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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 (Δ
_{SAS}) 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 (Δ
_{Z}) and the electron Coulomb energy, and are almost not affected by Δ
_{SAS}.

Quantum Hall (QH) effect [_{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 [

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 [

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) [_{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.

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:

H ^ = H ^ t + H ^ z + H ^ int ,

H ^ t = − Δ SAS 2 ∑ j σ ( c ^ j f σ + c ^ j b σ + c ^ j b σ + c ^ j f σ ) ,

H ^ Z = − Δ Z 2 ∑ j μ ( c ^ j μ ↑ + c ^ j μ ↑ − c ^ j μ ↓ + c ^ j μ ↓ ) ,

H ^ int = 1 2 ∑ j 1 − j 4 ∑ μ 1 − μ 4 ∑ σ 1 σ 2 V j 1 j 2 j 3 j 4 ( F μ 1 μ 2 μ 3 μ 4 ) × c ^ j 1 μ 1 σ 1 + c ^ j 2 μ 2 σ 2 + c ^ j 3 μ 3 σ 2 c ^ j μ 4 4 σ 1 , (1)

V j 1 j 2 j 3 j 4 ( F μ 1 μ 2 μ 3 μ 4 ) = e 2 4 π ε 1 2 a b δ ' ( j 1 + j 2 , j 3 + j 4 ) ∑ s ( q ≠ 0 ) e − i 2 π s ( j 1 − j 3 ) M [ 2 π q e − q 2 l B 2 / 2 F μ 1 μ 2 μ 3 μ 4 ] q x = 2 π a s , q y = 2 π b ( j 4 − j 1 ) ,

F f f f f = F b b b b = 1 , F f b b f = F b f f b = e − q d , otherwise F μ 1 μ 2 μ 3 μ 4 = 0 , (2)

where H ^ t , H ^ z and H ^ int represent the interlayer tunneling energy, Zeeman energy and two-body Coulomb interaction energy terms, respectively. c ^ j μ σ + ( c ^ j μ σ ) 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 H ^ int term are given by Equation (2) [_{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 φ r = c ^ j 1 μ 1 σ 1 + c ^ j 2 μ 2 σ 2 + ... c ^ j N e μ N e σ N e + | 0 〉 (e.g., {2f↑, 3b↑, 5f↓, 6b↓} = c ^ 2 f ↑ + c ^ 3 b ↑ + c ^ 5 f ↓ + c ^ 6 b ↓ + | 0 〉 in the four-electron system). The number of the basis vectors is calculated to be C 4 M N e , which gives 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 [_{y}, the dimension of the blocks are merely 1/M of that of the original H matrix [_{e} and MbeC. Then, the H matrix can be divided into independent M × C blocks with different combination of (J_{x}, J_{y}), J x , J y ∈ [ 0 , C − 1 ] , correspondingly, and its dimension is reduced to about 1/MC. The wave vector in the system is defined by k = ( k x , k y ) = ( J x 2 π / a , J y 2 π / b ) [_{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 E C = e 2 / 4 π ε l B , respectively.

The low-lying energy spectra of the ν = 2/3 bilayer QH system with the fixed d = 1.0 for several values of Δ_{SAS} and Δ_{Z} are presented in _{z}, and are measured from the ground-state energies indicated by arrows. Wave vectors k are normalized as kl_{B}, and (J_{x}, J_{y}) combinations contained in k are also represented. It

should be noted that k = 1.02/l_{B} states are two-fold degenerate because (0, 1) and (1, 0) are equivalent. Figures 1(a)-(c) represent the spectra at small Δ_{SAS} = 0.006. Actually, in these figures the marks of the ground state are almost doubly degenerate, composed of symmetry and antisymmetry states. While from the spectra at relatively large Δ_{SAS} = 0.2 in Figures 1(d)-(f), we find that all the ground states become non-degenerate states. On the other hand, the spin polarizations of the ground states are determined by Δ_{Z}. As Δ_{Z} is equal to a small value of 0.001 in _{z} = 0, being the SU states. When Δ_{Z} becomes as large as 0.05 in _{Z} has an intermediate value of 0.014 as shown in _{z}, thus they all may become the ground states in these crossover regions when Δ_{Z} is slightly changed.

