Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux
We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.
Recent advances in the field of topological phases and their potential application in implementing an intrinsically fault-tolerant quantum computer kitaev03 (); nayak08 () have revitalized interest in fractional quantum Hall (FQH) states as the most prominent examples of topologically ordered phases of matter wen90 (). Even though it has only been observed in (fermionic) solid-statesystems, the FQH effect can also exist for bosons wilkin00 (); cooper01 (); cooper08 (); morris07 (). Promising candidates are systems of interacting ultra-cold atoms where the necessary magnetic fields are simulated by rapid rotation madison00 (); abo01 () or by laser-induced synthetic gauge fields laserinduced (). At low temperatures when the filling fraction (the ratio of the particle density to the magnetic flux density ) is sufficiently small, one can expect to observe bosonic versions of the FQH effect cooper01 (). For example, the exact ground state of bosons with contact interaction at filling fraction is the Laughlin state wilkin00 (); paredes01 (), while at the ground state is in the same topological phase as the non-Abelian Moore-Read Pfaffian state mooreread (); cooper01 ().
A major advantage of optical and atomic systems over conventional solid-state systems is the possibility of creating and controlling quasiparticle excitations more naturally and with higher precision (e.g., by shining focused laser beams on the atomic gas) paredes01 (). A number of proposals suggest that the FQH regime for cold atoms can be most easily achieved using optical lattices LatticeFlux1 (); LatticeFlux2 (); sorensen05 (); gerbier10 (); cooper11 (). The question naturally occurs whether there is new physics that may arise for a system of interacting bosons in the FQH regime due to the effects of an underlying lattice. It has been shown that in the limit when the flux density , or equivalently the number of flux quanta per lattice plaquette, is small, one can ignore the existence of the lattice and treat the system in the continuum limit sorensen05 (); hafezi07 (). When is large, however, the presence of the lattice can potentially lead to new correlated states of matter that are absent in the continuum palmer06 (); palmer08 (); moller09 (). This is the limit we will focus on.
The starting point for our analysis of the many-body physics in this problem is the observation that when is close to a rational fraction, the lowest energy bands in the Hofstadter butterfly, a fractal structure realizing the single-particle energy spectrum of particles hopping on a lattice in a magnetic field hofstadter76 (), are reminiscent of Landau levels in the continuum. This resemblance can be formalized by mapping the single-particle states of the system to a continuum model when the flux density is near simple rational fractions palmer06 ().
The main result of this Letter is the following. In agreement with palmer06 (), we find that for flux per plaquette close to a rational fraction, with small integers, and sufficiently small, one can map the system to an effective continuum model with Landau levels and an added degree of freedom for the particles, a sub-band index or pseudospin, which can take possible values. However, in addition to the density-density interactions between bosons of different pseudospin found in palmer06 (), we find anomalous “pairing” interactions that do not conserve the number of particles of each pseudospin species. We find that these pairing terms, which become increasingly strong as is increased, are crucial in determining the possible ground states of the system for realistic values of the parameters of the problem.
As a detailed example, we consider the most (experimentally) realistic case and study several effective filling fractions . We find a new FQH state at , which does not exist without the pairing interactions but becomes stabilized by the increase in and the concomitant increase in these interactions. This new state is related to two copies of the non-Abelian Moore-Read Pfaffian state mooreread (); however, it is a topologically distinct phase of matter. In contrast, we find that the pairing terms destabilize the states predicted at fillings previously discussed by palmer06 (), palmer08 (), and moller09 (). We present detailed numerical evidence for our conclusions and argue that experiments are most likely to be in a regime where these pairing terms are important.
We consider bosons with onsite repulsive interaction, hopping on a two-dimensional square lattice, subject to a uniform perpendicular effective magnetic field. This system is described by a modified Bose-Hubbard Hamiltonian jaksch98 (),
where and are boson creation and annihilation operators on lattice site , is the hopping energy, and is the strength of the onsite interaction. Here is the phase acquired by a particle hopping from site to the neighboring site with being the vector potential, and we work in units where , and the effective electric charge coupled to is also set to unity. The kinetic term in the Hamiltonian then indicates that a particle hopping around a lattice plaquette acquires a phase of .
