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ladder systems have been the focus of much theoretical and experimental work over more than two decades . one theoretical motivation was to investigate the dimensional crossover between the well - known one - dimensional ( 1d ) physics and the two - dimensional ( 2d ) case , in the search of 2d non - fermi liquid physics . a second reason stems from experiments and the study of ladder compounds , such as the famous telephone number one sr@xmath1ca@xmath2cu@xmath3o@xmath4 which has a superconducting phase at high pression and for a small hole density . @xcite    the simplest ladder model is that of a two - leg ladder , made of two coupled fermionic chains . in stark contrast to the single chain case , the two - leg ladder system displays a superconducting phase with @xmath5-wave superconductivity for repulsive interactions which stems from the doping of a spin - gapped mott insulating phase at half - filling . @xcitethis gave the belief that the two - leg ladder problem already contains seeds of the rich physics of the cuprates . two - leg ladders have thus become over the years a fundamental system for the study of low - dimensional strongly correlated fermions . various exotic quantum phases have been predicted theoretically depending on the form of the coupling between the two chains and the filling.@xcite experimental realizations of two - leg ladder systems are clearly called for , in particular , to investigate the rich physics of the weak - coupling regime . in this respect , ultracold fermionic gases are a promising way to study two - leg ladder problems thanks to the high level of control on interchain hopping and interactions . @xcite the ladder geometry might be created by considering double - well optical lattices for instance.@xcite    a second possible way is to load a two - component fermi gas in a optical lattice and consider higher - lattice orbitals , typically the @xmath0-band , to simulate a fermionic two - leg ladder system . more precisely , we consider , in this paper , a two - component fermi gas which is loaded in a 1d optical lattice ( running along the @xmath6-direction ) with moderate strength of ( harmonic ) confining potential @xmath7 in the direction ( i.e. , @xmath8 ) perpendicular to the chain . @xcite it is assumed that all the @xmath9-level of the oscillator are fully occupied while the @xmath0-level , i.e. , @xmath10 , are partially filled . the resulting lattice fermionic model has been derived in refs . within the tight - binding approximation and takes the following form : @xmath11 ,     \label{pbandmodel}\end{aligned}\ ] ] where @xmath12 is the orbital index and @xmath13 is the  spin  index or internal components of the underlying cold atoms . in eq . ( [ pbandmodel ] ) , @xmath14 describes the density operator at site @xmath15 and a pseudo - spin operator for the orbital degrees of freedom has been defined : @xmath16 where @xmath17 are the pauli matrices and a summation over repeated indices is implied . model ( [ pbandmodel ] ) can be viewed as a two - chain fermionic system without interchain hopping but with density - density interchain interactions and pair - hopping processes between the two orbitals ( see fig . [ fig : p - band-2leg ] ) .    -band model ( [ pbandmodel ] ) . the orbital degrees of freedom plays the role of the legs of the ladder . besides the kinetic term with amplitude @xmath18 for each orbital , there are local interactions at each site @xmath15 : density - density intra - orbital ( amplitude @xmath19 , in red ) and inter - orbital ( amplitude @xmath20 , in green ) , orbital exchange ( amplitude @xmath20 , in blue ) and inter - orbital pair - hopping that breaks u(1)@xmath21 generically ( amplitude @xmath20 , in orange ) . [ fig : p - band-2leg ] ]    the continuous symmetry of model ( [ pbandmodel ] ) for general @xmath22 is : @xmath23 , where u(1)@xmath24 denotes the u(1 ) ( charge ) symmetry related to the conservation of the total number of atoms , while @xmath25 is the internal global su(2 ) ( spin ) symmetry of the two - component fermi gas . we note that the u(1)@xmath26 continuous symmetry for the orbital degrees of freedom is explicitly broken in eq . ( [ pbandmodel ] ) . however , the use of harmonic potential in the @xmath8 direction implies a constraint of the two coupling constants @xmath22 and the investigation of model ( [ pbandmodel ] ) along the harmonic line @xmath27 enjoys an u(1)@xmath26 symmetry corresponding to rotation along the @xmath28-axis in the orbital subspace . the u(1)@xmath21 symmetry is trivially realized when @xmath29 where the model reduces to two decoupled su(2 ) hubbard chains . finally , the @xmath30 case displays also an u(1)@xmath26 symmetry since it is directly related to @xmath29 after a redefinition of the orbital pseudo - spin operator . we consider , in this paper , the most general case where @xmath22 are not fine - tuned for two main reasons . on the one hand , departure from the harmonic line @xmath27 can be investigated by breaking the axial symmetry of the 2d harmonic trap with the introduction of quartic potentials . @xcite on the other hand , the harmonic line has been wrongly identified in refs .   and we want to make contact with their results obtained when @xmath31 .    at half - filling , it has been shown that model ( [ pbandmodel ] ) along the harmonic line with repulsive interaction describes a haldane phase @xcite of the spin-1 heisenberg chain . @xcite the latter is known to be the paradigmatic 1d topological phase protected by symmetry.@xcite for an attractive interaction , a haldane phase for the charge degrees of freedom @xcite has also been predicted in ref .   