Landau-level-mixing induced crystallization in the fractional quantum Hall regime
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the plane, where is the filling factor and is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking dependence, with the states away from the magic filling factors being much more susceptible to crystallization due to Landau level mixing than those at . Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for and also for in low density p-doped systems. We predict that in the vicinity of and , increasing LL mixing causes a transition not into an ordinary electron Wigner crystal but rather into a strongly correlated crystal of composite fermions carrying two vortices.
The search for two-dimensional Wigner crystalWigner (1934) in high magnetic fields has led to profound discoveries. The original ideaLozovik and Yudson (1975) was to induce a crystal state of electrons in two dimensions by effectively quenching their kinetic energy with the application of a strong transverse magnetic field, which drives them into the lowest Landau level (LL). While searching for the Wigner crystal, Tsui, Stormer and Gossard discovered Tsui et al. (1982) the Laughlin liquidLaughlin (1983). As we now know, over a range of filling factors the crystal phase is superseded by the formation of a topological quantum liquid of composite fermionsJain (2007, 1989); Lopez and Fradkin (1991); Halperin et al. (1993), manifesting through fractional quantum Hall (FQH) effect at and and Fermi seas at and . Theory suggested that the crystal should occur at sufficiently low filling factorsLam and Girvin (1984); Levesque et al. (1984), and extensive experimental work has been performed toward determining the phase boundary between the crystal and the liquidShayegan (2007); Fertig (2007); Jiang et al. (1990); Goldman et al. (1990); Paalanen et al. (1992); Manoharan and Shayegan (1994); Engel et al. (1997); Pan et al. (2002); Li et al. (2000); Ye et al. (2002); Chen et al. (2004); Csáthy et al. (2005); Sambandamurthy et al. (2006); Chen et al. (2006). For n-doped GaAs samples, in the limit of zero temperature, an insulating phase is seen for , and also for a narrow range of fillings between and . These features have persisted as the sample quality has significantly improved, indicating that the insulator is a pinned crystal rather than an Anderson-type single particle localized state. Direct evidence for a periodic lattice has been seen through commensurability oscillations Deng et al. (2016).
These observations are largely understood. Interestingly, while originally postulated to explain the liquid states, composite fermions turn out to be relevant also for the crystal phase at low . Nature appears to exploit both the composite fermion (CF) and the crystalline correlations to form a crystal of composite fermionsYi and Fertig (1998); Narevich et al. (2001); Chang et al. (2005); Archer et al. (2013), rather than an ordinary Wigner crystal of electronsMaki and Zotos (1983); Lam and Girvin (1984); Levesque et al. (1984). Theoretical studies have shown that the CF crystal is not only energetically favored over an electron crystalArcher et al. (2013) but is also an excellent representation of the crystal obtained in exact diagonalization studiesChang et al. (2005). There is growing experimental support for the CF nature of the crystalLiu et al. (2014); Zhang et al. (2015); Jang et al. (2017).
Despite substantial progress in our understanding, a striking puzzle has persisted since the early 1990s, namely a qualitative difference between the n-doped and p-doped GaAs systemsSantos et al. (1992a, b); Pan et al. (2005). In typical p-doped GaAs systems, the insulating phase is observed for filling factors below 1/3, and even between 1/3 and 2/5 for the lower density samples studied. In contrast, there is no sign of crystal for these filling factors in the n-doped samples with the same densities. These observations point to the importance of the role of LL mixing in establishing the crystal, because LL mixing is stronger in p-doped GaAs quantum wells due to the larger effective mass of holes. This motivated theoretical studies of the competition between FQH liquid and crystal states at Laughlin fractions 1/3, 1/5 and 1/7 through variationalZhu and Louie (1993); Price et al. (1993); Platzman and Price (1993), diffusionOrtiz et al. (1993), and path integral Monte CarloHe et al. (2005) methods.
