Two-leg fermionic Hubbard ladder system in the presence of state-dependent hopping
TTwo-leg fermionic Hubbard ladder system in the presence of state-dependent hopping
Shun Uchino and Thierry Giamarchi DQMP, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland (Dated: September 24, 2018)We study a two-leg fermionic Hubbard ladder model with a state-dependent hopping. We find that, contraryto the case without a state-dependent hopping, for which the system has a superfluid nature regardless of the signof the interaction at incommensurate filling, in the presence of such a hopping a spin-triplet superfluid, spin-density wave and charge-density wave phases emerge. We examine our results in the light of recent experimentson periodically-driven optical lattices in cold atoms. We give protocols allowing us to realize the spin-tripletsuperfluid elusive in the cold atoms.
PACS numbers: 67.85.-d,03.75.Ss,05.30.Fk
I. INTRODUCTION
Strongly correlated one-dimensional systems have attractedstrong attention over the past decades. In general, in suchsystems the excitations di ff er strongly from their higher di-mensional counterparts and for fermions are very di ff erentfrom the usual Landau quasiparticles occurring in a Fermi-liquid state [1]. Instead, many of the one dimensional systemsbelong to the universality class known to be the Tomonaga-Luttinger liquid [2].In particular, the system made of two coupled fermionicchains, namely, the two-leg ladder system, has been inten-sively studied in the past [2–13]. This system has been shownto exhibit superconductivity, not only for attractive interac-tions ( s -wave superconductivity), but also quite remarkablefor purely repulsive ones. In the latter case the superconduc-tivity is of d -wave symmetry. The d -wave superconductivityemerges by doping of a Mott insulating phase at half filling.While the one-dimensional system has been intensivelystudied as a first step towards other materials in higher dimen-sions, such as the high- T c superconductors, nowadays it is amajor subject in itself due to the relevance for some experi-ments, in particular in the field of cold atomic gases [14].Indeed, due to rapid advances in technology, cold atomsare a promising way to investigate the one dimensional sys-tems with an unprecedented level of control on the interchainhopping and interactions. Most of the atoms utilized in exper-iments have internal degrees of freedom, which correspond tohyperfine states when we focus on alkali species, already al-lowing to reproduction of models such as the Hubbard model[15, 16]. More recently, ladder systems have also been pro-duced, both for bosonic and fermionic states [17–21].In addition to simulating systems directly existing in con-densed matter physics, by using the unique manipulationsavailable in experiments, cold atoms also allow us to realizenew quantum states of matter.One such extension, which is the focus of this paper, is thetime modulation of optical lattices [22–26]. By applying sucha modulation with su ffi ciently high frequencies, it is possi-ble to tune the hopping matrix. This technique allows oneto control the hopping not just in strength but also in sign,since the renormalized hopping is essentially proportional toa Bessel function. In addition, by using the state-dependent optical lattice [27, 28] or applying a magnetic field one canalso control the hopping matrix element in a state-dependentmanner. In fact, such a setup has motivated several theoreticalstudies on existence or non-existence of exotic paired statesin the two-dimensional Hubbard model [29–32], and on thepresence of incommensurate density waves and segregationin the one-dimensional Hubbard model [33, 34].One may also expect the realization of an unconventionalsuperfluid in cold atoms by means of such a unique technique.To realize a superfluid in cold atoms, so far, it is necessaryto use a Feshbach resonance, since the typical temperature inthe experiments is of the order of a tenth of the Fermi tem-perature [14]. A weak-coupling BCS transition temperatureis extremely low compared to this temperature. A Feshbachresonance allows one to boost the interactions enough so that s -wave superfluidity can be routinely realized for attractiveinteractions. However, other symmetries are not