Monte Carlo studies of extensions of the Blume-Emery-Griffiths model
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Monte Carlo studies of extensions of the Blume-Emery-Griffiths model
C.C. Loois, G.T. Barkema and C. Morais Smith
Institute for Theoretical Physics, Utrecht University,Leuvenlaan 4,3584 CE Utrecht, The Netherlands (Dated: October 29, 2018)We extend the Blume-Emery-Griffiths (BEG) model to a two-component BEG model in order tostudy 2D systems with two order parameters, such as magnetic superconductors or two-componentBose-Einstein condensates. The model is investigated using Monte Carlo simulations, and thetemperature-concentration phase diagram is determined in the presence and absence of an externalmagnetic field. This model exhibits a rich phase diagram, including a second-order transition to aphase where superconductivity and magnetism coexist. Results are compared with experiments onCerium-based heavy-fermion superconductors. To study cold atom mixtures, we also simulate theBEG and two-component BEG models with a trapping potential. In the BEG model with a trap,there is no longer a first order transition to a true phase-separated regime, but a crossover to a kindof phase-separated region. The relation with imbalanced fermi-mixtures is discussed. We presentthe phase diagram of the two-component BEG model with a trap, which can describe boson-bosonmixtures of cold atoms. Although there are no experimental results yet for the latter, we hope thatour predictions could help to stimulate future experiments in this direction.
I. INTRODUCTION
Mixtures of He and He atoms exhibit a rich phasediagram, where besides a normal phase, there is a phasewhere He is superfluid, and a phase separated region ofsuperfluid He and normal He. In 1971, Blume, Emeryand Griffiths proposed a model to describe such mix-tures. They simplified the continuous phase of the su-perfluid order parameter such that it could acquire onlytwo values. Although they made this very rough ap-proximation and modelled the uniform system in a lat-tice, their results are very interesting. Qualitatively,they reproduced the right phases and the right ordersof the phase transitions. Furthermore, if disorder is in-troduced by placing the mixture into aerogel, after somemodifications, the model can still yield the experimen-tally observed phase diagram. Here, we generalize this model to a two-componentcase in order to describe systems with two order param-eters and study the problem numerically, using MonteCarlo simulations. The motivation for the model we areproposing is twofold. Firstly, we would like to study con-densed matter materials like heavy fermions, high- T c su-perconductors, and organic superconductors. In particu-lar, we want to study the interplay between magnetic andsuperconducting ordering in these materials. Both orderparameters are modelled as an Ising spin variable. Con-cerning the magnetism, we consider the ferro- and theantiferro-magnetic cases, and investigate also the effect ofan additional magnetic field. We find that in the absenceof a magnetic field, in the region where the two orderscoexist, the system is always phase separated. Whenwe add a magnetic field, we also find regions with mi-croscopic coexistence of the two phases. Secondly, wewant to study mixtures of cold atoms. Cold atoms haveemerged in recent years as an ideal simulator of con-densed matter systems. Because experiments with coldatoms are often carried out in a trap, we add a trapping potential to the model. This fact qualitatively changesthe physics of the problem. For the case of a single com-ponent BEG model in a trap, the results are comparedwith experimental and theoretical work on imbalancedFermi mixtures. For the case of the two-component BEGmodel, we make predictions for the phase diagram ofboson-boson mixtures.The outline of this paper is the following: in section II,we introduce the two-component BEG model, and inves-tigate it in the presence and absence of an external mag-netic field. The effect of a trapping potential is describedin section III. In section IV, we compare the results withmagnetic superconductors and cold atom systems. Ourconclusions are presented in section V. II. THE TWO-COMPONENTBLUME-EMERY-GRIFFITHS MODEL
The BEG model was originally proposed to de-scribe superfluidity. The phase diagram found byMonte Carlo simulations exhibits large similarities withthe phase diagram of He- He mixtures measured byexperimentalists. The main idea of studying superflu-idity with the BEG model relies on the U (1) symmetry-breaking of the ground-state wave function. For super-conductivity and Bose-Einstein condensation we have thesame symmetry breaking, hence we can try to modelthese phenomena in the same way.Several physical systems exhibit two unequal symme-try broken phases simultaneously. A general Hamiltoniandescribing this class of systems reads H = − J X
We investigate this model by Monte Carlo simulations.To determine the location of second-order phase transi-tions, we performed simulations at constant concentra-tion, in which the elementary moves were flips of s i and σ i or nonlocal spin exchanges. The location of the transitionis then obtained from the peak location of the magneticsusceptibility. The locations of first-order phase transi-tions are obtained from simulations at constant temper-ature, with as elementary moves local flips of s i and σ i ,as well as same-site replacements of s i by σ i and viceversa. A jump in the concentration c as a function ofthe anisotropy field D is then the signature of the phasetransition.All simulations are performed on lattices with approx-imately 40 ×
40 sites. Per point in the phase diagram,simulations were run over 3 · to 3 · Monte Carlosteps per site, depending on the correlation times.
