SU(3) and SU(4) singlet quantum Hall states at
We report on an exact diagonalization study of fractional quantum Hall states at filling factor in a system with a four-fold degenerate =0 Landau level and SU(4) symmetric Coulomb interactions. Our investigation reveals previously unidentified SU(3) and SU(4) singlet ground states which appear at flux quantum shift 2 when a spherical geometry is employed, and lie outside the established composite-fermion or multicomponent Halperin state patterns. We evaluate the two-particle correlation functions of these states, and discuss quantum phase transitions in graphene between singlet states with different number of components as magnetic field strength is increased.
Introduction:—The presence of internal degrees of freedom in the quantum Hall regime has often provided fertile ground for the emergence of new strongly correlated quantum liquid physics. Examples include the pioneering work of Halperin Halperin83 () in which he constructed multicomponent generalizations of the celebrated Laughlin states Laughlin83 (), the prediction of skyrmion quasiparticles Sondhi93 () in systems with small Zeeman splitting, and the identification of excitonic superfluidity MacDonald04 (); Jim2014 () in bilayer systems. Multicomponent fractional quantum Hall systems are often experimentally relevant thanks to the rich variety of two-dimensional electron systems that possess nearly degenerate internal degrees of freedom, for example spin Halperin83 (), layer Suen1992 () and/or sub-bands Liu2014 (); Liu2015 () in GaAs quantum wells, spin and/or valley in graphene Novoselov05 (), anomalous additional orbital indices in the Landau levels of few-layer graphene Novoselov06 (); Pablo11 (); Morpurgo15 (), valley in AlAs Bishop07 (), and cyclotron and Zeeman splittings that have been tuned to equality in ZnO Maryenko12 (); Maryenko14 (). In monolayer and bilayer graphene in particular, the nearly four-fold and eight-fold degenerate Landau levels have recently been shown to give rise to interesting examples of ground states with competing orders MacDonald06 (); Dean11 (); Young12 (); Yacoby12 (); Weitz (); Pablo14 (); Kharitonov_MLG (); Halperin13 (); Inti14 (); Wu14 (); Sachdev14 ().
A diverse toolkit of theoretical approaches that can be successfully applied to understand fractional quantum Hall states has accumulated over the nearly three decades of research. One of the most widely employed frameworks is that of composite fermions Jain (); Jain1989 (). The success of the composite fermion picture stems in part from its simplicity, since it allows fractional quantum Hall states of electrons to be viewed as integer quantum Hall states of composite fermions. An important success of the composite fermion approach is that it provides explicit trial wavefunctions that accurately approximate the ground states computed using exact diagonalization for the Jain sequence of filling fractions Jain (); Jain1989 (). The composite fermion picture can be generalized to account for a multicomponent Hilbert space, and it has been argued that it correctly captures the incompressible ground states of 4-component systems with SU(4) invariant Coulomb interactions Jain07 (); Jain12 (); Jain15 (). However, a detailed test of composite fermion theory in the SU(3) and SU(4) cases has been absent.
In this Letter we report on a striking deviation from the composite-fermion picture arising at filling fraction for three and four-component electrons residing in the Landau level and interacting via the Coulomb potential. This circumstance is relevant to the fractional quantum Hall effect in graphene Du09 (); Bolotin09 (); Dean11 (); Yacoby12 (), and also bilayer quantum wells Mong2015 (); Peterson2015 (). Employing exact diagonalization for the torus and sphere geometries we find that SU(3) and SU(4) singlets, in which electrons respectively occupy three and four components equally, have lower energy than the known single-component state and SU(2) singlet Zhang84 (); Xie89 () at the same filling factor. More specifically, we find that on the torus the ground state for electrons and flux quanta is a SU(3) singlet, and that for and the ground state is a SU(4) singlet. There are previous exact diagonalization studies of SU(4) Landau levels Regnault07 (); Regnault10 (); Jain07 (), but to our knowledge there is no previous report of the states we describe below.
On the sphere a shift occurs in the finite-size relationship between flux quanta and electrons compared to the torus . The shift is a quantum number that often distinguishes competing quantum Hall states associated with the same filling factor. In particular, under space rotational invariance, any two states that differ in their shift cannot be adiabatically connected and would thus belong to distinct quantum Hall phases Wen1992 (); Wen1995 (); Read2011 (). Our SU(3) and SU(4) singlets appear on the sphere at (, )=(7, 6) and at (, )=(10, 8) respectively, corresponding to a shift in both cases.
