Microscopic derivation of Hubbard parameters for cold atomic gases

Microscopic derivation of Hubbard parameters for cold atomic gases

Hans Peter Büchler Institute for Theoretical Physics III, University of Stuttgart, Germany
July 16, 2019

We study the exact solution for two atomic particles in an optical lattice interacting via a Feshbach resonance. The analysis includes the influence of all higher bands, as well as the proper renormalization of molecular energy in the closed channel. Using an expansion in Bloch waves, we show that the problem reduces to a simple matrix equation, which can be solved numerically very efficient. This exact solution allows for the precise determination of the parameters in the Hubbard model and the two-particle bound state energy. We identify the regime, where a single band Hubbard model fails to describe the scattering of the atoms as well as the bound states.

Cold atomic gases in optical lattices represent a perfect laboratory system for the quantum simulation of strongly correlated many-body systems described by Hubbard models greiner02 (); jaksch04 (); bloch08 (). Recently, experimental and theoretical efforts focus on the observation of a Fermionic Mott insulator jordens09 (); schneider08 (), and the ultimate goal towards the realization of magnetic and superconducting phases. The quantitative understanding of these experimental results and the comparison with the theoretical predictions require a precise knowledge of the parameters in the Hubbard model for cold atomic gases interacting with a Feshbach resonance leo08 (). In this letter, we present the solution to the two-particle problem in an optical lattice interacting via a Feshbach resonance, and provide a microscopic derivation of the parameters in the Hubbard model and the two-particle bound state energies.

The two-particle interaction potential between particles in ultra-cold atomic gases is well described by the pseudo-potential or in the presence of a Feshbach resonances within a two-channel model petrov05 (); bloch08 (). The two-particle problem within confined geometries has extensively been studied for the one-dimensional setup with strong transverse confining olshanii98 (); bolda03 (), and the harmonic trapping potential busch98 (). In addition, the influence of optical lattices has been studied for the deep lattices, where the influence of higher bands have been included semiclassicaly fedichev04 () or using the exact solution for a harmonic oscillator within each well of the lattice dickerscheid05 (); diener06 (); wouters06 (). Here, we analyze the two-particle problem interacting via a Feshbach resonance in a three-dimensional optical lattice and show that the equations can be efficiently solved numerically. The solution provides the exact scattering properties and bound state energies of two-particles in an optical lattice of arbitrary strength, see Fig. 1.

Figure 1: Exact bound state energies (red line) for two-atoms in an optical lattice of strength and . The additional bound states (dashed grey lines) are weakly coupled to atoms in the lowest Bloch band, i.e., . The green dotted line denotes the bound state energies predicted from the Hubbard model with the on-site interaction determined by Eq. (10); its deviations from the exact bound state indicate the break down of the Hubbard model. The blue dotted line denotes the bound states neglecting the correction .

From the exact scattering amplitude, we find the microscopic derivation for the interaction parameters in the Hubbard model. The simplest Hubbard model describes bosonic particles with creation (annihilation) operators (), and on-site interaction (extension to fermionic particles with spin is straightforward),


The hopping energies derive from a single particle band structure calculation, and are related to the dispersion relation in the lowest Bloch band . In turn, the on-site interaction is conventionally derived for weak interaction strengths and deep optical lattices by replacing the exact pseudo-potential by a -function interaction and restricting the system to the lowest Bloch band jaksch98 (); the latter step corresponds to introducing a short distance cut-off comparable to the lattice spacing . This approach is restricted to weak interactions with ; here, is the -wave scattering length. In the general situation, the precise derivation of the interaction potential in the Hubbard model for arbitrary interaction is obtained by comparing the exact scattering properties of two particles in an optical lattice with the scattering amplitude predicted from the Hubbard model. This approach is in analogy to the description of the interaction bewteen cold gases in free space in terms of a pseudo-potential: the strength of the pseudo-potential is fixed by the condition to reproduce the exact scattering properties.

In the following, the interaction between the two-particles is given by a Feshbach resonance, which can be conveniently described by the two-channel approach. Then, the Feshbach resonance is characterized by the detuning and the coupling between the open and closed channel, and gives rise to the scattering amplitude petrov05 ()


with the incoming momentum and the reduced mass. The scattering length and the effective range are experimentally accessible by measuring the bound state energy of the molecules across the Feshbach resonance claussen03 ().

