Canted Antiferromagnetic Order of Imbalanced Fermi-Fermi mixtures in Optical Lattices by Dynamical Mean-Field Theory
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Canted Antiferromagnetic Order of Imbalanced Fermi-Fermi mixtures in OpticalLattices by Dynamical Mean-Field Theory
Michiel Snoek , Irakli Titvinidze , and Walter Hofstetter Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The Netherlands Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at, 60438 Frankfurt/Main, Germany
We investigate antiferromagnetic order of repulsively interacting fermionic atoms in an opticallattice by means of Dynamical Mean-Field Theory (DMFT). Special attention is paid to the case ofan imbalanced mixture. We take into account the presence of an underlying harmonic trap, both ina local density approximation and by performing full Real-Space DMFT calculations. We considerthe case that the particle density in the trap center is at half filling, leading to an antiferromagneticregion in the center, surrounded by a Fermi liquid region at the edge. In the case of an imbalancedmixture, the antiferromagnetism is directed perpendicular to the ferromagnetic polarization andcanted. We pay special attention to the boundary structure between the antiferromagnetic and theFermi liquid phase. For the moderately strong interactions considered here, no Stoner instabilitytoward a ferromagnetic phase is found. Phase separation is only observed for strong imbalance andsufficiently large repulsion.
PACS numbers: 71.10.Fd, 75.50.Ee, 37.10.Jk, 67.85.Lm
I. INTRODUCTION
Ultracold atoms in optical lattices provide a versatilelaboratory for interacting quantum many body systems .One of the major challenges in this field is the experimen-tal investigation of quantum magnetism in atomic mix-tures. Impressive experimental progress in this directionhas already been made. The first important step in ex-periments with fermionic atoms in optical lattices was theexperimental observation of the Fermi surface . Recentexperiments with bosonic atoms directly observed corre-lated particle tunneling and superexchange , which arethe basic mechanisms underlying quantum antiferromag-netism. Moreover, strong experimental evidence for thefermionic Mott insulator state was obtained, both by thelocal probe of observing reduced double occupancy andthe global probe of observing a plateau in the cloud sizewhen the system is compressed . A recent experiment ina system of spin-1/2 fermions without optical lattice indi-cates a Stoner instability toward a ferromagnetic state forstrong repulsion . These are important steps on the waytoward realization of strongly correlated many-fermionstates . Currently the experimental temperatures arestill higher than the critical (N´eel) temperature, belowwhich antiferromagnetic order is predicted to develop .Most accurate theoretical estimates for the entropy perparticle below which long-range antiferromagnetic orderis expected yield a value of S/N ≈ ln(2) / , whereascurrent experiments reach an average entropy which isstill a factor 2 higher .Ultracold atomic system offer the unique possibilityto control the relative densities of the two spin compo-nents, as alreday has been demonstrated in experimentswithout the presence of an optical lattice . Experi-mentally, the density imbalance is precisely tunable bymeans of radiofrequency sweeps and stable due tothe suppression of spin-flip scattering processes in coldatomic gases. This realizes an imbalanced spin mixture, in which the SU (2)-symmetry is broken by an artificialmagnetic field. When the density of atoms correspondsto one particle per lattice site, the ground state of thissystem is expected to be a canted antiferromagnet, withantiferromagnetic order characterized by a N´eel vectordirected perpendicular to the applied field. However,experimentally ultracold atom systems are always con-fined by an external harmonic trapping potential, whichleads to an inhomogeneous system. If the total parti-cle number is sufficiently high, in the center of the trapa region with particle density per site close to one willdevelop, where antiferromagnetic order is stable at suf-ficiently low temperatures . The edges of the systemhave lower filling; they are Fermi liquid regions withoutspin order. If the total particle number is even higher,also in the trap center a Fermi liquid with particle den-sity higher than one or a band insulating state can exist.In that case antiferromagnetic order can be stable in ashell around this Fermi liquid . This poses interestingquestions regarding the nature and the stability of spinorder, which we will address in this paper by means of(Real-Space) Dynamical Mean-Field Theory.These issues have recently also been investigated