Fulde-Ferrell--Larkin-Ovchinnikov state in the dimensional crossover between one- and three-dimensional lattices
FFulde-Ferrell–Larkin-Ovchinnikov state in the dimensional crossover between one- andthree-dimensional lattices
Dong-Hee Kim and P¨aivi T¨orm¨a ∗ Department of Applied Physics and Centre of Excellence in Computational Nanoscience (COMP), Aalto University, FI-00076 Aalto, Finland
We present a full phase diagram for the one-dimensional (1D) to three-dimensional (3D) crossover of theFulde-Ferrell–Larkin-Ovchinnikov (FFLO) state in an attractive Hubbard model of 3D-coupled chains in a har-monic trap. We employ real-space dynamical mean-field theory which describes full local quantum fluctuationsbeyond the usual mean-field and local density approximation. We find strong dimensionality effects on the shellstructure undergoing a crossover between distinctive quasi-1D and quasi-3D regimes. We predict an optimalregime for the FFLO state that is considerably extended to intermediate interchain couplings and polarizations,directly realizable with ultracold atomic gases. We find that the 1D-like FFLO feature is vulnerable to thermalfluctuations, while the FFLO state of mixed 1D-3D character can be stabilized at a higher temperature.
The interplay between fermion pairing and magnetism isat the heart of understanding strongly correlated systemsranging from unconventional superconductors and ultracoldgases to neutron stars and quarks. BCS-type superconduc-tivity is suppressed by a large magnetic field exceeding theChandrasekhar–Clogston limit. However, it has been pro-posed as a paradigm of superconductivity in high magneticfields that it is possible for superconductivity and magnetismto coexist with exotic pairing mechanisms.
The Fulde-Ferrell–Larkin-Ovchinnikov (FFLO) state would arise withthis interplay, but it still remains elusive in spite of indirectexperimental evidence observed. The FFLO state is charac-terized by the Cooper pair carrying finite momentum causinga spatially modulated order parameter. One peculiar featureof this exotic phase is that apparently its stability is largelyaffected by the dimensionality of the system. It turns out thatthe three-dimensional (3D)-FFLO state occupies a thin areaof the mean-field phase diagram though the signature can bestronger in the systems that support nesting such as opticallattices and elongated traps. Indeed, only phase separationwas observed for ultracold gases in 3D traps.
On the otherhand, in the exact solution of a one-dimensional (1D) system,the FFLO character appears at any finite spin-polarization, while long-range order cannot exist in 1D. The experiment in 1D was consistent with the FFLO theory, although the stateremains unidentified.A natural question arising is whether one can combine thepromising 1D-FFLO features and long-range order providedby higher dimensions. This has been considered in previousmean-field and effective field theory studies for coupledcontinuum-1D gases, and for the Hubbard ladder. The trap-ping potential is essential in ultracold gas experiments, andthus in spin-polarized systems, one can expect a shell struc-ture of different phases along the trap. Therefore, beyond thelocal density approximation, the inhomogeneous superfluidand normal phases need to be treated in a unified frameworkby including full local quantum fluctuations. Neglecting lo-cal quantum fluctuations creates an apparent bias in favor ofthe superfluid state and against the normal state. Here, us-ing a real-space dynamical mean-field theory (DMFT), we in-vestigate the 1D-3D crossover problem within the anisotropicHubbard model in a trap.The dimensionality effect to the FFLO state that we are considering here differs from the two-dimensional (2D)-3Dcrossover studied in layered superconductors. There, thequasi-2D character minimizes the orbital pair breaking effectsin a magnetic field, and the Zeeman effect may then lead toa d -wave FFLO state. Here we consider s -wave pairing, andorbital effects are absent. Note that a particle-hole transfor-mation can bring in another interesting perspective by map-ping the FFLO state to the striped phase of the doped repul-sive Hubbard model. This emphasizes the importance of theFFLO state in the general context of high- T c superconductiv-ity.We perform a real-space variant of DMFT calculations onthe attractive Hubbard model of 3D-coupled chains, H = − t (cid:107) (cid:88) ilσ ( c † ilσ c ( i +1) lσ + h . c . ) − t ⊥ (cid:88) (cid:104) ll (cid:48) (cid:105) (cid:88) iσ c † ilσ c il (cid:48) σ + U (cid:88) il ˆ n il ↑ ˆ n il ↓ + (cid:88) ilσ ( V i − µ σ )ˆ n ilσ , where c † ilσ ( c ilσ ) creates (annihilates) a fermion