Unconventional pairings of spin-orbit coupled attractive degenerate Fermi gas in a one-dimensional optical lattice

Understanding novel pairings in attractive degenerate Fermi gases is crucial for exploring rich superfluid physics. In this report, we reveal unconventional pairings induced by spin-orbit coupling (SOC) in a one-dimensional optical lattice, using a state-of-the-art density-matrix renormalization group method. When both bands are partially occupied, we find a strong competition between the interband Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and intraband Bardeen-Cooper-Schrieffer (BCS) pairings. In particular, for the weak and moderate SOC strengths, these two pairings can coexist, giving rise to a new phase called the FFLO-BCS phase, which exhibits a unique three-peak structure in pairing momentum distribution. For the strong SOC strength, the intraband BCS pairing always dominates in the whole parameter regime, including the half filling. We figure out the whole phase diagrams as functions of filling factor, SOC strength, and Zeeman field. Our results are qualitatively different from recent mean-field predictions. Finally, we address that our predictions could be observed in a weaker trapped potential.

All the previous predictions are demonstrated in the framework of mean-field theory [12][13][14][15][16][17][18][19][20][21][22][23][24].In these calculations, the pairing is simply assumed to take place between two fermions with total center-of-mass momentum Q, which serves as a parameter to minimize the total free energy.In this regard, Q = 0 and Q = 0 correspond to the BCS and FFLO phases, respectively.Such that, the many-body Hamiltonian can be treated using standard Bogoliubov transformation.This fundamental picture has never been challenged even in low dimensional system, in which quantum fluctuation becomes significant and the mean-field theory becomes unreliable [29].
A representative example is to discuss the FFLO phase in imbalanced system [30][31][32][33], in which the basic pairing can still be well described by the above mechanism.Since the topological superfluids and associated MF depend strongly on the pairing symmetry, the true pairing in this new platform need to be examined more seriously, which is, however, still lacked.This work is devoted to addressing this fundamental issue in a 1D spin-orbit coupled optical lattice, using the exact density matrix renormalization group (DMRG) method.
Our results demonstrate that the relevant physics is completely modified by the SOC-induced triplet pairing [26][27][28].This new pairing can lead to a strong competition between the interband FFLO and intraband BCS pairings.In the weak SOC, we find that these pairings may coexist, when both bands are partially occupied near half filling.This exotic coexistence can generate a new quantum state, called the FFLO-BCS phase.Moreover, the predicted FFLO-BCS phase is characterized by a unique three-peak structure in pairing momentum distribution.In the intermediate and strong SOCs, the BCS superfluid can emerge in the whole parameter regime.Especially, the independent BCS pairing in each band can be formed, when both bands are partially occupied.These results shed great light on the realization of topological superfluids and associated MF in the spin-orbit coupled lattice.
Theoretical Model.We consider the following 1D Fermi-Hubbard model with an synthetic SOC [21,22] where c † ls and c ls are the creation and annihilation operators, with spin s =↑, ↓ (encoded by the hyperfine states), at lattice site l, n ls = c † ls c ls is the number operator, t is the spin-independent hopping, h is the Zeeman field along z direction, U is the on-site attractive interaction, and λ is the SOC strength.All these parameters can be tuned independently, which is the basic advantage of ultracold atom systems compared with solid materials [11].For example, the SOC strength can be tuned through a fast and coherent modulation of the Raman beams [34].
Unconventional pairings.Figure 1 shows the fundamental physical picture for pairings in an optical lattice with or without SOC.In the absence of SOC, the Hamiltonian (1) reduces to the well-studied Fermi-Hubbard model [36,37], in which the Zeeman field breaks the degeneracy of each band (the breaking of symmetry from SO(4) [38] at half filling and otherwise SU(2) to U(1)⊗U( 1)); however, both bands are still spin fully polarized.Consequently, the pairing can only be formed between two fermions at different bands due to the interparticle interaction; see Fig. 1a.This interband pairing gives rise to the FFLO phase [39,40].However, this picture is completely modified by SOC.The SOC-induced triplet pairing (see momentum-dependent spin polarization of each band in Fig. 1b-d) can generate the intraband BCS pairing, which competes with the original interband FFLO pairing.As a result, the true pairing depends strongly on the competition of these two pairings.When one of the band is fully occupied or fully unoccupied, the relevant physics is the well-known BCS and topological superfluids [12][13][14][15][16][17][18][19][20][21]; see Fig. 1b.The pairing is most intriguing when both bands are partially occupied near half filling.In the weak SOC, the spin in each band is just slightly tilted from z direction by SOC, such that the interband FFLO pairing is still favored; however, an inclusion of the intraband BCS pairing lowers the ground-state energy, and thus leads to a new quantum state, called the FFLO-BCS phase, in some proper parameter regimes.In the strong SOC, the spin in each band is almost spin polarized along x direction near the Fermi point when |λ sin(k)| ≫ h, which can be understood from the spin texture of the lower (-) and upper (+) bands (see Fig. 1d), In this case, the interband FFLO pairing becomes ineffective and the independent BCS pairing in each band is thus more energetically favorable.These fundamental pairings are totally different from the mean-field description.Hereafter all these pairings induced by SOC are called unconventional pairings.Phase diagrams.We now perform exact DMRG calculations with open boundary condition to unveil the intuitive in Fig. 1.In this work, the basic energy scale is chosen as t = 1, the on-site interaction is set to U = −4t, and the lattice length is set to L = 30 (see Supplementary Material).In our calculations, we retain 100 truncated states (which is sufficient) per DMRG block with the maximum truncation error ∼ 10 −5 t.We also perform standard mean-field calculations [21] to compare with the exact numerical results.
