Phases and collective modes of hardcore Bose-Fermi mixture in an optical lattice
We obtain the phase diagram of a Bose-Fermi mixture of hardcore spinless Bosons and spin-polarized Fermions with nearest neighbor intra-species interaction and on-site inter-species repulsion in an optical lattice at half-filling using a slave-boson mean-field theory. We show that such a system can have four possible phases which are a) supersolid Bosons coexisting with Fermions in the Mott state, b) Mott state of Bosons coexisting with Fermions in a metallic or charge-density wave state, c) a metallic Fermionic state coexisting with superfluid phase of Bosons, and d) Mott insulating state of Fermions and Bosons. We chart out the phase diagram of the system and provide analytical expressions for the phase boundaries within mean-field theory. We demonstrate that the transition between these phases are generically first order with the exception of that between the supersolid and the Mott states which is a continuous quantum phase transition. We also obtain the low-energy collective excitations of the system in these phases. Finally, we study the particle-hole excitations in the Mott insulating phase and use it to determine the dynamical critical exponent for the supersolid-Mott insulator transition. We discuss experiments which can test our theory.
pacs:67.60.Fp, 64.70.Tg, 73.22.Gk, 73.22.Lp
March 2, 2018
Recent experiments on ultracold trapped atomic gases have opened a new window onto the phases of quantum matter (1). A gas of Bosonic atoms in an optical or magnetic trap has been reversibly tuned between superfluid (SF) and insulating ground states by varying the strength of a periodic potential produced by standing optical waves (1). This transition has been explained on the basis of the Bose-Hubbard model with on-site repulsive interactions and hopping between nearest neighboring sites of the lattice. (2); (3). Further, theoretical studies of Bosonic atoms with spin and/or pseudospin have also been undertaken (4); (5). These studies have revealed a variety of interesting Mott (1) and supersolid (6) phases and superfluid-insulator transitions (3) in these systems. On the Fermionic side, the experimental studies have mainly concentrated on the observation of paired superfluid states (7) and the BCS-BEC crossover in such systems near a Feshbach resonance (8). More recently, it has been possible to generate mixtures of Fermionic and Bosonic atoms in a trap (9). Several theoretical studies followed soon, which established such Bose-Fermi mixtures to be interesting physical systems in their own right (10); (11), exhibiting exciting Mott phases in the presence of an optical lattice.
Many of the earlier studies of Bose-Fermi mixtures has been restricted to one-dimensional (1D) systems (12) or have concentrated on regimes where the coupling between Bosons and Fermions are weak (13). The existence of a supersolid (SS) phase in these system in such a weak coupling regime has also been predicted (14). Other works (10); (11) which have looked at the strong coupling regime have restricted themselves to integer filling factors of Bosons and Fermions (per spin) and have therefore not addressed the phenomenon of translational symmetry breaking and possible associated SS phases in the strongly interacting regimes of these systems. More recently, however, the authors of Ref. (15) have studied a mixture of spinless softcore Bosons with an on-site interaction and spin-polarized non-interacting Fermions at half-filling in a 3D optical lattice using dynamical mean-field theory (DMFT). Several interesting phases, including a SS phase of Bosons and charge-density wave(CDW) states of Fermions, have been found in Ref. (15).
In this work, we study a mixture of hardcore spinless Bosons and spin-polarized interacting Fermions in an optical lattice at half-filling using a slave-boson mean-field technique. We concentrate on the case where both the Bosons and the Fermions have a nearest-neighbor density-density repulsive interaction in addition to the usual on-site interaction term between them. We provide an analytical, albeit mean-field, phase diagram for the system and demonstrate that the ground state of such a system consists of four distinct phases, namely, a) a Mott insulating (MI) phase where both Fermions and Bosons are localized at the lattice sites, b) a metal+SF phase where the Fermions are in a metallic phase with a gapless Fermi surface and the Bosons are in a superfluid state, c) a SS phase where the Bosons are in the SS state while the Fermions are localized at the lattice site, and d) a CDW+MI phase of coexisting Fermions with weak density wave order along with Mott insulating Bosons. We show, within mean-field theory, that the transition between these phases are generically first order with the exception of that between the SS and the MI phases which is a continuous quantum phase transition. We also obtain the low-energy collective modes in the metal+SF, CDW+MI and the SS phases and demonstrate that they have linear dispersions with definite group velocities. Further, in the MI phase which has no gapless modes, we find the dispersion of the gapped particle-hole excitations and use it to determine the dynamical critical exponent for the continuous MI-SS transition. We also discuss realistic experiments which can test our theory.