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 _{SAS} is small.

_{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 _{gap} at the saturation points at large Δ_{SAS}, resulting from different excited states.

We plot the image of E_{gap} in the Δ_{SAS-ΔZ} plane with the choice of d = 1.0 in the

right side of the vertical dotted line in _{gap} represents the energy gap between the lowest first and second eigenstates for the non-degenerate ground states case. We also plot the image of E_{gapD} in the left side of the vertical dotted line in _{gapD} represents the energy gap between the second and third eigenstates for the degenerate ground states case. The SP and SU phases occupy the high- and low-Δ_{Z} regions with relative large E_{gap} (E_{gapD}) respectively. Because the phase transition between them is continuous in the finite size systems, in this paper, the crossover (CR) region is defined by the remaining part around the phase transition boundary where the values of E_{gap} (E_{gap}D) are less than one tenth of the average values of the E_{gap} (E_{gapD}) in the SP and SU phases. The solid lines in the _{z} = 0 and S_{z} = 2). The points A-E plotted in the figure are the experimental results of the phase boundaries (from N. Kumadaet al, Y. D. Zhenget al). The most significant feature in _{Z} from 0.009 to 0.014. Though the transition region slightly bends toward the low Δ_{Z} side when Δ_{SAS} increases, we can say generally it is only weakly dependent on Δ_{SAS}. This fact gives the reliable evidence that the SP-SU phase transitions are determined by the competition between Δ_{Z} and Δ_{C}. It should be emphasized that in the ν = 2/3 system, because the electrons only occupy the symmetric level, the factor Δ_{SAS} representing the

energy gap between the symmetric and antisymmetric levels, will not affect the phase transition boundary. For this reason, the Δ_{Z} holds a nonzero value in the transition region, even if Δ_{SAS} vanishes. Figures 3(c)-(f) present the SP, SU phases and CR region in the Δ_{SAS}-Δ_{Z} plane for several values of d. When d changes from 0.5 to 3.0, the whole CR region slightly shifts to the low Δ_{Z} side, and arrives to a limit about 0.005. Qualitatively, d has no great influence on the phase transition region.

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 φ 1 = c ^ 1f ↑ + c ^ 2 f ↑ + | 0 〉 , φ 2 = c ^ 1f ↑ + c ^ 2 b ↑ + | 0 〉 , φ 3 = c ^ 1b ↑ + c ^ 2 f ↑ + | 0 〉 , φ 4 = c ^ 1 b ↑ + c ^ 2 b ↑ + | 0 〉 and φ 5 = c ^ 3 f ↑ + c ^ 3 b ↑ + | 0 〉 , which belong to the SP phase. Using the Hamiltonian in Equation (1), we obtain a 5 × 5 block matrix calculated by the basis vectors above as follows:

[ H 00 1 ] = ( A 1221 − A 1212 − Δ Z − ( 1 / 2 ) Δ SAS − ( 1 / 2 ) Δ SAS 0 0 − ( 1 / 2 ) Δ SAS B 1221 − Δ Z − B 1212 − ( 1 / 2 ) Δ SAS B 2133 − ( 1 / 2 ) Δ SAS − B 1212 B 1221 − Δ Z − ( 1 / 2 ) Δ SAS − B 1233 0 − ( 1 / 2 ) Δ SAS − ( 1 / 2 ) Δ SAS A 1221 − A 1212 − Δ Z 0 0 B 2133 * − B 1233 * 0 B 3333 − Δ Z ) (3)

where A_{pqrs}_{ }( 〈 ϕ p f , ϕ q f | V | ϕ r f , ϕ s f 〉 or 〈 ϕ p b , ϕ q b | V | ϕ r b , ϕ s b 〉 ) and B_{pqrs}(= 〈 ϕ p f , ϕ q b | V | ϕ r b , ϕ s f 〉 or 〈 ϕ p b , ϕ q f | V | ϕ r f , ϕ s b 〉 represent the intralayer and interlayer Coulomb interaction energies, respectively. The subscripts p, q, r and s denote the momentum index in the y direction. ϕ j μ 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.