with spin σ = ↑ , ↓ at site i of chain l , the density ˆ n ilσ = c † ilσ c ilσ .We define the superfluid order parameter as ∆ = −(cid:104) c †↑ c †↓ (cid:105) .Throughout the calculations, the hopping t (cid:107) is set to unity.The dimensionality is thus tuned by the interchain coupling t ⊥ and varied from 1D ( t ⊥ = 0 ) to 3D ( t ⊥ = 1 ). Thechemical potentials µ ↑ and µ ↓ control the polarization P =( N ↑ − N ↓ ) / ( N ↑ + N ↓ ) by keeping the total particle number N ↑ + N ↓ ∼ . The harmonic potential trapping the gasesin a longitudinal chain is given as V i = 5 × − ( i − / .The on-site interaction U is selected to be the value corre-sponding to unitarity where the 3D two-body scattering lengthdiverges. The value of U varies from − . ( t ⊥ = 0 . ) to − . ( t ⊥ = 1 ), which is comparable to the half bandwidths.In order to treat an inhomogeneous phase along thechain, a site-dependent self-energy Σ i is considered withinDMFT. With translational invariance in transverse di-rections, the on-chain Green’s function is written for sites i, j and transverse momentum k ⊥ as [ G − ( k ⊥ ; iω n )] ij = [ iω n σ − (cid:15) k ⊥ σ − Σ i ( iω n )] δ ij − h (cid:107) ij , where ω n denotes the Matsubara frequency, σ is the Pauli ma-trix, h (cid:107) is the non-interacting part of the chain Hamiltonian, a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y the 2D energy dispersion (cid:15) k ⊥ ≡ − t ⊥ (cos k x + cos k y ) with k ⊥ = ( k x , k y ) , and the operators are in the Nambu basis.In the self-consistency of DMFT, the impurity Green’s func-tion is obtained as G i ( iω n ) = (cid:80) k ⊥ G ii ( k ⊥ ; iω n ) . While wehave a local but site-dependent self-energy term, the proce-dures can be compared to the chain-DMFT. The exact di-agonalization method is employed to solve the site-dependentimpurity problem. With the on-site interaction U = − , we have foundthat our calculations qualitatively reproduce the two impor-tant characteristics of strongly interacting 1D Fermi gases. First, at small polarizations, the trap edges are fully pairedwhile the trap center is partially polarized. The entire area be-comes polarized as the total polarization increases. Second, atfinite polarizations, the polarized trap center is associated withthe FFLO state exhibiting a spatially oscillating order param-eter.Clear features of getting away from the 1D limit are ob-served. For the studied finite interchain couplings, the pairingat the edges, namely the first 1D feature listed above, is eas-ily broken at small polarizations. This indicates that imme-diately away from 1D, one can observe fully polarized edgesmuch before the whole area gets polarized. In contrast, it turnsout that the other part of the 1D features, the emergence of apartially polarized center in fully paired vicinities, survives atsmall interchain couplings. Away from but close to the 1Dlimit, we typically find a shell structure of polarized edges,fully paired shoulders, and a partially polarized center.In Fig. 1, we present a real-space DMFT phase diagram ofthe Hubbard model of 3D-coupled chains at zero temperature.Considering only the phase at the trap center, we find that theemergence of the FFLO-type oscillating order parameter oc-cupies a large area of polarizations and interchain couplings.This wide coexistence area of a finite density difference and afinite order parameter extends all the way to the 3D limit, asopposed to the mean-field phase diagram on a 3D continuumwhere the FFLO phase occupies only a tiny area. Our zero-temperature phase diagram characterizes theemergence of three types of shell structures, as shown inFig. 1(b). First, area I shows quasi-3D features in the shellstructure where the fully paired superfluid (SF) core exists.There are FFLO-type oscillations found between the SF coreand the fully polarized edges. Second, area II shows an in-verted sequence: an FFLO core surrounded by fully pairedshoulders. The edges are polarized in this area, and small or-der parameter oscillations also exist at the interfaces betweenthe fully paired shoulders and the fully polarized edges. Whilethis indicates a mixture of 1D and 3D features, area II can beidentified as a quasi-1D phase because of the spatial patternof the FFLO oscillations emerging at the trap center. Area IIis only found at interchain couplings t ⊥ ≤ . . Third, areaIII has a two-shell structure where the FFLO-type oscillationsreside in the entire area of the partially polarized core, sur-rounded by fully polarized edges. Area III is found acrossthe whole range of interchain couplings at intermediate-highpolarizations below the transition to the normal phase.We find that the FFLO character evolves very differentlywith polarization in the quasi-1D and quasi-3D regimes. Fig- FF L O IIIIII