B. Zero magnetic field, H = 0 In the absence of a magnetic field, the Hamiltonian (1)has ferro-antiferromagnetic symmetry.First, we consider K = 1. In this case, J = J , andthe shape of the phase diagram must be symmetric underthe transformation c → − c . The results of the simula-tions are plotted in Fig. 1. We see that it indeed obeysthis symmetry and exhibits four phases: a superconduct-ing phase (S), where the order parameter m s is nonzero,a ferromagnetic phase (FM), where m fm ,σ is nonzero, aphase-separated regime (PS) where the spins and the an-gular phases have formed ordered clusters, and finallythe normal phase (N), in which there is neither ordernor phase separation. Analogous to the BEG model,the transition from the phase-separated regions to otherphases are first-order (dashed line), the other ones aresecond-order (continuous line). FIG. 1: (color online) Phase diagram, temperature (in unitsof J /k B ) versus concentration, in the absence of a magneticfield. N indicates the normal phase, S superconductivity, FMferromagnetism, and PS phase separation. Solid lines repre-sent second order phase transitions, dashed lines first orderones. Lines are guides to the eye. Snapshots of the simulationare shown. Black (white) represents σ i = 1 ( − s i = 1 ( − Second, we consider the case K = 0 .
1. The resultsof the simulations are plotted in Fig. 2. We can under-stand the results as follows: J is much smaller than J ,hence the spins will not pay much attention to the angu-lar phases, and the part of the phase diagram concerningthe spins will be very similar to the BEG model. Be-cause J is so small, the phases will only order at verylow temperatures (at zero concentration, the tempera-ture is ten times lower than the one at which the spinsorder at a concentration of one). If the concentrationis slightly raised from zero, the system is already in thephase separated regime. All the states with a nonzerophase have clustered, and are not diluted by states withnonzero spin. Therefore, the critical temperature in thephase separated region will approximately remain con-stant. Because the temperature at which the angular FIG. 2: (color online) Phase diagram in the absence of amagnetic field, for a relative coupling constant of K = 0 . phases order is lower than the temperature at whichphase separation begins, there is a phase separated re-gion in which the angular phases of the wavefunction arenot ordered, which may appear unexpected at first sight.The transition within the phase separated regime, fromthe region where the angular phases are not ordered tothe phase where they are ordered (superconductivity),is second-order. This is expected, because in the phaseseparated regime, all the phases have clustered, and thetransition will be comparable with the transition in theIsing model, which is also second-order. C. Adding a magnetic field: the antiferromagneticcase
If we apply a nonnegative uniform magnetic field tothe system, the ferro-antiferromagnetic symmetry is bro-ken. We choose to consider the antiferromagnetic casehere, because then there are two competing effects, themagnetic field tends to align the spins, whereas the ex-change interaction wants to order the spins antiferromag-netically. The magnetic field H will be measured in unitsof J .Kimel et al. have studied the antiferromagnetic BEGmodel in the presence of a magnetic field, using MonteCarlo simulations. Their results at zero temperaturesuggest that the behavior of the system should be sep-arated into three qualitatively distinct regions, namely H ∈ [0 , , H ∈ [2 ,
4] and H ∈ [4 , ∞ ]. We consider herethe cases K = 1 and K = 0 . H within eachof these intervals.
