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Quantum phase transition in multilayer fractional quantum Hall systems
Physica B 298 (2001) 142}145
Quantum phase transition in multilayer fractional quantum Hall systems Sei Suzuki*, Yoshio Kuramoto Department of Physics, Tohoku University, Sendai 980-8578, Japan
Abstract Energetically stable states are studied for the in"nite multilayer quantum Hall system. Two-dimensional incompressible quantum Hall states are stable for appropriate "llings in the limit of in"nite interlayer separation, while threedimensional compressible states are stabilized by interlayer couplings such as tunneling and Coulomb interaction in the case of small separation. Trial states are introduced for typical cases and their Coulomb energies are calculated with use of the variational Monte Carlo method and the Hartree}Fock approximation. On energetical grounds, possible quantum phase transitions are discussed in this system. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum phase transition; Incompressible states; Compressible states; Multilayer FQH system
Study on the quantum Hall e!ect has developed generalization into various directions, for the last two decades. For example multilayer quantum Hall systems attract much interests theoretically and experimentally. Tens of electronic layers as well as a few layers have been fabricated owing to recent development in semiconductor technology. Bulk multilayer structures are also realized in organic conductor samples such as BEDT-TTF and (TMTSF) X. The present study focuses on the in" nite multilayer electron system under strong magnetic "eld. Before quantitative analysis we give the following naive expectation for our system: if the interlayer distance is large and interlayer coupling is weak enough, two-dimensional character dominates the system and incompressible fractional quantum Hall (FQH) states become stable in the case of
* Corresponding author. Fax: #81-22-217-6445. E-mail address: [email protected] (S. Suzuki).
particular "lling factor favoring FQH state. On the other hand three-dimensional compressible states take over due to interlayer coupling for small interlayer separation. Hence, a quantum phase transition between incompressible and compressible states accompanies the change of dimensional character. Moreover, a transition between di!erent incompressible states may also take place depending on the interlayer coupling. The ground state of the multilayer system is mainly controlled by two parameters: interlayer distance d and tunneling energy t. In the limit of dPR, layers are completely independent of each other and the Laughlin state is stable for "lling factor per layer "1/q where q is an odd integer. Other fractional quantum Hall states such as Halperin states which involve interlayer correlation are stable in d&l case where l is the magnetic length. When tunneling energy is not negligible, the Bloch state perpendicular to the layer is a good picture for one electron state and then three-dimensional
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 2 8 8 - 5
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
compressible state will be stable. For these three typical cases of " we introduce trial states re spectively and discuss their energetical stability. As calculational method of the Coulomb energy, variational Monte Carlo method is used for two fractional quantum Hall states and the Hartree Fock approximation is used for the three-dimensional state. The details are given in Ref. [1]. As the model to express the system, magnetic "eld induced three-dimensional electron gas model is considered. Here the multilayer structure is contained by positive charge and con"nement potential, that is, we assume discrete distribution of horizontal uniform positive charge sheets with interval d along the magnetic "eld (z direction) and periodic potential which con"nes electrons tightly into the layer corresponding to the positive charge sheet. With regard to the kinetic term, we assume that magnetic "eld is so strong that only the lowest Landau level (LLL) is relevant. Then the kinetic term is constant as long as tunneling is negligible, while we suppose the cosine band is formed due to the tight-binding approximation when tunneling is not negligible. The three trial states are introduced as follows. The "rst one is two-dimensional without tunneling and expressed by direct product of the Laughlin's " state in the each layer. * " [* ], (1) J J J J where l and p are the layer and the particle in a layer indices, respectively. and describe the motion in the layer and along the z direction:
1 * " ( ! ) exp ! , (2) J J JN JNY JN 4l NNY N " (z !ld ), (3) JN J N where "x!iy is the complex coordinate in the (x, y) space, l "(c /eB) , B is the strength of the magnetic "eld, and we have chosen the circular geometry Laughlin [2] took. (z!ld ) is the one particle wave function expressing localization inside the layer l. We call * &multi-Laughlin state', J that is expected to be stable for a large interlayer distance: d
143
The second trial state is introduced as incompressible state containing the interlayer correlation, which has an advantage near d&l . It is represented by " J J and
(4)
" ( ! ) ( ! ) J JN JNY JN J>NY J NNY N NY 1 ;exp ! . (5) JN 4l N It is called &(1 1 1) state' in the present paper. The numbers (1 1 1) indicate the exponents of the Jastrow factor which controls the strength of the electron correlation. The "rst &1' is the exponent of the Jastrow factor between a given layer and the nearest lower layer, the second &1' is that within the layer, and the last &1' is that between the layer and the nearest upper layer. Generally, we can de"ne () state for our system. For example (1 3 1) state is possible for ", and (2 1 2) state is also possible. Physically, (2 1 2) state should be unstable because the interlayer correlation becomes larger than the intralayer one in this state. In the case of " only the (1 1 1) state is allowed within the subspace of () type states. The third one is introduced for the case with "nite tunneling. In this case electrons itinerate along the z direction as Bloch wave and the state with smallest kinetic energy along the z direction is the simplest candidate for a stable state. We call it &itinerant state' and it is expressed by the Slater determinant. In the second quantized representation we obtain
itinerant" a? 0, (6) HI H I WI$ where a? is the creation operator of an electron HI with quantum number j in the LLL (such as momentum or angular momentum) and wave number k. k is the Fermi wave number given by k " /d. $ $ Although we assume " in this paper, the itiner ant state can be de"ned for all 3[0, 1].
