Modulated pair condensate of -orbital ultracold fermions
We show that an interesting of pairing occurs for spin-imbalanced Fermi gases under a specific experimental condition—the spin up and spin down Fermi levels lying within the and orbital bands of an optical lattice, respectively. The pairs condense at a finite momentum equal to the sum of the two Fermi momenta of spin up and spin down fermions and form a -orbital pair condensate. This momentum dependence has been seen before in the spin- and charge- density waves, but it differs from the usual -wave superfluids such as He, where the orbital symmetry refers to the relative motion within each pair. Our conclusion is based on the density matrix renormalization group analysis for the one-dimensional (1D) system and mean-field theory for the quasi-1D system. The phase diagram of the quasi-1D system is calculated, showing that the -orbital pair condensate occurs in a wide range of fillings. In the strongly attractive limit, the system realizes an unconventional BEC beyond Feynman’s no-node theorem. The possible experimental signatures of this phase in molecule projection experiment are discussed.
pacs:03.75.Ss, 71.10.Fd, 37.10.Jk, 05.30.Fk
Pairing with mismatched Fermi surfaces has long fascinated researchers in the fields of heavy fermion and organic superconductors, color superconductivity in quark matter Casalbuoni and Nardulli (2004), and, most recently, ultracold Fermi gases with spin imbalance Giorgini et al. (2008); Sheehy and Radzihovsky (2007); Radzihovsky and Sheehy (2010); Parish et al. (2007). In a classic two-component model for superconductivity, the mismatch arises from the spin polarization of fermions in the same energy band. Its effect was predicted to produce intriguing, unconventional superfluids such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Fulde and Ferrell (1964); ffl (), deformed Fermi surface Müther and Sedrakian (2002); Sedrakian et al. (2005), and breached pair phases Liu and Wilczek (2003); Forbes et al. (2005). The limiting case of large spin imbalance was also studied to explore the formation of Fermi polarons Schirotzek et al. (2009). In parallel, the behavior of particles in the higher orbital bands of optical lattices, due to large filling factors, thermal excitations or strong interactions, is widely studied for novel orbital orderings of both bosons Liu and Wu (2006); Isacsson and Girvin (2005); Kuklov (2006) and fermions Zhao and Liu (2008a); Wu (2008) with repulsive interactions. Recently, interband pairing of unpolarized fermions was shown theoretically to give rise to Cooper pair density waves Nikolić et al. (2010).
In this article, we report a fermion pairing phase resulting from the interplay of Fermi surface mismatch and -orbital band physics. In such a phase, the pair condensate wave function is spatially modulated and has a -wave symmetry. This phase arises in an attractive two-component Fermi gas on anisotropic optical lattices under a previously unexplored condition of spin imbalance. Namely the majority () spin and the minority () spin occupy up to Fermi levels lying in the and bands, respectively. We show that pairings take place near the respective Fermi surfaces of the spin fermions in band and fermions in band. This induces a modulated -orbital pair condensate that differs from the usual -wave superfluids such as He. The state requires only an on-site isotropic contact interaction and the pair is a spin singlet, while the He -wave superconductivity has to involve anisotropic interaction and spin triplet. The modulation wave vector of the order parameter is , where , are Fermi momenta for spin and species, respectively. This momentum dependence is an unprecedented signature in superfluids other than the spin- and charge- density waves. In the strongly attractive limit, tightly bounded pairs condense at finite momentum , which realizes an unconventional Bose-Einstein condensate beyond Feynman’s no-node theorem Liu and Wu (2006); Isacsson and Girvin (2005); Wu (2009); Kuklov (2006); Radzihovsky and Choi (2009).
The system under consideration is at zero temperature and consists of two-component fermions in a three-dimensional (3D) cubic optical lattice with lattice constant , described by the Hamiltonian
Here is the fermionic field operator at with spin , is the lattice potential, is the chemical potential for spin fermions, and is the contact attraction which can be tuned by the Feshbach resonance. In particular, we consider the case where the lattice potential in the (parallel) direction is much weaker than the other two (transverse) directions, so the system behaves quasi-one-dimensionally.
