Behavior of fractional quantum Hall states in LLL and 1LL with in-plane magnetic field and Landau level mixing: A numerical investigation
1. IntroductionA two-dimensional electron gas system in a strong perpendicular magnetic field displays a host of collective ground states. The underlying reason is the formation of two-dimensional Landau levels (LLs) in which the kinetic energy is completely quenched. In a macroscopically degenerate Hilbert space of a given LL, the Coulomb potential between electrons is dominated, which makes the system strongly interacting. The different fractional quantum Hall (FQH) states are realized[1,2] for different specific interactions. For example, in the description of Haldane’s PPs[3,4] for interaction with rotational symmetry, the celebrated Laughlin[5] states at filling fraction ν = 1/3 and ν = 1/5 are dominated by V1 and {V1, V3} potentials respectively where Vm is the component in the effective interaction with two electrons having relative angular momentum m. Naturally, the FQH state will be destroyed while these dominant PPs are weakened by the realistic interaction in the systems, i.e., the increase of the Vm (m > 1) can decrease the V1. Therefore, the analysis of the effective interactions should be very helpful to investigate the properties of the FQH liquids.
The interaction between electrons can be tuned by various choices of “experimental knobs”, such as the layer thickness of the sample, the effects of the LL mixing from unoccupied LLs, the in-plane magnetic field and other external methods such as lattice strain or electric field. With considering these effects on the ideal electron–electron interaction, some of the inherent symmetry will be broken. For example, the LL mixing breaks the particle-hole symmetry, which has made the 5/2 state on the 1LL a mystery for more than two decades.[6,7] The in-plane magnetic field[8–12] for electrons or the in-plane component for dipolar fermions[13,14] introduce anisotropy and break the rotational symmetry of the system. In the absence of the rotational invariance,[15] we recently[15] generalized the pseudopotential description without conserving the angular momentum in which the anisotropy of the system is depicted by non-zero non-diagonal PPs Vm,n (m ≠ n). Some of the anisotropic interactions can be simply modeled by a few PPs. For instance we found the anisotropic interaction of the dipolar fermions in the FQH regime can be modeled by V1 + λ V1,2 in the LLL and V1 + V3 + λ1 V1,2 + λ2 V3,2 in the 1LL.[17] The effect of anisotropy introduced by an in-plane magnetic field has been quite consistent in the LLL. In the ν = 1/3 Laughlin state, the incompressibility generally decreases while increasing the anisotropy of the system.[18,19] However, people found that the experimental results on the 1LL are controversial. Different experimental results are observed in different samples.[10,11] Some of them reveal that increasing the in-plane B field stabilizes the FQH states and some of them destabilize the FQH states. Therefore, for the incompressible states, one needs a more accurate theoretical prediction on how the stability of the FQH states in 1LL (such as FQH states at ν = 7/3, 8/3 and ν = 5/2, 7/2, etc.) reacts in the presence of a tilted magnetic field. It was suggested that the width of the quantum well plays an important role in explaining these different results.[10]
In this paper, based on the pseudoptential description of the electron–electron interaction in a tilted magnetic field with a finite layer thickness, we numerically compare the behavior of the stability of the FQH states in the LLL and 1LL as varying the strength of the in-plane B field and layer thickness. The stability of the FQH states are described by the ground state energy gap, wave function overlap, and the static structure factor. We also introduce the effect of the Landau level mixing, especially for the FQH states in 1LL. The rest of this paper is arranged as follows. In Section 2, we review the single electron solution and analyze the PPs in different LLs with different parameters. In Section 3, the numerical diagonalizations for ν = 1/3 and ν = 7/3 states are implemented. The energy gaps as a function of the in-plane field at various parameters are compared. The effect of the LL mixing is also introduced in the end of this section. A summary and conclusions are given in Section 4.
2. Model and effective interactionWithout loss of generality, we assume that the in-plane B field is along the x direction. A general single particle Hamiltonian with a tilted magnetic field in a confinement potential can be written as
where the second line gives the harmonic potential along the
z axis, which mimics the layer thickness of the two-dimensional electron gas (2DEG). The smaller
ω0 means a larger thickness. After defining the canonical momentum as
πi =
Pi +
eAi, with
i = 1,2,3 along
x,
y,
z and
π4 =
mω0z, equation (
1) can be written as
with the following commutation relations:
where the three length scales are given by
, and
.
ℓ0 gives the characteristic width of the harmonic well. The cyclotron energies in a magnetic field are defined by frequency
and
. In order to diagonalize the Hamiltonian of Eq. (
2), we used the Bogoliubov transformation
[20] to write the Hamiltonian in the following form:
where the new decoupled operators (
X,
X†) and (
Y,
Y†) are some linear combinations of the canonical momentum
πis. The single particle Hilbert space is thus built from these two sets of decoupled ladder operators, and the LLs are now indexed by two integers
where |0⟩ is the vacuum state. In the limit of
ωx → 0, when
ω0 >
ωz, the operator (
X†,
X) raises and lowers the in-plane LLs, while (
Y†,
Y) raises and lowers the harmonic modes along the z-axis (or the subbands). The roles of
X and
Y are reversed for
ω0 <
ωz.