We start by considering the kinetic term of the Hamiltonian only. This is the well-known single-particle Hofstadter problem, which we review only briefly. We assume the lattice is in the -plane, with lattice spacing set to unity for simplicity, and choose the Landau gauge so that . The wave function becomes , where satisfies Harper’s equation and is the momentum in the -direction. Note that and are both integers.
Consider the case of , where a continuum approximation of the discrete Harper’s equation can be used for the low-energy eigenstates. In this limit, it is convenient to use a Wannier basis localized near minima of the cosine potential. These Wannier functions can be approximated by harmonic oscillator (Landau level) solutions with oscillator length (magnetic length) centered at , i.e., . The bandwidth of the lowest band arises from tunneling between adjacent minima of the potential and for small it scales as where the constant can be obtained by the WKB approximation Watson91 (). Note that for small , the bandwidth is much smaller than the band gap , making this limit of the Hofstadter problem an example of an (almost) flat Chern band FlatBand ().
Now let us consider flux densities close to a rational fraction. For simplicity, we focus on , for which Harper’s equation becomes . This form suggests a Wannier solution analogous to the above case, but with a two-site form factor to account for the rapidly oscillating factor . We propose the ansatz solution,
where we have defined and is a sub-band index. Thus for each momentum , there are two possible wave functions, which are spatially separated (due to the shift in the center of the oscillator) and also have their main weight on either the even or odd sites of the lattice. Choosing the value solves Harper’s equation to order and higher order terms can be added to to satisfy the equation to still higher order.
If we interpret the sub-band index as a quantum number representing a new degree of freedom, the low-energy bands in the lattice at are equivalent to the energy bands of a two-species system at , which is the continuum (Landau level) limit. Thus at we can understand as the effective flux density giving rise to an effective filling fraction defined as . Similarly, for general (with and coprime), solutions can be found and the system can be treated as a -species model with effective flux density palmer06 (); palmer08 (). With increasing , the bandwidth increases as whereas the band gap, while remaining proportional to , decreases with increasing .
In order to be in the FQH regime, it is necessary that the interaction energy be larger than the bandwidth (so that the interaction dominates over the kinetic energy). In addition, we would like the interaction to be smaller than the band gap so that all of the physics occurs within the lowest Landau band; however this requirement may not be crucial LLMixing (). Finally, the temperature must be less than the energy gap of the FQH state, which is typically set by the interaction energy (although it could also be set by the band gap if that is smaller). States competing with quantum Hall liquids include Bose-Einstein condensates: these describe the physics at , for example Moller10 (); Powell11 ().
Given these restrictions, and given that experimentally obtaining low temperatures will always be a challenge, it is clear that the FQH effect is most likely to be observed in the regime of intermediate where, the band gap is not too small and the bandwidth is not too large. Indeed, it is perhaps optimal to work in a regime where bandwidth and band gap are comparable. One can simply look at the Hofstadter spectrum to see where these inequalities are best satisfied hofstadter76 (). The most experimentally favorable case occurs for . Here, might be as large as before the bandwidth is on the order of the band gap, and the band gap may be as large as . This particular case has been studied extensively previously sorensen05 (); hafezi07 ().
The case of , which we focus on here, is also fairly favorable for the observation of FQH effect. The parameter can be as large as before the band gap is on the order of the bandwidth, and the band gap may be as large as about . While is still experimentally plausible, the cases of with have extremely tiny band gaps and, hence, seem less accessible. We note that despite the fact that these inequalities of energy scales are harder to satisfy for than for , the former has richer physics associated with the new quantum number, the sub-band index introduced above.