by means of a strong - coupling approach and density - matrix renormalization group ( dmrg ) calculations . @xcite in ref . , it has been shown that there is adiabatic continuity between weak and strong coupling regime and these two haldane phases are related by a spin - charge interchange symmetry . the half - filled @xmath0-band model ( [ pbandmodel ] ) along the harmonic line thus paves the way to realize experimentally non - trivial haldane phases in the context of cold fermions .    in this paper , by means of a low - energy approach and dmrg calculations , we map out the zero - temperature phase diagram of the @xmath0-band model ( [ pbandmodel ] ) for incommensurate filling and at quarter filling which best avoids three - body looses . in stark contrast to the conclusion of ref . for attractive interaction , we show that the @xmath0-band model ( [ pbandmodel ] ) away from half - filling does not behave as an attractive 1d hubbard chain with the formation of a 2@xmath32 cdw but instead is a representative of the physics of two - leg fermionic ladders . in particular , we find that the hallmark of its phase diagram is the emergence of various competing superconducting phases for attractive and repulsive interactions as well as a mott transition with the formation of fully gapped bond - ordering waves ( bow ) . the rest of the paper is organized as follows . in sec . [ sec : lowenergy ] , we present our low - energy approach based on a mapping onto majorana fermions and a one - loop renormalization group ( rg ) analysis . a zero - temperature phase diagram for model ( [ pbandmodel ] ) is then deduced in the weakly - interacting regime for incommensurate filling and at quarter filling with fermi vector @xmath33 ( @xmath34 being the lattice spacing ) . in sec . [ sec : dmrg ] , dmrg calculations are carried out to investigate the intermediate and large - coupling regimes to fully determine the phase diagram of the @xmath0-band model . we present then our concluding remarks in sec . [ sec : conclusion ] . in this section , we perform a weak - coupling approach to model ( [ pbandmodel ] ) away from half - filling . we focus on incommensurate filling and a special commensurate filling where @xmath33 . the starting point of this approach is the introduction of left - right moving dirac fermions from the continuum limit of the non - interacting hamiltonian ( [ pbandmodel ] ) with @xmath35 : @xmath36 with @xmath37 , @xmath38 , and @xmath39 . in stark contrast to the half - filled case of ref . , there is a  spin - charge  separation which strongly simplifies the analysis and the full hamiltonian of the @xmath0-band model ( [ pbandmodel ] ) decomposes into two commuting pieces : @xmath40 with @xmath41 = 0 $ ] , where @xmath42 governs the physical properties of the charge degrees of freedom and @xmath43 refers for all remaining non - abelian degrees of freedom , i.e. , spin , orbital and spin - orbital degrees of freedom . let us first consider the simplest part in eq . ( [ spinchargesep ] ) , i.e. , the charge degrees of freedom . for incommensurate filling , on general grounds , the charge part takes the form of a tomonaga - luttinger hamiltonian : @xcite @xmath44 , \label{luttbis}\ ] ] where @xmath45 and @xmath46 are the luttinger parameters . in this low - energy approach , the charge excitations are described by the bosonic field @xmath47 and its dual field @xmath48 . the explicit form of the luttinger parameters for the @xmath0-band model can be extracted from the continuum limit ( [ contlimitdirac ] ) after standard calculations . we find : @xmath49 where @xmath50 is the fermi velocity and @xmath51 . for incommensurate filling , no umklapp term appears and the charge degrees of freedom display metallic properties in the luttinger liquid universality class . @xcite however , at quarter filling , umklapp processes might be generated leading to an additional term in eq . ( [ luttbis ] ) . such perturbation can be found by means of higher orders of the rg calculations or from lattice symmetries . @xcite in this respect , the one - step translation symmetry expresses as follows in terms of the charge bosonic field : @xmath52 . the charge perturbation with the smallest scaling dimension which is compatible with the translation symmetry is then : @xmath53 . the full charge hamiltonian density for the commensurate filling @xmath33 becomes then equivalent to the quantum sine - gordon model at @xmath54 : @xmath44    -g_u \cos(\sqrt{16 \pi } \phi_c ) . \label{chargeham}\ ] ] this model is exactly solvable and the development of the strong - coupling regime with a charge gap @xmath55 corresponds to @xmath56 , while for @xmath57 the charge degrees of freedom remain gapless in the luttinger universality class . using the estimate ( [ luttingerparameters ] ) , we find the position of the critical line for a mott transition within the bosonization approach : @xmath58 such that @xmath59 . one has to be very careful with this estimate since the full expression of @xmath46 as a function of @xmath22 ( [ luttingerparameters ] ) is strictly valid only in the weak - coupling regime . one needs a complementary approach , i.e. , numerical calculations to conclude on the existence or not of a mott transition for the lattice model ( [ pbandmodel ] ) . note that the mechanism of the mott transition is very similar to the one which occurs in the repulsive u(4 ) hubbard chain @xcite or in the problem of spin-3/2 cold fermions . @xcite    for @xmath56 , the charge degrees of freedom are fully gapped and the strong - coupling regime of the sine - gordon model ( [ chargeham ] ) is described by the pinning of the charged field on the minima : @xmath60 @xmath0 being an integer . the sign of the umklapp coupling constant @xmath61 is difficult to fix but we expect @xmath62 from an higher - order perturbative expansion in the weak - coupling regime so that:@xmath63 . @xcite the dmrg calculations of sec .  [ sec : dmrg ] will shed light on the position of the pinning of the charge field from the determination of the mott - insulating phases of the @xmath0-band model . the low - lying excitations are massive kinks and antikinks which interpolate between the ground states of the quantum sine - gordon model ( [ chargeham ] ) . the charges associated to these charge excitations are @xmath64 in units of the electron charge . we have thus massive holon as low - lying excitations in the charge sector .      the low - energy physics of the remaining degrees of freedom , included in @xmath65 of eq . ( [ spinchargesep ] ) , can be inferred from a mapping onto six majorana fermions as in refs . a simple way to obtain this correspondence is through the introduction of four chiral bosonic fields @xmath66 from the bosonization of dirac fermions:@xcite @xmath67 where the bosonic fields satisfy the following commutation relation : @xmath68 = \frac{i}{4}\delta_{l l'}\delta_{\sigma \sigma'}. \label{commutator}\ ] ] the presence of the klein factors @xmath69 ensures the correct anticommutation of the fermionic operators . the klein factors satisfy the anticommutation rule @xmath70 and they are constrained so that @xmath71 , with @xmath72 . hereafter , we will work within the @xmath73 sector . the next step of the approach is to introduce a bosonic basis which singles out the different degrees of freedom , i.e. , charge , spin , orbital , and spin - orbital degrees of freedom:@xcite @xmath74    from these new bosonic fields , one can now consider a refermionization procedure by introducing six left and right moving majorana fermions through : @xmath75 where @xmath76 are again klein factors which ensure the adequate anticommutation rules for the fermions . using this correspondence , the interacting part of @xmath65 can be expressed in terms of these majorana fermions : @xmath77 with the identification : @xmath78    using standard calculations , the one - loop rg equations for model ( [ majomodel ] ) can de derived : @xmath79    as often in 1d , the rg equations enjoy some hidden discrete symmetries : @xmath80 which correspond to the existence of duality symmetries on the majorana fermions @xcite @xmath81 while the right - moving majorana fermions remain invariant . these chiral transformations for each majorana fermion correspond , in the continuum limit , to the well - known kramers - wannier duality symmetry of the underlying 1d ising model in a transverse field . the latter transformation maps the ordered phase to the disorder one which is ordered in terms of the disorder operator . @xcite starting from the ordered phase of the six underlying ising models in a transverse field , the transformations ( [ dualitiespband ] ) lead then to four possible different gapful phases . a numerical rg analysis of eqs . ( [ rgn=2 ] ) is necessary to determine which phases is indeed reached starting from the initial conditions ( [ couplingsn=2 ] ) .    ) . bottom : for each phase denoted i - iv , we give the dominant superconducting correlations ( see text ) as well as the duality symmetries which relate the phases between them . we display the central charge @xmath82 or @xmath83 which characterizes the different quantum phase transitions of the model.,title="fig : " ] 0.3 cm ) . bottom : for each phase denoted i - iv , we give the dominant superconducting correlations ( see text ) as well as the duality symmetries which relate the phases between them . we display the central charge @xmath82 or @xmath83 which characterizes the different quantum phase transitions of the model.,title="fig : " ]    we did this numerical analysis and , as depicted in fig . ( [ fig : phasediag_n=2 ] ) , it reveals that the rg flow goes in the strong - coupling regime in the far infrared ( ir ) along four asymptotic lines : @xmath84 along these rays , the interacting part of the effective hamiltonian ( [ majomodel ] ) simplifies as follows : @xmath85 with @xmath86 . using the duality symmetries ( [ dualitiespband ] ) , all these models reduces to the same model which takes the form of the so(6 ) gross - neveu ( gn ) model with interaction : @xcite @xmath87 this is an example of a dynamical symmetry enlargement ( dse ) where an so(6 ) symmetry , i.e. a higher symmetry than the initial symmetry of the @xmath0-band lattice model , emerges at low - energy . this phenomenon occurs in a large variety of models with marginal interactions in the scaling limit @xcite as , for instance , in the half - filled two - leg hubbard model @xcite and in the su(4 ) hubbard chain at half - filling @xcite , where an so(8 ) symmetry occurs at low energy . the emergence of an so(6 ) symmetry has also been obtained within the one - loop rg analysis of various models of doped two - leg ladders . @xcite      the main interest of this so(6 ) dse scenario stems from the fact that the isotropic rg ray is described by a massive integrable field theory ( [ gnso6 ] ) when @xmath88 . @xcite the development of the strong - coupling regime in the so(6 ) model leads to generation of a non - perturbative fermionic mass , i.e. the emergence of a spin - gap phase where all spin , orbital and spin - orbital excitations are fully gapped . the nature of the underlying electronic phases of the @xmath0-band model can then be inferred by a straightforward semiclassical approach of the so(6 ) model and the application of the duality symmetries ( [ dualitiespband ] ) . alternatively , one can perform a direct bosonic semiclassical approach of the different models in eq . ( [ so6raysapp ] ) using the identification ( [ refer ] ) . in the following , we proceed to this analysis by identifying the phases of fig . [ fig : phasediag_n=2 ] .      