The plan of rest of the paper is as follows. In Sec. II.1, we develop a slave-Boson mean-field theory for the system. This is followed by Sec. II.2, where the mean-field phase diagram is charted out. In Sec. III, we obtain the low-energy collective modes of the system in the metal+SF, CDW+MI and the SS phases. This is followed by Sec. IV, where we discuss the gapped particle-hole excitations of the MI phase and use it to determine for the SS-MI transition. We discuss relevant experiments which can test our theory and conclude in Sec. V
Ii Slave-Boson Mean-field Theory
ii.1 Mean-field equations
The Hamiltonian of the Bose-Fermi mixture in a -dimensional hypercubic lattice is described by the Hamiltonian
where () and () denote the annihilation and number operators for Fermions(Bosons) at site , and denote the nearest-neighbor interaction strengths and hopping amplitudes for the Fermions(Bosons) respectively, represents the amplitude for on-site interaction between the Fermions and the Bosons, and represents sum over nearest neighbor pairs on the lattice. In what follows, we shall study the Hamiltonian at half-filling. We note at the outset that this constraint of half-filling implies at each site.
To obtain an analytical understanding of the phases of these model, we first introduce a slave-boson representation for the Fermions: , where denotes annihilation operator for the slave-bosons and represents the annihilation operator for pseudo-fermions. We note that in this representation, the anticommutation relation for the Fermionic operator , where denotes the Kronecker delta, enforces the constraint on each site. In terms of these slave-bosons and pseudo-fermions, the Hamiltonians and in Eqs. 2 and 4 can be written as
where we have implemented the constraint using Lagrange multipliers at each site, and have used the fact that at each site . We note that the Hamiltonian is exact and is completely equivalent to (Eq. 1).
To make further progress, we proceed with mean-field approximation of . To this end, we first decompose the quartic hopping term in Eq. 5 using
where is the number of sites in the lattice and the hopping amplitudes and are given by
Here denotes average with respect to the ground state of the system. Next, we use a mean-field approximation for the constraint term and approximate the Lagrange multiplier field , where for sake of definiteness, we take to be even for sublattice sites. Such an ansatz for is motivated by the fact that it is the simplest mean-field ansatz that preserves the basic symmetries of the problem and, at the same time, allows for translational symmetry breaking in the pseudo-fermion sector. With these approximations, the mean-field Hamiltonian for the system can be written as
In writing Eq. 9, we have ignored the term since it merely renormalizes the chemical potential of the fermions and thus behave like a constant as long as we restrict ourselves to half-filling.
To obtain the ground state energy corresponding to this mean-field Hamiltonian, we now use a variational ansatz for the ground-state wavefunction
where denotes the Fermi wavevector for the pseudo-fermions and and are variational parameters which has to be determined by minimizing the ground state energy. Note that the variational wavefunction given by Eq. 10 has a two sublattice structure which allows for the possibility of translational symmetry broken phases. For the current system, where the Fermions and the Bosons both interact via nearest-neighbor density-density interaction terms, Eq. 10 is the simplest possible mean-field variational wavefunction which respects all the symmetries of the Hamiltonian. We would like to point out here that capturing the phases of a Bose-Fermi mixture with such interaction terms is beyond the scope of any single-site mean-field theory including single site DMFT.