The () state was originally introduced in Ref. [3].
144
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
For the two incompressible states, multi-Laughlin and (1 1 1) states, the energy expectation value is calculated numerically by the variational Monte Carlo (VMC) method. The relevant Hamiltonian consists of the Coulomb interaction only in these cases. Since the layers are independent of each other for the multi-Laughlin state, the expectation value per particle corresponds exactly to that of the original two-dimensional Laughlin state. Therefore, the energy is independent of the interlayer distance. It is not, however, the case with the (1 1 1) state. Namely the energy of the (1 1 1) state depends on the interlayer distance so that the energy decreases with decreasing the interlayer distance. This behavior is attributed to the fact that the e!ect of gain with regard to the interlayer Coulomb energy becomes larger with decreasing interlayer distance. The relevant Hamiltonian for the itinerant state consists of one particle kinetic part and the Coulomb interaction part. The Coulomb energy was obtained by the Hartree}Fock approximation. Since the direct interaction energy between electrons cancels exactly with the Coulomb energy of positive charge, only the exchange energy has "nite negative contribution. Therefore, the Coulomb energy of the itinerant state decreases with decreasing interlayer distance because the energy gain due to the exchange interaction increases with layers approaching each other. The calculation was performed for a "nite size system (42 particles in a layer and "ve layers with periodic boundary condition imposed along the z-axis) with respect to the (1 1 1) state, while the results on the other two states are obtained for the in"nite sized system (by extrapolation in multiLaughlin state case). In the case of (1 1 1) state the electronic correlations between the next nearest layers and other layers further apart should be small. Therefore, we consider that the system with "ve layers reasonably represent the situation of in"nite layers. The results on the Coulomb energy of the three trial states are given in Fig. 1. In the region of large d, the multi-Laughlin state is the lowest. On the other hand the (1 1 1) state is the lowest in the region of small d. There is no region where the itinerant state is the lowest. Hence we "nd that
Fig. 1. Comparison of Coulomb energies per particle for the three states. The result on the (1 1 1) state is shown for a "nite size system, while the other two results are shown for the in"nite size system. The (1 1 1) state has the lowest energy in the region of small inter-layer distance while the multi-Laughlin state becomes stable for large d. The crossing of the two states occurs near d&l . The itinerant state has no region where its Coulomb energy is the lowest.
the transition between the multi-Laughlin state and the (1 1 1) state takes place as long as the tunneling energy is negligible. Taking the tunneling energy into account, the energy of the itinerant state gets lower and the region where it is the lowest appears. Based on the above result a phase diagram on ground states of the present system is proposed as shown in Fig. 2, taking the tunneling energy t and the interlayer distance d as free parameters. It is summarized as follows: In the region d
(7)
where A and are constants. We put appropriate values into A and , namely A"0.2eH/l ,
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
Fig. 2. Proposed phase diagram in the plane of t and d. The solid lines are drawn on the basis of the VMC calculation for the (1 1 1) state of the "nite size system. The boundary with the itinerant state is determined from comparison with the Hartree}Fock calculation in the thermodynamic limit. The broken lines indicate formula (7) of t and d with appropriately chosen parameters.