Normal N o r m a l F F L O F u ll y - P a i r e d S F c o r e Δ a t c e n t e r ( n ↑ − n ↓ ) Δ ( x ) a t c e n t e r FFLO - SF - FFLOSF - FFLO - SF FFLO IV (a)(b) Interchain couplingPolarization Interchain coupling P o l a r i z a t i on FIG. 1. (Color online) Phase diagram of the 3D coupled-chainHubbard model. The particle densities n ↑ , ↓ and the order parameter ∆ are calculated as a function of the interchain coupling and thepolarization at zero temperature. (a) The oscillation amplitude ∆ of the order parameter and the density difference n ↑ − n ↓ at the trapcenter. The phase at the trap center can be divided into three: thefully-paired superfluid (SF) ( n ↑ = n ↓ , ∆ (cid:54) = 0 ), FFLO (oscillating n and ∆ ), and normal ( ∆ = 0 ) phases. (b) Phases I–III associatedwith the shell structures in the trap, explained in the text. ure 2 shows the shell structures with increasing polarization P at both sides of the 1D-3D crossover. In the quasi-1D regime[Fig. 2(a)], the evolution is dominated by the expansion of theFFLO core. In contrast, in the quasi-3D regime [Fig. 2(b)], thefully paired core shrinks with increasing P while the FFLO-type oscillations at the shoulders move toward the trap cen-ter. Near the crossover, these quasi-1D and quasi-3D featurescoexist. At the interchain coupling t ⊥ = 0 . , the oscilla-tions of the order parameter ∆ become significant at both thetrap center and edges, and the fully paired shoulders decreasefrom both sides as P increases. In addition, when P is in-creased to enter the area III, the oscillations of ∆ along thetrap show a distinct feature: At small t ⊥ , the amplitude of ∆ becomes spatially uniform, despite the trap, at intermediate P [see Fig. 2(c)]. This can be compared to the zero derivativeof the FFLO momentum as a function of the chemical po-tential. The uniform oscillations occur only in the quasi-1Dregime, raising the possibility that a peak signal of the FFLOmomentum is more visible in this regime.Our phase diagram suggests that the optimum spot for ob-serving the FFLO state is extended over a significantly largearea of the dimensional crossover. In particular, the largest n i ↑ , n i ↓ t ⊥ =0.2 P=0.04 i n i ↑ n i ↓ -150 -100 -50 0 50 100 150 P=0.06 i -150 -100 -50 0 50 100 150 P=0.13 i -0.2 0 0.2 ∆ i n i ↑ , n i ↓ t ⊥ =0.5 P=0.08 i n i ↑ n i ↓ -200 -100 0 100 200 P=0.14 i -200 -100 0 100 200 P=0.18 i -0.4 0 0.4 ∆ i ∆ i P=0.17 ∆ i P=0.20 ∆ i (a)(b) (c) t ⊥ =0.2 P=0.23
FIG. 2. Evolution of the FFLO oscillations in the 1D-3D crossover. The profiles of the particle densities n ↑ , ↓ and the order parameter ∆ along the chain sites i are presented for the two regimes of interchain couplings (a) t ⊥ = 0 . (quasi-1D) and (b) t ⊥ = 0 . (quasi-3D). In thequasi-1D regime, for small polarization P = 0 . , the FFLO-type oscillations at the center are surrounded by the fully paired shoulders. Theregion of oscillating ∆ develops from the center, expands toward the edges as P increases, and then emerges over the whole area for P = 0 . .On the contrary, in the quasi-3D regime, the oscillations are initially at the partially polarized intermediate regions and spread toward the centeras P increases. At finite interchain couplings, the far-edges of the trap are always polarized. In (c), the evolution of the oscillation envelope of ∆ is shown with increasing P at t ⊥ = 0 . in the quasi-1D regime. Uniform oscillations occur along the trap at an intermediate polarization. range of polarizations associated with the FFLO-featured ar-eas (II+III) at the trap center is found around interchain cou-pling t ⊥ ∼ . . Since we still have two control parame-ters, the onsite interaction and the temperature, the phase di-agram should be further generalized to discuss ultracold gasexperiments which are conducted at finite temperatures andwith a tunable atomic scattering length. We have examinedalso a stronger interaction U = − at small t ⊥ ’s, includ-ing the 1D limit ( t ⊥ = 0 ) where the 3D scattering approachfails. At small interchain couplings, we have found that themain effect of a stronger interaction is to increase the criti-cal polarization of the transition to the normal phase whilethe change in the phase II area is relatively small. In the 3Dlimit, stronger interactions tend to keep the fully paired SFcore even at larger polarizations, as observed in the previousexperiments in the BEC regime, which can make the 3D-FFLO area much smaller.The stability of the FFLO state at finite temperature is acrucial question. We have examined the behavior at a finitetemperature T = 0 . near the