The variational mean-field energy of the system can now be easily obtained and is given by
where we have used , , , and has to be determined from Eq. 8. The corresponding mean-field equations which determine the ground-state values of the variational parameters are given by and yields
Next, we evaluate the effective hopping amplitude and . To this end, first, let us consider the pseudo-fermion Hamiltonian . The energy spectrum of is given by , where , is the energy dispersion of free fermions in a hypercubic lattice in dimensions, and is the lattice spacing which we shall, from now on, set to unity. The density of states (DOS) corresponding to these fermions are therefore given by
where denotes the DOS of free fermions with tight-binding dispersion on a hypercubic lattice, , and we have used the relation . It is convenient to use Eq. 13 to express the expectation values over pseudo-fermion ground states in Eqs. 8 and 12 and one obtains
ii.2 Phase diagram
Eqs. 12 and 14 can be easily solved numerically to obtain the mean-field phase diagram for the system. However, before resorting to numerics, we provide a qualitative discussion of the nature of the phases.
We find that Eqs. 12 and 14 yields four distinct solutions which corresponds to four possible phases of the system. First, we find a MI phase with broken translational symmetry where both the fermions and the Bosons are localized. Such a phase corresponds to the solution (16)
Note that the divergence of corresponds to which in turn ensures that . Such a MI state corresponds to a intertwined checkerboard density-wave pattern where the Fermions are localized in sublattice ( for ) and the Bosons are localized in sublattice ( for ). The mean-field energy of this state is .
Second, we find a SS phase, where the Bosons are in a supersolid phase with coexisting density-wave and superfluid order and the Fermions are localized in a Mott phase. Such a state corresponds to the solution
Such a state has and and thus corresponds to a SS phase for the Bosons. Note that the realization of this state necessarily requires . For , and we recover the MI state where . The energy of the SS state is per site given by
Third, we find the MI+CDW state where the fermions show weak density-wave oscillations whereas the Bosons are localized in the MI state. This corresponds to the solution
where and are to be determined from a numerical solution of the mean-field equations
The energy of this state per site is given by
Finally, we find the state in which the superfluid Bosons coexist with metallic Fermions. This corresponds to the solution
Note that such a phase has and so that the Bosons are in an uniform superfluid state. Also, and in this state indicating that the Fermions are in a gapless uniform metallic state. The energy of this state per site is given by
The phase boundaries corresponding to these phases can be analytically computed using Eqs. 18, 21 and 23, provided and (which determines ) are obtained from numerical solutions of Eqs. 20 and Eq. 8. For the MI phase to occur, we must have which yields the conditions
Note that the condition which is necessary for the realization of the SS phase has to be simultaneously satisfied with the condition to make sure that the SS phase is actually a competing candidate to the MI state. The MI phase can indeed be realized in the parameter regime provided . The first condition shows that the MI phase is favored over the metal+SF phase for large and predicts a linear phase boundary in the plane with a slope of and intercept of between these two phases. Note that the MI phase always wins over the metal+SF phase if the nearest-neighbor interactions between the Bosons and Fermions are large compared to their hopping amplitudes making negative. The final condition indicates that the phase boundary between the MI and MI+CDW phases is independent of . The former phase is favored over the latter for larger and smaller .
Similarly for the SS phase to occur one needs and , which yields
We note that the SS phase is favored when the nearest-neighbor interaction between the Bosons are weak compared to their hopping amplitudes making positive and when is small enough so that . Also, from the conditions in Eq. II.2 (obtained using ), we note that for a given and , the boundary between the metal+SF and the SS phases is a parabola in the plane given by while that between the SS and the MI state is a line given by .
Finally, the condition for occurrence of the metal+SF phase is given by and is given by
The last condition in Eq. II.2 determines the phase boundary between the metal+SF and the MI+CDW phases which depends on value of . However, numerically, we find that for , and in this regime, the phase boundary between these phases occurs at for all and . Note that strictly at , the Fermionic state is metallic; however a CDW gap opens up in the Fermionic spectrum for an infinitesimal finite .
To verify the above-mentioned qualitative arguments and to find a precise phase diagram for the system, we numerically solve Eqs. 12 ,14, and 15, for and for representative values . We plot the ground state phase diagram as a function of and in Figs. 1 and 2. We find that the numerical results agree well with the qualitative arguments. Figs. 1 and 2 indicate that the phase boundary between the CDW+MI and MI phases is independent of as noted earlier. The linear and the parabolic nature of the phase boundaries between the MI and SS phases and the SS and metal+SF phases respectively can also be easily verified from the Figs. and are in accordance with the qualitative discussion. We note that one of the effects of nearest-neighbor repulsion between the Fermions is to enhance the SS phase which occupies a larger region of phase space in Fig. 2 () than in Fig. 1 (). Such an interaction, for , also favors the Mott phase over the CDW+MI phase as can also be seen from Figs. 1 and 2.