We investigate in this article the competition between liquid and crystal states treating LL mixing non-perturbatively using the fixed phase diffusion Monte Carlo (DMC) method of Ortiz, Ceperley and Martin (OCM) Ortiz et al. (1993). Two important aspects of our work are: a) we address the issue as a function of continuous filling , which is necessary for understanding the observed re-entrant phase transitions; and b) we use very accurate crystal and liquid wave functions. The FQH state at maps into a state of CFs at filling , which is in general not an integer. (CF refers to composite fermion carrying two quantized vortices.) We assume a modelArcher et al. (2013) in which the CFs in the partially filled level (i.e. Landau-like level of composite fermions) form a crystal. Although this state has a crystalline order, we refer to it as FQH liquid in the following, because the pinning of this crystal by disorder results in a quantized Hall resistance. (Such a crystal residing on top of a FQH state is called a type-II CF crystal, by analogy to the type-II superconductor which exhibits zero resistance when the Abrikosov flux lattice is pinned.) The insulating state is modeled as a pinned “type-I” crystal of electrons or composite fermions in which all particles form a crystal. Extremely accurate lowest LL (LLL) wave functions are available for these states, which we use to fix the phase of the wave function in the DMC method; this is important because the accuracy of the results depends sensitively on the choice of the phase. (We note that while we use the terminology “electron crystal” or “electron liquid,” our results below apply to both electron and hole systems.)
Following the usual convention, we quantify the strength of LL mixing through the parameter , which is the ratio of the Coulomb energy to the cyclotron energy. (Here, is the magnetic length, is the band mass, and is related to the standard parameter as .) Our principal result is the phase diagrams shown in Fig. 1. The most striking feature they reveal is the strong dependence of the phase boundary separating the FQH and the crystal phases. For example, the FQH effect at and 2/5 survives up to the largest value of () we have considered, but the electron crystal appears already at for certain in between 1/3 and 2/5, and at even lower values of for . Another notable feature is that in the vicinity of and 2/9, LL mixing induces a transition into the strongly correlated CF crystal rather than an electron crystal. (If we only considered the electron crystal, no transition into the crystal state would occur at and for up to .) In what follows, we give details of calculations leading to these phase diagrams, and discuss their connection to experiments.
Fixed phase DMC: The goal is to find the minimum energy by varying over the entire Hilbert space of states, where is the Hamiltonian for interacting two-dimensional electrons in a magnetic field and represents the particle coordinates . Because this is not feasible for fermions, we employ an approximate strategy called the fixed phase DMCOrtiz et al. (1993) wherein we search for the ground state in a restricted subspace. (The fixed phase DMC is closely related to the fixed node DMC.Melton and Mitas (2017)) Following OCM, we substitute where is real and non-negative. The above energy is then given by with . Now, keeping the phase fixed and varying gives us the lowest energy within the subspace of wave functions defined by the phase sector . This minimization is most conveniently accomplished by the DMC methodReynolds et al. (1982); Foulkes et al. (2001). In this approach, one views the imaginary time Schrödinger equation, , as a diffusion equation, where is interpreted as the probability distribution of the diffusing “walkers” and is an energy offset. Evolving this equation in imaginary time projects out the lowest energy state, which is the ground state provided that the initial trial wave function has a non-zero overlap with the ground state. DMC is a method for implementing this scheme through importance sampling, where “walkers” in the dimensional configuration space proliferate (die) in regions of low (high) potential energy according to certain standard rules, and converge into the probability distribution of the ground state in the limit . The fixed phase DMC produces the lowest energy in the chosen phase sector, and hence a variational upper bound for the exact ground state energy.
We perform our calculations in the spherical geometryHaldane (1983) in which electrons are confined on the surface of a sphere, with a flux passing radially through it, where is an integer and is the flux quantum. We use as the unit of length and as the unit of energy. The particle position is identified through the “spinor” coordinates and . Melik-Alaverdian, Bonesteel and Ortiz Melik-Alaverdian et al. (1997) have formulated the fixed phase DMC in the spherical geometry through a stereographic projection, and we will follow their method.
Trial wave functions: The accuracy of the energies obtained from fixed phase DMC is critically dependent on the choice of the phase . It was shown by Güçlü and Umrigar Güçlü and Umrigar (2005) that the phase of the wave function is not significantly altered by LL mixing, which suggests that it is a good approximation to take accurate LLL wave functions as the trial wave functions to fix the phase . We follow this approach below.
