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

Y. D. Zheng^{}, T. Sorita^{}

Research and Development Department, Mitsubishi Electric (China) Company Limited, Shanghai, China.

**DOI: **10.4236/jamp.2018.64060
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Research and Development Department, Mitsubishi Electric (China) Company Limited, Shanghai, China.

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}.

Keywords

Bilayer Fractional Quantum Hall Systems, Phase Diagrams, Phase Transition, Exactdiagonalization, Exactly Solvable Model

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Zheng, Y. 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. doi: 10.4236/jamp.2018.64060.

1. 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.

2. 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:

$\stackrel{^}{H}={\stackrel{^}{H}}_{\text{t}}+{\stackrel{^}{H}}_{\text{z}}+{\stackrel{^}{H}}_{\text{int}}$ ,

${\stackrel{^}{H}}_{\text{t}}=-\frac{{\Delta}_{\text{SAS}}}{2}{\displaystyle \underset{j\sigma}{\overset{}{\sum}}\left({\stackrel{^}{c}}_{jf\sigma}^{+}{\stackrel{^}{c}}_{jb\sigma}^{}+{\stackrel{^}{c}}_{jb\sigma}^{+}{\stackrel{^}{c}}_{jf\sigma}^{}\right)}$ ,

${\stackrel{^}{H}}_{\text{Z}}=-\frac{{\Delta}_{\text{Z}}}{2}{\displaystyle \underset{j\mu}{\overset{}{\sum}}\left({\stackrel{^}{c}}_{j\mu \uparrow}^{+}{\stackrel{^}{c}}_{j\mu \uparrow}^{}-{\stackrel{^}{c}}_{j\mu \downarrow}^{+}{\stackrel{^}{c}}_{j\mu \downarrow}^{}\right),}$

${\stackrel{^}{H}}_{\text{int}}=\frac{1}{2}{\displaystyle \underset{{j}_{1}-{j}_{4}}{\overset{}{\sum}}{\displaystyle \underset{{\mu}_{1}-{\mu}_{4}}{\overset{}{\sum}}{\displaystyle \underset{{\sigma}_{1}{\sigma}_{2}}{\overset{}{\sum}}{V}_{{j}_{1}{j}_{2}{j}_{3}{j}_{4}}^{}\left({F}_{{\mu}_{1}{\mu}_{2}{\mu}_{3}{\mu}_{4}}^{}\right)\times {\stackrel{^}{c}}_{{j}_{1}{\mu}_{1}{\sigma}_{1}}^{+}{\stackrel{^}{c}}_{{j}_{2}{\mu}_{2}{\sigma}_{2}}^{+}{\stackrel{^}{c}}_{{j}_{3}{\mu}_{3}{\sigma}_{2}}^{}{\stackrel{^}{c}}_{j{}_{4}\mu {}_{4}{\sigma}_{1}}^{}}}},$ (1)

$\begin{array}{l}{V}_{{j}_{1}{j}_{2}{j}_{3}{j}_{4}}^{}\left({F}_{{\mu}_{1}{\mu}_{2}{\mu}_{3}{\mu}_{4}}^{}\right)=\\ \frac{{e}^{2}}{4\pi \epsilon}\frac{1}{2ab}\delta \text{'}\left({j}_{1}+{j}_{2},{j}_{3}+{j}_{4}\right){\displaystyle \underset{s(q\ne 0)}{\overset{}{\sum}}{e}^{-i2\pi \frac{s\left({j}_{1}-{j}_{3}\right)}{M}}{\left[\frac{2\pi}{q}{e}^{-{q}^{2}{l}_{B}^{2}/2}{F}_{{\mu}_{1}{\mu}_{2}{\mu}_{3}{\mu}_{4}}^{}\right]}_{{q}_{x}=\frac{2\pi}{a}s,{q}_{y}=\frac{2\pi}{b}\left({j}_{4}-{j}_{1}\right)}}\end{array}$ ,