in the first phase , when @xmath90 and @xmath91 of fig . ( [ fig : phasediag_n=2 ] ) , the numerical analysis of the one - loop rg equations reveals that the rg flow is attracted in the far ir along a special ray ( i ) of eq . ( [ asymptotrg ] ) . this phase is described by the @xmath89 duality and the resulting physical properties of that phase are governed by the interacting hamiltonian @xmath92 of eq . ( [ so6raysapp ] ) . a straightforward semiclassical approach of the latter model leads to the following pinning of the bosonic fields : @xmath93 the electronic properties of phase i , which includes the harmonic line @xmath94 , depend also on the charge degrees of freedom that are decoupled . let us first consider the @xmath95 case . one has then a gapless charge excitation whereas all remaining spin , orbital , and spin - orbital excitations are fully gapped . all @xmath96 densities are short - ranged due to the pinning of the dual spin - orbital field @xmath97 . interestingly enough , the leading electronic instability in this phase turns out to be a bond - centered superconducting instability with order parameter : @xmath98 which is odd under the @xmath99 orbital exchange symmetry ( @xmath100 ) . a similar superconducting instability has been introduced in the study of doped two - leg electronic ladder in ref .  . in the continuum limit , we have : @xmath101 one can then bosonize this operator by means of the identification ( [ bosofer ] ) and the physical basis ( [ basebosons ] ) : @xmath102 . \nonumber \label{bcssboson}\end{aligned}\ ] ] taking account of the vacuum expectation values ( [ pinningi ] ) , we get @xmath103 so that the equal - time correlation function of the pairing operator has a power - law decay : @xmath104 when @xmath105 , the bond - centered bcs superconducting instability is strongly enhanced with respect to the non - interacting case . _ we have thus a dominant superconducting instability for repulsive interaction_.    while the @xmath96 densities are short - ranged in phase i , @xmath106 densities might compete with the superconducting instability ( [ bcss ] ) . many @xmath106 density terms can be written in the continuum description like @xmath107 for instance . the latter gives in the bosonized language : @xmath108 so that in the spin - gapped phase ( [ pinningi ] ) , one has @xmath109 , and therefore a power - law decay for its correlation function : @xmath110 since @xmath111 , one observes that the @xmath106 density correlation function ( [ 4kfcorrel ] ) decays much faster than the superconducting one ( [ bcsscorrel ] ) . in summary , the leading instability of phase i with @xmath95 is the bond - centered bcs superconducting one .    in the mott - insulating phase with @xmath112 , all excitations are fully gapped . the bond - centered bcs superconducting instability ( [ bcss ] ) is now short - ranged since the charge field @xmath47 is pinned ( [ pinnedcharge ] ) and thus its dual field @xmath48 is a strongly fluctuating field . in contrast , the @xmath106 density of eq . ( [ rho4kf1 ] ) can have now a non - zero expectation value , breaking spontaneously the translation symmetry . we thus expect a mott phase with a two - fold degenerate ground state at quarter filling @xmath113 . the nature of this phase depends on the sign of the coupling constant @xmath61 of the umklapp perturbation in model ( [ chargeham ] ) . in this respect , we introduce the following order parameters as in the study of the extended quarter - filled two - leg hubbard ladder : @xcite @xmath114 with @xmath115 . one can obtain a continuum and bosonized representation for these order parameters . the @xmath106 contribution of the latter plays a crucial role and can be derived using the results of refs . and we get for the leading contribution : @xmath116 with @xmath117 , @xmath118 . from these results , we find the bosonized descriptions of the order parameters ( [ orderparamottn=2 ] ) : @xmath119 taking into account that in phase i , we have @xmath120 , we have either a uniform @xmath106 bow or uniform @xmath106 cdw depending on the sign of the umklapp term in eq . ( [ chargeham ] ) : @xmath121 we expect in the weak - coupling regime that @xmath122 . a uniform @xmath106 bow is thus stabilized which is two - fold degenerate and breaks the translation symmetry . the latter phase is similar to the bow phase obtained in the quarter - filled spin-3/2 so(5 ) chain model . @xcite      in the second phase , when @xmath124 and @xmath125 , the one - loop rg flow is now attracted in the far ir by the asymptote ( ii ) of eq . ( [ asymptotrg ] ) . this phase is described by the @xmath123 duality . a straightforward semiclassical approach of model @xmath126 in eq . ( [ so6raysapp ] ) leads to the following pinning of the bosonic fields in this phase : @xmath127 in this phase , as we will see in sec . [ sec : dmrg ] , numerical results find that @xmath95 . all spin , orbital , and spin - orbital excitations are fully gapped in this phase and the charge degrees of freedom are gapless since @xmath95 . due to the presence of attractive interactions , one may call this gapless phase a luther - emery phase as in the spin-1/2 hubbard chain with @xmath128 . @xcite its physical nature is very different from that of phase i since the expectation values of the bosonic fields ( [ pinningii ] ) are different . as it can readily be seen from eq . ( [ pinningii ] ) , we have again the condensation of a dual field , here the orbital one @xmath129 , which implies