In the spherical geometry, a localized wave packet centered at is given by where are particle coordinates. (This is the delta function projected in the LLL.) The wave function for the type-I electron crystal Maki and Zotos (1983) is given by , and for the type-I CF crystal by:
where and are the spherical coordinates for the crystal lattice sites. Because it is not possible to fit a hexagonal lattice perfectly on the surface of a sphere, we choose our crystal sites that minimize the Coulomb energy of point charges on a sphere. This is the famous Thomson problemThomson (1904), and the positions have been evaluated numerically and available in the literatureWales and Ulker (2006); Wales et al. (2009); Tho (). As expected, the Thomson lattice has a triangular structure locally but contains some defects. We consider the thermodynamic limit to eliminate the contribution from defects. We also note that only those values of are allowed that produce a positive value for . In particular, for , we only have the electron crystal available (at , the CF crystal wave function in Eq. 1 becomes identical to the Laughlin liquid wave function) and for we can form crystals also with .
For the FQH state of electrons at flux we construct , where is the wave function of electrons at effective flux and is the LLL projection operator, which will be evaluated using standard methodsJain and Kamilla (1997a, b). When corresponds to an integer filling , we obtain wave function for electrons at . When , we will assume that the composite fermions in the topmost partially filled level form a Thomson crystal, and for , we will assume a crystal of CF holes in the lowest level. This assumption is expected to be very accurate when the density of CF particles or holes is small, and a good first approximation in the entire range we have considered. The density profiles for certain type-I and type-II crystals on the surface of a sphere are shown in Fig. 2.
Results: In the spherical geometry the relation between , and has the form , where is called the “shift.” We define the filling factor as which gives the correct shifts at and and is sufficient for our considerations. All energies quoted below are energies per particle, and are corrected for the fact that the density in the spherical geometry has an dependent deviation from the thermodynamic density; this corresponds to multiplication by . We find that the density correction makes the energies independent to a large extent.
It is crucial to obtain the thermodynamic value for the ground state energy of the electron or the CF crystal. This is complicated by the fact that the fixed phase DMC energy for , the only systems accessible to our fixed phase DMC calculation, shows substantial finite size fluctuations due to the inevitable presence of defects, thereby precluding a reliable extrapolation to the thermodynamic limit. (See Appendix for details.) Fortunately, we find (see Appendix) that the energy difference is very well behaved and nearly constant as a function of , leading to an accurate thermodynamic value using systems with up to . Furthermore, it is possible to obtain the thermodynamic limits of very precisely for , because here we only need to perform variational Monte Carlo and can access much larger . The quantity thus produces an accurate value for the CF crystal energy for non-zero . We use the same method to obtain the energy of the FQH liquid phase.
Fig. 3 shows the energies of the liquid and crystal states for a system with 96 particles, which is large enough that the results reflect the thermodynamic limit. For this purpose, we first obtain the energies of the liquid and crystal states by variational Monte Carlo at , and then add to it to obtain the values shown in the figure. To obtain the energy reduction due to LL mixing, we assume that is a smooth function of in the narrow filling factor ranges considered and therefore it is sufficient to determine only for and , and then use at arbitrary in the neighborhood.
One may ask why the crystal phase is favored by LL mixing. LL mixing allows the electron wave packet at each site to become more localized, which enhances the kinetic energy but reduces the interaction energy. In Fig. 4 we show the line shape of the localized wave packet as a function of for both the electron and the CF crystals. Fig. 4 also shows how the pair correlation function of the liquid changes as a function of . (The pair correlation function is evaluated by the so-called mixed-estimator: , where is the initial trial state, is the “final” ground state, and is the average density.) While the energies of both the liquid and crystal states are reduced, only a detailed and quantitatively reliable calculation can tell if and where a transition into a crystal takes place.
Comparison with Experiment: In n-type GaAs quantum wells, with and we have . For () we have () for cm and () for cm. For these values, both the 1/3 and 1/5 states are deep in the FQH regime. The same is true of 2/5 and 2/9. This is consistent with the observation that all best quality n-doped samples show these FQH states. Furthermore, an insulator is seen in between 1/5 and 2/9 as well as below 1/5 in all high quality samples.