${F}_{ffff}^{}={F}_{bbbb}^{}=1$ , ${F}_{fbbf}^{}={F}_{bffb}^{}={e}^{-qd}$ , otherwise ${F}_{{\mu}_{1}{\mu}_{2}{\mu}_{3}{\mu}_{4}}^{}=0$ , (2)

where
${\stackrel{^}{H}}_{\text{t}}$ ,
${\stackrel{^}{H}}_{\text{z}}$ and
${\stackrel{^}{H}}_{\text{int}}$ represent the interlayer tunneling energy, Zeeman energy and two-body Coulomb interaction energy terms, respectively.
${\stackrel{^}{c}}_{j\mu \sigma}^{+}$ (
${\stackrel{^}{c}}_{j\mu \sigma}^{}$ ) 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
${\stackrel{^}{H}}_{\mathrm{int}}$ term are 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
${\phi}_{r}={\stackrel{^}{c}}_{{j}_{1}{\mu}_{1}{\sigma}_{1}}^{+}{\stackrel{^}{c}}_{{j}_{2}{\mu}_{2}{\sigma}_{2}}^{+}\mathrm{...}{\stackrel{^}{c}}_{{j}_{N\text{e}}{\mu}_{N\text{e}}{\sigma}_{N\text{e}}}^{+}|0\rangle $ (e.g., {2f↑, 3b↑, 5f↓, 6b↓} =
${\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{3\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{5\text{f}\downarrow}^{+}{\stackrel{^}{c}}_{6\text{b}\downarrow}^{+}|0\rangle $ in the four-electron system). The number of the basis vectors is calculated to be
${C}_{4M}^{{N}_{\text{e}}}$ , 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 [17] [18] [19] [20]. Along the y axis, for instance,
${J}_{\text{y}}={\displaystyle \underset{n=1}{\overset{{N}_{\text{e}}}{\sum}}{j}_{\text{n}}}\left(\mathrm{mod}M\right)$ 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}),
${J}_{x},{J}_{y}\in [0,C-1]$ , correspondingly, and its dimension is reduced to about 1/MC. The wave vector in the system is defined by
$k=\left({k}_{x},{k}_{y}\right)=\left({J}_{x}2\pi /a,{J}_{y}2\pi /b\right)$ [17] [18] [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
${E}_{\text{C}}={e}^{2}/4\pi \epsilon {l}_{B}$ , respectively.

3. Numerical Results and Discussion

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 Figure 1. The energies are plotted by different marks indicating different total spins S_{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

Figure 1. Energy spectrum of finite size ν = 2/3 bilayer QH systems (four electrons) for several values of Δ_{SAS} and Δ_{Z} at fixed d = l_{B}. Eigenenergies are in unit of (l_{B} is the magnetic length), and are plotted by different marks indicating different total spins S_{z} (leaving space between the neighboring S_{z} in the same k for clarity). The wave number k, normalized as kl_{B}, is shown together with (J_{x}, J_{y}). The energies are measured from those of the ground states indicated by arrows. (a) Δ_{SAS} = 0.006, Δ_{Z} = 0.001. (b) Δ_{SAS} = 0.006, Δ_{Z} = 0.05. (c) Δ_{SAS} = 0.2, Δ_{Z} = 0.014. (d) Δ_{SAS} = 0.2, Δ_{Z} = 0.001. (e) Δ_{SAS} = 0.2, Δ_{Z} = 0.05. (f) Δ_{SAS} = 0.2, Δ_{Z} = 0.014.

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 Figure 1(a) and Figure 1(d), the ground states have the total spin S_{z} = 0, being the SU states. When Δ_{Z} becomes as large as 0.05 in Figure 1(b) and (e), the ground states turn out to be the SP states with Sz = 2. The ground states have k = 0, as implies that they are homogeneous states. Both SP and SU ground states are QH states, where the lowest excited states have nonzero energy gap: They compose SP and SU phases. When Δ_{Z} has an intermediate value of 0.014 as shown in Figure 1(c) and Figure 1(f), there are several states with almost equivalent lowest gap energies but different S_{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 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(a) and Figure 2(b) are similar, while the only difference between them is the values of E_{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