We now turn to consider the effect of the interaction term in the Hamiltonian Eq. (1). Using any basis of (single-particle) states with corresponding creation and annihilation operators and , the interaction may be decomposed as,
For , we use the basis for the lowest band, i.e., with the Gaussian form as described above (properly normalized). In this limit we may convert the sums into integrals, then, projected to the lowest energy band, we obtain , where the function is defined to be unity if the argument is an integer multiple of and is zero otherwise. The Gaussian factor enforces so that total momentum must be zero, not just . This derived form of the interaction is precisely what we expect for continuum bosons in the lowest Landau band with short-range contact interaction Chakraborty ().
For , we can use the basis with given by Eq. (2). Projecting to the lowest Landau level, we correspondingly find,
where the matrix results from summing over the discrete form factors in the expressions for the corresponding wave functions. Note that the functional form of the interaction Eq. (5) is identical to contact interactions for a continuum Landau level up to the band index dependent matrix out front once we redefine the momentum as . In terms of these new variables, the Gaussian enforces which now allows (allowed by ) if .
Given the precise resemblance to a two-species continuum model, we may employ exact diagonalization, which is well established as a numerical technique for the study of interacting particles in continuum Landau levels Chakraborty (). We chose to formulate the effective problem obtained from our preceding analysis in a finite spherical geometry haldane83 (); haldane85 (), which eliminates edge effects and thus allows direct access to the physics of the bulk. This problem is defined entirely within the lowest energy band by the interaction, Eq. (5), as the kinetic energy is assumed to have small bandwidth compared to the interaction scale .
Following palmer06 () we switch to a new basis , where (or ‘up’ () and ‘down’ () in the current text). We refer to this new form of the sub-band index as pseudospin. Using this basis, the nonzero elements of the transformed matrix become . In addition, we also find two extra “pairing” terms: . These terms correspond to , which resemble umklapp scattering processes, and do not conserve pseudospin . These terms indicate that a pair of pseudospin ups (downs) can be annihilated while a pair of pseudospin downs (ups) are created thus suggesting that a BCS-type pairing bcs (); generalPairing () could occur between particles with the same pseudospin. While these pairing terms vanish in the limit of , as mentioned above, the experimentally relevant regime is likely to be at finite where these terms will be important.
The novel twist for lattice bosons near is the emergence of an umklapp scattering term between the two emergent species. Consequently, the total pseudospin is not conserved, and we need to take into account the full Hilbert space containing all possible distributions of particles into the two sub-bands. We examine the spectrum of the Hamiltonian Eq. (5) for the occurrence of incompressible ground states that are characterized by translational invariance (angular momentum on the sphere) and a finite FQH gap. Our search yields four candidates at effective filling fractions and where shifts ().
For the effective filling fraction , we find that the energy gap between the ground state and the first excited state rapidly opens up as one increases [see Fig. 1(a)]. This indicates that the umklapp pairing terms, and , which are the only terms in Eq. (5) that change with , are responsible for producing an incompressible state at this filling factor. On the contrary, the energy gaps at effective filling fractions and close as increases, indicating that the pairing terms destabilize the corresponding incompressible states [see Fig. 1(b)]. At , as pointed out in palmer06 (), the 221 state is an exact ground state, and this remains true even in the presence of the pairing terms. However, as increases, this gap also closes, as excited states are sensitive to the pairing terms [see Fig. 1(b)].
Interestingly, the energy gaps of the and states have a very similar magnitude and dependency on , which may be unexpected in the current formalism. This, however, is a natural conclusion in the composite fermion (CF) theory for the lattice, which explains both these states by the same energy gap in the CF spectrum moller09 (). Furthermore, our expansion of the effective model to linear order in predicts that the gap of these states closes near the value, , predicted by CF theory moller09 (). The case of is clearly very different. Here we find that the ground state at small has significant overlap of with the non-Abelian spin singlet (NASS) state NASS () for particles. However, this state is very fragile and we cannot ascertain that it persists in the thermodynamic limit.
