Numeric and Analytic Investigation on Phase Diagrams and Phasetransitions of the ν = 2 / 3 Bilayer Fractional Quantum Hall Systems

The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.


Introduction
Quantum Hall (QH) effect [1] [2] [3] [4] [5], which rivals superconductivity in its fundamental significance, has attracted a great deal of experimental and theoretical interest since its discovering.Especially, fractional QH (FQH) systems exhibit a variety of many body quantum phenomena, due to the complete domination of the electron Coulomb interactions.In a more complicated case, the bilayer QH systems with both spin and layer (pseudospin) degrees of freedom, four sub energy levels are formed in the lowest Landau level (LLL), and the ground states are to be determined by various factors such as interlayer/intralayer Coulomb energies (Δ C ), Zeeman energy (Δ Z ), interlayer tunneling energy (Δ SAS ) and bias energy (Δ bias ), etc.One expects reasonably that there exist rich quantum phases and many novel properties in the systems [6] [7].
The bilayer QH systems with filling factor ν = 2/m should be of the same type for any odd integer m.In this type of systems, the spin and pseudospin indices compete with each other, and the ground states are quite nontrivial because there are several ways to fill electrons into two sub energy levels in the LLL.Up to now, most of theoretical studies have focused on the ν = 2 bilayer integer QH (IQH) systems based on Hartree-Fock analysis [8] [9] [10] [11] [12] [13], effective bosonic spin theory [14] [15], exact-diagonalization calculations [16] as well as effective quantum field theory, and have identified three phases, the ferromagnetic (FM), the symmetric (SYM) and the canted antiferromagnetic (CAF) phases in the ground states.On the other hand, the ν = 2/3 bilayer QH system is a typical bilayer FQH system of this type, and can be regard as a best example of strongly correlated two-dimensional electron system for investigating the interplay of those entangled energy factors indicated above.However, even the basic problems, the ground states and the basic phase diagram of the ν = 2/3 system are still leaved uninvestigated from the theoretical viewpoint.Because of the existence of the degeneracy in the ground state as indicated below, in many cases, the ν = 2/3 bilayer QH system cannot simply be mapped to the ν = 2 bilayer QH system based on the composite-fermion picture, and must be investigated directly by the microscopic theories and the numerical calculations.
Motivated by the present situation mentioned above, as the first step, in this paper, we employ the numerical and traditional analytic methods to investigate the finite size bilayer FQH systems.We report on some basic features of the ground states in theν = 2/3 bilayer QH systems at the layer balanced point (Δ bias = 0) and provide evidential quantitative results from exact-diagonalization (ED) [16] [17] [18] [19] [20] numerical calculations and analytic approaches carried out at ν = 2/3.These features are essentially different from those in the ν = 2 systems, and reflect the special characteristics of the bilayer FQH systems. 1) At ν = 2/3, because the number of electrons is less than that of Landau sites in one sub energy level, in the small Δ SAS limit, the degeneracy of the ground states occurs depending on the relative strength of the intralayer and interlayer Coulomb energies.Contrarily, at ν = 2, because two sub energy levels in the LLL are filled by electrons, the ground states are non-degenerate even if Δ SAS vanishes.2) At least, the spin-polarize (SP) and spin-unpolarized (SU) phases exist in the ν = 2/3 bilayer QH systems.The phase transitions between them are continuous in the finite size systems, and are determined by the competition between Δ Z and Coulomb energy Δ C , not that between Δ Z and Δ SAS , as in the ν = 2 systems.The experimental results so far seem to support the conclusions above.

Exact-Diagonalization Method
We choose a finite size system with rectangular geometry for ED calculations.
Periodic boundary conditions are imposed on the rectangular cell of area a×b along the x and y axes, with the periodicities a andb, respectively.For simplicity, Landau level mixing and finite thickness of the system are not considered.
Within the LLL, the Hamiltonian at the layer balanced point is described by Equations (1) and (2) as follows: )

B
x y j j j j s j j i q l M s q q s q j j a b V F e j j j j e e F ab q , otherwise given by Equation (2) [17] [18], where l B means the magnetic length, d is the layer separation, q x (q y ) indicates the single-electron wave numbers in x(y) direction.We define M as the degeneracy of the single Landau level.Only when j 1 = j 2 (mod M), δ'(j1, j 2 ) = 1, otherwise, δ'(j 1 , j 2 ) = 0.The total number of electrons in the system is defined by N e .The N e -electron basis vector is expressed by Along the y axis, for instance, ( ) is conservative, hence the H matrix as well as the basis vectors can be divided into M independent blocks corresponding to M different values of J y , the dimension of the blocks are merely 1/M of that of the original H matrix [17] [18].Let the common factor of N e and MbeC.Then, the H matrix can be divided into independent M × C blocks with different combination of (J x , J y ), x y J J C ∈ − , correspondingly, and its di- Journal of Applied Mathematics and Physics mension is reduced to about 1/MC.The wave vector in the system is defined by [19].Since the z-component of the total spin (S z ) is conserved in the system, each block above can be divided further into N e + 1 independent blocks keeping S z from −N e /2 to N e /2, respectively.
In the ED calculations, we compute the matrix elements of these blocks and diagonalize them numerically.The aspect ratio a/b is fixed at 1.0.In this study, a finite size ν = 2/3 bilayer QH systems containing four electrons is chosen to execute the ED calculations.Henceforth, in the numerical results, the length and energy units are selected by l B and Coulomb energy scale , respectively.