byother methods. For homogeneous systems describedby the hole-doped Hubbard model, both commensu-rate and incommensurate spin-density-waves have beenpredicted . By mapping to an effective spin model,the critical temperature for canted antiferromagnetic or-der was calculated and topological excitations of imbal-anced mixtures were studied . A Hartree-Fock staticmean-field theory for balanced mixtures in a trap pre-dicts that antiferromagnetism can coexist with param-agnetic states in various spatial patterns, for exampleantiferromagnetism in the center of the trap surroundedby a hole-doped atomic liquid or antiferromagnetism in aring with a Fermi liquid in the center and at the edge .For imbalanced mixtures, this approach predicts cantedorder perpendicular to the (artificial) magnetic field upto moderate values of the repulsion . Very recently theHartree-Fock approach has also been applied to largerrepulsion: in addition to canted antiferromagnetism, acritical interaction was found, beyond which the Stonerinstability drives a ferromagnetic transition at the edgeof the system, where the particle density is lower thanhalf-filling .A Real-Space Dynamical Mean-Field (R-DMFT)study of antiferromagnetism in a harmonic trap has alsobeen performed, but so far without allowing for the pos-sibility of canted antiferromagnetic order . For thecase of an imbalanced mixture, this constraint lead tothe prediction of phase separation between the majoritycomponent in the center and the minority component atthe edge for sufficiently strong repulsive interactions andlarge values of the imbalance .Here we perform a full R-DMFT study, which includesthe possibility of canted order. Unlike static Hartree-Fock mean-field theory, DMFT is a non-perturbativemethod which is reliable both for strong and weak in-teractions in sufficiently high dimensions. Local correla-tions are included exactly . R-DMFT thereby takesthe inhomogeneity induced by the presence of a harmonictrap into account in a fully consistent way.For imbalanced systems we indeed observe canted an-tiferromagnetic order. We consider weak to moderatelystrong interactions, for which no Stoner instability to-ward spontaneous ferromagnetism is found: in the case ofa balanced mixture the wings of the systems are alwaysparamagnetic. Only upon applying a finite amount ofimbalance, the system gets polarized and ferromagneticorder starts to develop. We generally also do not ob-serve the phase-separation scenario for large imbalance.Instead, the canted antiferromagnetic order allows fora continuous transition between balanced antiferromag-netism order and fully imbalanced ferromagnetic order.Only for large values of the interaction and strong imbal-ance phase separation occurs.We compare our full R-DMFT results with calcula-tions based on a local density approximation in combi-nation with DMFT, in which the harmonic trap is in-corporated by a spatially varying chemical potential. Asfor the balanced case we find that the total densityis well approximated by the local density approximation,but a strong proximity effect is observed for the anti-ferromagnetic order: the staggered antiferromagnetismas obtained by the full R-DMFT calculation extends toregions where the local density approach predicts a para-magnetic solution. II. MODEL
Repulsively interacting fermions in a sufficiently deepoptical lattice are well described by the single-band Hub- bard Hamiltonian in the tight-binding approximation H = − J X h ij i ,σ ˆ c † iσ ˆ c jσ + U X i ˆ n i ↑ ˆ n i ↓ + X iσ ( V i − µ σ )ˆ n iσ , (1)where ˆ n iσ = ˆ c † iσ ˆ c iσ , and ˆ c iσ (ˆ c † iσ ) are fermionic annihi-lation (creation) operators for an atom with spin σ atsite i , J is the hopping amplitude between nearest neigh-bor sites h ij i , U > µ σ isthe (spin-dependent) chemical potential and V i = V r i is the harmonic confinement potential. We also define¯ µ ≡ ( µ ↑ + µ ↓ ) and ∆ µ ≡ µ ↑ − µ ↓ as the average chemi-cal potential and difference in chemical potential, respec-tively. Although ∆ µ acts as a magnetic field, experimen-tally the imbalance is not induced by a physical magneticfield, but by directly controlling the difference in particlenumber. The parameters of this model can be tuned inexperiments by changing the intensity of the optical lat-tice and via Feshbach resonances . In the following, wetake the lattice constant to be a = 1. III. METHOD