so easily at-tainable. A p -wave Feshbach resonance is unstable due tothe atom-molecule and molecule-molecule inelastic collisions[35]. Therefore, the realization of an unconventional super-fluid with cold atoms is a highly challenging issue.In this paper, we show how one can realize a spin-tripletsuperfluid in a two-leg Hubbard ladder system. In the pres-ence of a state-dependent hopping, the d -wave pairing state inthe normal ladder is replaced by a spin triplet superfluid anda spin density wave (SDW) state. We also discuss the caseof an attractive interaction which would lead in the absenceof state dependent hopping to s -wave superconductivity andwhich gives an incommensurate charge density wave (CDW)in the presence of state-dependent hopping.With a ladder system we thus show that we can obtain aspin-triplet state with purely local ( s -wave) repulsive interac-tions, which is an attainable situation in experiments. In a sin-gle chain such a state would have demanded an extended Hub-bard model with on-site repulsion and nearest-neighbor attrac-tion of the same order of magnitude [36], something which isat the moment out of reach in cold atomic systems.This paper is organized as follows. Section II discussesthe Hamiltonian we propose and its low-energy descriptionby means of the bosonization technique. In Sec. III, the pos-sible phases are determined by using a renormalization groupanalysis. In Sec. IV, we discuss the properties of the strong-coupling limit in the system and experimental protocols to- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n ward its realization. Section V is the Conclusion. Technicaldetails can be found in the Appendix. II. HAMILTONIAN ( " , ) (0, " ) (0, " ) ( " , ) ! " FIG. 1. Band structure of the two-leg fermionic Hubbard ladder ofatoms with spin down (a) without the state-dependent hopping and(b) with state-dependent hopping as t KÒ “ ´ t KÓ . In each case, thereare four di ff erent points at the Fermi level. If the repulsive interactionis added, the latter leads to a spin-triplet superfluid while the formerleads to a d -wave superfluid at incommensurate filling. The bandstructure of atoms with spin up does not change in the presence ofthe state-dependent hopping. We study two-component fermions confined in the two-legladder geometry. Our starting point is the following two-legHubbard ladder model: H “ ´ t (cid:107) N ÿ j “ ÿ s “Ò , Ó ÿ p “˘ p c : j , s , p c j ` , s , p ` h . c . q´ ÿ j , s t K s p c : j , s , c j , s , ´ ` h . c . q ` U ÿ j , p n j , Ò , p n j , Ó , p , (1)where t (cid:107) and t K s are respectively the hopping matrices alongthe chain and rung directions, and j and p indicate the chainand ladder indices. Here, the on-site Hubbard U can corre-spond to both repulsive and attractive interactions, which in-deed can be realized experimentally. We focus on a system atincommensurate filling since we are interested in the stability of the superfluids in the presence of the state-dependent hop-ping, in particular, in the presence of such a hopping along therung direction. The e ff ect of the state-dependent chain hop-ping has been partially discussed in Refs. [30, 31, 33]. In thissection and section III, we discuss the weak-coupling limit toanalyze the possible phases using a field theory analysis. Inour model, this condition implies t (cid:107) " | U | , t K s .To deal with the system in the weak-coupling limit cor-rectly, we first move to the bonding and anti-bonding repre-sentation for the fermion operators: c j , s , p π q “ r c j , s , ` p´q c j , s , ´ s{ ? , (2)which allows to diagonalize the hopping terms. While in theabsence of the rung hopping, the bonding and anti-bondingbands are energetically degenerate, these are split in the pres-ence of the the rung hopping. In the absence of the state-dependent rung hopping, the splitting is independent of thestates (or spins), and therefore, there are four di ff erent pointsat the Fermi level as can be seen from Fig. 1. In the pres-ence of the state-dependent rung hopping, however, the split-ting starts to depend on the states and leads to eight di ff erentpoints at the Fermi level. At the same time, at t KÒ “ ´ t KÓ ,the four point structure at the Fermi level is recovered eventhough in this case the degeneracies occur