To determine the nature of transition between the different phases, we plot the ground state values as a function of for and in Fig. 3. Such a plot clearly shows that the transition between the metal+SF and the SS phases is, within the mean-field theory considered here, first order and is accompanied by a jump in the value of . In contrast, the SS-MI transition turns out to be continuous. A similar plot of ground state values as a function of for and , shown in Fig. 4, indicates that the transition between the CDW+MI and the MI phase is also discontinuous and is accompanied by a jump in the ground state value of .
Next, we compare our phase diagram with that obtained from DMFT in Ref. (15). This can be done in the regime of large positive (which correspond to ) which was the case treated in Ref. (15). We find that the two phase diagrams qualitatively agree in the sense that both yield SS and MI phases in these limit. The difference lies in the fact that our mean-field predicts a second-order transition between the two phases whereas DMFT yields a narrow region of coexistence. This is presumably an effect of quantum fluctuation which is not captured within the mean-field theory. In addition, we also find a region of metal+SF phase at low which was not seen in Ref. (15).
Finally, we would like to point out that the slave-boson mean-field phase diagram obtained above yields qualitatively correct phase diagram, but not a quantitatively correct one. This can be most clearly seen by noting that our Hamiltonian reduces to an effective Falicov-Kimball (FK) model (17) in the limit . This is most easily seen by writing our starting Hamiltonian (Eq. 1) for
At half-filling the Bosons are localized in the B sublattice so that . Thus the Fermions have a CDW instability even for an infinitesimal due to nesting for half-filling on a square lattice. Consequently, the FK model at half-filling is insulating for any infinitesimal , as known from several earlier studies (18). Such a CDW instability, which can be easily captured by weak-coupling mean-field theory, is not straightforward to obtain in our strong coupling slave-boson mean-field approach which predicts a finite critical for the transition from metal+SF to the MI phase. Note however that the slave-boson mean-field theory does predict a CDW+MI state for weak , but for small negative , as can be seen from Fig. 1. This indicates that the phase diagram obtained has qualitatively, but not quantitatively, correct features.
Iii Collective Modes
The phases of the Bose-Fermi mixture discussed in the previous section allows for two types of excitations. The first type is the low-energy gapless collective modes that are present in the metal+SF, CDW+MI, and the SS phases of the system. These are the gapless Goldstone modes corresponding to the Boson and the slave-Boson fields. For all of these modes, the pseudo-Fermion sector remain gapped and do not contribute to their dispersion. The gapped modes corresponds to particle-hole excitations in the CDW+MI, SS and the Mott states. In this section, we concentrate on the gapless collective modes in the CDW+MI, SS, and the metal+SF phases.
To obtain the dispersion, we first consider a time-dependent variational wave-function
where , , , and are space-time dependent fields. Note that in the static limit, the ground-state of the system corresponds to , for sites, and , so that reduces to .
The Lagrangian can now be computed using the variational wave-function and one obtains
where is the Fermion number density different on and sublattices and all time dependence of the fields are kept implicit for the sake of clarity. To obtain the collective modes, we now write , , and expand the Lagrangian to quadratic order in , , and . Then a variation of this Lagrangian with respect to , , and and consequent adjustment of values of the parameters and following Ref. (19), yields the equations for the low-energy collective modes (we set from now on)
where we have taken the Fourier transform of all the fields, , and and are given by
It is important to note that Eqs. 30 and 31 holds when while Eqs. 32 and 33 holds when . Thus none of these equations are valid in the MI phase which do not support any low-energy collective modes. The gapped modes of the MI phases will be obtained in the next section.