1. H=1.5
First, we considered a magnetic field in the interval[0 , H = 1 .
5. Both for K = 1 and K = 0 . H = 0. This behavior was expected fromthe phase diagram of the single-component BEG modelat zero temperature. Because the magnetic field tries toalign the spins, the antiferromagnetic transition temper-ature is lower than in the absence of a magnetic field.
2. H=2.5
In the usual BEG model, the first-order phase tran-sition disappears in the presence of a magnetic field H ∈ [2 , σ i = 0 at everysite, and a checkerboard phase, where one sublattice has σ i = 0 at every site, and the other one σ i = −
1. Thereis also a transition between the checkerboard state, andan antiferromagnetic phase, but this transition is absentat nonzero temperature. For K = 1, the behavior of the two-component BEGmodel is still very similar to the case H = 0. For K = 0 . T intermediateregime, we also simulated the problem at a relative cou-pling strength of K = 0 .
5. In Fig. 4, we clearly observethat there is a region where antiferromagnetism and su-perconductivity coexist, without true phase separation ,since the first-order phase transition has disappeared.What is also interesting is that at zero temperature thisregion begins at a nonzero concentration, and ends ata concentration smaller than one. When there is phaseseparation, this coexistence region always begins at c = 0and ends at c = 1. c H=2.5 K=0.1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) AF (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) T / J B k S N
FIG. 3: Phase diagram at a magnetic field H = 2 . K = 0 .
1. N denotes the normalphase, S superconductivity and AF antiferromagnetism.
FIG. 4: (color online) Phase diagram at a magnetic field H =2 . K = 0 .
5. There is a regionwhere superconductivity and antiferromagnetism coexist, butwhere there is no true phase separation.
3. H=5
In the original BEG model, when the magnetic field isincreased to a value higher than H = 4 at zero tempera-ture, antiferromagnetism totally disappears because thespins tend to align with the magnetic field. The systemis therefore magnetized, but not because of the nearest-neighbor interactions. Therefore, this is not really ferro-magnetism, but for the sake of simplicity, we denote itlike this. For the case of K = 1, we observe a phase withferromagnetic and superconducting ordering, and a fer-romagnetic phase (not shown). For K = 0 .
1, we find an-other interesting phase, namely a ferromagnetic checker-board phase, consisting of two sublattices, see Fig. 5. Atthe first sublattice, all sites are randomly occupied byphases with a value of s i = 1 or s i = −
1. At the secondone, all sites are occupied by the spin that is favored bythe magnetic field, σ i = −
1. This phase is most likely tooccur at a concentration of c = 0 . FIG. 5: (color online) Phase diagram at H = 5 and K = 0 . III. ADDING A TRAP POTENTIALA. The Blume-Emery-Griffiths model
Because experiments with cold atoms are often car-ried out in a trap, we will add a harmonic potential tothe original BEG Hamiltonian, to describe mixtures offermions and bosons in a trap. In general, the potentialfelt by the bosons is different from the one felt by thefermions, what implies that we must include two terms, a b X i ( x i + y i ) σ i + a f X i ( x i + y i )(1 − σ i ) . (7)Here, x i and y i are the horizontal and vertical distancesof site i , measured from the center of the lattice, in latticeunits, and a b and a f measure how much the bosons (thestates with σ i = ± σ i = 0) feel the influence of the trap. If a b = a f , thisterm is constant, and the phase diagram is not modified.We will consider the case a b > a f , which is the mostrelevant experimentally. Using the hard core constraint σ i + s i = 1, we can then rewrite this term and add it tothe BEG Hamiltonian, thus obtaining H = − J X
1, thetransition temperatures approach the transition temper-ature of the Ising model for almost all concentrations, asexpected. When the states with σ i = ± FIG. 6: (color online) Phase diagrams of the BEG model witha trapping potential. N denotes the normal, unordered state, C the condensed phase, in which the sites with σ i = ± σ i = 1, red σ i = −
1, and white σ i = 0. will not speak of a superfluid state, but of a condensedstate, because we now consider bosons in general.It is important to estimate at which temperature thesystem starts to feel the influence of the trapping po-tential. Let us assume that a cluster of size m feels thepotential when the energy difference between the statewith this cluster in the center and in the corner of thelattice is of the order k B T /J . For a lattice of size L ,this estimation results in maL J ∼ k B TJ . (9)In this approximation, a single particle ( m = 1) in a lattice of size L = 41 will start to feel the potential if k B T /J ∼ a . For a/J = 0 . a/J = 0 .
