Origin of the $\nu=1/2$ fractional quantum Hall effect in wide quantum wells

The field of fractional quantum Hall effect [1] (FQHE) has been the birthplace for a web of spectacular phenomena, exotic emergent particles, and nontrivial states, all arising as a result of the interaction between electrons. The FQHE is a rare example of a strongly correlated state for which we not only have a qualitative understanding of a large part of the prominent phenomenology but have achieved a detailed microscopic description that is quantitatively accurate [2, 3]. Nonetheless, the origin of a few experimentally observed states remains unsettled. This article aims to report on our theoretical investigations of one such state, namely the FQHE state at filling factor $\nu=1/2$ observed in wide quantum wells (WQWs) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], the origin of which has been a topic of debate ever since its discovery. There are two motivations for our study. First, this observation is in stark contrast to the state at half-filling in narrow quantum wells, which is established to be a Fermi sea of composite fermions (CFs) [15, 2, 3]. The FQHE thus arises due to changes in the interaction arising from finite quantum well width, and thus constitutes an important challenge for our quantitative understanding of the FQHE. Second, the physical origin of the observed state can be potentially very interesting.

A promising two-component state is the Halperin $(3,3,1)$ state [16], which can be relevant because a very WQW behaves as a two-component system. [There is little doubt that the $1/2$ FQHE observed in real double-layer systems [17], observed at the same time as the $1/2$ state in WQWs, is the two-component Halperin $(3,3,1)$ state [18]. Our focus in this article is on WQWs, not double layer systems.] However, another promising candidate is the one-component Pfaffian state, which is a paired state of composite fermions [19, 20]. This state is believed to be responsible for the FQHE at $\nu=5/2$ [21], and one can ask if the changes in the inter-electron interaction due to finite width may stabilize this state at $\nu=1/2$ as well. The Halperin $(3,3,1)$ state supports Abelian quasiparticles, whereas the Pfaffian is believed to support non-Abelian quasiparticles. The latter has motivated many interesting theoretical and experimental studies of the $5/2$ FQHE. If the 1/2 state in WQWs turns out to be the Pfaffian state, that would provide another venue where non-Abelian quasiparticles may be investigated.

While the $1/2$ FQHE in WQWs has often been interpreted in terms of the $(3,3,1)$ state, arguments can also be given in favor of a one-component state. We provide here a summary of experimental results and their implications for the nature of the state:

• •

  In a double layer system, which consists of two layers separated by a distance $d$, the situation is relatively clear [22, 23, 18, 24, 25, 26, 27, 28, 29]. For zero layer separation, the two-component system of spin polarized electrons is formally equivalent to a single layer system of spinful electrons with zero Zeeman splitting. Here the state is a layer singlet Fermi sea of composite fermions [22, 30, 31]. In variational calculations [24, 25, 18] this state survives in the range $d/l_{B}\lesssim 1$. The $(3,3,1)$ state is predicted to occur for layer separations $1\lesssim d/l_{B}\lesssim 3$[24, 18], in general agreement with experiments. For layer separations $d/l_{B}\gtrsim 3$ two uncoupled CF Fermi seas (CFFSs) are formed in each layer, with composite fermions now binding four vortices [24, 18]. In contrast, the 1/2 FQHE in WQWs is seen when the width is approximately $2.6-8$ $l_{B}$[4, 5, 32, 6, 33, 34, 8, 9, 10, 35, 12, 13, 11]. Although not conclusive, this points against the two-component $(3,3,1)$ state.

• •

  For quantum well widths and densities where the 1/2 FQHE is observed in WQWs, the behavior of FQHE states surrounding it is often consistent with single layer physics. In particular, the standard Jain sequences $n/(2n\pm 1)$ [36] are observed. Recently, Mueed et al. [37] have directly measured, from commensurability oscillations, the Fermi wave vector of composite fermions in the vicinity of filling factor $1/2$ and found that the Fermi sea is a one-component state. The fact that the states in the immediate vicinity of $\nu=1/2$ are one-component states makes it plausible that the $\nu=1/2$ FQHE also has a one-component origin. If not, it would be important to understand what is special about $\nu=1/2$ that makes a two-component state favorable.