In Fig. 1, we plot the energies of the lowest three generalized LLs as a function of ωx for two different layer thicknesses with ω0 = 5.0 and ω0 = 2.0. For both cases, we find that the lowest two LLs are getting closer to each other while increasing ωx and thus the in-plane B field strongly mixes the lowest two LLs. On the other hand, with comparing the results for two different ω0s, the larger layer thickness (smaller ω0) makes the third LL more closer to the other two LLs. It demonstrates that the effect of the LL mixing should be more profound for small ω0 and large ωx.
With the exact solution of the single particle Hamiltonian, we can compute the form factor of the effective two-body interaction while projecting to the LLL. We now look at the full density-density interaction Hamiltonian with a bare Coulomb interaction
where
Vq = 1/
q2 is the Fourier components of the 3D Coulomb interaction, and
ρq = ∑
i e
i q · r is the density operator. The part relevant to the Landau level form factor is thus given by
where
,
, and
r3 are the coordinates of the cyclotron motion for an electron in a
B field. By integrating out the component
q3, we then obtain the effective two-dimensional interaction
One should first note that equation (
8) can be integrated exactly in the LLL (i.e.,
m =
n = 0). The result is as follows:
in which
and
is the Dawson integral.
and
below are defined in Ref. [
20]. In the limit of infinitesimal sample thickness
ω0 → ∞,
. For the higher LLs, we calculate the effective interaction for LL (1, 0) and (0, 1):
By using the following identities:
The
,
are obtained as follows:
is obtained by switching
by
.
For a two-body interaction without rotational symmetry, we recently found[16] that a generalized pseudopotential description can be defined by
where the normalization factors are
and
for
n = 0 or
λn = 1 for
n ≠ 0. They satisfy the orthogonality
thus the effective two-body interaction including the anisotropic ones can be expanded as
with the coefficient
In Fig. 2, we plot the ratio of the first two dominant pseudopotentials c1,0/c3,0 as a function of the in-plane B field ωx with different thicknesses ω0. It is interesting to see that the c1,0/c3,0 in LLL monotonically decreases as increasing ωx; however, things are different in 1LL. The ratio increases for small tilting and reaches its maximum at some specific value of ωx before decreasing at large tilting. From the analysis of the PPs, we therefore have a conjecture that the Laughlin state at ν = 1/3 is smoothly weakened by the in-plane B field. However, in 1LL, the FQH state at ν = 7/3 or ν = 8/3 will be stabilized with a small tilted field and finally destroyed by a large tilting field.
3. Numerical diagonalizationIn this section, we systematically study the FQH state at ν = 1/3 on LLL and 1LL in a torus geometry with the effect of the tilted magnetic field. Since the particle hole symmetry of the two-body Hamiltonian, the energy spectrum at ν = 2/3 (8/3) is the same as that for ν = 1/3 (7/3). We therefore only consider the FQH state at ν = 1/3 in LLL and ν = 7/3 in 1LL. In the following, we set ωz = 1 for simplicity. In Fig. 3, we plot the energy spectrum for 9 electrons at a 1/3 filling on different LLs without and with the tilted magnetic field. The three-fold degeneracy and large c1,0 PPs as shown in Fig. 2 on both LLs demonstrate that the Laughlin state describes the ground state very well. A small tilting of the magnetic field does not break the ground state degeneracy except varying the energy gap. However, when we compare the energy spectrums between the cases with ωx = 0 and ωx = 1.8, it is interesting to see that the gap of the 1/3 state is reduced but the gap of the 7/3 state is enhanced by this in-plane field.
As being indicated by the PPs and the energy spectrum, we compare the energy gap of the 1/3 and 7/3 FQH states as a function of the in-plane B field ωx. The energy gap is defined as the energy difference between the ground state and the lowest excited state (which corresponds to the minimal energy of the magneto-roton excitation) in the spectrum. The results are shown in Fig. 4. We compare the behavior of the energy gap as a function of the in-plane B field for two different ω0s. The energy gap of the 1/3 state always monotonically decays as increasing ωx as shown in Figs. 4(a) and 4(b). However, the gap for the 7/3 state has different tendencies for different ω0s. When ω0 = 5.0 as shown in Fig. 4(c), similar to the LLL, the gap for the largest two systems (9 and 10 electrons) still monotonically decreases as increasing ωx; however, for the case of ω0 = 2.0, we find the energy gap increases and reaches to its maximum for a small ωx and deminishes for a large ωx. Besides the energy gap, we also consider the wave function overlap under the effects of the in-plane magnetic field. Here we use the exact Laughlin state as the reference wave function, which is obtained by diagonalizing the V1 Hamiltonian. Figure 5 shows that the overlap between the 1/3 ground state and the Laughlin state monotonically decays as increasing ωx when ω0 = 2.0. However, for the ground state of ν = 7/3, the overlap is enhanced for small tilting, which is consistent to the energy gap calculation.