The two particles in the open channel are described by the wave function with and the position of the particles. In order to capture the above characteristics of a Feshbach resonance, it is enough to describe the closed channel by a single molecular state . Then, the Schrödinger equation for the energy eigenstates reduces to


where the single particle physics is described by the Hamiltonians with the optical lattices and the molecular mass . Furthermore, we have introduced the relative and center of mass coordinates . The properties of the Feshbach resonance are determined by the coupling strength and the bare detuning , while accounts for a regularization of the coupling with cut-off . In the limit , the bare detuning entering the microscopic theory is related to the physical observable detuning via with the renormalization .

The periodic structure of the optical lattice is characterized by the lattice vectors . The single particle properties are then fully determined by the Bloch wave functions , , and with the corresponding band energies , , and ; here, , and are the quasi-momentum, while , , and characterize the different Bloch bands. In the following, we measure energies with respect to the ground state energy of two-particles in the lowest Bloch band, i.e., . The discrete translation invariance provides the conservation of the total quasi-momentum . Then, the general solution with fixed total quasi-momentum can be written as

and . Inserting this expansion in Eq. (3), we obtain


Here, we have introduced the notation , and the characteristic coupling energy with the volume of the unit cell. The dimensionless coupling elements reduce to

with the notation , while denotes the quantization volume. Substituting the bare detuning with the physical detuning by adding on both sides of Eq. (4) the renormalization , we obtain


The matrix describes the shift of the Feshbach resonance due to the change in dispersion relation of the particles in the open channel; this phenomena is in analogy to the lamb shift of atoms in a cavity brune94 (). It takes the form ( denotes the volume of the Brioulline zone)

The quantities and are the energies and coupling parameters for the system in absence of an optical lattice. The first term in the above equation describes the influence of higher bands, while the second term appears from the renormalization. The divergent parts in the two terms cancel each other, and remains finite in the limit . This behavior can be easily understood: for large Bloch bands, the influence of the optical lattices vanishes and the coupling elements and energies reduce to the values of the free system. Then, the terms in the bracket cancel each other, and the summation over the Bloch bands converges. For a finite short distance cut-off , the corrections vanishes with ; i.e., the convergence is very slow in the number of Bloch bands.

In the following, we discuss the setup with a three-dimensional cubic lattice with lattice spacing and recoil energy . For equal particle species and far detuned optical lattice, the relative strengths of the lattice potentials naturally satisfy . We focus on a wide Feshbach resonance; the generalization to a narrow Feshbach resonance is straightforward. A wide Feshbach resonance is obtained in the limit with a fixed -wave scattering length , and the energies and in the first term in Eq. (5) can be dropped.

Figure 2: Bound state energies (red line) for different lattice depths: (a) For weak optical lattices with , the Bloch bands in three-dimensions nearly overlap. (b) For deep optical lattices with , the bound states energies are compared with the energies obtained by replacing the optical lattice by a harmonic well with trapping frequency (dashed blue line).

Bound states: The equation for the energies of the repulsive and attractive bound states reduces to the eigenvalue equation


with the molecular wave function of the bound state, and ; the numerical solution is shown in Fig. 1 and Fig. 2. For a fixed value of the -wave scattering length and quasi-momentum , there appear now several bound states. This behavior is in strong contrast to the free system, where for a fixed center of mass motion only a single bound state exists for a repulsive scattering length . Here, the appearance of several bound states is a consequence of the reduced translation symmetry: molecular states and two-particles states in the open channel differing in center of mass motion by a reciprocal lattice vector are coupled the periodic lattice. These coupling strength are given by the overlaps .

The matrix elements are fully determined by the single particle properties such as the Bloch wave functions and Band structure, and can be efficiently determined numerically. For the three-dimensional cubic lattice above, the single particle wave function separate for each space direction, i.e., , and it is therefore sufficient to determine the Bloch wave functions and energies for a one-dimensinal setup; the Bloch wave functions are determined using reciprocal lattice vectors. The integration and the summation over the different Bloch bands is performed. While the integration converges very quickly using unit cells, the limiting factor in accuracy for the matrix elements is the slow convergence with the number of Bloch bands: the restriction to a finite number of Bloch bands in the summation corresponds to introducing a high energy cut-off . Consequently, the summation converges with , and a finite size scaling analysis can be performed; the number of Bloch bands included in this analysis was . Then, the matrix elements can be calculated with an accuracy better than ; the convergence has been extensively tested for varying number of unit cells , , and .