that the @xmath96-cdw operator is a strongly fluctuating order . as in phase i , the only possible cdw quasi - long range order in this phase is a @xmath106 cdw . however , the @xmath130 cdw of eq . ( [ rho4kf1 ] ) becomes now short - range since the orbital dual field @xmath131 condenses in phase ii . in this respect , we have to consider another @xmath106 cdw instability : @xmath132 . the latter can be directly expressed in terms of the bosonic field : @xmath133 using the expectation values ( [ pinningii ] ) , we deduce the leading asymptotics of the equal - time correlation function of the @xmath106 cdw operator : @xmath134 as it will be seen in the next section , we have @xmath135 and , one does not expect , on general ground , that this @xmath106-cdw will be the dominant instability in this phase . as in phase i , a superconducting instability turns out to be the leading one . the latter is defined by the following order parameter : @xmath136 which is odd under the @xmath99 orbital symmetry ( @xmath100 ) and antisymmetric with respect to the spin degrees of freedom ( @xmath137 ) . the order parameter ( [ bcs- ] ) can be expressed directly in terms of the bosonic fields : @xmath138 . \nonumber \label{bcs - boson}\end{aligned}\ ] ] using the expectation values ( [ pinningii ] ) , we get @xmath139 , so that the equal - time correlation function of the pairing operator ( [ bcs- ] ) has a power - law decay : @xmath140 which dominates the @xmath106 cdw ones ( [ 4kfcorrelbis ] ) when @xmath95 .      the next phase , defined by @xmath142 and @xmath125 , is described by the asymptote ( iii ) of the one - loop rg flow . the harmonic line of the @xmath0-band model with attractive interaction belongs to this phase which is described by the @xmath141 duality with interacting hamiltonian @xmath143 of eq . ( [ so6raysapp ] ) . the bosonic fields of the bosonization approach are now pinned to the values : @xmath144 a spin - gap is formed and a gapless @xmath145 phase emerges in this attractive regime with @xmath146 since the umklapp term can not gap out the charge degrees of freedom when @xmath147 . in close parallel to the previous cases , one can determine the nature of the leading electronic instability of this luther - emery phase by means of the bosonization approach combined with the pinning ( [ pinningiii ] ) . the dominant @xmath106 cdw is the one of phase ii , given by eq . ( [ rho4kf2 ] ) with the power - law behavior ( [ 4kfcorrelbis ] ) . the relevant superconducting instability for phase iii is defined by : @xmath148 which is even under the @xmath99 orbital symmetry ( @xmath100 ) and antisymmetric with respect to the spin degrees of freedom ( @xmath137 ) . in terms of the bosonic fields , it reads as follows : @xmath149 . \nonumber \label{bcstboson}\end{aligned}\ ] ] using the vacuum expectation values ( [ pinningiii ] ) , we immediately get @xmath150 , so that the equal - time correlation function of this pairing operator has a power - law decay : @xmath151 since @xmath147 , we conclude that the superconducting instability ( [ bcs+ ] ) dominates the @xmath106-cdw ordering ( [ 4kfcorrelbis ] ) . _ the luther - emery phase is thus governed by a superconducting instability ( [ bcs+ ] ) in stark contrast to the @xmath96-cdw phase predicted by the dmrg study of ref .  _ . the physics of the @xmath0-band model with attractive interaction is thus not similar to that of 1d attractive hubbard model as emphasized in ref .  . the last phase of the @xmath0-band model corresponds to the region where @xmath142 and @xmath91 . the numerical analysis of the rg flow shows that , here , the one - loop rg flow is attracted in the far ir by the special line ( iv ) of eq . ( [ asymptotrg ] ) . the resulting phase is described by the @xmath152 duality with interacting hamiltonian @xmath153 of eq . ( [ so6raysapp ] ) which leads to the following pinning for the bosonic fields of the bosonization approach : @xmath154 as before , this pinning leads to the formation of a gapless @xmath145 phase when @xmath95 where the charge degrees are the only critical modes of the problem . we now consider the standard @xmath5-wave superconducting instability of the two - leg electronic ladder to determine the nature of phase iv : @xmath155 which is even under the @xmath99 orbital symmetry ( @xmath100 ) . the bosonized expression of this superconducting instability reads as follows : @xmath156 . \nonumber \label{bcsdboson}\end{aligned}\ ] ] from this expression , we observe that this operator is a fluctuating order , i.e. , has short - ranged correlation , in the previous phases , while , in phase iv , one has from the pinning ( [ pinningiv ] ) : @xmath157 . the @xmath5-wave superconducting instability ( [ bcsd ] ) becomes dominant in phase iv with the power - law behavior : @xmath158 from the pinning ( [ pinningiv ] ) , we find that the @xmath96-cdw operator is short - ranged while the @xmath106 cdw of phase ii , given by eq . ( [ rho4kf2 ] ) , has a power - law behavior in phase iv : @xmath159 with subleading exponent when @xmath95 with respect to the superconducting instability ( [ bcsdcorrel ] ) .      