To obtain the ground state properties of this system,we apply R-DMFT . Within R-DMFT the self-energy is taken to be local (which is exact in the infinite-dimensional limit ) but allowed to depend on the lat-tice site, i.e. Σ ijσ ( iω n ) = Σ ( i ) σ ( iω n ) δ ij , where δ ij is aKronecker delta. The lattice sites are described by localeffective actions, each representing an effective Ander-son impurity model, which are coupled via the real-spaceDyson equation for the Green’s function. Details of themethod have been published previously .In the present paper we use Exact Diagonalization(ED) of the Anderson Hamiltonian to solve thelocal impurity actions. Within ED the spectral func-tion is represented by a finite number of delta peaks.Whereas this is sufficient for a faithful representation ofthe zero-temperature spectral function, we found it tolead to unphysical behavior at finite temperature, espe-cially away from half-filling. Therefore we restrict our-selves in this article to the low-temperature limit andonly investigate ground state properties. The multigridHirsch-Fye quantum Monte Carlo method has provento be a very efficient solver at finite temperatures fora balanced mixture , but canted antiferromagneticorder is probably harder to obtain within this method.Also when using the Numerical Renormalization Group(NRG) method to solve the Anderson Hamilto-nian, it is problematic to describe the canted off-diagonalspin order. In contrast, the ED-method we use here isvery flexible, which also allows to incorporate off-diagonalcanted spin order in a straightforward manner. However,since S z in this case is not a good quantum number forthe individual spin components, the size of the Hilbertspaces to be diagonalized is significantly enlarged, whichleads to far more time-consuming numerics compared tothe balanced case. FIG. 1: (Color online) Real-space spin profile for
U/J = 10,∆ µ/J = 0 . V /J = 0 .
05. The vertical componentsof the arrows symbolize the local magnetization in the z -direction h ˆ S zi i , while the horizontal components correspondto the (staggered) local magnetization in the x -direction h ˆ S xi i .For all data in this paper we took the filling at the centerequal to one, which is induced by the choice of the chemicalpotential equal to ¯ µ = U/
2. To obtain the ground state ex-pectation values a small fictitious temperature of
T /J = 0 . For very large repulsion, ED can run into unphysicalinstabilities. Therefore we consider only moderately largeratios of
U/J here.In the practical implementation of ED a small but fi-nite temperature is used to generate the T = 0 data,in order to obtain discrete Matsubara frequencies. Wechose this temperature equal to T /J = 0 .
02 for the two-dimensional data and
T /J = 0 .
025 for three dimensions.
IV. RESULTS
We now apply the R-DMFT method to spin- fermions in a two-dimensional square lattice and a three-dimensional cubic lattice with harmonic confinement.We focus on the density distributions of the two species n iσ = h ˆ n iσ i ( σ = ↑ , ↓ ), the total density n i tot = n i ↑ + n i ↓ ,and the local spin expectation values S αi = h ˆ c † iβ σ αβγ ˆ c iγ i ,in which σ α ( α = x, y, x ) are the Pauli matrices. Herewe have set ~ = 1.In the case of a balanced mixture the Hamiltonian is SU (2)-symmetric. This means that the staggered mag-netization can point in any direction. We have chosenit to point in x -direction. In the case of an imbalanced -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r ∆µ /J = 0 U/J=5a) n tot S z S x -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r ∆µ /J = 0.5 U/J=5b) n tot S z S x -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r U/J=10c) n tot S z S x -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r U/J=10d) n tot S z S x -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r U/J=20e) n tot S z S x -0.5-0.25 0 0.25 0.5 0.75 1 0 5 10 15 20r U/J=20f) n tot S z S x FIG. 2: (Color online) Radial profiles for the ground state of two-dimensional
Fermi-Fermi mixtures for different values of U and ∆ µ . Plotted are the total density h ˆ n i i = h ˆ n i ↑ + ˆ n i ↓ i ,and the expectation values of the spin in z -direction h ˆ S zi i andin x -direction h ˆ S xi i . The left column is for a balanced mixture(∆ µ/J = 0), the right column is an imbalanced mixture with∆ µ/J = 0 .
5. The other parameters are ¯ µ = U/ U/J = 5, V /J = 0 . U/J = 10, V /J =0 . U/J = 20, V /J = 0 .
1. Here and in the following, spinexpectation values are plotted in units of ~ . mixture, the SU (2)-symmetry is spontaneously brokenby the chemical potential difference, which acts as an ar-tificial magnetic field in the z -direction. In reaction, thestaggered magnetization orders perpendicular to this, i.e.in the xy -plane. The remaining U (1) symmetry is also inthis case in our calculations broken by a small initial nu-merical perturbation, resulting in alignment of the spinsalong the x -direction, such that h ˆ S y i = 0 in all resultspresented here.In all the calculations reported here we have chosen¯ µ = U/
2, such that the system is at half-filling at thecenter.