For two-component electrons the composite fermion picture allows two competing trial wavefunctions at Jain (); Davenport (). One is a fully spin polarized state that approximates the particle-hole conjugate of the Laughlin state. The second is a SU(2) spin singlet, constructed from the integer quantum Hall ferromagnet by flux attachment Jain93 (); Jain (). This state approximates the singlet ground state of the SU(2) symmetric Coulomb interaction Zhang84 (); Xie89 (). No new competing states are expected at upon increasing the number of components from two to three and four. Jain07 (); Jain12 (); Jain15 (). Our findings indicate that this expectation breaks down.
Another way to construct multicomponent wavefunctions is to follow Halperin’s approach Halperin83 () in which one requires that the wavefunction vanishes with power () when pairs of particles in the same (different) component approach each other. A four-component Halperin wavefunction arises naturally at with and . This state is not an exact singlet because it does not satisfy Fock’s cyclic condition Jain (). This alone does not rule out this wavefunction as a legitimate trial state, because one could still imagine it to be adiabatically connected to the exact singlet when exact SU(4) symmetry is relaxed. However, this Halperin wavefunction has a shift , which differs from the shift of the SU(4) singlet discovered numerically. Therefore, the two states can not be adiabatically connected in a system with rotational invariance. For the three-component case there are no multi-component Halperin wavefunctions at .
A possible strategy to construct trial wavefunctions for the new singlet states, detailed in the Supplemental Material, starts from a SU() singlet state at an integer filling . is the Slater determinant state in which fold degenerate lowest Landau levels are fully occupied. SU(3) and SU(4) singlets with the desired filling and shift are then obtained by multiplying the Slater determinant by appropriate Jastrow-type factors. Even within this rather general strategy, we have not found fully satisfactory trial wavefunctions that display similar short distance correlations with the states found in exact diagonalization. We hope our work can stimulate future studies that fully elucidate these new singlet states.
Energy spectra:— We consider the Coulomb interaction Hamiltonian projected to a component Landau level(LL):
Because the Coulomb interaction is independent of flavors, the Hamiltonian is SU(4) invariant. Since SU(3) is a subgroup of SU(4), the SU(3) spectrum is embedded in the current problem. Below we use the magnetic length and the Coulomb energy as length and energy units. Eigenstates of may be grouped into SU(4) multiplets. Within a multiplet, states are connected to each other by SU(4) transformations. A multiplet can be labeled by its highest weight state Georgi (). Here are the number of electrons in each component with . A SU() singlet () has a highest weight given by and for , and is invariant under the SU() transformation within the occupied components.
By applying periodic boundary conditions on a torus, magnetic translational symmetry can be used to classify many-body states Haldane (). Fig. 1 shows energy as a function of momentum at filling factor . In Fig. 1(a), and are respectively 9 and 6, and the ground state is a SU(3) singlet that has zero momentum, implying that it is a translationally invariant quantum fluid state. The first excited state at zero momentum is the well-known SU(2) singlet Zhang84 (); Xie89 () described in the introduction. The third excited state at zero momentum is the single-component particle-hole conjugate state of the Laughlin state.
In Fig. 1(b), and are increased to 12 and 8 respectively, and the ground state is a SU(4) singlet at zero momentum. The first and second excited states at zero momentum, labeled by and , are very close in energy. The particle-hole conjugate of the Laughlin state has a higher energy and is buried deep in the continuum.
To determine the shift of the singlets on the sphere, we vary while keeping fixed. Fig. 2 shows the ground state energy on the sphere as a function of at (Fig. 2(a)) and (Fig. 2(b)). For (Fig. 2(a)), the ground state at is a SU(2) singlet, which is the composite-fermion singlet with and . At , the ground state is our new SU(3) singlet at with . Note that a SU(3) singlet also appears at , which we identify as a composite-fermion SU(3) singlet with and . The analysis of Fig. 2(b) is similar. We identify the SU(4) singlet at and to with shift .
