exchanged moleculeq� =q�1+q�2, �=q�1

2 / �2M�+q�22 / �2M�−2A. Here, we take into

account the binding energy—the energy of an incoming atom

is −A+q� i2 / �2M�. We now consider the limit q� i→0 and

work in the broad-resonance limit h�2 →�. The vacuum in the

molecule phase is characterized by m�2 =0, and we end up

with

��,eff − ��,0 =4�

M3/2�− 2A

. �A13�

One infers for the scattering length of atoms in the moleculephase

a =M��,eff�− 2A,0��

4�=

1�− 2MA

+ a0. �A14�

The relation between our rescaled quantities and the scat-tering amplitudes is given by

�� = 2MkF��, �� =�

Z�2 , ��,eff = 2MkF��,eff,

h�,02 =

4M2

kFh�

2 , A =A

�F, m�

2 =m�

2

�F. �A15�

With MB=2M, the ratio between molecular and atomic scat-tering length �aM=aB, aMkF=�� / �4��� therefore reads

aM

a=

2��/Z�2

��,eff��,q��=

2��

��,eff�− 2A,q��. �A16�

Using the definition of the renormalized Yukawa coupling

h�2 = h�

2 /Z�, we may express

��,eff�− 2A,0� − ��,0 =8�

�− A

= −h�

2

4A

= −h�

2

2�M

.

�A17�

In the last two expressions, we have used the leading termsin Eqs. �41� and �42� for K=�−A→0.

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