between p π, Òq and p , Óq and between p , Òq and p π, Óq (see Fig. 1). Then, theinteraction term plays the role of hybridization between thebonding and anti-bonding bands, which is essential to lead tonontrivial states of matter in the system.We now consider the continuum limit to use the bosoniza-tion. The fermion in the continuum limit ψ can be expressedwith conjugate phase fields φ and θ as [2] ψ sqr p x q “ ? πα η sq e irk F x e ´ i r r φ sq p x q´ θ sq p x qs , (3)with the Fermi momentum k F , index q “ π for the bond-ing and anti-bonding bands, index r “ ´ ff parameter α , and the phase fields φ sq and θ sq to be conjugate. Here, we explicitly introduce the Kleinfactor η , which guarantees the correct anti-commutation rela-tion of the fermions and is also important to obtain correct ex-pressions for the bosonized Hamiltonian and correlation func-tions. By substituting (3) into (1), one may obtain the follow-ing low-energy e ff ective Hamiltonian: H “ ÿ µ “ ρ,σ ÿ ν “˘ ż dx π „ u µν K µν p ∇ θ µν q ` u µν K µν p ∇ φ µν q ` ż dx p πα q r cos 2 φ σ ` t g cos p φ σ ´ ´ δ σ ´ x q ` g cos p φ ρ ´ ´ δ ρ ´ x qu` cos 2 θ ρ ´ t g cos p φ σ ´ ´ δ σ ´ x q ` g cos 2 φ σ ` u ´ cos 2 θ σ ´ t g cos p φ ρ ´ ´ δ ρ ´ x q ` g cos 2 φ σ ` us , (4)where we introduced for φ fields, φ ρ ` “ p φ Ò ` φ Ó ` φ Ò π ` φ Ó π q , (5) φ ρ ´ “ p φ Ò ` φ Ó ´ φ Ò π ´ φ Ó π q , (6) φ σ ` “ p φ Ò ´ φ Ó ` φ Ò π ´ φ Ó π q , (7) φ σ ´ “ p φ Ò ´ φ Ó ´ φ Ò π ` φ Ó π q , (8) and similar relations for θ fields. To obtain the above, weneglect the umklapp scatterings since the system at incom-mensurate filling is concerned. For our original Hamiltonian,we find δ ρ ´ “ K ρ ´ p t KÒ ` t KÓ q{ u ρ ´ , δ σ ´ “ K σ ´ p t KÒ ´ t KÓ q{ u σ ´ , g i “ U p i “ , , ¨ ¨ ¨ , q . In addition, u µν and K µν are the velocity and the Tomonaga-Luttinger parameter,respectively. We also note that to obtain the above bosonizedHamiltonian (4), we adopt the following convention on theordering of the Klein factors: η Ò η Ó η Ó π η Ò π “ . (9) III. RENORMALIZATION GROUP ANALYSIS
Based on the bosonized Hamiltonian (4), we now determinethe possible phases in this model. To this end, we employthe renormalization group (RG) approach in the bosonizedHamiltonian [2]. By performing the scaling of the cut-o ff p α Ñ α “ α e dl q , one may obtain the set of the RG equationsat the one-loop level (quadratic with respect to the couplingconstants), which is given by (see Appendix) dK σ ´ dl “ ´ K σ ´ J p δ σ ´ α qr y ` y s ` J p δ ρ ´ α q y ` y , (10) dK σ ` dl “ ´ K σ ` r J p δ σ ´ α q y ` J p δ ρ ´ α q y ` y ` y s , (11) dK ρ ´ dl “ ´ K ρ ´ J p δ ρ ´ α qr y ` y s ` J p δ σ ´ α q y ` y , (12) dy dl “ p ´ K σ ´ ´ K σ ` q y ´ y y , (13) dy dl “ p ´ K ρ ´ ´ K σ ` q y ´ y y , (14) dy dl “ p ´ K σ ´ ´ { K ρ ´ q y ´ y y , (15) dy dl “ p ´ K σ ` ´ { K ρ ´ q y ´ y y J p δ σ ´ α q , (16) dy dl “ p ´ K ρ ´ ´ { K σ ´ q y ´ y y , (17) dy dl “ p ´ K σ ` ´ { K σ ´ q y ´ y y J p δ ρ ´ α q , (18) d δ σ ´ dl “ δ σ ´ ´ K σ ´ J p δ σ ´ α qr y ` y s α , (19) d δ ρ ´ dl “ δ ρ ´ ´ K ρ ´ J p δ ρ ´ α qr y ` y s α , (20)where the initial values are given as y i p q “ U {p π v F qp i “ , , ¨ ¨ ¨ , q , K ρ ´ p q “ K σ ´ p q “ K ρ ` “ { a ` U {p π v F q , K σ ` p q “ { a ´ U {p π v F q with theFermi velocity v F . We note that since there is no cosine termwith respect to φ ρ ` and θ ρ ` , which are decoupled from theother phase fields, K ρ ` does not flow up to this order of ap-proximation. In addition, J n p n “ , q is the n th order Besselfunction, which plays a role in controlling the relevance of thecorresponding cosine terms. Thus, one may classify the fixedpoints into the following three cases:(a) δ ρ ´ Ñ 8 , δ σ ´ Ñ , (b) δ ρ ´ Ñ 8 , δ σ ´ Ñ 8 , (c) δ ρ ´ Ñ , δ σ ´ Ñ 8 . ! (cid:3) " (cid:3) (cid:3) FIG. 2. Possible phases for the repulsive Hubbard interaction, U ą d -wave superfluid (a), spin-density wave (b), spin-triplet super-fluid along the z direction (c). The arrows and ellipses (shaded el-lipses) indicate the spins and spin-singlet pairing (spin-triplet pair-ing, especially, | S z “ y “ | ÒÓy ` | ÓÒy ), respectively. Since U ą