A. Results for two dimensions
In Fig. 1 a typical real-space spin configuration isshown for an imbalanced two-dimensional system. Herewe have chosen to label the spatial coordinates by x and a) U/J = 10 ∆µ /J = 1.2 n ↓ |S x | 0 0.1 0.2 0.3 0 5 10 15 20r b) U/J = 20 ∆µ /J = 0.6 n ↓ |S x | 0 0.05 0.1 0 5 10 15 20r c) ∆µ /J = 1.6 n ↓ |S x | 0 0.05 0.1 0 5 10 15 20r d) ∆µ /J = 0.9 n ↓ |S x | FIG. 3: (Color online) Radial distribution for the minoritydensity h n i ↓ i and the in-plane staggered order h ˆ S xi i (here plot-ted in absolute value) for large imbalance and different valuesof the repulsion: U/J = 10 (left column) and
U/J = 20 (rightcolumn) and ¯ µ = U/
2. Other parameters: a) V /J = 0 . µ/J = 1 .
2; b) V /J = 0 .
07, ∆ µ/J = 0 .
6; c) V /J = 0 . µ/J = 1 .
6; d) V /J = 0 .
07, ∆ µ/J = 0 . z , such that the spin direction and spatial direction canbe identified. Results of our R-DMFT calculation on thetwo-dimensional square lattice for the radial density andspin profiles are shown in Fig. 2, both for the balancedsystem and for the situation that imbalance is inducedby a nonzero chemical potential difference ∆ µ .
1. Antiferromagnetic region
We first turn our attention to the antiferromagneticregion in the center, where the particle density is at half-filling. Our results show that imbalance reduces antifer-romagnetic order in the center, and tilts it out of the xy -plane by inducing a nonzero ferromagnetic z -componentof the spin. The effect of imbalance becomes larger withincreasing interactions. This can be understood from thefact that for large U/J the local spins in the insulatingregion interact via a Heisenberg Hamiltonian H = J ex X h ij i ˆ S i · ˆ S j − ∆ µ X i ˆ S zi . (2)Here J ex = 2 J /U , such that with increasing U the ex-change coupling between the spins decreases and becomesweaker relative to the applied chemical potential differ-ence.We do not observe any sign of a Stoner instability inthe paramagnetic wings for the values of U/J consideredhere, which would mean a spontaneous ferromagnetic po-larization for equal chemical potentials of the spin com-ponents. In contrast, we observe that only upon the application of a finite chemical potential difference fer-romagnetic order is induced in the wings. Although wecannot establish the critical interaction above which theStoner instability occurs, we thus find that this value isconsiderably shifted upwards compared with the valueobtained within the Hartree Fock analysis, where spon-taneous ferromagnetism was observed for even smallervalues of
U/J than considered here . It is indeed well-known that the Hartree-Fock approximation underesti-mates the critical interaction for spontaneous ferromag-netism by more than an order of magnitude . WithinDMFT the dynamical screening of the local repulsion isfully accounted for and only for extremely large on-siterepulsion a ferromagnetic ground state was found on thehomogeneous cubic lattice . However, this limit is ex-perimentally hard to reach, because the associated criti-cal temperature for spin order is very low.
