Itinerant Ferromagnetism in a Fermi Gas ofUltracold Atoms
Gyu-Boong Jo, et al.
Science 325, 1521 (2009);
DOI: 10.1126/science.1177112
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REPORTS

References and Notes
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26. J. J. Yang et al., Nat. Nanotechnol. 3, 429 (2008).

Itinerant Ferromagnetism in a Fermi
Gas of Ultracold Atoms
Gyu-Boong Jo,1* Ye-Ryoung Lee,1 Jae-Hoon Choi,1 Caleb A. Christensen,1 Tony H. Kim,1
Joseph H. Thywissen,2 David E. Pritchard,1 Wolfgang Ketterle1
Can a gas of spin-up and spin-down fermions become ferromagnetic because of repulsive
interactions? We addressed this question, for which there is not yet a definitive theoretical answer,
in an experiment with an ultracold two-component Fermi gas. The observation of nonmonotonic
behavior of lifetime, kinetic energy, and size for increasing repulsive interactions provides strong
evidence for a phase transition to a ferromagnetic state. Our observations imply that itinerant
ferromagnetism of delocalized fermions is possible without lattice and band structure, and our data
validate the most basic model for ferromagnetism introduced by Stoner.
agnetism is a macroscopic phenomenon with its origin deeply rooted inquantum mechanics. In condensedmatter physics, there are two paradigms formagnetism: localized spins interacting via tunneling and delocalized spins interacting via an
exchange energy. The latter gives rise to itinerant ferromagnetism, which is responsible forthe properties of transition metals such as cobalt,
iron, and nickel. Both kinds of magnetism involve strong correlations and/or strong interactions and are not yet completely understood. For
localized spins, the interplay of magnetism withd-wave superfluidity and the properties of frustrated spin materials are topics of current research.
For itinerant ferromagnetism (1–7), phase transition theories are still qualitative.
We implemented the Stoner model, a textbook Hamiltonian for itinerant ferromagnetism(8), by using a two-component gas of free fer-

M

1

Massachusetts Institute of Technology–Harvard Centerfor Ultracold Atoms, Research Laboratory of Electronics,
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 2Department of Physics,
University of Toronto, Toronto, Ontario M5S1A7, Canada.
*To whom correspondence should be addressed. E-mail:gyuboong@mit.edu

mions with short-range repulsive interactions,which can capture the essence of the screened
Coulomb interaction in electron gases (8). However, there is no proof so far that this simple
model for ferromagnetism is consistent when thestrong interactions are treated beyond mean-fieldapproaches. It is known that this model fails in
one dimension, where the ground state is singlet
for arbitrary interactions, or for two particles inany dimension (3). In our work, cold atoms wereused to perform a quantum simulation of thismodel Hamiltonian in three dimensions, and weshowed experimentally that this Hamiltonian
leads to a ferromagnetic phase transition (2).
This model was also realized in helium-3 (9), but
the liquid turn into a solid phase and not into aferromagnetic phase at high pressure. It has also
been applied to neutrons in neutron stars (10).
To date, magnetism in ultracold gases has
been studied only for spinor (11, 12) and dipolar(13) Bose-Einstein condensates (BECs). In thesecases, magnetism is driven by weak spindependent interactions, which nevertheless determine the structure of the condensate becauseof a bosonic enhancement factor. In contrast,
here we describe the simulation of quantum magnetism in a strongly interacting Fermi gas.

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27. Y. Muraoka, Z. Hiroi, Appl. Phys. Lett. 80, 583 (2002).
28. H.-T. Kim et al., N. J. Phys. 6, 52 (2004).
29. I. Shadrivov, SPIE Newsroom doi:10.1117/
2.1200811.1390 (2008).
30. V. M. Shalaev, Nature Photon. 1, 41 (2007).
31. This work is supported by the U.S. Department of Energy(DOE), the Air Force Office of Scientific Research(AFOSR), and ETRI. Work at UCSD on VO2 was supported
by DOE–Basic Energy Sciences and the metamaterials
work was supported by AFOSR and ETRI. M.D.
acknowledges partial support from NSF. H.K.
acknowledges research support from a project of the
Ministry of Knowledge Economy in Korea.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/1176580/DC1
Materials and Methods
Fig. S1
Table S1
References
20 May 2008; accepted 4 August 2009
Published online 20 August 2009;10.1126/science.1176580
Include this information when citing this paper.