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
${E}_{\text{C}}={e}^{2}/4\pi \epsilon {l}_{B}$ and l_{B}, respectively.

right side of the vertical dotted line in Figure 3(a), where E_{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 Figure 3(b), where E_{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 Figure 3(a) and Figure 3(b) give the upper and lower limits of the CR regions, and the dashed lines between them indicate the spin-flipping positions (between S_{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 Figure 3(a) and Figure 3(b) is that, the SP-SU phase transition region (or the CR region) is limited in a narrow areawith Δ_{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

Figure 3. Image plot of energy gaps E_{gap} between the lowest first and second eigenstates (a) and E_{gapD} between second and third eigenstates (b) in the Δ_{SAS}-Δ_{Z} plane in finite size ν = 2/3 bilayer QH systems. Symbols SP, SU and CR denote the spin polarized, unpolarized phases and the crossover region, respectively. The solid lines give the upper and lower limits of the CR regions, and the dashed lines between them indicate the spin-flipping positions. The points A-E are the experimental results of the phase boundaries in the ν = 2/3 bilayer QH systems; (c)-(f) SP, SU and CR regions in the Δ_{SAS}-Δ_{Z} planes for several values of d: (c) d = 0.5; (d) d = 1.0; (e) d = 2.0; (f) d = 3.0. The energy and d are in units of
${E}_{\text{C}}={e}^{2}/4\pi \epsilon {l}_{B}$ and l_{B}, respectively.

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.

4. 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
${\phi}_{1}^{}={\stackrel{^}{c}}_{\text{1f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle $ ,
${\phi}_{2}^{}={\stackrel{^}{c}}_{\text{1f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle $ ,
${\phi}_{3}^{}={\stackrel{^}{c}}_{\text{1b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle $ ,
${\phi}_{4}^{}={\stackrel{^}{c}}_{1\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle $ and
${\phi}_{5}^{}={\stackrel{^}{c}}_{3\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{3\text{b}\uparrow}^{+}|0\rangle $ , 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:

$\begin{array}{l}\left[{H}_{\text{00}1}\right]=\\ \left(\begin{array}{ccccc}{A}_{1221}-{A}_{1212}-{\Delta}_{\text{Z}}& -\left(1/2\right){\Delta}_{\text{SAS}}& -\left(1/2\right){\Delta}_{\text{SAS}}& 0& 0\\ -\left(1/2\right){\Delta}_{\text{SAS}}& {B}_{1221}-{\Delta}_{\text{Z}}& -{B}_{1212}& -\left(1/2\right){\Delta}_{\text{SAS}}& {B}_{2133}\\ -\left(1/2\right){\Delta}_{\text{SAS}}& -{B}_{1212}& {B}_{1221}-{\Delta}_{\text{Z}}& -\left(1/2\right){\Delta}_{\text{SAS}}& -{B}_{1233}\\ 0& -\left(1/2\right){\Delta}_{\text{SAS}}& -\left(1/2\right){\Delta}_{\text{SAS}}& {A}_{1221}-{A}_{1212}-{\Delta}_{\text{Z}}& 0\\ 0& {B}_{2133}^{*}& -{B}_{1233}^{*}& 0& {B}_{3333}-{\Delta}_{\text{Z}}\end{array}\right)\end{array}$ (3)