In Table 1, we compare the Coulomb energies between the SU(3) and SU(4) singlets and the SU(2)singlet at Ne_12 (). In graphene Zeeman energy favors the SU(2) singlet which can have full spin polarization. Ideally, one would observe a transition from the new singlet states discovered here as the magnetic field is increased. The absence of an apparent transition in current experiments Yacoby12 () might be explained by screening Misha (); Sodemann2014 () and Landau level mixing effects Nayak2013 (); Sodemann2013 () which tend to weaken effective interaction strengths, reducing the critical fields to values where it is challenging to observe the fractional quantum Hall effect.
The largest system size we have attempted is on a torus with . For this number of electrons it is impossible to construct exact SU(3) or SU(4) singlets. We restricted the numerical calculation to 3-fold degenerate LLs, and found that a multiplet labeled by has a lower energy than the SU(2) singlet. This adds to evidence that the SU(2) singlet predicted by composite fermion theory is not the ground state in LLs with more than two components. We hope that future studies will be able to extend our study to larger system sizes.
Pair Correlation functions:— We now discuss the spatial correlation functions that describe the probability of finding two electrons at certain distance from each other. We have found that our new SU(3) and SU(4) singlets have similar short-distance correlations to the conventional SU(2) singlet and single component state at , and the long-distance correlations are different. The flavor-dependent spatial correlation function is defined by
where is the area of the 2D system, and is the number of electrons in flavor state .
Figs. 3(a) and (b) plot of states along the diagonal line of the torus, i.e. along . As required by the Pauli exclusion principle, vanishes as . It turns out that is very small, but not exactly zero, at for the singlets. In graphene, SU(4) symmetry is weakly broken by short-range interactions that arise from lattice-scale Coulomb interactions and electron-phonon interactions. The short-range interactions are typically modeled by a function potential Kharitonov_MLG (). Since the probability for two electrons to spatially overlap is small in these singlets, the short-range interactions should have an negligible effect on these states Halperin13 (); Inti14 (); Wu14 ().
At small electron separation, is similar in all singlet states, and likewise , with smaller than as shown in Fig. 3(a) and (b). We note that the four-component Halperin wavefunction with and has the opposite behavior, i.e. for small . This is another distinct feature between the Halperin wavefunction and the exact SU(4) singlet, besides the difference in the shift.
The similarities between the pair correlation functions of different singlet states at small do not extend to larger distances. For the SU(2) singlet, reaches a maximum at the maximum particle separation, while reaches its maximum closer. The opposite behavior applies for SU(3) and SU(4) singlets at the system sizes we are able to study, as illustrated in Fig. 3.
To get a deeper understanding of the small behavior of , we consider the relative-angular-momentum (RAM) correlation function :
where Jain () projects electrons and onto a state of RAM . contains the same information as and can be more physically revealing:
where is the wave-function for a state of a RAM Jain (). At small electron separation , is mainly determined by with small ,
The approximation in the second line of Eq. (5) follows from the fact that is always extremely small for states we consider. Values of are displayed in Fig. 3(c). Like the pair correlation functions, has similar values in all singlet states for both and . As proved in the Supplemental Material, in any singlet state. This property explains why is smaller than at small .
The energy per electron of a SU() singlet can be decomposed into contributions from interactions in different angular momenta channel:
where is the th Haldane pseudopotential of the Coulomb interactionJain (), and the term takes into account the contribution from the neutralizing background. For the SU() singlets described above, is approximately zero, while decreases as increases from 2 to 3 or 4. This analysis sheds light on why SU(3) and SU(4) singlets have lower energy than the SU(2) singlet at .
Summary:— By diagonalizing the Coulomb interaction Hamiltonian for electrons in multicomponent Landau levels, we have discovered translationally invariant SU(3) and SU(4) singlet ground states at filling factor . We have found these states in systems containing and electrons respectively, on both sphere and torus geometries. Both states on the sphere have shift . The pair correlation function of these states is similar to that of composite fermion SU(2) singlet state at short electron separation, and becomes different at large distances.
Our findings are striking because the states we have discovered do not fit into either the composite fermion or the multicomponent Halperin state patterns. These singlets are candidates to join the handful of important states that do not fit the simple composite fermion paradigm, such as the Pfaffian state MR1991 () and Read-Rezayi states Read-Rezayi (). It is remarkable that this novel physics occurs in the lowest Landau level where past experience has suggested that composite fermions best describe Coulomb interaction incompressible states.