An important recent development in cold
atom science has been the realization of superfluidity and the BEC–Bardeen-Cooper-Schrieffer(BCS) crossover in strongly interacting, twocomponent Fermi gases near a Feshbach resonance (14). These phenomena occur for attractive
interactions for negative scattering length and forbound molecules (corresponding to a positive
scattering length for two unpaired atoms). Verylittle attention has been given to the region of
atoms with strongly repulsive interactions. One
reason is that this region is an excited branch,which is unstable against near-resonant threebody recombination into weakly bound molecules. Nevertheless, many theoretical papers
have proposed a two-component Fermi gas neara Feshbach resonance as a model system for itinerant ferromagnetism (15–22), assuming that the
decay into molecules can be sufficiently suppressed. Another open question is the possibilityof a fundamental limit for repulsive interactions.
Such a limit due to unitarity or many-body physics may be lower than the value required for thetransition to a ferromagnetic state. We show that
this is not the case and that there is a window ofmetastability where the onset of ferromagnetism
can be observed.
A simple mean-field model captures many
qualitative features of the expected phase transition but is not adequate for a quantitative description of the strongly interacting regime. The
total energy of a two-component Fermi gas of
average density n (per spin component)
in an3
volume V is given by EF 2Vn 10½ð1 þ hÞ5=3 þ
2kF að1 þ hÞð1 − hÞg, where EF
ð1 − hÞ5=3  þ 3p
is the Fermi energy of a gas, kF is the Fermi wave
vector of a gas, a is the scattering length characterizing short-range interactions between the twocomponents, and h = Dn/n = (n1 − n2)/(n1 + n2)
is the magnetization of the Fermi gas. The local
magnetization of the Fermi gas is nonzero when
the gas separates into two volumes, where thedensities n1 and n2 of the two spin states differ

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ble spectra could easily push this effect to higher
frequencies that are beneficial for a variety ofpractical applications (30).

1521

by 2Dn. We studied an ensemble in which thenumber of atoms in each spin state is conserved.
This is equivalent to a free electron gas at zeroexternal magnetic field where the total magnetization is zero. The interaction term represents
any short-range spin-independent potential. Whenthe gas is fully polarized, it avoids the repulsive
interaction but increases its kinetic energy by afactor of 22/3. The phase transition occurs whenthe minimum in energy is at nonzero magnetization (Fig. 1A) at kFa = p/2. This onset waspreviously discussed in the context of phase separation in a two-component Fermi gas (15–18).
Figure 1B shows several consequences of thephase transition for a system at constant pressure. First, for increasing repulsive interactions,the gas expands, lowering its density and Fermi
energy; kinetic energy is therefore reduced.
When the gas enters the ferromagnetic phase,kinetic energy increases rapidly because of thelarger local density per spin state. Furthermore,
the volume has a maximum value at the phasetransition. This can be understood by noting thatpressure in our model is (2/3)Ekin/V + Eint/V,
where Ekin is kinetic energy and Eint is interactionenergy. At the phase transition, the system increases its kinetic energy and reduces its interaction energy, thus reducing the pressure. Thismaximum in pressure at constant volume turns
into a maximum in volume for a system held atconstant pressure or in a trapping potential. We
have observed three predictions of this model: (i)the onset of local magnetization through the
suppression of inelastic collisions, (ii) the minimum in kinetic energy, and (iii) the maximum inthe size of the cloud. These qualitative features
are generic for the ferromagnetic phase transitionand should also be present in more-advanced
models (19).
We start with an atom cloud consisting of an
equal mixture of 6Li atoms in the lowest two
hyperfine states, held at 590 G in an optical
dipole trap with additional magnetic confinement (23). The number of atoms per spin state isapproximately 6.5 × 105, which corresponds toa Fermi temperature TF of ~1.4 mK. The effective temperature T could be varied between
T/TF = 0.1 and T/TF = 0.6 and was determined
immediately after the field ramp by fitting the
spatial distribution of the cloud with a finite
temperature Thomas-Fermi profile. We definek∘F as the Fermi wave vector of the noninteracting gas calculated at the trap center. Applying
the procedure discussed in (24) to repulsive interactions, we estimate that the real temperature
is approximately 20% larger than the effective
one. The effective temperature did not dependon k∘F a for kFo a < 6. At higher temperatures,
additional shot-to-shot noise was caused bylarge fluctuations in the atom number. Fromthe starting point at 590 G, the magnetic field
was increased toward the Feshbach resonance at
834 G, thus providing adjustable repulsive interactions. Because of the limited lifetime of thestrongly interacting gas, it was necessary to ap-