We expand , where is the th band Wannier function at lattice site with the annihilation operator in Wannier basis. As a result, we obtain the usual attractive Hubbard model with nearest-neighbor hopping between th site with orbital band and th site with orbital band
and on-site attraction between orbitals
The lowest two energy bands are the and band (the and band are much higher in energy because of tighter confinement in the transverse directions). For brevity the band is simply called band in the following. By filling fermions with spin to the band and spin to the band, the Hamiltonian becomes
Here, and denote the nearest neighboring lattice sites in parallel and transverse directions. and are the hopping amplitudes along the parallel direction for the - and -band fermions respectively, while are the hopping amplitudes in transverse directions. () is the annihilation operator at lattice site for -band (-band ) fermions. are the number operators, and are the corresponding chemical potentials. is the attractive on-site interaction between - and -band fermions and can be tuned by changing the scattering length using Feshbach resonance. is related to the band gap. In the tight binding region we assume , and consequently the -band fully filled spin fermions are dynamically inert and not included in the .
Iii DMRG calculation for 1D case
First we consider the pairing problem in the simplest case of 1D (), which is schematically shown in Fig. 1(a). The two relevant Fermi momenta are (for -band fermions) and (for -band fermions). From a weak coupling point of view, to pair fermions of opposite spin near their respective Fermi surfaces, the Cooper pairs have to carry finite center-of-mass momentum (CMM) due to Fermi surface mismatch. Furthermore, in order for all Cooper pairs to have roughly the same CMM, the only choice is to pair fermions of opposite chirality. Note that the dispersion of band is inverted with respect to the band, so pairing occurs between fermions with momenta of the same sign but opposite group velocities. These elementary considerations show that the CMM of the pair should be approximately the sum of two Fermi momenta,
This result differs from that of the usual one-dimensional spin imbalanced fermions within the same band, where the FFLO pair momentum is the difference, , as found in a two-leg-ladder system Feiguin and Heidrich-Meisner (2009).
Mean-field theory and weak coupling consideration can provide only a qualitative picture for 1D problems. To unambiguously identify the nature of the ground state, we use density matrix renormalization group (DMRG) to compute the pair correlation function. In the numerical calculations, we used parameters as the unit of energy, , , , in which the ratio between and is chosen according to typical tight-binding bandwidth ratio. is tunable with Feshbach resonance and in the following calculation we will focus on foo (). The truncation error is controlled in the order of or less. Equation (5) predicts . Figure 1(b) shows the pairing correlation function in real space as a function of for a chain of sites with open boundary condition, where the indices and are real space positions. Since the system only has algebraic order, decays with according to a power law. On top of this, however, there is also an obvious oscillation. A curve fit with formula , shown in Fig. 1(b), yields a period of , which is in good agreement with the wave number given by Eq. (5) before. The Fourier transform of the pair correlation function
is peaked at (to be plotted in Sec. VI). These features of the pair correlation function are the signature of the existence of the CMM pairing in our system Feiguin and Heidrich-Meisner (2007); Zhao and Liu (2008b).
Iv Meanfield analysis for quasi 1d case
Now we move on to the quasi-1D system where a weak transverse hopping is added. We carry out a mean-field analysis of Hamiltonian by introducing the - pairing order parameter
where means the ground-state expectation value. Two different trial ground states are investigated, the exponential wave , which is analogous to the Fulde-Ferrell phase and the cosine wave , which is analogous to the Larkin-Ovchinnikov phase. and are determined self-consistently by minimization of ground-state free energy . Transverse hopping introduces a small Fermi surface curvature and spoils the perfect nesting condition as in the pure 1D problem above. However, the curvature is small for weak . Thus, we expect pointing almost along the parallel direction, , in order to maximize the phase space of pairing.
The mean-field Hamiltonian for the exponential wave can be diagonalized in momentum space by standard procedure. We get the ground state energy
with the self-consistent gap equation for
Here, is lattice momentum, is the total number of sites, is a step function, and is the eigenenergy of the Bogoliubov quasiparticles. As evident from these formulas, the pairing occurs between an -band fermion of momentum and a -band fermion of momentum with dispersion and , respectively.