• •

  A phase diagram has been constructed as a function of the filling factor and $\Delta_{\rm SAS}$, the gap between the symmetric and antisymmetric subbands [34]. The island of the 1/2 FQHE state straddles the boundary where many nearby FQHE states make a transition from a one-component state to an insulator, presumably a double layer crystal. However, the 1/2 FQHE island is contiguous, i.e. it is either all one component or two-component.

• •

  The effect of asymmetry in the charge distribution is complex but worth mentioning here. An early work on 80 nm wide QW by Suen et al. [5, 6] reported a monotonic decrease in the strength of the FQHE at $\nu=1/2$ as the charge distribution is made asymmetric, with the FQHE state disappearing at approximately 10% imbalance. This may arise from either two-component nature or complicated changes in the effective interaction. Subsequently, Shabani et al. [8, 9] found that in a 55 nm quantum well an asymmetry of the charge distribution favors FQHE at $1/2$. This suggests a one-component nature of the FQHE here. Numerical studies also show that in such asymmetric quantum wells around certain widths the one-component Pfaffian wave function has a large overlap with the ground state, although the $(3,3,1)$ state is also competitive [38, 11, 39].

We next briefly review the theoretical studies of the 1/2 FQHE in WQWs and also provide a summary of the main results arising from the present study. In particular, we indicate how the theoretical phase diagram is sensitive to the various assumptions that go into the calculation.

The problem has been addressed by exact diagonalization (ED) [23, 29, 40, 39, 41]. ED can often deal with only very small systems and is thus not likely to capture the thermodynamic behavior. This is especially the case for WQWs, for which the width may become comparable to the available lateral dimension of the system. The energy orderings of states are often seen to change as the system size increases.

This issue has also been investigated by variational Monte Carlo (VMC)[42, 43, 44, 38, 45, 18]. During the course of this work, we have determined the phase diagram of $\nu=1/2$ in a WQW using the VMC method, shown in Fig. 1. The $(3,3,1)$ state is stabilized in a part of the phase diagram that qualitatively agrees with experiments. The phase boundary between one component CFFS and the $(3,3,1)$ state is consistent with earlier calculations [45].

However, the VMC calculations make the following assumptions. (i) The effect of finite width is incorporated through a transverse wave function for electrons, which modifies the interactions between them (see Eq. 19). The transverse wave function is evaluated in local density approximation (LDA) at zero magnetic field [46], and it is assumed that it remains unaltered at a strong perpendicular magnetic field. Given that the nature of the transverse wave function depends on the state that the electrons form in two dimensions (for example, at zero magnetic field LDA assumes a Fermi sea state of electrons), one may wonder to what extent this assumption is valid. (ii) The phase boundary between the one- and two-component states depends sensitively on $\Delta_{\rm SAS}$, i.e. the gap between the symmetric and antisymmetric subbands. One uncritically uses its value obtained at zero magnetic fields. However, this gap is typically very large compared to the Coulomb energy differences between the competing states, and even a few percent change in $\Delta_{\rm SAS}$ can substantially shift the phase boundaries.

The VMC calculation also does not incorporate the effect of Landau level mixing (LLM) directly. We have further investigated the role of LLM within the VMC method through a two-dimensional (2D) fixed-phase diffusion Monte Carlo (DMC) method developed by Ortiz, Ceperley and Martin  [47, 48], which itself is a generalization of the standard DMC method [49] to find ground states in the presence of broken time-reversal symmetry. In this method, we allow for LLM for electrons interacting with the effective interaction derived from LDA at zero magnetic field. We refer to this as “2D-DMC.” We find that, at this level of approximation, the phase diagram is substantially altered and neither the (3,3,1) nor the Pfaffian state is stabilized for a significant range of parameters (see Fig. 2). However, a conceptual difficulty with this method is an uncontrolled double-counting, because mixing with higher bands has already been incorporated through the modification of the transverse wave function, which, in a sense, is akin to LLM at a finite magnetic field. (At finite magnetic fields, it is LLM that leads to a modification of the form of the transverse wave function.) This study nonetheless shows the importance of LLM, indicating that the results from neither VMC nor 2D-DMC are fully reliable.