For p-doped samples, the larger value of makes the situation more interesting. The band mass of holes in GaAs can be 3-6 times larger than that of electrons, implying 3-6 times larger for the same densities. (The hole band structure is more complicated due to the presence of multiple bands, and the hole mass also depends on density as well as the confinement potential. For example, see Ref. Habib et al. (2004).) For hole doped samples have () for cm and () for cm, assuming hole band mass 3 (6) times larger than the electron band mass. This implies that for low density p-doped samples, an insulating crystal state can occur in between 1/3 and 2/5 and also below , although the 1/3 and 2/5 state should remain fractional quantum Hall (FQH) liquids. These predictions are in qualitative and good semi-quantitative agreement with experimental observations. At (which lies between 1/3 and 2/5), corresponds to cm ( cm) for a hole mass of 3 (6) times the electron band mass. In Santos et al.Santos et al. (1992b) a transition is seen at approximately cm. Much lower values of are required to produce a crystal for .
Because the critical value of is determined by extremely tiny energy differences, we find the agreement between theory and experiment to be satisfactory. It also provides an a posteriori justification for our phase choice. It is worth stressing that the phase boundaries in the filling factor region including and depend sensitively on modeling the crystal as a CF crystal; if one used the electron crystal instead, the phase transition would occur at much higher values of .
To estimate how finite transverse width of the quantum wells influences the results, we have carried out similar calculations for quantum wells of finite width. To take into account of finite transverse width, we assume a quantum well of width with a transverse wave function , where the transverse coordinate . The effective 2D interaction is then given by where . Our calculations predict (see Appendix) that the transition into the crystal state is pushed to approximately 20% higher values for . For yet larger , e.g. at , we cannot determine the transition positions sufficiently well (see Appendix), because the linear extrapolation of energy is not very reliable for and also because the liquid and crystal energies remain very nearly equal for a large range of . For such widths, the wave function also develops double hump structureManoharan et al. (1996), which we have not considered in this work for simplicity.
In summary, we have studied the effect of LL mixing on the phase boundary separating the FQH liquid and the electron / CF crystal using a fixed phase DMC approach. Our results demonstrate that the phase boundary is strongly dependent on the filling factor. In particular, we show that the enhanced LL mixing in low density p-doped GaAs systems can cause the crystal to be stabilized just below 1/3 as well as between 1/3 and 2/5, thus explaining the insulating behavior observed in these systems. We also find that in the vicinity of , increasing LL mixing induces a transition into the CF crystal rather than the electron crystal.
We are grateful to Ajit Balram and Mansour Shayegan for very useful discussions, and acknowledge financial support from the DOE Grant No. DE-SC0005042.
Appendix A Additional Details
Figs. 5-8 show thermodynamic extrapolations for various energies at certain selected filling factors. (The analysis for other filling factors is very similar.) For the liquid energies at zero width, it is possible to obtain the thermodynamic energies by a direct extrapolation of the finite results, as seen in the top left panels of Figs. 5-8. This is not the case for liquids in quantum wells of finite width, and crystals in either zero of finite width quantum wells. For these, we obtain thermodynamic extrapolations separately for (i) , where large systems are accessible; and (ii) which provides a reliable linear extrapolation even with particles.
Figs. 9 and 10 show the thermodynamic energies as a function of for several different filling factors of interest. Fig. 9 reveals that while the liquid remains the ground state at 1/3 and 2/5 all the way up to , a level crossing transition takes place at nearby fillings, such as and . Fig. 10 also compares the energies of the electron crystal and the CF crystal at and .
All figures also show results for three different widths: , and , where is the magnetic length. As noted in the main text, we assume a quantum well of width with a transverse wave function , where the transverse coordinate . The effective two-dimensional interaction is then given by where .
Fig. 11 shows how finite width modifies the phase diagram in the vicinity of . It has not been possible to obtain the finite width phase diagram reliably in the vicinity of , because, as seen in Fig. 10, the energies of the liquid and the crystal remain very close for a large range of .
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