The block [ H 00 1 ] can be diagonalized analytically through the conventional method, and we derive all eigenvalues and eigenstates of the matrix characteristic equation [ H 00 1 ] ( A φ 1 A φ 2 A φ 3 A φ 4 A φ 5 ) T = E ( A φ 1 A φ 2 A φ 3 A φ 4 A φ 5 ) T . The five exact eigenvalues (E_{1} - E_{5}) of the block [ H 00 1 ] are given by

E 1 = ( A 1221 − A 1212 + B 1221 − B 1212 ) − Δ 1 2 − Δ Z , E 2 = A 1221 − A 1212 − Δ Z ,

E 3 = ( A 1221 − A 1212 + B 1221 − B 1212 ) + Δ 1 2 − Δ Z ,

E 4 = ( B 3333 + B 1221 + B 1212 ) − Δ 2 2 − Δ Z ,

E 5 = ( B 3333 + B 1221 + B 1212 ) + Δ 2 2 − Δ Z , (4a)

with

Δ 1 = ( A 1221 − A 1212 − B 1221 + B 1212 ) 2 + 4 Δ S A S 2 ,

Δ 2 = ( B 3333 − B 1221 − B 1212 ) 2 + 8 | B 1233 | 2 . (4b)

Since the relationship A 1221 − A 1212 < B 1221 − B 1212 holds always in the system, one obtains provided Δ_{SAS} = 0, exhibiting two degenerate ground states with equivalent eigenvalues E_{1} and E_{2}. The eigenstates in this case are simply c ^ 1 f ↑ + c ^ 2 f ↑ + | 0 〉 or c ^ 1 b ↑ + c ^ 2 b ↑ + | 0 〉 implying that electrons are restricted within one of the layers.

When Δ_{SAS} departures from zero, E_{1} is also separated from E_{2} taking a lower value. Thus, the degeneracy of the ground states is solved gradually. However, when Δ_{SAS} is sufficiently small, E_{1} and E_{2} are almost equal, and can be regarded as degenerate. In this case, the difference between E_{1} and E_{2} can be approximated by

| E 1 − E 2 | = Δ S A S | Δ S A S A 1221 − A 1212 − B 1221 + B 1212 | . (5)

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 interlayer Coulomb energies, because |E_{1} − E_{2}| is in proportion to the product of Δ_{SAS} and the ratio of Δ_{SAS} and | ( A 1221 − A 1212 ) − ( B 1221 − B 1212 ) | . Equation (5) reliably gives an analytic interpretation of the numerical results presented _{1} and E_{2} are expressed by

Ψ E 1 = a ( c ^ 1 f ↑ + c ^ 2 f ↑ + | 0 〉 + c ^ 1 b ↑ + c ^ 2 b ↑ + | 0 〉 ) + b 1 c ^ 1 f ↑ + c ^ 2 b ↑ + | 0 〉 + b 2 c ^ 1 b ↑ + c ^ 2 f ↑ + | 0 〉 + b 3 c ^ 3 f ↑ + c ^ 3 b ↑ + | 0 〉 ,

Ψ E 2 = ( 2 / 2 ) ( c ^ 1 f ↑ + c ^ 2 f ↑ + | 0 〉 − c ^ 1 b ↑ + c ^ 2 b ↑ + | 0 〉 ) . (6)

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.

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.

This research was supported in part by Grants-in-Aid for the basic research and development of Mitsubishi Electric (China) Company Limited.

Zheng, Y.D. and Sorita, T. (2018) Numeric and Analytic Investigation on Phase Diagrams and Phasetransitions of the ν = 2/3 Bilayer Fractional Quantum Hall Systems. Journal of Applied Mathematics and Physics, 6, 667-676. https://doi.org/10.4236/jamp.2018.64060