0, the on-site pairing is discouraged and the interchain pairingis selected by the many-body e ff ect for t KÒ « ˘ t KÓ . The SDW staterealized has the alternate occupation in spin on the two legs. First, let us consider the case (a), which corresponds tothe limit t KÒ « t KÓ . In this case, the terms proportionalto g , g can be dropped due to the rapid oscillation of thecosines. Thus, the RG equations reduce to ones without thestate-dependent hopping [2], since this limit also allows us todo the substitutions, J p δ σ ´ α q “ J p δ ρ ´ α q “
0. TheRG analysis shows the fixed point is given by g Ñ ´8 , g Ñ 8 , g Ñ 8 , g Ñ U ą g Ñ ´8 , g Ñ ´8 , g Ñ g Ñ 8 for U ă
0. While regardlessof the sign of the interaction, φ ρ ´ , φ σ ` , and φ σ ´ are gapped,these minimums are di ff erent for opposite signs of the interac-tion. It turns out that the minimum can be determined by thefixed point. Then, the dominant correlations are the d -wavesuperfluid for U ą ff er-ent chains and the s -wave superfluid for U ă O DSF p j q “ ÿ p p c j , Ò , p c j , Ó , ´ p ´ c j , Ó , p c j , Ò , ´ p q„ e ´ i θ ρ ` p cos φ ρ ´ sin φ σ ` sin φ σ ´ ´ i sin φ ρ cos σ ` cos φ σ ´ q , (21) O SSF p j q “ ÿ p p c j , Ò , p c j , Ó , p ´ c j , Ó , p c j , Ò , p q„ e ´ i θ ρ ` p cos φ ρ ´ cos φ σ ` cos φ σ ´ ` i sin φ ρ ´ sin φ σ ` sin φ σ ´ q , (22)respectively [2]. In contrast to the single chain Hubbardmodel, we have for the ladder a superfluid regardless of sign ! (cid:3) " (cid:3) (cid:3) FIG. 3. Possible phases for the attractive Hubbard interaction, U ă
0: bonding s -wave superfluid (a), charge density wave (b),anti-bonding s -wave superfluid (c). The di ff erence of the dashedcurves is that the s -wave superfluid (a) occurs for the bonding band ofthe Cooper pairs while the superfluid (c) occurs for the anti-bondingband of the Cooper pairs. The CDW (b) has the alternate occupationon the two legs. of the interaction.Let us next consider the case (b), where both of the runghoppings t K ρ ” t KÒ ` t KÓ and t K σ ” t KÒ ´ t KÓ are relevant andthe substitutions J p δ ρ ´ α q “ J p δ σ ´ α q “ ff ects of g , g , g , g can be dropped due to thelarge oscillations. By solving the RG equations under theseconditions, the fixed points are shown to be g Ñ 8 , g Ñ 8 for U ą g Ñ ´8 , g Ñ ´8 for U ă
0. Thus, we see θ ρ ´ , φ σ ` , θ σ ´ are going to be gapped. From the fixed pointanalysis, we find that the following SDW and CDW operatorsare relevant for U ą U ă
0, respectively: O SDW π p j q “ ÿ p p p c : j , Ò , p c j , Ò , p ´ c : j , Ó , p c j , Ó , p q„ e ´ i φ ρ ` p sin θ ρ ´ cos φ σ ` cos θ σ ´ ´ cos θ ρ ´ sin φ σ ` sin θ σ ´ q , (23) O CDW π p j q “ ÿ p p p c : j , Ò , p c j , Ò , p ` c : j , Ó , p c j , Ó , p q„ e ´ i φ ρ ` p cos θ ρ ´ cos φ σ ` sin θ σ ´ ´ sin θ ρ ´ sin φ σ ` cos θ σ ´ q , (24)where π indicates the di ff erence of the densities on the twolegs. The presence of the state-dependent rung hopping as | t KÒ { t KÓ | (cid:44) U ą U ă
0, respectively. Such emer-gences of the density wave states in the single chain systemare natural, since one of the spin components is reluctant tohop between di ff erent sites, However, now we impose the spindependence only for the rung direction. Thus, the emergenceof the SDW or CDW in our model is less trivial.Let us finally consider the case (c), which can be realizedwhen t KÒ « ´ t KÓ and therefore the substitutions J p δ ρ ´ α q “ J p δ σ ´ α q “ g , g can bedropped in a manner similar to the other cases. By solving theRG equations, we find the fixed points to be g Ñ ´8 , g Ñ g Ñ 8 , g Ñ 8 for U ą g Ñ ´8 , g Ñ g Ñ´8 , g Ñ ´8 for U ă
0, and therefore, φ ρ ´ , φ σ ` , θ σ ´ are gapped. In accordance with the fixed points, the dominantcorrelations are shown to be the spin-triplet superfluid alongthe z direction for U ą s -wave superfluid for U ă O TSF z p j q “ ÿ p p c j , Ò , p c j , Ó , ´ p ` c j , Ó , p c j , Ò , ´ p q„ e ´ i θ ρ ` p cos φ ρ ´ cos φ σ ` cos θ σ ´ ´ i sin φ ρ ´ sin φ σ ` sin θ σ ´ q , (25) O SSF π p j q “ ÿ p p p c j , Ò , p c j , Ó , p ´ c j , Ó , p c j , Ò , p q„ e ´ i θ ρ ` p sin φ ρ ´ sin φ σ ` cos θ σ ´ ` i cos φ ρ ´ cos φ σ ` sin θ σ ´ q , (26)respectively. We first focus on the emergence of the dom-inant fluctuation of the spin-triplet superfluid for U ą d -wave superfluid operator has the form as c j , Ò , c j , Ó , ´ c j , Ò ,π c j , Ó ,π , the spin-triplet superfluid occurring isgiven as c j , Ò , c j , Ó ,π ` c j , Ó , c j , Ò ,π . To understand the mechanism,we first point out that such a sign inversion in the rung hop-ping can be achieved by introducing the Peierls phases bothin charge and spin sectors by π {