2. Boundary structure
We now turn our attention to the boundary structurebetween the antiferromagnetic core and the paramagneticwings. The results in Fig. 2 show a local maximum of theferromagnetic polarization in the z -direction. This max-imum appears at the location where the antiferromag-netic order in the center disappears. The minority den-sity shows a local minimum at this point (cf. Fig. 3a,b),whereas the majority density has a local maximum. Thisfeature was also observed in the Hartree Fock analysis .The maximum appears because the antiferromagnetic or-der reduces the density difference of the two componentscompared to the paramagnetic situation: a smaller den-sity difference leads to a larger sublattice magnetizationand hence a lower energy. This mechanism suppresses theoccurrence of phase separation in the trap center, whichwas found previously in the case where canted order wasexcluded . In contrast, the possibility of canted orderallows a continuous transition between the limiting casesof an antiferromagnetic phase with equal populations ofthe two species and a fully polarized ferromagnetic phasein which only one of the two species is still present.An interesting structure emerges in the paramagneticouter region for large imbalance: the density of minorityatoms shows a second maximum (cf. Fig. 3) originat-ing from the strong repulsion which pushes them to theoutside. This ring-like structure is the remnant of phaseseparation found before . However, in this case it isno true phase separation, since also minority atoms arepresent in the central antiferromagnetic region. Only forlarge values of U/J in the limit of very strong imbal-ance, this outer ring of minority atoms survives whenthe antiferromagnetic order in the center disappears, im-plying true phase separation. In particular for
U/J = 10we do not observe this scenario, in contrast to the casewhere canted antiferromagnetic order was not included .When canted order is accounted for, the outer ring ofminority atoms disappears before the antiferromagnetic a) n tot S z |S x | 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10r b) n tot S z |S x | 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10r c) n tot S z |S x | 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10r d) n tot S z |S x | 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10r e) n tot S z |S x | 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10r f) n tot S z |S x | FIG. 4: (Color online) Radial profiles for the ground state of three-dimensional
Fermi-Fermi mixtures for
U/J = 10, ¯ µ = U/ µ . Plotted are the total density h ˆ n i i = h ˆ n i ↑ + ˆ n i ↓ i , and the expectation values of the spinin z -direction h ˆ S zi i and the absolute value of the expectationvalue in x -direction |h ˆ S xi i| . The points denote results of thefull R-DMFT calculation, whereas the solid lines are obtainedwithin the LDA (TFA) approximation combined with DMFT.The values for ∆ µ are: a) ∆ µ/J = 0; b) ∆ µ/J = 0 .
4; c)∆ µ/J = 0 .
8; d) ∆ µ/J = 1 .
2; e) ∆ µ/J = 1 .
6; f) ∆ µ/J = 2.The trap parameter is chosen as V /J = 0 . order in the center vanishes, as visible in the data in Fig.3a) and c). For U/J = 20 still phase separation occurs,as shown Fig. 3b) and d): for large imbalance the anti-ferromagnetic order in the center breaks down, but thering of minority atoms surrounding the phase separatedcentral region with only majority atoms is still present.Note that this phase separation cannot be identified withthe Stoner instability, because it only happens for largechemical potential difference. Moverover, it means thatthe complete central region, including the part where thedensity is at half-filling, is fully polarized, whereas the mi-nority atoms are located in a shell around this at low ma-jority density. In contrast, the Stoner instability favors ascenario, where the region at half-filling still supports an-tiferromagnetic order, whereas the edges are completelypolarized . B. Results for three dimensions
Results for the ground state (
T /J →
0) in three di-mensions are presented in Fig. 4. We compare the full R-DMFT results with a local density approximation (LDA)(previously also denoted as a Thomas-Fermi approxima-tion (TFA) ), where the trap is modeled within DMFTby a locally varying chemical potential combined with thehomogeneous density of states of the cubic lattice. In or-der to facilitate the comparison, the absolute value of thestaggered spin expectation value in x -direction is plottedin Fig. 4. This makes it impossible to distinguish ferro-magnetic and antiferromagnetic order in the figure; how-ever, as in Fig. 2 the in-plane xy -spin order described by S x is always staggered. For the total density we observegood agreement between the LDA+DMFT and the fullR-DMFT results; the only difference being that the LDAresults show a small discontinuity, which is smoothenedin the R-DMFT calculation. In contrast, the agreementbetween the LDA and R-DMFT results for the spin or-der parameters is far less good, as also observed for thecase of balanced mixtures . In particular, antiferro-magnetic spin order extends much further into the regionwith total density lower than half-filling (i.e. n i < z -direction is also visible in the data in Fig. 4,although it is more pronounced in the LDA curve thanin the R-DMFT data, and in general less pronouncedthan in two dimensions. This is because the ratio of theinteraction to the band-width U/ zJ ( z being the num-ber of neighbors) is chosen smaller here. This is alsothe reason that phase separation is not observed for thecase of strong imbalance, even though large values of ∆ µ are considered. The minority atoms still form a shell forlarge imbalance, but this shell disappears before the an-tiferromagnetic order in the center is destroyed. In orderto observe phase separation, a stronger repulsion wouldbe needed. Also, a Stoner instability is not observed inthe three dimensional case for the value of the repulsionchosen in this case.Our data in three dimensions are summarized in Fig.5. Here we plot global observables as a function ofthe chemical potential difference: the total magnetiza-tion in x and z direction S x,z tot = P i h ˆ S x,zi i and thestaggered magnetization in x and z direction S x,z stag = P i ( − i x + i y h ˆ S x,zi i , normalized to the total particle num-ber N tot = P i h ˆ n i ↑ + ˆ n i ↓ i . Fig 5 shows that the totalmagnetization always points in the z -direction; which isalso the direction in which it is induced experimentallyvia the spin imbalance. This means that the system isnot frustrated and, for the parameters considered here,the interesting scenarios like the one pursued in Ref. are not realized. ∆ µ S ztot /N tot S zstag /N tot S xtot /N tot S xstag /N tot FIG. 5: (Color online) Global observables for the three-dimensional system obtained by R-DMFT at
U/J = 10,¯ µ = U/ V /J = 0 .
