4.3 Mixed-fluid Models of the Phase Boundary
4.3.2 Continuous Crossover
As an alternative, the finite width of the transition can be understood without in- voking a first-order phase transition. For example, Simon, Rezayi, and Milovanovic (SRM) have suggested that instead of a true phase transition, a continuous crossover separates the correlated and uncorrelated phases [100]. SRM construct a set of wave functions for the νT = 1 bilayer system in which some number of the electrons in the
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consist of electrons bound to correlation holes only within their own layers. The CBs are constructed from electrons bound to one correlation hole in its own layer and to another correlation hole in the other layer. While the CFs fill up a Fermi sea, the CBs will eventually condense into the same state. The CBs provide the necessary interlayer correlations associated with theνT = 1 phase while the composite fermions
lack such interlayer correlations.
As the system evolves from the completely uncorrelated phase at d/` =∞ (con- sisting only of composite fermions) to the correlated phase at d/` = 0 (consisting only of composite bosons), the composite fermions are one-by-one transformed into composite bosons. The growing number of composite bosons leads to a continuous increase in interlayer correlations. SRM find numerical evidence that this variation in composite boson number occurs over a range of d/`, which would naturally ex- plain the residual width of the phase transition even in the zero temperature limit [65]. They also construct a Chern-Simons transport theory that arrives at the same semicircle law for drag resistivities (equation (4.1)) as found by Stern and Halperin. A distinguishing feature of this view is that, unlike the first-order scenario, the two types of quasiparticles are permitted tointermix spatially. As we shall see, this leads to observable consequences when the Zeeman energy is increased and the composite fermion phase becomes fully spin polarized.
In order to interpret our results using this view of the phase boundary, we first create a simple mean-field model of the continuous crossover proposed by Simon, Rezayi, and Milovanovic. We assume that the CBs are fully condensed and fully spin polarized. The CFs fill up two Fermi seas that correspond to the two spin states and are displaced in energy from each other by the Zeeman energy. Let fCF denote the
fraction of electrons in the CF phase, withf↑ occupying the spin-up Fermi sea andf↓
occupying the spin-down Fermi sea. We use the constraint fCF = f↑+f↓. Ignoring
any interactions among the various flavors of composite particles or any dependence of the CF effective mass on fCF, we write the total energy per electron as
E = 1 2EF0(f 2 ↑ +f↓2)− 1 2EZ(f↑−f↓) + (1−f)(C− 1 2EZ). (4.2)
CFs. The second term is the contribution from the Zeeman energies of the two spin species of CFs. The third term includes the energyCof each condensed CB as well as the Zeeman energy of each spin polarized CB. In this model,C is a phenomenological parameter that represents the net Coulomb energy cost associated with converting a CF into a CB. Presumably, C includes contributions from both intralayer and interlayer interactions and thus is expected to be a function of d/`. That is, as the effective interlayer separation becomes smaller it becomes energetically more cost effective to lower interlayer interactions by forming a CB from a CF, even if that gives rise to an increase of intralayer interactions. The simplest assumption is thatC
varies linearly with d/` during the transition region.
To obtain the ground state, we minimize E and obtain the following solutions for
f↑ and f↓:
f↑ =C/EF0, (4.3)
f↓ = (C−EZ)/EF0. (4.4)
Thus, ifEZ < C, CFs of both spins are present andfCF = (2C−EZ)/EF0. However,
if EZ > C, the CFs are fully spin polarized (f↓ = 0) and fCF = C/EF0. A mixed
phase (0< fCF <1) will exist over a range of C in both the partially and fully spin
polarized CF regimes. In the partially spin polarized regime, contours of fixed fCF
will satisfy the condition C = (fCFEF0+EZ)/2 and thus will rise with EZ. This is
illustrated in upper-left half of figure 4.6 for the contours associated with fCF = 1/4
and fCF = 3/4.
However, when the Zeeman energy reaches EZ = C (denoted as a dashed line in
figure 4.6), the CF phase becomes fully spin polarized, and the contours will become independent of Zeeman energy. This is indicated in the lower-right half of the phase diagram in figure 4.6. The knee in each contour should occur at EZ =fCFEF0 and
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!
d /!
!
fCF
=
14
!
fCF
=
43
!
EZ
/(e2
/"!)
Par$ally spin-‐polarized CF phase Fully spin-‐polarized CF phaseFigure 4.6: Depiction of phase diagram within the coexistence picture. At the dashed line, the Zeeman energy EZ is equal to the Fermi energy of the composite fermion
phase. Above the dashed line, the composite fermion phase is partially spin polarized. Below the dashed line, the composite fermion phase is fully spin polarized. Contours corresponding to composite fermion fractions fCF = 34 and fCF = 14 are shown as
Figure 4.6 also illustrates how we can understand the broadening of the transition region at highEZwithin the coexistence picture. If we use the range 1/4< fCF <3/4
to define the transition width ∆C, we find that the width ∆C = EF0/2 in the fully
spin polarized regime is twice as large as ∆C =EF0/4 in the partially spin polarized
regime. For a linear relation between C and d/`, one would also expect ∆(d/`) to grow by a factor of two between the low and high Zeeman regimes. This qualitatively agrees with our data near T = 0, but is lower than the observed factor of∼3 for the change in the transition width. A better comparison between this simple model and our Coulomb drag data might be obtained if there existed a theory relating fCF and
drag. However, such a theory remains lacking.
Before ending this section, we also note the possibility that both the first order and the continuous crossover pictures could provide faithful descriptions of the phase boundary, but under different regimes. That is, the phase transition could be first- order at low Zeeman energy, when there is a difference in spin polarizations for the two phases. However, as suggested by Zou and collaborators [139], the phase transition does not have to be first-order in the fully spin polarized regime. There, either a second-order transition or continuous-crossover might be allowed.