As was mentioned above, the presence of the umklapp pairing terms in the Hamiltonian, and , suggest a BCS-type pairing between particles of the same pseudospin. To further investigate the effect of these terms on the nature of the ground state at we propose an explicit trial wave function. We use the common conventions for studying the FQH effect: we adopt the symmetric gauge and write coordinates of particles on the plane in dimensionless complex form . Our trial wave function is given in the coordinates of bosons of ‘up’ type and bosons of ‘down’ type (with both and assumed to be even):
As written, the up and down bosons are assumed distinguishable. Expanding this wave function in the original bosonic basis, one obtains an expression that is fully symmetric in all coordinates.
Here, the two Pfaffian factors (Pf) are antisymmetrized sums over pairs of particles with the same pseudospin with the antisymmetrizing operator. This Pfaffian form is precisely the real space form of a BCS pairing wave function, which indicates that particles with the same pseudospin form pairs. Without the Jastrow factors this type of pairing is analogous to a He-A phase with a -vector in the -plane bcs (). As in the case of other paired Hall states, the topological properties are only trivially altered by restoring the Jastrow factors bcs ().
Note that the wave function, Eq. (6), is the exact ground state of , the sum of a three-body delta-function interaction for particles with the same pseudospin and a two-body delta-function interaction between particles of opposite pseudospin.
To check the validity of this trial wave function (obtained as the ground state of on the sphere), we calculate its overlap with projection of the ground state of Eq. (5) onto the sector with . As is increased and the gap opens up, we find increasing overlap between our trial state and the exact ground state [see Fig. 1(c)]. The overlap for is above for particles, which is an excellent indicator of the validity of our proposed wave function. Note that, although outside the regime of validity for our model, at , Eq. (6) is nearly an exact ground state of the two-body interaction, Eq. (5), to an accuracy of about . We also find that the inverse dependence of the paired wave function in Eq. (6) is optimal, as introducing variational parameters to change its shape generalPairing () does not increase the overlap significantly.
We have also studied the quasihole spectrum of Eq. (6) in the presence of additional flux. For the model Hamiltonian, the quasihole spectrum is precisely that of two decoupled Moore-Read layers — the quasiholes of each layer corresponding to the so-called half-quantum vortices of He-A. However, for our Hamiltonian of interest, Eq. (5), the umklapp pairing terms lock the direction of the -vector, thus requiring that the quasiholes pair between layers, confining the half quantum vortices and leaving the system with effectively Abelian excitations. To establish with clarity that this is the correct physics we have been able to predict the entire low-energy quasihole spectrum of the He-A model, using a generalization of the approach introduced in read96 (), which precisely matches the low-lying spectrum of the microscopic Hamiltonian, Eq. (5), for every case we could numerically access. These results will be presented elsewhere.
Signatures for our proposed state can be derived from a range of experimental probes for the detection of quantum Hall states in cold gases, such as measurements of groundstate incompressibility cooper05 (), noise correlations altman04 (); palmer08 (), and possibly a direct measurement of quasihole statistics paredes01 ().
The methods described here can be generalized to flux density although, as discussed above, FQH effect with larger is likely to be harder to realize in experiments. In this case there would be a sub-band index and the umklapp terms of the interaction would allow nonconservation of this sub-band index via , which could lead to new pairing terms and possibly new physics.
To summarize, we have shown that anomalous pairing (umklapp) interaction terms are crucial to the physics of FQH effect for interacting bosons on a lattice at flux density . We find that the pairing terms greatly modify the ground state at various effective filling fractions . At , we demonstrate that these terms stabilize a new paired FQH state, which is effectively two coupled copies of the Moore-Read Pfaffian state. At and , we find that the incompressible states are destabilized by the pairing terms.
Acknowledgements.Discussions with S. Adam, E. Ardonne, N. R. Cooper, M. Hafezi, L. Mathey, R. Palmer, M. Peterson, and especially E. Tiesinga are gratefully acknowledged. The authors acknowledge the hospitality of Nordita and the Aspen Center for Physics and support from NIST/NRC (L.H.), Trinity Hall Cambridge, the Newton Trust, and the Leverhulme Trust under Grant ECF-2011-565 (G.M.) and EPSRC Grant EP/I032487/1 (S.H.S.).
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