15 as a function of the chemicalpotential difference ∆ µ . Shown are the total magnetizationin x and z -direction S x,z tot and the staggered magnetization in x and z -direction S x,z stag , all normalized to the particle number N tot V. CONCLUSIONS
We have solved the fermionic Hubbard model with re-pulsive interactions in a harmonic trap by means of Real-Space Dynamical Mean-Field Theory, with a focus on theground state properties of imbalanced spin populations.We considered systems with a density of one particle perlattice site in the center, for which a shell structure ap-pears: an insulating regime in the center, in which anti-ferromagnetic order can arise, is surrounded by a Fermiliquid regime without magnetic order. Imbalance leadsto canted antiferromagnetic order in the center, with an interesting boundary structure to the paramagnetic edge,which we investigated in detail.Due to the possibility of canting, the antiferromag-netism turns out to be very stable against imbalance:only for sufficiently large values of the repulsion and atlarge imbalance phase separation occurs. In this regimethe central region becomes completely polarized: onlythe majority component is present and forms a band in-sulator in the center. The minority atoms organize them-selves in a shell around this central plateau.We do not observe a Stoner instability toward sponta-neous ferromagnetic order for the moderately large valuesof the repulsion considered here. The critical interactionto observe this phenomenon is thus necessarily relativelylarge, leading to a small critical temperature for ferro-magnetic order.Canted antiferromagnetic order is more challengingto detect experimentally than antiferromagnetic orderin the z -direction, because many proposed detectionschemes are only sensitive to staggered order which isdiagonal in the basis of the physical particles constitut-ing the system. This limitation does, however, not applyto Bragg scattering , which is therefore the experimen-tal method of choice to detect canted antiferromagneticorder. Acknowledgments
We thank I. Bloch, E. Demler, W. Ketterle, A. Koetsierand H.T.C. Stoof for useful discussions. This work wassupported by the Nederlandse Organisatie voor Weten-schappelijk Onderzoek (NWO) and the German ScienceFoundation DFG via Forschergruppe FOR 801, Sonder-forschungsbereich SFB-TRR 49 and the DIP project BL574/10-1. I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008). M. K¨ohl, H. Moritz, T. St¨oferle, K. G¨unter, and T.Esslinger, Phys. Rev. Lett. , 080403 (2005). S. F¨olling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A.Widera, T. M¨uller, and I. Bloch, Nature , 1029 (2007). S. Trotzky, P. Cheinet P, S. F¨olling, M. Feld, U. Schnor-rberger, A.M. Rey, A. Polkovnikov, E.A. Demler, M.D.Lukin, and I. Bloch, Science , 295 (2008). R. J¨ordens, N. Strohmaier, K. G¨unter, H. Moritz, and T.Esslinger, Nature , 204 (2008). U. Schneider, L. Hackerm¨uller, S. Will, Th. Best, I. Bloch,T. A. Costi, R. W. Helmes, D. Rasch, and A. Rosch, Sci-ence , 1520 (2008). G.-B. Jo, Y.-R. Lee, J.-H. Choi, C. A. Christensen, T. H.Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle,Science , 1521 (2009). W. Hofstetter, J.I. Cirac, P. Zoller, E. Demler and M.D.Lukin, Phys. Rev. Lett. , 220407 (2002). F. Werner, O. Parcollet, A. Georges, and S.R. Hassan,Phys. Rev. Lett. , 056401 (2005). A. Koetsier, R.A. Duine, I. Bloch, and H.T.C. Stoof, Phys.Rev. A , 023623 (2008). L. De Leo, C. Kollath, A. Georges, M. Ferrero, and O.Parcollet, Phys. Rev. Lett. , 210403 (2008). S. Wessel, Phys. Rev. B 81, 052405 (2010). R. J¨ordens et al. , Phys. Rev. Lett. , 180401 (2010). M.W. Zwierlein, A. Schirotzek, C.H. Schunck, and W, Ket-terle, Science , 492 (2006). G. B. Partridge, W. Li, R. I. Kamar, Y. Liao, and R. G.Hulet, Science , 503 (2006). M. Snoek, I. Titvinidze, C. T˝oke, K. Byczuk, and W. Hof-stetter, New J. Phys. , 093008 (2008). E.V. Gorelik, I. Titvinidze, W. Hofstetter, M. Snoek andN. Bl¨umer, Phys. Rev. Lett. , 065301 (2010). B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. , 1564(1989). H.J. Schulz, Phys. Rev. Lett. , 1445 (1990). J.K. Freericks and M. Jarrell, Phys. Rev. Lett. , 186(1995). A. Koetsier, F. van Liere, and H. T. C. Stoof, Phys. Rev.A , 023628 (2010). B.M. Andersen and G.M. Bruun, Phys. Rev. A , 041602(2007). T. Gottwald and P.G.J. van Dongen, Phys. Rev. A ,033603 (2009). B. Wunsch, L. Fritz, N.T. Zinner, E. Manousakis, and E.Demler, Phys. Rev. A , 013616 (2010). D. Vollhardt
Correlated Electron Systems , vol 9, ed. V. J.Emery, (Singapore: World Scientific) p 57 (1993). Th. Pruschke, M. Jarrell, and J.K. Freericks,
Adv. in Phys. , 187 (1995). A. Georges, G. Kotliar, W. Krauth, and M.J. Rozenberg,Rev. Mod. Phys. , 13 (1996). G. Kotliar and D. Vollhardt,
Phys. Today (3), 53 (2004). V. Dobrosavljevic and G. Kotliar, Phil. Trans. R. Soc.Lond. A , 57 (1998). M. Potthoff and W. Nolting, Phys. Rev. B , 2549 (1999). M.-T. Tran, Phys. Rev. B , 205110 (2006). M.-T. Tran, Phys. Rev. B , 245122 (2007). Y. Song, R. Wortis and W. A. Atkinson, Phys. Rev. B ,054202 (2008). R.W. Helmes, T.A. Costi, and A. Rosch, Phys. Rev. Lett. , 056403 (2008). A. Koga, T. Higashiyama, K. Inaba, S. Suga, and N. Kawakami, Phys. Rev. A , 013607 (2009). D.-H. Kim, J. J. Kinnunen, J.-P. Martikainen, and P.T¨orm¨a, e-print arXiv:1009.5676. N. Bl¨umer and E. V. Gorelik, Comput. Phys. Commun. , 115 (2011). W. Metzner and D. Vollhardt, Phys. Rev. Lett. , 324(1989). E. M¨uller-Hartmann,
Z. Phys. B , 211 (1989). M. Caffarel and W. Krauth, Phys. Rev. Lett. , 1545(1994). Q. Si, M. J. Rozenberg, G. Kotliar, and A. E. Ruckenstein,Phys. Rev. Lett. , 2761 (1994). E. V. Gorelik and N. Bl¨umer, Phys. Rev. A , 051602(R)(2009). G. Wilson, Rev. Mod. Phys. , 773 (1975). R. Bulla, T. Costi, and Th. Pruschke, Rev. Mod. Phys. ,395 (2008). D. Vollhardt, N. Bl¨umer, K. Held, and M. Kollar, in
Pro-ceedings of the Workshop on Ground-State and Finite-Temperature Band Ferromagnetism, Berlin/Wandlitz, 4-6 October 2000 , Lecture Notes in Physics, Vol. 580,(Springer, Berlin, 2001), pp. 191-207. Th. Obermeier, Th. Pruschke, and J. Keller, Phys. Rev. B , R8479 (1997). T. A. Corcovilos, S. K. Baur, J. M. Hitchcock, E. J.Mueller, and R. G. Hulet, Phys. Rev. A81