"1.0/l for a case shown as t , and A" 0.3eH/l , "0.6795/l for another case shown as t . The parameters for t with B"10¹ are chosen to reproduce the band width of 4t" 2.5 meV"0.179eH/l at d"0.226 As "2.8l which seems appropriate for a GaAs/AlGaAs superlattice [4]. The change of parameter according to the
145
formula (7) (t or t in Fig. 2) corresponds to applying a uniaxial pressure. Then as shown in Fig. 2 single transition (t ) or double transitions (t ) take place depending on the parameters A and . We should remark whether other states get energetical stability than the three states introduced. For compressible states, inhomogeneous states with electron correlations such as the charge-density wave might have less energy than itinerant state which does not include electron correlations. For incompressible states, it is di$cult to consider other possible states as long as only the two-body correlation is important. In conclusion, we introduced three trial functions including both compressible and incompressible ones for the in"nite multilayer electron system under strong magnetic "eld. By calculating their Coulomb energies, we obtained the phase diagram in the plane of the tunneling energy and the interlayer distance. The quantum phase transitions between the trial states are consequently proposed.
References [1] S. Suzuki, Y. Kuramoto, Phys. Rev. B 62 (2000) 1921. [2] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [3] X. Qiu, R. Joynt, A.H. MacDonald, Phys. Rev. B 40 (1989) 11943. [4] H.L. StoK rmer, J.P. Eisenstein, A.C. Gossard, W. Wiegmann, K. Baldwin, Phys. Rev. Lett. 56 (1986) 85.
Quantum phase transition in multilayer fractional quantum Hall systems Sei Suzuki*, Yoshio Kuramoto Department of Physics, Tohoku University, Sendai 980-8578, Japan
Abstract Energetically stable states are studied for the in"nite multilayer quantum Hall system. Two-dimensional incompressible quantum Hall states are stable for appropriate "llings in the limit of in"nite interlayer separation, while threedimensional compressible states are stabilized by interlayer couplings such as tunneling and Coulomb interaction in the case of small separation. Trial states are introduced for typical cases and their Coulomb energies are calculated with use of the variational Monte Carlo method and the Hartree}Fock approximation. On energetical grounds, possible quantum phase transitions are discussed in this system. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum phase transition; Incompressible states; Compressible states; Multilayer FQH system
Study on the quantum Hall e!ect has developed generalization into various directions, for the last two decades. For example multilayer quantum Hall systems attract much interests theoretically and experimentally. Tens of electronic layers as well as a few layers have been fabricated owing to recent development in semiconductor technology. Bulk multilayer structures are also realized in organic conductor samples such as BEDT-TTF and (TMTSF) X. The present study focuses on the in" nite multilayer electron system under strong magnetic "eld. Before quantitative analysis we give the following naive expectation for our system: if the interlayer distance is large and interlayer coupling is weak enough, two-dimensional character dominates the system and incompressible fractional quantum Hall (FQH) states become stable in the case of
* Corresponding author. Fax: #81-22-217-6445. E-mail address: [email protected] (S. Suzuki).