The primary motivation of our work is to develop a technique that circumvents some of the above issues and treats finite width and LLM effects directly at a large magnetic field. Specifically, we use a three-dimensional (3D) version of the fixed phase DMC method, referred to below as “3D-DMC,” or simply as “DMC.” The most important advantage of the 3D-DMC method is that it directly gives the ground state energy (as well as the form of the transverse wave function) at a high magnetic field, automatically including the effects of finite width and LLM. No reference is made to zero magnetic field in our calculation. Of course, this method also makes an approximation, namely the choice of fixed phase, and all of our conclusions are subject to the validity of our choice of the phase. (We use the accurate lowest Landau level wave functions to fix the phase, which has been found to give good agreement with experiments in the past [50, 51, 52].) There are other practical difficulties with our method. One is that the required computation time does not allow treatment of very large systems; we have studied systems with up to about 25 particles. The second is that it does not allow treatment of two-component states with non-zero $\Delta_{\rm SAS}$. For two-component states, we assume that $\Delta_{\rm SAS}=0$, i.e. the wave function strictly vanishes at the center. This should be a decent approximation for sufficiently large widths and densities where $\Delta_{\rm SAS}$ is small.

The phase diagram obtained from 3D-DMC calculations is shown in Fig. 3. The light purple region shows the part of the phase diagram where the energies of the one-component CFFS and the one-component Pfaffian states are so close that we cannot distinguish between them within numerical uncertainty [although both of these energies are lower than the energy of the two-component $(3,3,1)$ state]. Given that experiments show an incompressible state here, we believe that the one-component Pfaffian state is the most likely possibility. Nonetheless, in light of the approximations made in the calculation, a definitive confirmation can come only from experiments, and we hope that our study will motivate further experimental studies of this state.

We have also studied several other candidate wave functions at $\nu=1/2$ but found them not to be relevant for the issue at hand.

Additionally, our 3D-DMC study yields the form of the transverse wave function directly in the presence of a high perpendicular magnetic field. Here, the double layer nature of the ground state for large widths or densities arises due to LLM. We find that, surprisingly, the form of the transverse wave function of the lowest symmetric band is not particularly sensitive to the nature of the 2D state; we find very similar forms for $\nu=1$, $1/3$, and $1/5$, as discussed later. Furthermore, also surprisingly, we find that the transverse wave function obtained from our 3D-DMC is also close to that obtained from LDA at zero magnetic field. Nonetheless, our phase diagram with 3D-DMC method is very different from that obtained from VMC.

A recent work [53] has concluded that switching on tunneling in a bilayer favors the Pfaffian state. The model for quantum well considered in Ref. [53] is different from ours.

The plan of our paper is as follows. In Sec. II, we briefly review the fundamentals of the FQHE on a torus and give explicit forms of wave functions that are involved in our calculation. In Sec. III we report our VMC studies on the topic. We next introduce the general principles of the DMC method in Sec. IV. After that, we present our 2D-DMC and 3D-DMC investigations individually. We discuss our results in the end and more technical details can be found in the appendices.

All calculations are performed in the torus geometry, except those presented in Appendix A. Throughout this work, we assume parameters appropriate for GaAs, with dielectric constant $\epsilon=12.6$ and band mass $m=0.067m_{e}$, where $m_{e}$ is the electron mass in vacuum. The magnetic length is denoted $l_{B}=\sqrt{\hbar c/eB}$ where $B$ is the magnetic field.

We shall include in our study several different states at filling factor $\nu=1/2$, which we now list. We primarily use the torus geometry for our study, because the CFFS can be constructed on a torus with explicit wave vector configuration. (On the sphere one must approach the CFFS by taking the limit $n\to\infty$ for Jain states at $\nu=\frac{n}{2n+1}$ [54, 55, 31], which requires going to very large systems that are not accessible to DMC.) We also give the VMC results in the spherical geometry in Appendix A for comparison. We start this section by reviewing some basics of FQHE on a torus[56, 57, 58, 59, 60].