01 thisresults in k B T /J ∼
80 and k B T /J ∼
8, respectively, inboth cases much higher than the temperatures we areinterested in, because ordering starts around k B T /J ≈ .
4. Therefore, the single particles will experience theinfluence of the trap in the entire temperature range ofFig. 6 ( b ) and ( c ). For a/J = 0 . k B T /J ∼ .
8. However, for higher temperatures thesystem already orders, and therefore there are some largeclusters that according to Eq. (9) will feel the potentialalready at much higher temperatures. This reasoning isin agreement with the snapshots in Fig. 6 ( a ). For a/J =0 . σ i = ± a/J = 0 . a/J = 0 .
01, we indeed see the influence of the trap forall temperatures, even in the disordered state.
B. The two-component Blume-Emery-Griffithsmodel
Analogous to the previous subsection, we will also adda trapping potential to the two-component BEG model.In the latter, both the states with σ i = ± s i = ± a σ X i ( x i + y i ) σ i + a s X i ( x i + y i ) s i (10)to the Hamiltonian. Because at every lattice site σ i + s i = 1, we can rewrite this term and add it to the two-component BEG Hamiltonian, to get H = − J X
1. For K = 0 .
1, the right part of the first-order phase transition disappears and the left one be-comes second-order (see Fig. 7), whereas for K = 1 bothleft and right parts of the first-order phase transition areconverted into second-order (see Fig. 8). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) K=0.1 c (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) c N N (a) (cid:0)(cid:0)(cid:1)(cid:1) T / J B (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) k T / J B k (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (b) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) =0.01 a/J =0.001 a/J + S C σ C σ C σ C+ S C σ C (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) FIG. 7: (color online) Phase diagrams of the two-componentBEG model with a trapping potential. N denotes the normal,unordered state, C σ and C s the phases where the bosons rep-resented by the state with σ i = ±
1, respectively s i = ± σ i = ±
1, and red andblue s i = ± In the limit of a → ∞ , all the sites with σ i = ± s i = ± K = 1, we see indeed that the transitiontemperatures for both species approach the Ising transi-tion temperature. For K = 0 .
1, because J is ten timessmaller than J , one of the species will order at the Isingtransition temperature, and the other one at one tenthof the Ising transition temperature.To find the temperature at which the system startsto feel the presence of the trap, we can make the sameanalysis as in subsection III A. Also here, we see in thesnapshots of Figs. 7 and 8 that for a/J = 0 . a/J = 0 .