Numerical Results and Discussion
The low-lying energy spectra of the ν = 2/3 bilayer QH system with the fixed d = We introduce the most important basis states (MIBSs) of the ground state to investigate the properties of them.We find that two degenerate ground states in Figure 1(b) have the same representative basis vectors in the MIBSs.They are {2f↑, 3f↑, 5f↑, 6f↑}, {2b↑, 3b↑, 5b↑, 6b↑}, {1f↑, 2f↑, 3f↑, 4f↑} and {1b ↑, 2b↑, 3b↑, 4b↑}, with the amplitudes of (0.6279, 0.6269, 0.3250, 0.3245) and (0.6269, −0.6279, 0.3245, −0.3250) for two degenerate ground states, respectively.Although the absolute value of the amplitudes are almost equal, the signs of the second and the fourth components are opposite, implying that they represent the symmetry and antisymmetry states, respectively.This phenomenon generally appears in the other degenerate ground states when Δ SAS is small.
Figure 2 demonstrates the energy gaps E gap between the lowest two eigenstates as a function of Δ SAS for several values of d in the ν = 2/3 bilayer QH system for Δ Z = 0.001 and 0.05.The lowest two eigenstates are the symmetric and antisymmetric states.As expected, they are almost degenerate in the small Δ SAS regions.
With the increase of Δ SAS , E gap smoothly widen, implying that the degeneracy of two states is resolved gradually, and finally tend to saturation points where the energies of the antisymmetric states exceed those of excited states created on the symmetric states.On the other hand, we notice that, when d changes from 0 to 10, the degenerate regions expand gradually and get to the limits.It is probably because the interlayer Coulomb energies exponentially decrease when d increases, while the intralayer Coulomb energies are independent from d.It is conjectured that the large difference between the intralayer and interlayer Coulomb energies will increase the degeneracy of the ground states.The general features in Figure 2

Analytic Investigation Using an Exactly Solvable Model
The degeneracy of the ground states when Δ SAS is small can be investigated analytically by a two-electron model in the SP phase.With the help of the symmetries mentioned previously, the H matrix of this model can be divided into nine independent blocks.We write down five basis vectors corresponding to the block with the J y (J x ) value of 0 and the S z value of 1 as where A pqrs ( represent the intralayer and interlayer Coulomb interaction energies, respectively.The subscripts p, q, r and s denote the momentum index in the y direction.jµ φ is the single-electron wave function in the LLL.A 1221 , for instance, is the direct Coulomb energy between two electrons in sites 1 and 2, while A 1212 is the exchange energy between them.

001
H can be diagonalized analytically through the conventional method, and we derive all eigenvalues and eigenstates of the matrix characteristic equation [ ]( ) ( ) The five exact eigenvalues (E 1 -E 5 ) of the block [ ] 001 H are given by ( ) .
It is obvious that the degeneracy of the ground states is increased not only by the small Δ SAS but also by the large difference between the intralayer and inter- For sufficiently small Δ SAS , since coefficients b 1 − b 3 are all close to zero, the two degenerate ground states with energies E 1 and E 2 are symmetric and antisymmetric states, as argued previously.

Conclusion
In summary, we have studied a typical bilayer FQH system with finite size using the numerical and analytic methods and provided evidential quantitative results.
We have found some basic features of the ground states at the layer balanced point in the ν = 2/3 bilayer QH systems taking advantage of the ED calculations and the analysis of an exactly solvable two-electron system.When Δ SAS is small, the degeneracy of the ground states occurs depending on the relative strength of the intralayer and interlayer Coulomb energies.SP-SU phase transitions are continuous in the finite size systems, and are determined by the competition between Δ Z and Coulomb energy Δ C , almost not affected by Δ SAS .These features exhibit the essential difference between the ν = 2/3 bilayer FQH systems and the ν = 2 bilayer IQH systems, and the peculiar characteristics generally existing in most bilayer FQH systems.The ED numerical method and the exact-solution method employed in this paper also can be considered to be valuable in studies of other bilayer FQH systems.
tunneling energy, Zeeman energy and two-body Coulomb interaction energy terms, respectively.ˆj c µσ + ( ˆj c µσ ) denotes electron creation (annihilation) operator.j > 0 is the y-direction momentum (Landau site) index (y-direction momentum j y = j − 1) , μ = f, b is the layer index that labels the front and back layers, and the spin denotation σ = ↑, ↓ represents the up and down spins.The coefficients in the int Ĥ term are the dimension of the Hamiltonian matrix (H matrix).Diagonalization process of the H matrix can be simplified by reducing its dimension with the help of several symmetries in the system.Owing to the translational symmetry along the x (y) axis, the Hamiltonian conserves the total momentum J x (J y ) in the x (y) direction[17] [18]  [19][20].

Figure 2 .Figure 3 .
Figure 2. Energy gaps Egap between the lowest two eigenstates as a function of Δ SAS for several values of d in finite size ν = 2/3 bilayer QH systems (a) When Δ Z = 0.001; (b) When Δ Z = 0.05.The energy and d are in units of 2 C 4 B E e l πε the SP phase.Using the Hamiltonian in Equation (1), we obtain a 5 × 5 block matrix calculated by the basis vectors above as follows: Coulomb energies, because |E 1 − E 2 | is in proportion to the product of Δ SAS and the ratio of Δ SAS and ( gives an analytic interpretation of the numerical results presented Figure 2. On the other hand, the eigenstates corresponding to E 1 and E 2 are expressed by