Scattering amplitude: Next, we analyze the scattering states in the lowest Bloch band;


with an incoming wave at relative momentum and center of mass momentum in the lowest Bloch band. Then the generalization of the -wave scattering length in free space is obtained via at low energy of the incoming wave. Again, the scattering amplitude is fully determined by the matrix . Introducing the notation for the eigenvectors of the matrix with eigenvalues , the scattering amplitude reduces to ()


with the width of each scattering resonance. The crossing of each bound state with the lowest Bloch band, see Fig. 1, gives rise to a pole in the scattering amplitude and describes a scattering resonance. Except for the first resonance, the couplings are in general weak and the scattering amplitude is dominated by the lowest eigenvalue and width . However, these additional resonances can give rise to characteristic loss features for cold atoms in an optical lattice at large -wave interactions.

In the following, we will now compare this exact value for the scattering amplitude for two particles in an optical lattice with the predictions from the Hubbard model Eq. (1). For an on-site interaction the scattering solution in the Hubbard model takes the form winkler06 ()


where describes the scattering amplitude in the Hubbard model; with . For nearest neighbor hopping and low scattering energies , reduces to with . The effective on-site interaction is therefore completely fixed by the condition, that the Hubbard model reproduces the exact two-particle properties, i.e., , and we obtain


The parameters for different strengths of the optical lattice are shown in Table 1. The contribution describes the dominant part for weak interactions, while the correction becomes relevant for stronger interactions. It is important to stress, that this derivation of the on-site interaction is valid for arbitrary values of the -wave scattering length , and gives rise to a finite value for . However, its validity is restricted to low scattering energies: first, additional interaction terms beyond the on-site interaction can play an important role and will account for the full momentum dependence of the scattering amplitude . Second, the bound state energies can be strongly modified by additional terms, which are not included in a single band Hubbard model. A test for the validity of the Hubbard model is therefore the comparison with the exact bound state energy and the repulsive/attractive bound states predicted from the Hubbard model. The bound states within the Hubbard model are determined by poles in the scattering amplitude , i.e., . A comparison with the exact bound state energies is shown in Fig. 1, and we find already very strong deviations for at : the validity for the description of bound states in the Hubbard model is limited to very weak interactions.

4 0.0855 -4.188 2.412
8 0.0308 -26.82 5.954
12 0.0122 -101.3 9.483
16 0.00533 -303.0 12.63
20 0.00249 -788.2 15.50
Table 1: Effective parameter in the Hubbard model with hopping . The on-site interaction is given by Eq. (10).

Finally, for deep optical lattices , the width of the lowest Bloch band provides a small parameter characterized by the the hopping energy . As a consequence, for all energies of the order of the band width , the first term in the matrix dominates

while all the remaining terms from higher Bloch bands as well as the renormalization provide a contribution . Consequently, the results reduces to the well known approach jaksch98 () for the derivation of the Hubbard parameters, where the influence of higher bands are neglected and the pseudo potential is replaced by a -function. Then, the momentum dependence of the interaction potential in the Hubbard model reduces to

This term also accounts for contributions such as nearest-neighbor interactions and correlated hopping jaksch98 (); duan05 (); werner05 (). For increasing interactions, the shift in Eq. (10) becomes important, and in addition the bound state energies start to deviate from the predictions within the Hubbard model. The crossover from the two regimes can be self-consistently checked: the higher bands become relevant as soon as becomes in the range of the Bloch band separation , while the renormalization requires , i.e., . In the limit of deep optical lattices, the first condition is always more stringent and reduces to , which has previously been suggested jaksch98 (). In order to derive Hubbard models which reproduce the bound state as well as the scattering states, the influence of the higher bands as well as the renormalization have to be included; in contrast to recent attempts to include the influence of higher bands alone duan05 ().

We would like to thanks L. Tarruel, T. Esslinger, M. Troyer, P. Zoller, and A. Muramatsu for fruitful discussions. Support from the DFG within SFB/TRR21 and DRAPA OLE program is acknowledged.


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