The symmetry breaking and phase transition of the FQH states can be described by the projected static structure factor, which is defined as[21]
where |0⟩ is the ground state and
Ri is the guiding center coordinate of the
i′-th particle. In Fig.
6, we plot the static structure factor of the 7/3 state with different in-plane magnetic fields. When the in-plane field is small, i.e.,
ωx = 1.8 as shown in Fig.
6(c) and its lateral view (a), the system has strong anisotropy between
qx and
qy directions. With a strong in-plane field
ωx = 7.0, as explained in Fig.
4, a phase transition has occurred and the system enters into a compressible state. From Figs.
6(b) and
6(d), we find there are only two peaks in the structure factor which characterizes the ground state is a charge density wave (CDW) state in a large tilted field.
As shown in Fig. 1, the lowest two LLs are very close to each other and the in-plane magnetic field reduces the gap of the two lowest LLs. Therefore, the neglection of the LL mixing in the previous calculation may not be a good approximation. Generally, the strength of the LL mixing is defined as the ratio of characteristic Coulomb interaction and kinetic energy in the magnetic field:
The Landau level mixing is a virtual excitation process of the electrons hopping between the occupied and unoccupied LLs. Recently, several research groups
[22–25] calculated the LL mixing corrections on the PPs in a perturbation way. The main contribution of the LL mixing can be classified into a correction on the two-body and three-body interactions. Here we consider the effects of the LL mixing in the tilted magnetic field system. With a magnetic field in the
x–
z plane, we can generally decompose the two-body Hamiltonian into
x and
z components:
In the language of the PPs, the first term contributes the diagonal PPs
Vm,m and the second term contributes the off-diagonal PPs
Vm,n ≠ m. In the following, we consider the LL mixing correction on the diagonal part of the two-body interaction:
 | |
In our model, the strength of the confinement ω0 in the z direction cannot be converted to the layer thickness directly. Therefore, we approximately use the two-body LL mixing correction with a typical layer thickness lB[22–25] in HLLM. The parameter λ describes the strength of the LL mixing, which has the same role of the κ. Here we have a factor
which is the proportion of the kinetic energy in the z direction (because of ωz = 1). We should note that the correction is calculated in an isotropic system, so the way of introducing the LL mixing is just an approximation. The results are depicted in Fig. 7. We set the system parameters at ω0 = 2.0 for the ν = 7/3 state. As shown in Fig. 4(d), this parameter corresponds to the case of increasing the gap by small tilting. When the LL mixing is small, as shown in Fig. 7(a) with λ = 0.05, we find the ground state energy gap still has a bump as a function of the ωx. However, when the LL mixing is strong enough, such as λ = 5.0 as shown in Fig. 7(b), the bump disappears and the gap monotonically reduced by the in-plane field as that in the LLL. Therefore, we conclude that in a strong LL mixing, the stability of the 7/3 state is no longer enhanced by the in-plane field.
4. Summaries and discussionsIn conclusion, we systematically study the stability of the FQH state at ν = 1/3 and ν = 7/3 with the effects of the in-plane magnetic field. By exactly solving the single particle Hamiltonian in a tilted magnetic field and harmonic confinement along the z direction, we obtain the effective two-body interaction in the lowest three Landau levels. With expanding these effective interactions by the generalized pseudopotentials, we find the c1,0/c3,0 behaviors differently in the LLL and 1LL, which indicate different behaviors under tilting. The results of the numerical exact diagonalization are consistent with the PPs analysis. By comparing the ground state energy gap and the wavefunction overlap, we conclude that the stability of the FQH at 1/3 is monotonically reduced by the in-plane B field. However, for the 7/3 FQH state on the 1LL, we find a small tilting magnetic field can stabilize the state when ω0 is small, such as increasing the energy gap and wave function overlap. From the calculation of the static structure factor, we observe the anisotropy of the system for small tilting and finally a phase transition into a CDW-like state occurs in large tilting. Our numerical results are qualitatively consistent with those of the experimental observations[10,11] in which the enhancements of the FQH on the 1LL were indeed observed in a small in-plane magnetic field. However, with considering the effect of the LL mixing correction on the two-body interaction, we find that a strong LL mixing correction can diminish and finally erase the enhancements of the gap. Therefore, we conjecture that the LL mixing should be small in those experiments.
Here we should note that we just consider the two-body correction of the LL mixing in our calculation. The two-body interaction does not break the particle–hole symmetry of the system. Therefore, all the results for 1/3 (7/3) are the same as those for the 2/3 (8/3) state. The breaking down of the particle-hole symmetry needs the three-body terms of the LL mixing, which will be included in a future study. The three-body or higher order interaction are from the higher correction of the LL mixing. The three-body interaction breaks the particle–hole symmetry, which may result in different affects on the particle–hole conjugate states, such as 7/3 and 8/3 FQH states. Our study of the energy gap with considering the two-body correction should be qualitatively correct since the three-body correction is much smaller than the two-body correction.