The cosine wave is spatially inhomogeneous. A full mean-field analysis requires solving the Bogoliubov-de Gennes equation to determine the gap profile self-consistently. Here we are interested only in computing the free energy for the ansatz to compare with the exponential wave case. Thus, it is sufficient to numerically diagonalize the full Hamiltonian Eq. (4) for a finite size lattice. We introduce a vector of dimension
where is the discrete momentum in the direction. The components of obey anticommutation relation , where labels the corresponding operator component of . The Hamiltonian takes the compact form . Since is real and symmetric, it can be diagonalized by an orthogonal transformation to yield eigenvalues . The new operators automatically obey the fermionic anticommutation relationship . We get the ground state energy,
and the gap equation,
Here, labels the eigenenergy, and , labels the matrix elements corresponding to the original , operators in the gap equation.
The parameters used in the mean-field calculations are the same as in the 1D case with small ’s added, and we still expect that the order parameter has the momentum around as before. By self-consistently solving for and , in the case , the ground state is the cosine wave phase with and . The ground state energy per site is , lower than the noninteracting value . When , the ground state is also the cosine wave phase with and . The ground state energy per site is , lower than the noninteracting value . These results confirm that (i) the cosine wave state has lower energy than the exponential wave state, (ii) the order parameter has the momentum close to the prediction of Eq. (5), and (iii) larger transverse hopping tends to destroy the -orbital pair condensate since the energy gain for larger transverse hopping is much smaller than for smaller transverse hopping.
An interesting feature of the -orbital pair condensate in quasi-1D is the possible existence of Fermi surfaces with gapless energy spectrum. We monitor the fermion occupation number, i.e. and for increasing transverse hopping. The results are shown in Fig. 2. For small , they take the usual BCS form and vary smoothly from 1 (red) to 0 (blue) across the bare Fermi surface (with interaction turned off), as shown in Figs. 2(a) and 2(c) for . For larger transverse hopping, sharp Fermi surfaces characterized by a sudden jump in and appear. This is clearly shown in Figs. 2(b) and 2(d) for as the occupation number changes discontinuously from 1 (red) to 0 (blue). It can be understood qualitatively as follows. As increases, the original Fermi surfaces acquire a larger curvature in the transverse directions and the pairing condition in Eq. (5) cannot be satisfied everywhere anymore. Therefore in some regions fermions are not paired and Fermi surfaces survive. One should also note that the calculation is based on the assumption that , which predicts that is in the parallel direction. This prediction should fail as increases beyond certain critical values.
V Phase Diagram
Now, we systematically explore the phases of our system for general band filling and spin imbalance. Since we have - and - bands with different bandwidths, we introduce two dimensionless quantities for the chemical potentials and
Thus, for a non-interacting system, control the filling for the and -band fermions respectively. We then define the quantities
as the parameters controlling the average filling and polarization in the phase diagram. The phase at is the same as the state at , since the transformation gives , and the mean-field Hamiltonian with is identical to Hamiltonian with via a particle-hole transformation up to a constant.
We have four possible phases in such a system as shown in Fig. 3. As before, we ignored the inert fully filled band of spin fermions. We consider the band of spin fermions and band of spin fermions. When one of these two bands is empty and the other is filled, the pairing does not happen and we call it normal phase I (N1) as in Fig. 3(a). When one of these two bands is fully filled and the other is partially filled, the pairing also does not happen since the fully filled band is inert. We call it normal phase II (N2) as in Fig. 3(b). When both of them are partially filled, fermions near Fermi surfaces from the two bands will be paired and the system is in superfluid phases as shown in Figs. 3(c) and 3(d). In the superfluid regime, when is small, the pairing momentum prefers and we call it commensurate -orbital pair condensate (CpPC). It is a special case of the -orbital pair condensate, where the occupation numbers of -band spin fermions and -band spin fermions are the same. It is similar to the conventional unpolarized pairing (BCS), where the spin fermions and spin fermions have the same population. However, in BCS pairing the CMM of the pair has the property , while here . To understand the momentum preference, note that in conventional BCS case, the two species of fermions have the same energy spectrum and the pairing is between two fermions with opposite momenta, which leads to the CMM of pair . Here, the structure of energy spectrum of band is different from band. The equal occupation numbers mean , which gives rise to , as shown in Fig. 3(c). At last, when is large, the pairing momentum stays at a general and the occupation number for the two species of fermions differ. We call it incommensurate -orbital pair condensate (IpPC) as shown in Fig. 3(d).