1522

ply the fastest possible field ramp, limited to
4.5 ms by eddy currents. The ramp time is approximately equal to the inverse of the axial trap
frequency (23) and therefore only marginallyadiabatic. Depending on the magnetic field during observation, either atoms or atoms andmolecules were detected by absorption imaging
as described in fig. S1 (25).
The emergence of local spin polarization canbe observed by the suppression of (either elastic
or inelastic) collisions, because the Pauli exclusion principle forbids collisions in a fully polarized cloud. We monitored inelastic three-bodycollisions, which convert atoms into molecules.
The rate (per atom) is proportional to f(a,T)n1n2
or f(a,T) n2(1 − h2) and is therefore a measureof the magnetization h. For kFa << 1, the rate
coefficient f(a,T) is proportional to a6 max(T,TF)(26). This rate can be observed by monitoring
the initial drop in the number of atoms during
the first 2 ms after the field ramp. We avoided
longer observation times, because the increasing
molecule fraction could modify the properties ofthe sample.
A sharp peak appears in the atom loss ratearound kFo a ≅ 2.5 at T/TF = 0.12 (Fig. 2), indicating a transition in the sample to a state withlocal magnetization. The gradual decrease is consistent with the inhomogeneous density of thecloud, where the transition occurs first in thecenter. The large suppression of the loss rate
indicates a large local magnetization of the cloud.
The kinetic energy of the cloud was determined by suddenly switching off the optical trap
and the Feshbach fields immediately after thefield ramp and then imaging state |1〉 atoms atzero field using the cycling transition after aballistic expansion time of Dt = 4.6 ms. The kinetic energy was obtained from the Gaussianradial width sx as Ekin = [(3msx2)/(2Dt 2)] wherem is the mass of the 6Li atom. A minimum of
the kinetic energy at kFo a ≅ 2.2 for the coldest
temperature T/TF = 0.12 nearly coincided with