We shall model the confinement potential as a quantum well with the infinite depth and a width of $W$. In some circumstances the finite depth is also considered, but in general this does not cause any significant difference because the GaAs quantum wells we discuss in this article (and also those in experiments) are generally very deep. The problem is modeled via a VMC calculation which includes an effective two-dimensional interaction, defined as follows:

Here $\psi(w)$ is the transverse wave function, $w$ is the transverse coordinate, and $r$ is a two-dimensional vector. In the simplest approximation, the subband wave functions are taken as the single-particle solutions of a quantum well problem $\psi_{S}(w)=\sqrt{\frac{2}{W}}\cos\left(\frac{\pi w}{W}\right)$ and $\psi_{A}(w)=\sqrt{\frac{2}{W}}\sin\left(\frac{2\pi w}{W}\right)$, where $S$ and $A$ refer to symmetric and antisymmetric. In this approximation the subband gap is $\Delta_{\rm SAS}=\frac{3\pi^{2}}{2}\frac{\hbar^{2}}{mW^{2}}$, where $m$ is the band mass of the electron. A better approximation for $\psi(w)$ is obtained by LDA at zero magnetic field, where one assumes a Fermi liquid state in the 2D plane [46, 72]. In this and the next section that introduces the 2D fixed-phase DMC, we use the LDA form for $\psi(w)$. We denote the lowest two subbands as $\psi_{S}$ and $\psi_{A}$, in which $S$ represents the symmetric subband and $A$ represents the anti-symmetric subband. The typical LDA density profiles of the lowest two subbands are shown in Fig. 4. Before going further, let us discuss how the occupation of the subbands changes as one tunes the subband gap. When $\Delta_{\rm SAS}$ is much larger than the Fermi energy, as is the case for either very small $W$ or small densities, only the lowest subband is occupied. In the limit when the lowest two bands are approximately degenerate ($\Delta_{\rm SAS}\approx 0$), which happens at large $W$ or at large densities, two-component states are possible, where the two components are linear combinations of the two subbands. Because the system tends to form two layers at large widths, we choose the left-right bases as (Fig. 5):

More generally, we can choose $\psi_{\theta}=\frac{1}{\sqrt{2}}(\psi_{S}+e^{i\theta}\psi_{A})$ and $\psi^{\prime}_{\theta}=\frac{1}{\sqrt{2}}(\psi_{S}-e^{-i\theta}\psi_{A})$. However, because the systems becomes a bilayer for sufficiently wide quantum wells or large densities, we expect that $\theta=0$ will produce the lowest energy.

Similarly to Eq. 19, we define the effective interactions as follows. For one-component states, only the lowest symmetric subband is used for defining the effective interactions whereas for two-component states, both intra-component interaction and inter-component interaction are needed:

The densities are defined as $\rho_{\text{S}}=\left|\psi_{S}\right|^{2}$ and $\rho_{\text{L,R}}=\left|\psi_{\text{L,R}}\right|^{2}$.

We mention here two caveats. First of all, we consider states for which either only the lowest subband is occupied, or the two lowest subbands are equally occupied. All of our trial wave functions, namely the single-component CFFS, the single component-Pfaffian, the pseudo-spin singlet CFFS, the uncoupled $1/4+1/4$ CFFS, and the $(3,3,1)$ satisfy this requirement. In principle, we can also consider a partially polarized CFFS, which will have an unequal occupation of two subbands, but we have not done so (because it significantly enhances the calculational difficulty). All other states considered here cannot be partially polarized. Second, the value of $\Delta_{\rm SAS}$ is relevant for transitions from a single-component to a two-component state. $\Delta_{\rm SAS}$ is typically very large compared to the Coulomb energy differences between the relevant states. We determine the value of $\Delta_{\rm SAS}$ from the LDA calculation (Fig. 6).