dimensional crossover at theinterchain coupling t ⊥ = 0 . . A low temperature algorithm isused. We find that finite temperature can stabilize the polar-ized superfluid (pSF) state at small polarizations while it de-stroys the quasi-1D FFLO character. Figure 3 shows compar-isons between the density and order parameter profiles at twodifferent temperatures T = 0 . and T = 0 . These profilesindicate that area II with the FFLO core melts at T = 0 . intothe pSF state, similar to the results in 1D. In the case of shellstructures, the polarization from the FFLO area is easily redis-tributed in the trap at finite temperature to create a BCS-typeorder parameter with extra majority particles accommodatedas thermal quasiparticles. In contrast, it turns out that area IIIwith a trap-wide FFLO character at higher polarizations is notaffected by the finite temperature examined.A comparison with the previous mean-field results in 1Dand 3D lattices shows the drastic effects of local quantumfluctuations: (1) Our results indicate that the large size of the FFLO area predicted by mean-field theory in 3D lattices may have been overestimated. In the 3D limit, we find thatthe FFLO state is broken near local polarization . at a trapcenter that is approximately a quarter filled, which contraststo the mean-field values ∼ . ( | U | = 5 . ) and ∼ . ( | U | = 6 ) at the same filling. This enhancement of the nor-mal state can be understood in light of the fact that includingthe particle-hole channel was shown to reduce pairing signif-icantly in lattices: DMFT includes full local quantum fluc-tuations causing such higher order effects. (2) In 1D lattices,mean-field theory predicted only a 3D-like shell structure n i ↑ , n i ↓ T=0.05 P ≅ pSF in i ↑ n i ↓ T=0
FFLO i -0.2 0 0.2 ∆ i n i ↑ , n i ↓ T=0.05 P ≅ FFLO in i ↑ n i ↓ -150-100 -50 0 50 100 150 T=0
FFLO i -0.2 0 0.2 ∆ i (a)(b) FIG. 3. Polarized superfluid phase at finite temperature. For in-terchain coupling t ⊥ = 0 . , the profiles at temperature T = 0 . are compared with those at T = 0 for the particle densities n ↑ , ↓ andthe order parameter ∆ along the chain sites i . At polarization (a) P (cid:39) . , corresponding to phase II, the structure of the FFLO coresurrounded by fully paired shoulders found at T = 0 is completelychanged into the polarized superfluid phase with a uniform order pa-rameter at T = 0 . . The order parameter oscillations at the edgesare still observed at T = 0 . . In contrast, at a higher polarization(b) P (cid:39) . , corresponding to phase III, similar FFLO characteris-tics are identified at both temperatures T = 0 . and T = 0 . with a fully paired center and polarized edges. In contrast, wefind the reversal of the shell structure, similar to what has beenpredicted by continuum mean-field study. (3) In addition,our characterization of the FFLO state does not find the well-separated domain wall between the sign changes of the or-der parameter predicted by the mean-field theories in lattices and in continuum. Our findings, the drastic decrease of crit-ical polarization, the shell structure reversal, and the absenceof domain walls, emphasize the importance of local quantumfluctuations in lattices, regardless of the dimensionality.Our findings are directly applicable to future experiments.Recent experiments realized a weakly coupled 2D array of1D tubes with ultracold Li gases, observing the density pro-file characteristics of attractively interacting spin-polarized1D Fermi gases. This system can be further extended towardthe quasi-1D regime by adjusting the optical lattice potentials,and if needed, discreteness along the tube direction can be re-alized by 3D lattices.
Although it is nontrivial to experi-mentally reveal the oscillating order parameter, several meth-ods have been recently suggested to detect the FFLO state par-ticularly in 1D systems, see and references therein. OurDMFT calculations have shown that the FFLO phase occupiesa significant area of the phase diagram throughout the 1D-3D crossover. In addition, our calculations at finite temperaturehave found that the FFLO character is more stable at inter-mediate rather than small polarizations where the phase be-comes the polarized superfluid at low temperature. Our find-ings will help identify the presence of the FFLO state in ultra-cold atomic gases. In a general perspective, by using beyond-mean-field calculations, we have investigated the dimension-ality effects on the existence of the FFLO state in 1D-3D lat-tice systems. This may potentially help to understand also thepuzzling repulsive Hubbard model counterparts such as thestriped phase.
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