The phase diagrams from DMRG calculations are presented in Figs.2a-c, in which the filling factor is defined as n = N/L, with N the total number of particles in the optical lattice.The detailed criteria for determining different phases are presented in Supplementary Material.Without SOC (λ = 0), the normal gas (NG), BCS, and FFLO phases can be found in different parameter regimes [36]; see Fig. 2a.In the presence of SOC, both the interband FFLO and intraband BCS pairings become effective in the Hamiltonian (1), and a strong competition between them occurs, which gives rise to nontrivial quantum phases.Especially, in the weak SOC, we predict a new quantum state, called the FFLO-BCS phase; see Fig. 2b.The presence of both pairings breaks the exact solvability of the Hamiltonian (1) even in the mean-field level.In the intermediate and strong SOCs (λ > 0.28t; see Fig. 2c and Fig. 3), the interband FFLO pairing is completely suppressed, and only the BCS phase can emerge in the whole parameter regime, when both bands are partially occupied.This result is also in contrast to that from mean-field prediction presented in Fig. 2d, in which the FFLO phase always exists even in the stronger SOC regime (λ = t).
Notice that the Hamiltonian (1) has the particle-hole symmetry, c iσ → (−1) i c † iσ , where ↑ =↓ and verse versa.This symmetry ensures that the relevant physics in low filling factor (n < 1) is identical to that in high filling factor (n > 1), i.e., all the phase diagrams are symmetric with respect to µ = 0 (half filling at n = 1).For this reason, the FFLO-BCS and FFLO phases can only be observed around half filling, which corresponds to the largest "effective filling" in the optical lattice.
In Fig. 3, we plot the phase diagram in the h − λ plane at the half filling (n = 1), which further confirms that the interband FFLO pairing can be suppressed by the intraband BCS pairing.However, the situation for the FFLO-BCS phase is quite different.The FFLO-BCS phase is firstly enhanced by SOC in the weak SOC (λ ∈ [0, 0.13]t), due to conversion from the FFLO phase to the FFLO-BCS phase, and then it is gradually suppressed by the intraband BCS pairing in the intermediate SOC (λ ∈ [0.13, 0.28]t).When λ > 0.28t, the FFLO-BCS phase completely disappears in the whole parameter regime; see also Fig. 2c.These basic observations agree with the picture for pairings presented in Fig. 1.To see this result more clearly, we plot the ground-state energy E g as a function of λ in the inset of Fig. 3.This result shows explicitly that the transition between different phases lowers the ground-state energy.
Pairing correlation and pairing momentum distribution.The novel pairings in different phases can be identified directly from the pairing correlation function P (l, j) = c † l↓ c † l↑ c j↑ c j↓ , and the pairing momentum distribution P (k) = l,j P (l, j)e ik(l−j) /2L [30][31][32][33].These two quantities should be combined together to understand the unique features of each phase.In Fig. 4, we plot the pairing correlation function and spin polarization for the FFLO-BCS and BCS phases.Due to existence of the finite-momentum pairing in the FFLO-BCS phase, the pairing correlation exhibits strong oscillation in both magnitude and sign, and . Moreover, the local spin polarization also exhibits a similar oscillating behavior.These features are similar or identical to those in the FFLO phase without SOC [32], and their tiny difference can not be distinguished solely from the spatial pairing correlation.In contrast, for the BCS phase, the pairing correlation function shows a power decay, with respect to |l − j|, without node.No obvious spin polarization can be (the oscillation of spin polarization near the two ends is attributed to edge effects).