The energies of one-component states relative to the CFFS are shown in Fig. 7. As one can see, the CFFS remains the lowest energy state for all parameters, although the Pfaffian comes as close as $0.001\frac{e^{2}}{\epsilon l_{B}}$ at densities greater than $2\times 10^{11}\text{cm}^{-2}$. This conclusion is also supported by exact diagonalization studies of finite systems in the spherical geometry. In Appendix D, we show the overlap between the exact ground state of the LDA interaction with the one-component CFFS and the Pfaffian states, and find that in the entire region of parameter space that we considered, the one-component CFFS always has a very high overlap with the exact ground state and thus is superior to the Pfaffian. [Note: We have also performed the energy comparison in the spherical geometry, where we see a different result, namely that the Pfaffian state has lower energy in the thermodynamic limit for some parts of the phase diagram. We believe that the torus results are more reliable because the thermodynamic extrapolation on the sphere is less accurate for finite widths. See Appendix A for further discussion.]

The energies of the two-component states, namely the Pseudo-spin singlet CFFS, $(3,3,1)$ and the uncoupled $1/4+1/4$ CFFS, relative to the $(3,3,1)$ are shown in Fig. 8. A transition from the singlet CFFS to the Halperin $(3,3,1)$ occurs at very low densities, followed by a second transition into the uncoupled $1/4+1/4$ CFFS (Fig. 8). This behavior is similar to that found in earlier VMC calculations on the zero-width bilayer systems[24]. The phase diagrams for one and two-component states separately are shown in Fig. 10.

Fig. 9 shows the energies of all states. We add $\frac{1}{2}\Delta_{\text{SAS}}$ to the energy per particle for each two-component state, because half of all particles occupy the second subband. We quote all energies relative to the $(3,3,1)$ state in Fig. 9, which yields the phase diagrams in Fig. 1. In general, one can see the ground state of the system is in a one-component state when the carrier density is low and the system makes a transition into a two-component state at high density. This result is qualitatively consistent with the experiments [9, 4, 7], and favors the possibility that the observed incompressible state is the $(3,3,1)$ state.

There are significant differences, however. As noted earlier, the lower phase boundary is very sensitive to $\Delta_{\rm SAS}$. However, the upper theoretical phase boundary ought to be more reliable, and its deviation from the experimental phase boundary is thus significant. We note, however, that the calculation, so far, does not include LLM or disorder.

In the following sections, we will use the fixed-phase DMC method to evaluate the phase diagram. The general DMC is a standard Monte Carlo method designed to obtain the ground state of the many-body Schrödinger equation[73, 49] by a stochastic method. By setting time to an imaginary variable $(t\to t=-i\uptau)$, the Schrödinger equation takes the form

where $\mbox{\boldmath$R$}=\left(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2},\dots,\mbox{\boldmath$r$}_{N_{e}}\right)$ is the collective coordinate of the system and $E_{T}$ is a constant energy offset. When $\Psi\left(\mbox{\boldmath$R$},\uptau\right)$ is real and non-negative, one can interpret the above equation as a diffusion equation, with $\Psi\left(\mbox{\boldmath$R$},\uptau\right)$ interpreted as the density distribution of randomly moving walkers. The energy offset $E_{T}$ controls the population of random walkers. Starting from an initial trial wave function $\Psi_{T}$, as the walkers diffuse stochastically, the distribution gradually converges to a stable distribution that represents the ground state (provided $\Psi_{T}$ has a non-zero overlap with the ground state). More details can be found in Ref. [49,74] .

The applicability of the DMC method relies on the assumption that the ground state is real and non-negative. However, this condition is not satisfied in a system with broken time-reversal symmetry, which is the case in the presence of a magnetic field. To overcome this difficulty, the fixed-phase DMC method has been proposed[47, 48]. The key idea is to write the wave function as

and determine the $\left|\Psi(\mbox{\boldmath$R$})\right|$ that gives the lowest energy for a fixed phase $\phi(\mbox{\boldmath$R$})$ by DMC method. This amounts to solving the Schrödinger equation

with

The diffusion equation is often efficiently solved by an importance sampling method. The so-called guiding function is defined as

where $\Psi_{T}$ is the trial wave function. Instead of solving Eq. 24, we have an equivalent equation:

where $\nabla=\left(\nabla_{1},\nabla_{2},\dots,\nabla_{N}\right)$ is the dN-dimensional (in d space dimensions) gradient operator, $\mbox{\boldmath$v$}_{D}\left(\mbox{\boldmath$R$}\right)$ is the dN-dimensional drift velocity defined by

and

is the local energy. We give their explicit forms in Appendix E based on Ref. [47].