2. Then, what is important forthe pairing is the Peierls phase in the spin sector. In fact, it hasbeen shown in Ref. [37] that such a Peierls phase causes thespin rotation of the fermions for one of the chains and trans-forms a spin-singlet into a spin-triplet pairing. For U ă
0, onthe other hand, the di ff erence between the s -wave superfluidsin Eq. (26) and in Eq. (22) is that if we treat the Cooper pairsoccurring in each chain as the bosons, the superfluid in theabsence of the state-dependent hopping occurs for the bond-ing band of the bosons while the superfluid in the presenceof it occurs for the anti-bonding band of the bosons. Com-pared with the situation from the spin-singlet to spin-tripletpairings for U ą
0, the important ingredient for this changeof the s -wave superfluids for U ă IV. DISCUSSIONA. Strong coupling limit
So far, we have discussed the weak-coupling limit by meansof the bosonization and RG analysis, it is also interesting tosee what happens in the strong-coupling limit in which naivelya similar phase diagram may be expected.For the U ą ffi cult to depicta general phase diagram analytically since a faithful e ff ectiveHamiltonian has yet to be known except for commensuratefilling such as half filling. In addition, the rung hopping isa relevant perturbation, which prevents one from starting atthe single chain Hubbard model where the Bethe ansatz ap-proach is available. At the same time, the previous numericalanalyses in the absence of the state-dependent hopping showthat the d -wave superfluid state emerges even in the strong-coupling limit [2, 10–13]. In addition, since the hybridizationamong the four di ff erent Fermi points by the on-site repulsiveinteraction shown in Fig. 1 (a) is an essential ingredient ofthe d -wave state, we obtain the d -wave superfluid not only for t (cid:107) " t K but also for t (cid:107) « t K in which the numerical calcu-lation has been performed [10–13]. Therefore, the presenceof the spin-triplet superfluid in the strong-coupling limit canalso be shown with the argument in Sec. III. Namely, by usingthe canonical transformations c j , s , Ñ a j , s , c j , s , ´ Ñ ˘ a j , s , ´ where the sign is ` for s “Ò and ´ for s “Ó , the Hamilto-nian with t KÒ “ ´ t KÓ is mapped onto one with t KÒ “ t KÓ ,that is, a normal two-leg fermionic Hubbard ladder can be ob-tained. Accordingly, the operator of the spin-triplet superfluidis transformed into that of the d -wave superfluid. Therefore,once we confirm the emergence of the d -wave superfluid inthe normal two-leg fermionic Hubbard ladder system, we seethat the spin-triplet superfluid occurring in t KÒ « ´ t KÓ is ro-bust. We also note that the essence of the spin-triplet super-fluid is the manipulation on the rung hopping, and thus, noth-ing happens and the d -wave superfluid remains even if such amanipulation on the hopping is performed for the chain direc-tion. Thus, to see the spin-triplet superfluid, the manipulationon the hopping along the rung direction is required. Anotherinteresting but remaining issue may be the possibility of seg-regation in the limit t Ò “ t Ó “ U ă ff ective Hamiltonian approach. To see this, we first performthe so-called particle-hole transformation [41] in this model.Then, the original model is mapped onto the system with U ą ff ectiveHamiltonian is shown to be H “ J (cid:107) ÿ j p (cid:126) S j , ¨ (cid:126) S j ` , ` (cid:126) S j , ´ ¨ (cid:126) S j ` , ´ q ´ h ÿ j p S zj , ` S zj , ´ q` J xy K ÿ j p S xj , S xj , ´ ` S yj , S yj , ´ q . ` J z K ÿ j S zj , S zj , ´ , (27)where J (cid:107) “ t (cid:107) {| U | , J xy K “ t KÒ t KÓ {| U | , J z K “ p t KÒ ` t KÓ q{| U | , and h is a magnetic field corresponding to fillingin the original attractive model. By performing bosonization for the above Hamiltonian [2], one may obtain H e ff “ ÿ µ “ s , a ż dx π ˆ u µ K µ p ∇ θ µ q ` u µ K µ p ∇ φ µ q ˙ ` p πα q ż dx r J xy K cos p ? θ a q ` J z K cos p ? φ a qs , (28)where φ s p a q “ r φ ` p´q φ ´ s{ ? p , φ p p p “ ˘ q , and similar relations for the θ field. The original spin fields and phase fields are related as S zp p x q “ ´ ∇ φ p p x q{ π `p´ q x cos p φ p p x qq{p πα q and S ` p p x q “ e ´ i θ p p x q rp´ q x ` cos 2 φ p p x qs{ ? πα . Since J (cid:107) " J K is con-cerned, we can determine the Tomonaga-Luttinger parametersas K s , a “ K ˆ ¯ K J z K π u ˙ . (29)Here K and u are the Tomonaga-Luttinger parameter andvelocity in the single chain Heisenberg model, respectively.The Tomonaga-Luttinger parameter K can be determined bymeans of Bethe ansatz, and it is known that the possible rangeis 1 { ď K ď