from the duality symmetries ( [ dualitiespband ] ) , we can , as well , discuss the different quantum phase transitions that occur in the @xmath0-band model by investigating self - dual manifolds . however , fig . [ fig : phasediag_n=2 ] reveals that the transitions belong to special lines of the lattice model : @xmath160 and @xmath29 . from eq . ( [ pbandmodel ] ) , the @xmath29 describes two decoupled quarter - filled spin-1/2 hubbard chains with @xmath19 coupling constant . when @xmath161 , a spin - gap is formed and therefore the phase ii / phase iii transition is critical with @xmath162 gapless charge modes . when @xmath90 , all degrees of freedom are gapless since the quarter - filled spin-1/2 hubbard chains is known not to exhibit a mott transition . @xcite the phase i / phase iv transition is thus critical with central charge @xmath83 . the transition between phase i and phase ii is located along the @xmath163 line of the @xmath0-band model . from the definition of the @xmath0-band model ( [ pbandmodel ] ) , one can show that the latter line corresponds to two decoupled repulsive quarter - filled spin-1/2 hubbard chains which does not have relevant umklapp process . a @xmath83 behavior should occur along this quantum phase transition . finally , the last transition between phase iii and phase iv belongs to the @xmath164 line which takes the form of two decoupled attractive quarter - filled spin-1/2 hubbard chains with a @xmath82 quantum critical behavior . all these results can be derived by investigating self - dual manifolds of the rg eqs . ( [ rgn=2 ] ) . since the transition lines are located on special high symmetry lines , one would expect the location of these transitions to remain universal beyond the weak coupling regime where this analysis is valid . this will be indeed confirmed numerically in sec . [ subsec : qpt_num ] . a summary of the phases and quantum phase transitions , obtained from the low - energy approach , can be found in the bottom of fig . [ fig : phasediag_n=2 ] . we will now determine the phase diagram of the @xmath165 @xmath0-band model at quarter filling ( i.e. 1 particle per site ) using numerical simulations with the dmrg algorithm . this will allow to go beyond the one - loop rg analysis done in fig . [ fig : phasediag_n=2 ] and check the analytical predictions for realistic intermediate or strong couplings @xmath166 ( we fix @xmath167 as the unit of energy ) . moreover , numerical data are needed to determine the numerical value of the luttinger parameter @xmath46 which allows to compute the dominant correlations . typically , we have used open boundary conditions ( obc ) and lengths @xmath168 and @xmath169 , keeping up to 4000 states when computing correlations in order to keep a discarded weight below @xmath170 in most regions , although simulations where @xmath171 were found to be more difficult to converge ( discarded weight around @xmath172 , see discussion below ) . for practical purpose , we have mapped the @xmath0-band model onto an _ equivalent _ ( pseudo)spin-1/2 ( where the pseudo spin corresponds to the orbital ) fermionic models on a @xmath173-leg ladder , and we have implemented the abelian u(1 ) symmetry corresponding to the conservation of particles spin .    our main result is presented in fig .  [ fig : phasediag_pband_n2_n1 ] where we plot the phase diagram vs @xmath174 obtained by computing various correlation functions on @xmath169 systems . we will present the numerical data below , but we can already discuss the different phases . first of all , the ( quantum phase ) transition lines are found to be @xmath29 and @xmath175 , which is expected by symmetry since they correspond to special lines of the model , see sec . [ subsec : qpt ] . a detailed analysis will be given in sec . [ subsec : qpt_num ] . second , it is remarkable that phases ii , iii and iv found in the weak - coupling analysis are confirmed to exist in a wide range of parameters . last , we have computed the luttinger parameter @xmath46 using the dominant correlation , which is always of some bcs type , in each of these phases . overall , we observe a rather good agreement between the numerical phase diagrams obtained by solving rg equations ( fig . [ fig : phasediag_n=2 ] ) or the full microscopic model using dmrg ( fig . [ fig : phasediag_pband_n2_n1 ] ) , except for the large mott phase at this commensurate filling . for incommensurate filling , we predict that the rg phase diagram would be identical to the numerical one .    . numerical values correspond to the luttinger parameter @xmath46 in each region . ] we will now present some numerical data that were used to compute this numerical phase diagram . we have relied mostly on computing superconducting correlation functions ( in various channels ) as well as density ones . to avoid spurious effects due to obc , we have chosen to compute correlations as @xmath176 so that we can determine if they decay algebraically or are short - ranged . in figs . [ fig : corr_u1_m6_u2_6]-[fig : corr_u1_m6_u2_m2]-[fig : corr_u1_6_u2_m6 ] , such correlations are plotted and correspond respectively to phases ii , iii and iv . in all cases , we emphasize that dominant correlations are found to be the superconducting ones , in different channels ( see below ) . indeed , the luttinger parameter @xmath46 is always larger than 1/2 in the critical phases , otherwise a mott phase is stabilized . some values of @xmath46 are given on the phase diagram in fig . [ fig : phasediag_pband_n2_n1 ] . in particular , for phases ii , iii and iv , we observe an adiabatic continuity from weak to strong coupling , while phase i on the contrary is much reduced and replaced instead by mott phase at intermediate coupling and polarization at strong coupling .      