The corresponding pairing momentum distribution P (k) is plotted in Fig. 5a.When the SOC strength λ is smaller than a critical value λ c = 0.08t (see Fig. 3), two distinct peaks at ±Q ( = 0) can be observed.This phase is referred as the FFLO phase, according to our basic criteria in Supplementary Material and the results in Refs.[30][31][32][33].When increasing λ, the dip at zero momentum turns to a peak, while the other two peaks at ±Q remain almost unchanged.This is a strong signature for transition from the FFLO phase to the FFLO-BCS phase, where d 2 P (k)/dk 2 | k=0 = 0 serves as a basic criterion to determine the corresponding boundary between these two phases.This new phase is characterized by a unique three-peak structure in the pairing momentum distribution.As further increasing λ, we see a sudden change of three peaks to one peak at around λ ∼ 0.21t, which marks a transition from the FFLO-BCS phase to the BCS phase.This process is accompanied by a sudden jump of the population imbalance m = (N ↑ −N ↓ )/N , although the ground-state energy is still a smooth function (see inset of Fig. 3).It signals a crossover between  4), in which the population imbalance m(λ) is directly obtained from DMRG calculations.The difference between Eq. ( 4) and DMRG calculations is attributed to a finite size effect.(c) Q as a function of imbalance m.The boundary between difference phases is defined as the critical population imbalance mc.(d) mc as a function of the SOC strength.The symbols at λ = 0 represent the critical boundary from the superfluid phase to the NG at the half filling; see Franca et al. [50].In all subfigures, h = 1.5t and n = 1 are used.
different phases.This phase transition can therefore be detect directly by measuring both the pairing momentum distribution [42,43] and population imbalance [44][45][46].In Fig. 5b, we plot the center-of-mass momentum Q as a function of the SOC strength λ.We find that the SOC slightly decreases Q, and then increases it.Here we develop a simple model to better understand the behavior in the weak SOC.The Fermi points for two bands are assumed to have momenta ±k 1 and ±k 2 , respectively.These values are governed by the following equations: The center-of-mass momentum is determined by Here we adopt the simplified model in free space, with which the analytical expression can be obtained.We do not expect quantitatively modification of our conclusion by replacing k by sin(k) for a lattice model.Since the interband FFLO pairing can only emerge in the weak SOC, we can employ the Taylor expansion to obtain (see Supplementary Material), where A 2 = (3 + m 2 )n 2 π 2 /24h 2 and A 4 = (15 + 50m 2 − m 4 )n 4 π 4 /1920h 4 .We can recover the well-known result Q = nmπ by setting λ = 0 in Eq. ( 4) [30][31][32][33].We extract m(λ) from exact DMRG calculations, and find that Eq. ( 4) can be used to describe the evolution of Q in the weak SOC; see Fig. 5b.Since the SOC provides a new channel for spin flipping (the breaking of symmetry from U(1)⊗U(1) to U(1)), m is first decreased by SOC and so is Q.However, in the FFLO-BCS phase, the strong competition between the interband FFLO and intraband BCS pairings leads to an increasing of m when λ ∈ [0.11, 0.16]t, and Q is also increased accordingly.
In addition, without SOC, any population imbalance can give rise to the FFLO phase.This basic conclusion is also completely modified by SOC.We find that a finite population imbalance is required to realize the interband FFLO pairing in our model; see Fig. 5c.The relationship between m c , which is defined as the critical population imbalance between different phases, and the SOC strength λ is presented in Fig. 5d.Obviously, m c (λ = 0) = 0, as expected.
Discussion and conclusion.-Wediscuss the relevance of our results to topological superfluids and associated MF.In this work, we can not determine the topological boundary due to the absence of anomalous pairings [47].However, some valuable conclusions can still be made from these exactly numerical results.The topological BCS phase can occur when (2t + |µ|) 2 + ∆ 2 > h 2 and h 2 > (2t − |µ|) 2 + ∆ 2 [21,28], where µ is the chemical potential and ∆ = c i↑ c i↓ = 0 is the order parameter.These inequalities provide very stringent constraint for the realization of topological superfluids.On one hand, we hope ∆ can be as large as possible, which is essential for a large energy gap to protect the MF.On the other hand, the strong Zeeman field may destroy the pairing [48,49].Thus, there exists a balance between these two competing effects.The topological phase is best to be realized at the condition that ∆ 2 ≫ (2t ± |µ|) 2 , or equivalently, 2t ∼ |µ|.This regime can be realized by properly choosing the on-site interaction U in the strong SOC.This work resolves the most crucial problem related for this application, that is, we show the relevant physics is the trivial BCS and topological BCS superfluids in the whole parameter regime.We do not need to worry about the potential formation of the FFLO phase, which may destroy the topological phase.The topological phase can still be observed even in the case that both bands are partially occupied.Notice that the half filling condition at µ = 0 is impossible to support topological phase (see Fig. 2d), thus the best regime for achieving the topological phase should deviate a bit from this special point.