1, where K “ { K “ ? θ a and cos ? φ a have thescaling dimensions of p K a q ´ and 2 K a , respectively. Thus,we see that cos ? θ a is ordered for K a ą { K a ă {
2. As can be seen from Eqs.(28) and (29), K a ą {
2, and we expect that θ a is orderedexcept for the limit J xy K Ñ φ a is ordered. To specifythe ground state in the spin language, let us introduce bond-ing and anti-bonding spin operators as (cid:126) S “ (cid:126) S ` (cid:126) S ´ and (cid:126) S π “ (cid:126) S ´ (cid:126) S ´ , respectively. Then, one finds that the bonding(anti-bonding) transverse spin-spin correlation x S ` p r q S ´ p qypx S ` π p r q S ´ π p qyq is dominant for θ a to be gapped with J xy K ă p J xy K ą q , while the anti-bonding longitudinal spin-spin cor-relation x S z π p r q S z π p qy is dominant for φ a to be gapped [2].Now, we can determine the dominant correlation in the orig-inal model by using the particle-hole transformation again.Since by this transformation S ´ Ñ ÿ p pc j , Ò , p c j , Ó , p “ O SSC π , (30) S ´ π Ñ ÿ p c j , Ò , p c j , Ó , p “ O SSC , (31) S z π Ñ ÿ p , s pc : j , s , p c j , s , p “ O CDW π , (32)we conclude that the s -wave superfluid is dominant exceptfor t KÒ t KÓ Ñ s -wave pairing state is realized for t KÒ t KÓ ą s -wave pairing state isrealized for the opposite sign case. Thus, the phase struc-ture is compatible with the weak-coupling analysis while inthe weak-coupling limit the region of the s -wave superfluidis rather narrow but in the strong-coupling limit the situationis reversed. This may be explained by the observation thatthe pairing gap becomes larger as the attractive interaction isincreased and the pairing in the s -wave superfluid essentiallyoccurs in a single site, and therefore the introduction of thesmall state-dependent rung hopping may not cause the disap-pearance of the superfluid correlation. B. Experimental protocol
We now discuss the realization of our model and its groundstates in cold atoms.In order to realize the two-leg ladder geometry, we can con-sider an optical superlattice [17, 18]. By using this technol-ogy, we can obtain a system where there are a number of two-leg ladders, each of which is weakly coupled by some hop-ping parameter. To ensure the one-dimensional character inthe system, this hopping parameter should be much smallerthan a temperature [2]. Then, the two-leg ladder system in theabsence of state-dependent hopping is obtained. As anotherroute to realize such a ladder geometry, one may also utilizeinternal degrees of freedom in an atom as discussed in Refs.[43, 44].On the other hand, the Hubbard interaction U can be tunedby selecting atomic species and by changing lattice depth.Typically, K and Li have been utilized to realize the sys-tem for U ą U ă
0, respectively [14]. In addition, theFeshbach resonance is available to change the strength andsign of the interaction.The most important ingredient in the system discussed is astate-dependent hopping. When it comes to a positive state-dependent hopping, heteronuclear mixtures such as Li ´ K[45] and Li ´ Yb [46] are available. However, since weare also interested in a state-dependent hopping whose sign isdi ff erent between spin up and down, another scheme is neces-sary.To this end, we start with the two-leg Hubbard ladder sys-tem in the absence of a state-dependent hopping. To obtain astate-dependent rung hopping, we consider adding the follow-ing time-dependent term in the Hamiltonian: ÿ s “Ò , Ó A s cos p ω t q ÿ p “˘ pn j , s , p . (33)If A Ò “ A Ó , the above time-varying linear potential can beobtained with a sinusoidal shaking of an optical lattice alongthe rung direction [24]. In order to exactly obtain the abovetime-dependent term for A Ò (cid:44) A Ó , a sinusoidal shaking of astate-dependent optical lattice [27, 28] or of a magnetic fieldgradient [51] can be