in this region of the phase diagram , our numerical results , shown in fig . [ fig : corr_u1_m6_u2_6 ] for instance when @xmath177 and @xmath178 , are in perfect agreement with the rg predictions and dominant bcs@xmath179 predictions . however , numerics is needed to compute the luttinger parameter @xmath46 and determine whether pairing correlations dominate over density ones . using eqs . ( [ 4kfcorrelbis])-([bcs - correl ] ) , our numerical fits for both bcs@xmath179 and cdw correlations are compatible with a single @xmath180 , for instance 0.65 for the parameters set chosen on the plot . all the other correlations are short - ranged as expected . the leading instability in this phase is therefore the bcs@xmath179 superconducting one . , @xmath178 ) and @xmath169 obtained with dmrg . this corresponds to our phase ii . fitting the dominant bcs@xmath179 correlation with eq . ( [ bcs - correl ] ) leads to @xmath181 , which is also consistent with the @xmath106 cdw correlation function ( see eq . ( [ 4kfcorrel ] ) ) . ] the phase iii region is particularly interesting since it could be achieved experimentally using the harmonic trapping scheme ( i.e. @xmath182 ) with attractive interactions . as seen in fig . [ fig : corr_u1_m6_u2_m2 ] , we have found that the dominant correlations are of the pairing type , in the bcs@xmath183 channel . in all this region , they can be fitted using eq . ( [ bcs+correl ] ) to get @xmath184 ( values are given in the phase diagram ) . in such a case , the dominant density correlations are _ uniform _ and decay as @xmath185 as expected .    , @xmath186 ) for @xmath169 obtained with dmrg . this corresponds to our phase iii . fitting the dominant bcs@xmath183 correlation with eq . ( [ bcs+correl ] ) leads to @xmath187 . ] note that this dominant superconducting correlation function was not computed in ref . , where only cdw signal was discussed . as a result , the dominant instability in this luther - emery phase is a superconducting one , in stark contrast to the @xmath96-cdw phase predicted earlier  @xcite .      in the region corresponding to phase iv , our numerical results in fig . [ fig : corr_u1_6_u2_m6 ] indicate that dominant correlations are of the pairing type again , but in a different bcs@xmath188 channel , as expected from the low - energy analysis . in the whole region , we can fit these correlations using eq . ( [ bcsdcorrel ] ) to extract @xmath46 , or equivalently we could fit the subleading cdw correlations with eq . ( [ 4kfcorrelfin ] ) , although it would be less precise . some values of @xmath46 are given in fig . [ fig : phasediag_pband_n2_n1 ] and are always larger than 1/2 so that no mott transition is found .    , @xmath189 ) and @xmath169 obtained with dmrg . this corresponds to our phase iv . fitting the dominant bcs@xmath188 correlation with eq . ( [ bcsdcorrel ] ) leads to @xmath181 , which is also consistent with the @xmath106 cdw correlation function ( see eq . ( [ 4kfcorrel ] ) ) . ]      the region @xmath190 , which should correspond to phase i ( see fig . [ fig : phasediag_n=2 ] ) according to rg solution is more involved to analyze . we have plotted on fig . [ fig : bow ] the finite - size scaling of the @xmath191 bow , i.e. the kinetic energy difference measured in the middle of the chain . data are given along the harmonic line @xmath182 . extrapolations are compatible with a finite , albeit small , value for intermediate interactions ( for instance @xmath192 , @xmath193 ) , while it seems to vanish in the weak and strong coupling regimes . it is known that the weak coupling is difficult numerically since the non - interacting starting point correspond to two decoupled fermionic chains with a large total central charge @xmath83 . for instance , the mott transition in the su(@xmath194 ) hubbard model at filling @xmath195 has been discussed quite extensively to occur for a _ critical @xmath196 when @xmath197 based on bosonization and quantum monte - carlo results  @xcite as well as dmrg ones  @xcite , while some older dmrg simulations had indicated a _ vanishing _ @xmath198  @xcite . since the charge gap is expected to open in an exponentially way , this is clearly a difficulty for any numerical technique . however , this regime is perfectly suited for bosonization and weak - coupling rg : indeed , since the luttinger parameter @xmath199 in the non - interacting case , umklapp processes are irrelevant so that finite interactions are necessary to enter the mott phase . thus , our interpretation is the following : ( i ) for small interaction parameters , rg analysis should be valid and phase i with dominant bcs@xmath200 correlations is expected with @xmath180 so that umklapp processes are irrelevant ; ( ii ) for intermediate interactions , a mott phase occurs with @xmath191 bow and exponentially decaying bcs correlations at large distance ; ( iii ) for very large interactions ( i.e. @xmath201 and @xmath202 ) , we have noticed that the ground - state is ferromagnetically polarized ( hence degenerate ) : the polarized ground - state can be simply understood as two decoupled spinless fermionic chains ( one for each orbital ) , which energy is independent of both @xmath19 and @xmath20 . intuitively , such a state could be stabilized at large @xmath203 s since other competing states will have higher energies . still , our numerics is clear on this point as can be checked by comparing ground - state energies in different sectors ( varying the number of particles per spin ) , or measuring @xmath199 from charge correlations ( data not shown ) . bow order parameter along the harmonic line @xmath182 . ]      a simple way to obtain information on the phases , or on their phase transitions , is through the measurements of block von neumann entanglement entropy which is known to scale as  @xcite : @xmath204 where @xmath205 is the central charge and @xmath206 the conformal distance . on a finite system with obc , due to friedel oscillations , the fitting can be more involved and one can use for instance the knowledge of the local kinetic bond energies to get more reliable results ( see fig . 11 of ref .   for instance ) . our numerical results should be compared to the analytical predictions made in sec . [ subsec : qpt ] .    