Acknowledgments:—IS is thankful to Xiao-Gang Wen for illuminating discussions. Work at Austin was supported by the DOE Division of Materials Sciences and Engineering under Grant DE-FG03-02ER45958, and by the Welch foundation under Grant TBF1473. IS is supported by a Pappalardo Fellowship. We thank the Texas Advanced Computing Center (TACC) and IDRIS-CNRS Project 100383 for providing computer time allocations.
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Appendix A Proof of for odd
The second quantized form of the RAM correlation function is
where () is an electron creation (annihilation) operator with denoting the orbital index. The matrix element is
where is the single particle wavefunction for orbital .
We can define a correlation function that is conjugate to ,
The RAM projector has the property,
which leads to:
An immediate consequence is that for even , which is expected from Pauli exclusion principle.
A SU() singlet state is invariant under a unitary transformation,
where is a unitary matrix () that is independent of the orbital index . By making use of this invariance and noting that particle number in each flavor is a good quantum number, we arrive at the following constraints:
where denotes the expectation value with respect to the singlet state . The two constraints in Eq. (13) are imposed by an arbitrary unitary matrix , and give rise to an identity,
Appendix B for particle-hole conjugates of single component Laughlin states
Consider the spinless Laughlin state at filling , , and its particle-hole conjugate at filling , where is an odd integer. The expectation value of evaluated in can be shown to be related to that evaluated in as follows:
The expression is obtained using .
Appendix C Trial wavefunctions
One strategy to construct trial SU() singlets at is to start from a SU() singlet state at filling . is a Slater determinant state in which fold degenerate lowest LLs are fully occupied. In analogy with the flux attachment procedure, we can multiply by appropriate Jastrow-type factors. We note that the following SU(3) and SU(4) singlet wavefunctions and have Fermi statistics, filling factor and shift :
Here denotes the complex coordinate of the th electron. Pf indicates a Pfaffian factor, like the one appearing in the Moore-Read wavefunction MR1991 (). and are wavefunctions for the single-component particle-hole conjugates of the and Laughlin states respectively. The Jastrow-type factors, that appear in square brackets are chosen to be completely symmetric functions of all the particle coordinates. Because have Fermi statistics and shift , and the conjugate Laughlin states have Fermi statistics and shift , the Pfaffian factor is required both to restore Fermi statistics and to increase the shift by 1.
In counting the filling factor and shift, we have used the rule that shifts and inverse of filling factors of holomorphic functions are additive under wavefunction multiplication:
where is a wavefunction with filling factor and shift .
Since is a factor of and due to the antisymmetrization property, the Pfaffian factor does not lead to divergences in the trial wavefunctions. However, from the previous section one can conclude that the particle-hole conjugate of any single component Laughlin state has a finite probability of being in RAM with , and, hence, that these Jastrow factors do not vanish in the limit . This implies that the full wavefunctions, and , do not vanish when pairs of particles of different flavors approach each other. This appears to be in conflict with the qualitative behavior displayed by the pair correlations of the states found in exact diagonalization, depicted in Fig. 3 of the main text.
Appendix D states in 4-fold degenerate Landau levels
The particle-hole symmetry in 4-fold degenerate LLs provides a one-to-one mapping between eigenstates at filling factor and those at . Therefore, we can focus on states. In the main text, we presented a detailed analysis of states. Here, we will discuss states with =1, 4 and 5 based on exact diagonalization (ED) study on torus.
At , we performed an ED study with up to 15 and the ground state multiplet is represented by a single-component state, for which the Laughlin state is a good approximation.
Before discussing and 5/3 states, we first recall two useful mappings studied in Ref. Inti14, , which generates states at from seed states at . Mapping-I is the particle-hole conjugation restricted to two-components,
where and are the Coulomb energies per flux quantum and . Mapping-II attaches a fully occupied LL to a seed wavefunction with three components or less,
At , ED on torus with of 6 and 9 shows that the ground state multiplet is represented by a SU(2) singlet , which can be generated from the SU(2) singlet at by Mapping-I.
At , the ground state multiplet with is represented by (9222), which is connected to (2220) state at through Mapping-II. We are not able to perform ED for every sector in the Hilbert space when is increased to 12 or 15. However, we can still make the following two predictions based on ED results at and Mapping-II. One prediction is that (12, 3, 3, 2) has very similar energy as (12, 4, 4, 0) for . Another one is that (15, 4, 4, 2) has a lower energy compared to (15, 5, 5, 0) for . These finite-size results tend to suggest that is not the ground state at in SU(4) LLs.