18 SEPTEMBER 2009

the onset of local polarization (Fig. 3). The peak in
the atom loss rate occurs slightly later than theminimum of kinetic energy, probably because
f(a,T) increases with a (22). Because the temperature did not change around kFo a ≅ 2.2, the increase in kinetic energy is not caused by heating
but by a sudden change in the properties of thegas, which is consistent with the onset of ferromagnetism. The observed increase in kinetic energy is approximately 20% at T/TF = 0.12, smaller
than the value (22/3 − 1) = 0.59 predicted for a
fully polarized gas. This discrepancy could be
due to the absence of polarization or partial polarization in the wings of the cloud. Also, it ispossible that the measured kinetic energy of thestrongly interacting gas before the phase transition
includes some interaction energy if the Feshbach
fields are not suddenly switched off. For the current switch-off time of ~100 ms, this should be
only a 5% effect, but the magnetic field decay
may be slower because of eddy currents.
Finally, Fig. 4 shows our observation of a
maximum cloud size at the phase transition, inagreement with the prediction of the model. The
cloud size may not have fully equilibrated, because
our ramp time was only marginally adiabatic, but
this alone cannot explain the observed maximum.
The suppression of the atom loss rate, theminimum in kinetic energy, and the maximum
in cloud size show a strong temperature dependence between T/TF = 0.12 and 0.22. The properties of a normal Fermi gas approaching theunitarity limit with k∘F a >> 1 should be insensitive
to temperature variations in this range; therefore,
the observed temperature dependence provides
further evidence for a transition to a new phase.
At higher temperature (e.g., T/TF = 0.39 asshown in Fig. 3), the observed nonmonotonic
behavior becomes less pronounced and shifts tolarger values of kFo a for 3 ≤ koF a ≤ 6. For all threeobserved properties (Figs. 2 to 4), a nonmonotonic
behavior is no longer observed at T/TF = 0.55 (27).
One interpretation is that at this temperature andFig. 1. Ferromagnetic phase transition at T = 0, according to themean-field model described in the
text. The onset of itinerant ferromagnetism occurs when the energy
as a function of magnetization flips
from a U shape to a W shape (A).
(B) Enthalpy, volume, and kinetic
energy, normalized to their values
for the ideal Fermi gas, and magnetization as a function of the interaction parameter kFa. kF is definedby the density of the gas. The dottedline marks the phase transition.

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REPORTSabove, there is no longer a phase transition. In amean-field approximation, a ferromagnetic phasewould appear at all temperatures but for increasing
values of kFo a. Our observations may imply thatthe interaction energy saturates around kFo a ≈ 5.

The spin-polarized ferromagnetic state should
not suffer from inelastic collisions. However,typical lifetimes were 10 to 20 ms, which were
probably related to a small domain size and threebody recombination at domain walls.

Fig. 2. Atom loss rate as a probe
for local spin polarization, fordifferent temperatures. T/TF = 0.55
(triangles, dashed curve), T/TF =0.22 (open circles, dotted curve),
and T/TF = 0.12 (solid circles, solid
black curve). The curves are guides
to the eye, based on the assumption of a loss rate that saturates forincreasing a in the normal state. The
shaded area around the phasetransition at T/TF = 0.12 highlights
the same region as in Figs. 3 and 4.

Fig. 3. Kinetic energy ofa repulsively interacting
Fermi gas determined for
different interaction parameters kFo a and temperatures. The measured
kinetic energy is normalized by the Fermi energy
EFo of the noninteracting
Fermi gas at T = 0, calculated at the trap centerwith the same number ofatoms per spin state. Eachdata point represents theaverage of three or fourmeasurements.