In the SS phase where and , the collective mode corresponds to the low-energy excitations of the Bosons and are given by Eqs. 32 and 33. A simple set of standard manipulations of these equations yield the dispersion of the collective modes , where
Note that for low momentum, we get a gapless linearly dispersing collective mode with velocity . Similarly for the MI+CDW phase, where and , the collective mode corresponds to the low energy excitations of the pseudo-bosons and can be obtained by solving Eqs. 30 and 31. Since the pseudo-Fermion sector is always gapped in this phase (), the collective mode here corresponds to the density-wave mode of the real Fermions. These modes have linear dispersion where
Thus for low momenta, we again get a gapless linearly dispersing collective mode with velocity .
Finally for the metal+SS phase, all the Eqs. 30..33 hold and they need to be solved simultaneously. In this phase since , we find that and . Solving these equations, one finds two collective modes with linear dispersions where are given by
These collective modes result from the hybridization of the Bogoliubov modes of the Bosons and the density-wave modes for the metallic Fermions. This fact can be easily checked by putting in Eq. 37 by which one can retrieve these modes with velocities for the Bosons and for the Fermions. As increases, the hybridization between these modes become stronger until the velocity touches zero at which is precisely the condition for the metal+SF phase to become unstable to the SS phase.
Iv Gapped Modes in the MI phase
The MI phase, in contrast to the other three phases of the system, do not support a gapless mode. The lowest-lying excitations of such a state with conserved number density are particle-hole excitations. Such excitations, are of two types. The first type, shown in second panel of Fig. 5, involves particle and hole excitations that spatially well-separated while the second type, shown in second panel of Fig. 6, involves particle and hole excitations in nearest-neighbor sites which forms a dipole. In what follows, we first compute the energies of both these excitations using perturbation theory up to second order in which are supposed to small in the MI phase.
Such an energy estimate can be easily carried out by strong-coupling perturbation theory developed in Ref. (20) in context single species Bose-Hubbard model. The generalization is largely trivial, except for one important detail. In the standard Bose-Hubbard model studied in Ref. (20), any particle/hole excitation could have lowering of energy via nearest-neighbor hopping which is process. In contrast, as can be seen from Figs. 5 and 6, it is not possible for the particle-hole or dipole excitations to directly hop to the next site since such a direct hop always take us out the low energy manifold of states in the Mott limit. In particular, we note that any kinetic energy gain of the particle-hole or dipole excitation must occur via hopping of the partice/hole to the second-neighbor sites and hence necessarily leads to energy gain.
We first compute the excitation energy of the Bosonic(Fermionic) particle-hole pair when they are far apart. The on-site energy of creating such a pair is while the energy-lowering due to hopping of each of the particle and the hole is given by . The energy of the Mott state to second order in perturbation theory is so that the excitation energy of the particle-hole pair is
We note that in the limit of large , where the mean-field results are expected to be accurate, we have
The Mott state is destabilized in favor of the SS phase when .
Next, we compute the excitation energy of the Bosonic/Fermionic dipole state. We are going to do this in the limit of . We note at the outset that once such a dipolar excitation is created, it remains stable, i.e., the hole can not hop away arbitrarily far away from the particle. It can be easily verified from Fig. 6 that such hoppings generate higher-energy end states and takes one out of the low energy manifold of states.
The on-site energy cost for creating such an excitation is while the hopping process, shown schematically in Fig. 6, necessarily involves hopping of both Fermions and Bosons and leads to an energy gain of . Thus the net energy of such a dipole excitation is given by
We note that for , where our mean-field theory holds, it is always energetically favorable to create particles and holes well-separated since . However, the dipolar excitations may becomes favorable in low dimension and for large . In this case, for and in this limit, the dipolar excitations would be preferred in destabilizing the MI phase. We shall not discuss this issue here any further since this is clearly beyond the scope of our mean-field theory.
Finally, we compute the dispersion of the gapped particle-hole excitations within mean-field theory where the particle and the hole are spatially well-separated and do not interact. To this end, we temporarily relax the constraint of conservation of particle-number and consider the energy of excitations of adding a particle and a hole to the Mott state. The physical particle-hole excitation energy can then be computed from . To compute the energy of these particel/hole excitations, we adapt a time-dependent variational approach as done in Ref. (19) for single species Bosons in an optical lattice. We begin with the variational wave-function of Bosons and slave Bosons(Fermions) given by
where denotes sublattice indices. The coefficients and satisfy the normalization condition and . We note that at equilibrium ,