01, the system always feels the influence ofthe trap, and for a/J = 0 . a ), we see thatthere is a phase C s in which the bosons represented by s i = ± σ i = ± σ i = s i = 0,to verify the stability of this phase, when we relax theconstraint that every lattice site must be occupied by oneof the bosons. Note that for small enough concentrations,this phase will always occur, since the bosons s are hardlydiluted by the bosons σ . IV. COMPARISON WITH EXPERIMENTSA. Magnetic superconductors
FIG. 9: Phase diagram of CeCo(In − x Cd x ) . The figure isextracted from Ref. 8. There are several examples of Cerium-based supercon-ductors, for example CeCoIn and CeIrIn , as well as an-tiferromagnets that contain this element, like CeRhIn ,CeCoCd , CeRhCd , and CeIrCd . Let us considerCeCoIn and CeCoCd . These two materials have twoelements in common, Ce and Co, and differ in the thirdelement. By doping CeCoIn with Cd on the In site,we can change the superconductor CeCoIn into an an-tiferromagnet. There are more of these Cerium-basedpairs, and therefore, this class of materials is appropri-ate for studying the interplay between superconductivityand magnetism.Let us consider the heavy fermion superconductorCeCoIn , with Cadmium doping on the In-site. This ma-terial has the highest superconducting transition temper-ature ( T c = 2 . K ) of all heavy fermions, and its electronicstructure is quasi-2D. Nicklas et al. and Pham et al. determined the antiferromagnetic and superconductingonset temperatures of this material as a function of dop-ing by elastic neutron scattering, specific heat, and resis-tivity measurements. Their results are plotted in Fig. 9.For experimental details we refer the reader to Ref. 8.The phase diagram of CeCo(In − x Cd x ) shows three or-dered phases: a superconducting phase, a commensurateantiferromagnetic phase, and a region where supercon-ductivity and antiferromagnetism microscopically coex-ist. δ δ NFL
AF SAF+Scontrol parameter T e m p e r a t u r e FIG. 10: Schematic phase diagram of unconventional super-conductors in temperature-control parameter space. AF de-notes antiferromagnetism, S superconductivity and NFL anon-Fermi liquid. Experimentally, antiferromagnetism oftendisappears abruptly at some critical value δ of the controlparameter, although one would expect a magnetic quantumcritical point at some value δ of the control parameter. It is interesting to observe that in this material antifer-romagnetism suddenly disappears at the point where theonset temperatures for superconductivity and antiferro-magnetism are equal. This feature, however, may changein the presence of an applied magnetic field. In Fig. 10we see a schematic phase diagram of unconventional su-perconductors, in temperature-control parameter space.In the case of CeCo(In − x Cd x ) , the control parameterwould be doping. Another example of such a parameter is pressure. Park et al. determined the phase diagramof CeRhIn in temperature-pressure space with and with-out a magnetic field. Without a magnetic field, they alsofound this abrupt disappearance of the incommensurateantiferromagnetic order at δ . However, when they ap-plied a field of 33 KOe, the line of the magnetic orderingtemperature went smoothly down to zero at δ . Such aphase diagram shows many similarities with Fig. 4 if weidentify pressure with inverse concentration in our model.Indeed, for an external magnetic field of H = 2 . K = 0 . − x Cd x ) ,see Fig. 11 and Ref. 9. For this material, it is not clear ifthere is a region where superconductivity and magnetismcoexist. If there is such a region, it is in a small dopinginterval. The phase diagram of this material strongly re-sembles the phase diagram of the two-component BEGmodel with an external magnetic field of H = 2 . K = 0 .
1, see Fig. 3. Al-though this experiment was also carried out without anexternal magnetic field, we only find similarities with ourmodel in the presence of a magnetic field.
FIG. 11: Phase diagram of the heavy fermionCeIr(In − x Cd x ) . The figure is extracted from Ref. 9. B. Cold atom systems
In 2006, two experimental groups, the group ofKetterle at MIT, and the group of Hulet at RiceUniversity, have performed experiments with imbal-anced ultracold Li atoms in a trap, and obtained con-tradictory results. The MIT group measured a transitionbetween a normal and a superfluid phase at a polariza-tion of P ≈ .
70, whereas the group at Rice Universityobserved a transition between two superfluid phases at P ≈ .