To determine the phases, we minimize the free energy as a function of the pairing amplitude and pairing momentum by mean-field analysis using the cosine wave function as outlined in the previous section. When the minimum is realized at , it is normal phase. When is finite, there are two possibilities. When , it is CpPC. When , it is IpPC. For the transition between superfluid and normal phase, and the transition between CpPC and IpPC, the behaviors of free energy show that the phase transitions are first order in a lattice system. Between the superfluid and normal phases, near the phase transition, changes suddenly from to finite, and the free energy shows two local minima at and . Between CpPC and IpPC, the pairing momentum changes from to discontinuously, and the free energy as a function of also has two local minima at and . Thus, they are first-order phase transitions according to our mean field analysis. Therefore, we can determine the phase boundaries between normal phase and superfluid phase by monitoring changing from zero to finite. We can also monitor changing from to to determine the phase boundaries between CpPC and IpPC.
In Fig. 4, we present a phase diagram for . The x’s in Fig. 4 show the data points for the phase boundary obtained from the numerical procedure, and by connecting them we get the phase boundaries. An illustrative physical understanding about this phase diagram is as follows. In Fig. 4, when chemical potential difference is small and the two bands are still partially filled to ensure the pairing, the system tends to stay in CpPC where . It is similar to the conventional BCS superfluid case. As becomes larger, as long as the average filling is not too large or small and the two bands are still both partially filled, the pairing persists despite the spin imbalance and the system is in IpPC. If gets more and more negative, the average filling becomes smaller and smaller, and at certain , band of spin fermions will be empty and the system will become N1 without pairing. Similarly, when is large and positive, the average filling is very high and at certain , the band of spin fermions will be fully occupied, and the system becomes N2 without pairing. The almost straight phase boundaries in Fig. 4 between IpPC and normal phases indicate that these phase transitions are due to the change of band occupation as empty partially filled fully filled. In Fig. 4, the phase boundary between IpPC and N1 corresponds to the critical condition that the band of spin fermions is partially filled while the band of spin fermion becomes empty, and the almost straight phase boundary corresponds to the condition that (but, as before, this is only an approximate argument due to the presence of interaction). Similarly, the almost straight phase boundary between IpPC and N2 corresponds to the condition that the band of spin fermions becomes fully filled, while the band of spin fermions is partially filled, or . All the phase transition lines in Fig. 4 are mean field results, and these straight lines are expected to be corrected by quantum critical fluctuations. The phase diagram shows that the -orbital pair condensate happens in large parameter regimes and is closely related to the band and orbital properties in the optical lattice systems.
Vi Signature of the -orbital pair condensate in Molecule Projection Experiment
The -orbital pair condensate phase can inspire important experimental signatures for finite momentum condensation of bosonic molecules in higher orbital bands. By fast sweeping the magnetic field (and thus the interaction) from the BCS region to the deep BEC region across a Feshbach resonance, the BCS pairs are projected onto Feshbach molecules, which can be further probed for example by time-of-flight images Liu and Wu (2006). The bosons produced effectively reside in band and are stable, since by Pauli blocking the filled -band fermions will prevent the the -wave bosons from decaying Liu and Wu (2006). Here, we use a simple model Diener and Ho (2004); Altman and Vishwanath (2005) to evaluate the momentum distribution of molecules after projection
where is the molecular wave function, and the correlation function can be evaluated within mean field theory Altman and Vishwanath (2005). For fast sweeps, the molecular size is small compared to lattice constant and its wave function can be approximated by a delta function in real space (a constant in momentum space). By this assumption, is the same quantity as in Eq. (6). Figure 5(a) shows the of -wave Feshbach molecules and a peak is located at . Figure 5(b) shows from Eq. (6), based on the DMRG results shown in Fig. 1(b).The time-of-flight experiment is predicted to distribute peaks corresponding to that in Fig. 5. Note that for the 1D problem (Fig. 5(b)), the delta-function peak is replaced by a cusp characteristic of power law due to the lack of long range order.
We thank Chungwei Lin for helpful discussions. This work is supported by ARO Grant No. W911NF-07-1-0293.
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