in fig . [ fig : entropies_ll](a ) , we have plotted typical data in phases ii , iii and iv . by removing oscillations using the bond kinetic energy as an additional fitting parameter , we can obtain smooth functions that perfectly agree with the expected behavior ( [ eq : svn ] ) with @xmath145 . obtained for @xmath169 and various @xmath166 . oscillations have been removed using the bond kinetic energies @xmath207 and fitting parameter @xmath208 ( equal respectively to @xmath173 , @xmath209 and @xmath210 ( see text ) . in all cases , we can extract a central charge @xmath211 as expected . ( b ) similar data along the quantum phase transitions which can be fitted with @xmath212 ( using @xmath213 ) in the attractive case and with @xmath214 ( using @xmath215 and @xmath216 , plus a shift for clarity ) in the repulsive case using smaller @xmath168 system.,title="fig : " ]   obtained for @xmath169 and various @xmath166 . oscillations have been removed using the bond kinetic energies @xmath207 and fitting parameter @xmath208 ( equal respectively to @xmath173 , @xmath209 and @xmath210 ( see text ) . in all cases , we can extract a central charge @xmath211 as expected . ( b ) similar data along the quantum phase transitions which can be fitted with @xmath212 ( using @xmath213 ) in the attractive case and with @xmath214 ( using @xmath215 and @xmath216 , plus a shift for clarity ) in the repulsive case using smaller @xmath168 system.,title="fig : " ]    now , considering the expected quantum phase transition along the lines @xmath175 and @xmath29 , we can clearly identify different behaviors for attractive vs repulsive interactions ( as expected since there will be a finite spin gap or not respectively ) . in fig . [ fig : entropies_ll](b ) for attractive interactions , we can fit our entanglement entropies data perfectly with @xmath82 using @xmath169 . in the repulsive case , our numerical data are not converged on the same size keeping up to @xmath217 states , so that we plot instead data obtained with @xmath168 and @xmath217 . in this case , we can not remove entirely the oscillations in the data , but we can get a very good fit using the expected @xmath83 central charge . overall , we get an excellent agreement with the theoretical predictions . we have presented a comprehensive study of the most general model relevant for one - dimensional @xmath0-band two - component fermionic cold gases with local interactions only . we have concentrated here on incommensurate filling and quarter - filling .     using a state - of - the - art low - energy approach , supplemented with a one - loop rg numerical analysis , we have found that generically , the charge sector decouples from the spin - orbital one which is gapped . as a consequence , most of the phase diagram is occupied by standard luttinger liquid phases with a single gapless charge mode ( hence a central charge @xmath145 ) . nevertheless , we have clarified the nature of the dominant instability and have found that it is always of some bcs superconducting kind , in one of the following channels : bcs@xmath200 , bcs@xmath179 , bcs@xmath183 and bcs@xmath188 , see eqs . ( [ bcss])-([bcs-])-([bcs+])-([bcsd ] ) respectively . in particular , an interesting bond - centered superconducting instability emerges along the harmonic line for the repulsive interaction . the nature of the phase transitions between these four superconducting phases is also elucidated and found to behave with central charges @xmath82 or @xmath83 . our numerical simulations do confirm that phases with dominant bcs@xmath179 , bcs@xmath183 and bcs@xmath188 extend from weak to strong coupling and occupy large regions in the phase diagram . in particular , for attractive interactions and harmonic trapping , bcs@xmath183 correlations are the dominant ones , different from the @xmath96-cdw phase predicted earlier  @xcite . in the last region where the bond - centered superconducting instability bcs@xmath200 is expected at weak coupling , our dmrg data at quarter - filling ( i.e. , one particle per site ) indicate that a mott phase intervene with fully gapped bond - ordering waves at intermediate coupling , and spontaneous polarization at strong coupling . we have also numerically confirmed the nature of all the quantum phase transitions present in this model . the @xmath0-band two - component fermion mode , studied in this paper , is thus an interesting model to explore superconducting instabilities of doped two - leg fermionic ladder systems as well as the occurence of a mott transition . given the recent progress in realizing su(@xmath194)-symmetric fermi gases  @xcite , we hope that some of the phases discussed here could be realized experimentally and probed using local spectroscopy techniques or by measuring short - range correlations . the authors would like to thank k. totsuka for a related collaboration and useful discussions . numerical simulations have been performed using hpc resources from genci  tgcc , genci  idris ( grant x2015050225 ) and calmip ( grant 2015-p0677 ) . the authors would like to thank cnrs for financial support ( pics grant ) .
the zero - temperature phase diagram of @xmath0-orbital two - component fermionic system loaded into a one - dimensional optical lattice is mapped out by means of analytical and numerical techniques . 
 it is shown that the @xmath0-band model away from half - filling hosts various competing superconducting phases for attractive and repulsive interactions . at quarter filling , we analyze the possible formation of incompressible mott phases and in particular for repulsive interactions , we find the occurrence of a mott transition with the formation of fully gapped bond - ordering waves .