In a realistic experiment, the trapped potential always exists.This means that in the Hamiltonian (1) we should add another term V ( 2 L−1 ) 2 l (l − l c ) 2 (n l↑ + n l↓ ), where V is the trapping potential depth and l c = L/2.In the presence of the trapped potential, the particle-hole symmetry breaks down.Thus, the phase diagrams are not symmetric with respect to the half filling.However, the FFLO-BCS phase still remain, which can be verified also by the DMRG calculations when V = 5t.In addition, in such a case, the predicted three-peak structure of the pairing momentum distribution still exists at the center of the trapped potential.It implies that the FFLO-BCS phase is an intrinsic quantum state of the Hamiltonian (1), not the conventional phase separation; see also Fig. 4 for spin polarization.
We illustrate our basic idea in a lattice with length L = 30.We have verified that the similar conclusions can also be drawn for a much longer lattice L = 60 or 80. Generally, for a much longer lattice, more complicated center-of-mass momenta are admitted and more complicated FFLO-BCS phase is thus expected in the mixed pairing regime.However, the basic conclusions of unconventional pairing are unchanged.
To conclude, we have shown, using exact DMRG calculations, that the true pairing in an optical lattice can be completely modified by SOC, due to the induced triplet pairing.Especially, this system admits an exotic coexistence of the interband FFLO and intraband BCS pairings in the weak SOC coupling.However, in the intermediate and strong SOCs, the intraband BCS pairing dominates, and thus the relevant physics is the conventional BCS and nontrivial topological BCS phases in the whole parameter regime.These results shed great light on the realization of topological superfluids and associated MFs in the spin-orbit coupled lattice.

FIG. 1 :
FIG.1:(Color online).Unconventional pairings in an optical lattice with or without SOC.In all subfigures, the filled regimes represent the occupied quantum states, the dashed lines mark all the allowed pairings, and the arrows stand for the momentum-dependent spin polarization.(a) Without SOC, each band is fully polarized and any population imbalance can lead to the interband FFLO pairing.(b) In the presence of SOC, the intraband BCS pairing dominates when the chemical potential just occupies the lowest band.This condition is most relevant to achieve topological superfluids and MFs in the conventional mean-field description.(c-d) The interband FFLO and intraband BCS pairings for weak (c) and strong (d) SOC when both bands are partially occupied near half filling.In (c), the spin in each band is almost polarized along z direction.This situation allows the coexistence of these two pairings.However, in (d), the spin in each band is almost fully polarized along x direction when |λ sin(k)| ≫ |h|, and only the intraband BCS pairing becomes effective.

FIG. 3 :
FIG. 3: (Color online).Phase diagram in the h − λ plane at the half filling (n = 1).The inset shows the ground-state energy Eg as a function of the SOC strength λ at fixed h = 1.5t.

FIG. 4 :
FIG. 4: (Color online).Pairing correlation function (left) and spin polarization (right).(a-b) Basic features of the FFLO-BCS (a) and BCS (b) phases when h = 1.3t and n = 1.In the right column, the dotted line marks the local spin difference (diff.), which is defined as sz = n l↑ − n l↓ .The behaviors of the FFLO phase are always similar or identical to these of the FFLO-BCS phase, and thus are not displayed here.

FIG. 5 :
FIG. 5: (Color online).Pairing momentum distribution, center-of-mass momentum, and critical population imbalance.(a) The pairing momentum distribution P (k) for the different SOC strengths.(b) The symbols represent the obtained center-of-mass momentum Q, and the solid line is the prediction from Eq. (4), in which the population imbalance m(λ) is directly obtained from DMRG calculations.The difference between Eq. (4) and DMRG calculations is attributed to a finite size effect.(c) Q as a function of imbalance m.The boundary between difference phases is defined as the critical population imbalance mc.(d) mc as a function of the SOC strength.The symbols at λ = 0 represent the critical boundary from the superfluid phase to the NG at the half filling; see Franca et al.[50].In all subfigures, h = 1.5t and n = 1 are used.