where A_{pqrs}_{ }(
$\langle {\varphi}_{p\text{f}},{\varphi}_{q\text{f}}|V|{\varphi}_{r\text{f}},{\varphi}_{s\text{f}}\rangle $ or
$\langle {\varphi}_{p\text{b}},{\varphi}_{q\text{b}}|V|{\varphi}_{r\text{b}},{\varphi}_{s\text{b}}\rangle $ ) and B_{pqrs}(=
$\langle {\varphi}_{p\text{f}},{\varphi}_{q\text{b}}|V|{\varphi}_{r\text{b}},{\varphi}_{s\text{f}}\rangle $ or
$\langle {\varphi}_{p\text{b}},{\varphi}_{q\text{f}}|V|{\varphi}_{r\text{f}},{\varphi}_{s\text{b}}\rangle $ represent the intralayer and interlayer Coulomb interaction energies, respectively. The subscripts p, q, r and s denote the momentum index in the y direction.
${\varphi}_{j\mu}$ 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
$\left[{H}_{\text{00}1}\right]$ can be diagonalized analytically through the conventional method, and we derive all eigenvalues and eigenstates of the matrix characteristic equation
$\left[{H}_{\text{00}1}\right]{\left(\begin{array}{ccccc}{A}_{{\phi}_{1}}& {A}_{{\phi}_{2}}& {A}_{{\phi}_{3}}& {A}_{{\phi}_{4}}& {A}_{{\phi}_{5}}\end{array}\right)}^{\text{T}}=E{\left(\begin{array}{ccccc}{A}_{{\phi}_{1}}& {A}_{{\phi}_{2}}& {A}_{{\phi}_{3}}& {A}_{{\phi}_{4}}& {A}_{{\phi}_{5}}\end{array}\right)}^{\text{T}}$ . The five exact eigenvalues (E_{1} - E_{5}) of the block
$\left[{H}_{\text{00}1}\right]$ are given by

${E}_{1}=\frac{\left({A}_{1221}-{A}_{1212}+{B}_{1221}-{B}_{1212}\right)-\sqrt{{\Delta}_{1}}}{2}-{\Delta}_{Z}$ , ${E}_{2}={A}_{1221}-{A}_{1212}-{\Delta}_{Z}$ ,

${E}_{3}=\frac{\left({A}_{1221}-{A}_{1212}+{B}_{1221}-{B}_{1212}\right)+\sqrt{{\Delta}_{1}}}{2}-{\Delta}_{Z}$ ,

${E}_{4}=\frac{\left({B}_{3333}+{B}_{1221}+{B}_{1212}\right)-\sqrt{{\Delta}_{2}}}{2}-{\Delta}_{Z}$ ,

${E}_{5}=\frac{\left({B}_{3333}+{B}_{1221}+{B}_{1212}\right)+\sqrt{{\Delta}_{2}}}{2}-{\Delta}_{Z}$ , (4a)

with

${\Delta}_{1}={\left({A}_{1221}-{A}_{1212}-{B}_{1221}+{B}_{1212}\right)}^{2}+4{\Delta}_{SAS}^{2},$

${\Delta}_{2}={\left({B}_{3333}-{B}_{1221}-{B}_{1212}\right)}^{2}+8{\left|{B}_{1233}^{}\right|}^{2}$ . (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
${\stackrel{^}{c}}_{1\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle $ or
${\stackrel{^}{c}}_{1\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle $ 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

$\left|{E}_{1}-{E}_{2}\right|={\Delta}_{SAS}^{}\left|\frac{{\Delta}_{SAS}^{}}{{A}_{1221}-{A}_{1212}-{B}_{1221}+{B}_{1212}}\right|.$ (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
$\left|\left({A}_{1221}-{A}_{1212}\right)-\left({B}_{1221}-{B}_{1212}\right)\right|$ . Equation (5) reliably 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

${\Psi}_{{E}_{1}}^{}=a\left({\stackrel{^}{c}}_{1\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle +{\stackrel{^}{c}}_{1\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle \right)+{b}_{1}{\stackrel{^}{c}}_{1\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle +{b}_{2}{\stackrel{^}{c}}_{1\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle +{b}_{3}{\stackrel{^}{c}}_{3\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{3\text{b}\uparrow}^{+}|0\rangle $ ,

${\Psi}_{{E}_{2}}^{}=\left(\sqrt{2}/2\right)\left({\stackrel{^}{c}}_{1\text{f}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{f}\uparrow}^{+}|0\rangle -{\stackrel{^}{c}}_{1\text{b}\uparrow}^{+}{\stackrel{^}{c}}_{2\text{b}\uparrow}^{+}|0\rangle \right)$ . (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.

5. 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.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of interest.

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