utilized. Then, an essential point is thatwhen (cid:126) ω " t (cid:107) , t K , | U | , we may perform the time average ofthe above oscillation term, which causes the renormalizationof the hopping parameter as t K Ñ t K J p A s {p (cid:126) ω qq [23, 26].Since the argument of the Bessel function is now state depen-dent, the Hamiltonian (1) can be obtained. We note that theBessel function can take a negative value, which allows us toconsider a negative hopping parameter. Indeed, such a nega-tive hopping parameter by the time-dependent oscillation termhas been observed in Refs. [24, 25] A particularly interesting challenge is to make the spin-triplet superfluid realized around t KÒ « ´ t KÓ for a repulsive U . Here, we note that « implies t K ρ { T ! T isa temperature. In this case, the e ff ect of t K ρ can be droppedin the Hamiltonian, and the e ff ective Hamiltonian reduces toone of t KÒ “ ´ t KÓ [2]. In the two-leg fermionic ladder sys-tem at incommensurate filling, we have a one charge gap andtwo spin gaps, which are exponentially small for the weak-coupling limit but are of the order of the exchange energy forthe strong-coupling limit ( See Refs. [10–13] for numericalestimations for the strong-coupling limit). Since the gaps areessential to characterize the spin-triplet superfluid state, thetemperature should be smaller than them as well as the otherHamiltonian parameters t (cid:107) , U , t K , s . Thus, the dominant spin-triplet superfluid correlation should show up at the tempera-ture satisfying these conditions. We note that a similar argu-ment for the realizations of the other phases is also possible.When it comes to the spin-triplet superfluid, we can alsoutilize the technique of synthetic gauge fields [47]. Recently,by using a Raman laser and lattice driving [48–52], it is possi-ble to introduce the Peierls phase in the hopping parameter. Ifsuch a Peierls phase has a state dependency, which is indeedpossible experimentally, the hopping parameter is modified as t Ñ te i Φ s , where Φ s is the Peierls phase. When Φ Ò “ ´ Φ Ó ,such a hopping term can also be regarded as a spin-orbit cou-pling. As explained in Sec. III, the essence of the spin-tripletsuperfluid is the emergence of the state-dependent Peierlsphase along the rung direction as t K Ñ t K e is Φ . Thus, the spin-triplet superfluid realized with this manipulation is essentiallythe same mechanism as one discussed in this paper.Finally, we give a few comments on the experimental ob-servability of the superfluids. An important feature is the pres-ence of the gaps, which may be measured by rf spectroscopy[53]. However, the measurement of gaps alone is not enoughto distinguish di ff erent superfluids. One of the possible solu-tions to this problem is to use the particle-hole transformation[41]. Then, the Hamiltonian for U ą U ă d -wave and spin-triplet superfluids are transformed intothose of staggered spin-flux and bond SDW phases, respec-tively [37]. The properties of such phases may be captured bythe local addressing of the flux [48, 54] and spin correlations[20, 55] or by spin-sensitive Bragg scattering of light [56]. V. CONCLUSION
We have examined a two-leg fermionic Hubbard laddermodel in the presence of a state-dependent hopping. We havefocused on a case where such a hopping exists for the rungdirection. This system can be treated as the minimal construc-tion of the physics of mixed dimensions [57] if the rung hop-ping in one of the states (spins) is zero since another runghopping plays a role in connecting di ff erent chains. Due tothe state-dependent hopping, the original d -wave and s -wavesuperfluids realized in the normal two-leg fermionic Hubbardladder model for the repulsive and attractive interactions, re-spectively become unstable. We have demonstrated that in-stead the spin-triplet superfluid, SDW, and CDW states be-come stable depending on the ratio t KÒ { t KÓ . In particular, ourproposal shows a spin-triplet superfluid state for purely localinteractions can be realized. We have also discussed the ex-perimental protocol and observability toward the spin-tripletsuperfluid. ACKNOWLEDGEMENT
SU thanks A. Tokuno for fruitful discussions on the real-ization of our model. This work was supported by the Swiss National Foundation under division II.