The accuracy of the DMC energy depends on the choice of the phase $\phi(\mbox{\boldmath$R$})$. In this paper, our initial DMC trial wave functions will be our candidate trial wave functions described earlier. (In the case of 3D-DMC, these will also include the transverse wave function.) Each trial wave function identifies a specific phase $\phi_{T}$. The DMC algorithm then produces the lowest energy state for each choice of the trial wave function.

We stress that the DMC calculation automatically includes LLM. In fact, it is a non-perturbative method for treating LLM, which has been shown in past studies to give rather accurate results[75, 47, 76, 77, 48, 78, 51, 50, 79].

We implement a 2D fixed-phase DMC study of the problem where we obtain the lowest energy using DMC while setting $V(\mbox{\boldmath$R$})$ in Eq. 25 to $V_{\rm eff}(\mbox{\boldmath$R$})$ introduced in Eq. 19. This allows for LLM in a model where electrons confined to 2D are interacting via $V_{\rm eff}(\mbox{\boldmath$R$})$. As noted above, the phase is fixed by the trial wave functions described above.

As shown in Fig. 11, the comparison between the one-component CFFS and the Pfaffian state is very similar to that from VMC calculation and no transition occurs into the Pfaffian state (Fig. 14(a)). Meanwhile, the result for two-component states is quite different from the VMC result (Fig. 12). We find that the uncoupled $1/4+1/4$ CFFS is very efficient in lowering its energy in the presence of the LLM. In contrast to the VMC result, the system makes a transition from the pseudo-spin singlet CFFS directly into the $1/4+1/4$ CFFS state for most parameters (Fig. 14(b)). For very large widths and low densities, we find a small region of $(3,3,1)$ state.

When both one-component and two-component states are considered, the resulting phase diagram is shown in Fig. 2. The one-component CFFS makes a transition into the uncoupled two-component $1/4+1/4$ without going through an incompressible state, except in a small region where the well-width is large. We note here that the extrapolation of the 2D-DMC is less linear for the two-component states, which leads to a larger statistical error of about $2\times 10^{10}\text{cm}^{-2}$ for the density where the phase transition occurs.

The transverse trial wave function for the one-component states are chosen to be:

where $W$ is the width of the quantum well and $\alpha$ is a parameter introduced to improve the converging speed. Empirically we find the program to be most efficient and stable when $\alpha$ is tuned from $0.2$ to $0.8$ when the well width ranges from $2l_{B}$ to $10l_{B}$. However, one should keep in mind that the choice of $\alpha$ is a technical matter; as long as the number of iterations is large enough, any choice of $\alpha$ leads to the same result because the fixed-phase DMC solves for the lowest energy state within a given phase sector independent of the initial wave function.

For two-component states, before coming to 3D-DMC, it is necessary to address a significant difficulty. In general, one needs to evaluate the energy expectation of a given two-component state [e.g. Halperin $(3,3,1)$ state] by fully anti-symmetrizing the wave function. For two-component states in a single layer with real spin, the Coulomb interaction does not depend on the spin index, and all the cross-terms produced by anti-symmetrization vanish, and one can treat the two components as two sets of distinguishable particles, which greatly simplifies the calculation. This, however, is not true for the present case since the Coulomb interaction explicitly depends on the transverse coordinates and the cross-terms are nonzero. Here, one must include all the permutation-terms to fully anti-symmetrize the wave function. This is impractical for systems with greater than 10 particles because there are $\frac{N_{e}!}{(N_{e}/2)!(N_{e}/2)!}$ inter-component permutations. A special case is when the two transverse bases have no overlap. In this case, all cross-terms vanish and one can calculate the energy expectation as if the two components were two distinguishable sets of particles. We, therefore, use a transverse trial wave function for the two components to be strictly spatially separated, i.e. one basis function is strictly confined in the left half of the quantum well while the other component in the other half. In other words, our basis is given by:

where

represents the left- and right-components. The 3D trial wave function for the $(3,3,1)$ state is constructed as:

The other two-component states are constructed similarly, with the in-plane part replaced by the corresponding wave functions.