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We were unsuccessful in imaging ferromagnetic domains using differential in situ phasecontrast imaging (28). A signal-to-noise level of~ 10 suggests that there were at least 100 domains in a volume given by our spatial resolutionof ~3 mm and by the radial size of the cloud. Thisimplies that the maximum volume of the spin domains is ~5 mm3, containing ~50 spin-polarizedatoms. We suspect that the short lifetime prevented
the domains from growing to a larger size andeventually adopting the equilibrium texture of theground state, which has been predicted to havethe spins pointing radially outward, like a hedgehog(20, 22). All our measurements are sensitive only
to local spin polarization and are independent of
domain structure and texture.
The only difference between our experimentand the ideal Stoner model is a molecular admixture of 25% (Fig. 4). The molecular fraction
was constant for kFo a > 1.8 for all temperatures andtherefore cannot be responsible for the suddenchange of behavior of the gas at kFo a ≅ 2.2 at thecoldest temperature T/TF = 0.12. This prediction
was confirmed by repeating the kinetic energy
measurements with a molecular admixture of 60%.
The minimum in the kinetic energy occurred at thesame value of kFo a within experimental accuracy.
For a comparison of the observed phase transition at kFo a ≅ 2.2 to the theoretical predictions, theideal gas kFo has to be replaced by the value for theinteracting gas, which is smaller by ~15% becauseof the expansion of the cloud (Fig. 4), resulting in a
critical value for kFa ≅ 1.9 T 0.2. At T/TF = 0.12, thefinite temperature correction in the critical value for
kFa is predicted to be less than 5% (19). The
observed value for kFa is larger than both the meanfield prediction of p/2 and the second-order prediction of 1.054 at zero temperature (19). Depending on the theoretical approach, the phase transition
has been predicted to be first or second order. This
could not been discerned in our experiment becauseof the inhomogeneous density of the cloud.
It has been speculated (19) that earlier experiments on the measurement of the interaction energy (29) and radio frequency spectroscopy of Fermi
gases (30) showed evidence for the transition to aferromagnetic state at or below kFa = 1. This interpretation seems to be ruled out by our experiment.
Our work demonstrates a remarkable asymmetry between positive and negative scattering
length. Early work (15) predicted that for kF|a| =p/2, both an attractive and a repulsive Fermi gas
become mechanically unstable (against collapse
and phase separation, respectively). In an attractive Fermi gas, however, the mechanical instability does not occur [due to pairing (31)], in
contrast to our observations in a repulsive Fermi
gas. This suggests that the maximum total repulsive energy [in units of 3/5(2Vn)EF] is larger
than the maximum attractive energy |b| of 0.59
(32) that is realized for infinite a (23).
The interpretation of our results in terms ofa phase transition to itinerant ferromagnetismis based on the agreement with the predictionof simplified models [Fig. 1, (15–22)]. Future

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Fig. 4. Maximum in volume at the phase transition. (A) Axial sizeand chemical potential of the cloud for various temperatures. The
chemical potential m is determined from the measured cloud size, sz,
as m = [1/2]mwz2sz2. (B) Number of particles including both atoms
and molecules right after the field ramp. This result shows that 25%of the atoms were converted into molecules during the field ramp,
and this fraction stayed constant for kFo a > 1.8, where the phasetransition was reached. This molecule fraction was independent of
temperature.

work should address how the observed signatures are modified by strong interactions andcorrelations. Additional insight can be obtainedby varying the magnetic field ramp time over awide range and studying the relaxation toward
an equilibrium state (33).
Heisenberg and Bloch’s explanation for ferromagnetism was based on exchange energy;
that is, the Pauli principle and spin-independent
repulsive interactions between the electrons.
However, it was unknown what other “ingredients” were needed for itinerant ferromagnetism.
It was not until 1995 (6, 7) that a rigorous proof
was given that, in certain lattices, spin-independent
Coulomb interactions can give rise to ferromagnetism in itinerant electron systems. Our finding
suggests that Heisenberg’s idea does not require alattice and band structure but already applies to afree gas with short-range interactions. Our experiment can be regarded as quantum simulation of a
Hamiltonian for which even the existence of aphase transition was unproven. This underlines thepotential of cold atom experiments as quantum
simulators for many-body physics.
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27. The interpretation of the loss rate is complicated because
f(a,T ) is unkown for kFa ≥ 1. The three-body rate f(a,T ) is
expected to be unitarity saturated for kFa >> 1 (34). Thelines in Fig. 2 indicate that the observed loss rate isconsistent with unitarity saturation and a sudden drop at
the phase transition, which occurs at large values of kFa
at higher temperature.
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A. Paramekanti, L. J. LeBlanc, and G. J. Conduit for useful
discussions; T. Wang for experimental assistance; andA. Keshet for development of the computer control system.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/325/5947/1521/DC1
Materials and Methods
SOM Text
Fig. S1
References
1 June 2009; accepted 21 July 200910.1126/science.1177112

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