APPENDIX: RENORMALIZATION GROUP EQUATIONS
In this Appendix, we wish to outline the derivation of therenormalization group equations in a similar way as Ref. [2].We first consider the following correlation function: R p r ´ r q “ x T τ e i φ σ ` p x ,τ q e ´ i φ σ ` p x ,τ q y (34)where T τ denotes the time-ordered product. By expanding theabove correlation function in terms of g i up to third order, weobtain R p r ´ r q « e ´ K σ ` F p r ´ r q ` p S q ` p T q , (35)where F p r q “ ln p r { α q , p S q “ ˆ g p πα q v F ˙ ÿ (cid:15) ,(cid:15) “˘ ż d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ σ ´ p (cid:126) r q´ φ σ ´ p (cid:126) r q´ δ σ ´ p x ´ x qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ σ ´ p (cid:126) r q´ φ σ ´ p (cid:126) r q´ δ σ ´ p x ´ x qqs y ı ` ˆ g p πα q v F ˙ ÿ (cid:15) ,(cid:15) “˘ ż d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ ρ ´ p (cid:126) r q´ φ ρ ´ p (cid:126) r q´ δ ρ ´ p x ´ x qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ ρ ´ p (cid:126) r q´ φ ρ ´ p (cid:126) r q´ δ ρ ´ p x ´ x qqs y ı ` ˆ g p πα q v F ˙ ÿ (cid:15) ,(cid:15) “˘ ż d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ı ` ˆ g p πα q v F ˙ ÿ (cid:15) ,(cid:15) “˘ ż d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p θ σ ´ p (cid:126) r q´ θ σ ´ p (cid:126) r qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p θ σ ´ p (cid:126) r q´ θ σ ´ p (cid:126) r qqs y ı , (36)and p T q “ ´ g g g ˆ p πα q v F ˙ ÿ (cid:15) ,(cid:15) ,(cid:15) “˘ ż d r d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ σ ´ p (cid:126) r q´ φ σ ´ p (cid:126) r q´ δ σ ´ p x ´ x qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ σ ´ p (cid:126) r q´ φ σ ´ p (cid:126) r q´ δ σ ´ p x ´ x qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ı ´ g g g ˆ p πα q v F ˙ ÿ (cid:15) ,(cid:15) ,(cid:15) “˘ ż d r d r d r ” x e i r φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r q` (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ ρ ´ p (cid:126) r q´ φ ρ ´ p (cid:126) r q´ δ ρ ´ p x ´ x qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ´x e i p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq y x i r (cid:15) p φ σ ` p (cid:126) r q´ φ σ ` p (cid:126) r qq` (cid:15) p φ ρ ´ p (cid:126) r q´ φ ρ ´ p (cid:126) r q´ δ ρ ´ p x ´ x qq` (cid:15) p θ ρ ´ p (cid:126) r q´ θ ρ ´ p (cid:126) r qqs y ı . (37)In the above, x¨ ¨ ¨ y denotes the average without the cosine terms, that is, one with the Tomonaga-Luttinger Hamiltonian. Whenwe focus on p T q , the dominant contributions come from (cid:126) r “ (cid:126) r ` (cid:126) r or (cid:126) r “ (cid:126) r ` (cid:126) r for the term proportional to g g g andfrom (cid:126) r “ (cid:126) r ` (cid:126) r or (cid:126) r “ (cid:126) r ` (cid:126) r for one proportional to g g g with a small r . Therefore, by expanding around (cid:126) r “
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