09. Here, P measures the imbalance between thespin-up and the spin-down atoms, P = N ↑ − N ↓ N ↑ + N ↓ . (12)Gubbels et al. have set up a theoretical model to de-scribe these imbalanced Fermi mixtures and determined ageneral phase diagram in temperature-polarization spacethat can explain the observations of both groups. Thetopology of their phase diagram shows large similaritieswith the phase diagram of the BEG model. We can un-derstand this resemblance as follows. In the BEG model,the concentration c is the fraction of lattice sites with σ i = 0, and thus the fraction of the system that cannotcondense. The polarization P is a measure for the differ-ence of the atoms in the spin-up and the spin-down state,and thus for the number of fermions that remain afterthe others have paired. The atoms with spin up and spindown will form pairs, and such a pair can be describedas a boson. Therefore, the polarization is also a measurefor the fraction of the system that cannot condense, andthe concentration can be mapped onto the polarization.We can identify the paired atoms, the preformed bosons,with the states σ i = ±
1, and the remaining fermions with σ i = 0, see Fig. 12. (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) N (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) SF + phase separationSF (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) T / J B k c FIG. 12: Phase diagram of the original BEG model, obtainedby Monte Carlo simulations. N denotes the normal phase, SF superfluidity. Lines are guides to the eyes. The transitionbetween the normal and the superfluid state is second-order,the transition to the phase separated regime is first-order. The experiments with Li are carried out in a trap,and the theoretical model of Gubbels et al. only includesthe presence of the trap by using the local density ap-proximation. Now, we would like to compare their phasediagram with our results of the BEG model with a trap-ping potential, in Figs. 7 and 8. Although in the case ofimbalanced fermions the frequency ω of the optical trapfelt by the pairs of fermions (bosons) and the remainingunpaired fermions is the same, the mass of the bosonsis twice as large, and the potential constant a b is thuslarger than a f . This means that the comparison mustbe made with the BEG model in a trap. In this model,the first-order phase transition, measured by a jump inthe concentration as a function of the anisotropy field D has disappeared, thus there is no true transition toa phase-separated regime. However, if we inspect thesnapshots, we see that for low enough temperatures, orlarge enough trapping potential, there still is a clear sep-aration between the condensed bosons and the fermions,suggesting some kind of effective phase separation. We note that in experiments, phase separation is measuredby inspecting the radii of the clouds of the atoms in thedifferent hyperfine states, and not by a jump in some or-der parameter. Our results thus suggest that the mea-sured different radii are not per se an evidence of a truethermodynamic phase separation. Further experimentsare required to clarify this issue.Although our model describes qualitatively the exper-imentally observed phases, it cannot capture the fine de-tails of recent experimental results. Studies by Shin etal. indicate that there is no superfluid phase, or phase-separated phase for polarizations above P ≈ .
36. By aquantum Monte Carlo approach, Lobo et al. predict aphase transition between a normal and a superfluid stateat a polarization of P ≈ .
39 at zero temperature, andGubbels and Stoof recovered this results using a Wilso-nian renormalization group theory. V. CONCLUSIONS
We simulated a two-component extension of the BEGmodel without an external magnetic field and determinedthe phase diagram in the concentration-temperaturespace. In the region where magnetism and superconduc-tivity coexist, the system is always phase separated. Weadded a magnetic field to our model, and considered theantiferromagnetic case. In this case, we also find phasediagrams with true coexistence of two ordered phases.These diagrams are comparable with the phase diagramof doped heavy fermions in the presence of a magneticfield.In order to describe cold atom systems, we added atrapping potential to the BEG model, and our extensionof this model. The added potential changes the phaseseparation regime conceptually. We cannot speak any-more about true phase separation, but more about acrossover to a phase separated region. We argue thatthe BEG model with a trapping potential can be used tomodel imbalanced Fermi mixtures. However, there arestill quantitative differences with experiments, which ourmodel is not able to cover. We also made predictionsfor the phase diagram of boson-boson mixtures based onour simulations of the two-component BEG model witha trapping potential. Although there is no available ex-perimental data on boson-boson mixtures, we hope thatour work can motivate further studies in this direction.
VI. ACKNOWLEDGEMENTS
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