In Appendix F.1 we test the regime of validity of our approximation (that the right and left components are non-overlapping) for a system of four particles, for which we can implement full antisymmetrization. We find that our approximation becomes excellent near the upper phase boundary in Fig. 3.

This work concerns the nature of the FQHE at $\nu=1/2$ in wide quantum wells. We have evaluated the phase diagram of states at $\nu=1/2$ as a function of the quantum well width and the carrier density at three different levels of approximation.

Figure 1 shows the phase diagram obtained by a variational Monte Carlo calculation. In this calculation, we evaluate an effective 2D interaction with the help of a transverse wave function calculated by LDA at zero magnetic field. A shortcoming of this method is the assumption that the transverse wave function and $\Delta_{\rm SAS}$ evaluated at zero magnetic field remain valid at finite magnetic fields as well. This is of particular concern for the phase boundaries separating one- and two-component states, because these phase boundaries depend sensitively on $\Delta_{\rm SAS}$, which is a relatively large energy, and also a rapidly varying function of the quantum well width and density. For that reason, in Fig. 1 the phase boundary separating the $(3,3,1)$ and $1/4+1/4$ CFFS states is more reliable than that separating the one-component CFFS and the $(3,3,1)$ states.

The VMC calculation also does not include the effect of LLM directly. Figure 2 includes LLM for electrons interacting with an effective 2D interaction within a fixed phase DMC calculation.

The principal result of the present work is given in Fig. 3, which is the phase diagram obtained from a 3D fixed phase DMC. This method produces the ground state energy directly at a finite magnetic field, including, in principle, the effect of finite width and LLM. This suggests, although does not prove, that the incompressible state observed in experiments is the one-component Pfaffian state.

A technical difficulty of the 3D-DMC method is that for two-component states we must assume that the transverse wave function vanishes at the center of the quantum well. One may question if this affects comparisons between one- and two-component states. Fortunately, this is an excellent approximation near the upper phase boundary of Fig. 3, which separates the single component “CFFS/Pfaffian” state from the two-component “$1/4+1/4$ CFFS” state. That gives us some degree of confidence that the transition from the two-component $1/4+1/4$ CFFS occurs into the one-component Pfaffian state. Nonetheless, a definitive confirmation must await further experimental studies. In particular, thermal Hall measurements, which have shown half-quantized value at $5/2$ [84], can convincingly reveal whether the FQHE state here has a non-Abelian origin.

We also note that we do not consider the anti-Pfaffian state, which is the hole partner of the Pfaffian state [81, 82, 83]. These two are degenerate in energy in the absence of LLM, but LLM is expected to select one of them. We have not investigated this issue here, both because the anti-Pfaffian is harder to deal with numerically, and because the energy differences are expected to be small compared to the Monte Carlo uncertainty.

Before ending we list other assumptions made in our study. We do not consider the crystal phase. Previous theoretical studies of possible states in an ideal bilayer [85, 18] (i.e. two 2D layers separated by a distance $d$) did not find any crystal states, but a crystal may occur in wide quantum wells [45]. Such a crystal might be responsible for the fact that the experiments see an insulator on the either side of the FQHE state, rather than the compressible $1/4+1/4$ CFFS state. Of course, an alternative possibility is that disorder, omitted in our study, may turn the $1/4+1/4$ CFFS into an insulator. Experimental studies in better quality samples can clarify the situation.

Acknowledgement: We thank Mansour Shayegan for many insightful discussions. The work was made possible by financial support from the U.S. Department of Energy under Award No. DE-SC0005042. The VMC and DMC calculations were performed using Advanced CyberInfrastructure computational resources provided by The Institute for CyberScience at The Pennsylvania State University. A. C. B. acknowledges the Science and Engineering Research Board (SERB) of the Department of Science and Technology (DST) for funding support via the Start-up Grant SRG/2020/000154. Computational portions of exact diagonalization research work were conducted using the Nandadevi supercomputer, which is maintained and supported by the Institute of Mathematical Science’s High-Performance Computing Center. Some of the numerical diagonalizations were performed using the DiagHam package, for which we are grateful to its authors.