particular "lling factor favoring FQH state. On the other hand three-dimensional compressible states take over due to interlayer coupling for small interlayer separation. Hence, a quantum phase transition between incompressible and compressible states accompanies the change of dimensional character. Moreover, a transition between di!erent incompressible states may also take place depending on the interlayer coupling. The ground state of the multilayer system is mainly controlled by two parameters: interlayer distance d and tunneling energy t. In the limit of dPR, layers are completely independent of each other and the Laughlin state is stable for "lling factor per layer "1/q where q is an odd integer. Other fractional quantum Hall states such as Halperin states which involve interlayer correlation are stable in d&l case where l is the magnetic length. When tunneling energy is not negligible, the Bloch state perpendicular to the layer is a good picture for one electron state and then three-dimensional
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 2 8 8 - 5
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
compressible state will be stable. For these three typical cases of " we introduce trial states re spectively and discuss their energetical stability. As calculational method of the Coulomb energy, variational Monte Carlo method is used for two fractional quantum Hall states and the Hartree Fock approximation is used for the three-dimensional state. The details are given in Ref. [1]. As the model to express the system, magnetic "eld induced three-dimensional electron gas model is considered. Here the multilayer structure is contained by positive charge and con"nement potential, that is, we assume discrete distribution of horizontal uniform positive charge sheets with interval d along the magnetic "eld (z direction) and periodic potential which con"nes electrons tightly into the layer corresponding to the positive charge sheet. With regard to the kinetic term, we assume that magnetic "eld is so strong that only the lowest Landau level (LLL) is relevant. Then the kinetic term is constant as long as tunneling is negligible, while we suppose the cosine band is formed due to the tight-binding approximation when tunneling is not negligible. The three trial states are introduced as follows. The "rst one is two-dimensional without tunneling and expressed by direct product of the Laughlin's " state in the each layer. * " [* ], (1) J J J J where l and p are the layer and the particle in a layer indices, respectively. and describe the motion in the layer and along the z direction:
1 * " ( ! ) exp ! , (2) J J JN JNY JN 4l NNY N " (z !ld ), (3) JN J N where "x!iy is the complex coordinate in the (x, y) space, l "(c /eB) , B is the strength of the magnetic "eld, and we have chosen the circular geometry Laughlin [2] took. (z!ld ) is the one particle wave function expressing localization inside the layer l. We call * &multi-Laughlin state', J that is expected to be stable for a large interlayer distance: d
143
The second trial state is introduced as incompressible state containing the interlayer correlation, which has an advantage near d&l . It is represented by " J J and
(4)
" ( ! ) ( ! ) J JN JNY JN J>NY J NNY N NY 1 ;exp ! . (5) JN 4l N It is called &(1 1 1) state' in the present paper. The numbers (1 1 1) indicate the exponents of the Jastrow factor which controls the strength of the electron correlation. The "rst &1' is the exponent of the Jastrow factor between a given layer and the nearest lower layer, the second &1' is that within the layer, and the last &1' is that between the layer and the nearest upper layer. Generally, we can de"ne () state for our system. For example (1 3 1) state is possible for ", and (2 1 2) state is also possible. Physically, (2 1 2) state should be unstable because the interlayer correlation becomes larger than the intralayer one in this state. In the case of " only the (1 1 1) state is allowed within the subspace of () type states. The third one is introduced for the case with "nite tunneling. In this case electrons itinerate along the z direction as Bloch wave and the state with smallest kinetic energy along the z direction is the simplest candidate for a stable state. We call it &itinerant state' and it is expressed by the Slater determinant. In the second quantized representation we obtain
itinerant" a? 0, (6) HI H I WI$ where a? is the creation operator of an electron HI with quantum number j in the LLL (such as momentum or angular momentum) and wave number k. k is the Fermi wave number given by k " /d. $ $ Although we assume " in this paper, the itiner ant state can be de"ned for all 3[0, 1].
The () state was originally introduced in Ref. [3].
144
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
For the two incompressible states, multi-Laughlin and (1 1 1) states, the energy expectation value is calculated numerically by the variational Monte Carlo (VMC) method. The relevant Hamiltonian consists of the Coulomb interaction only in these cases. Since the layers are independent of each other for the multi-Laughlin state, the expectation value per particle corresponds exactly to that of the original two-dimensional Laughlin state. Therefore, the energy is independent of the interlayer distance. It is not, however, the case with the (1 1 1) state. Namely the energy of the (1 1 1) state depends on the interlayer distance so that the energy decreases with decreasing the interlayer distance. This behavior is attributed to the fact that the e!ect of gain with regard to the interlayer Coulomb energy becomes larger with decreasing interlayer distance. The relevant Hamiltonian for the itinerant state consists of one particle kinetic part and the Coulomb interaction part. The Coulomb energy was obtained by the Hartree}Fock approximation. Since the direct interaction energy between electrons cancels exactly with the Coulomb energy of positive charge, only the exchange energy has "nite negative contribution. Therefore, the Coulomb energy of the itinerant state decreases with decreasing interlayer distance because the energy gain due to the exchange interaction increases with layers approaching each other. The calculation was performed for a "nite size system (42 particles in a layer and "ve layers with periodic boundary condition imposed along the z-axis) with respect to the (1 1 1) state, while the results on the other two states are obtained for the in"nite sized system (by extrapolation in multiLaughlin state case). In the case of (1 1 1) state the electronic correlations between the next nearest layers and other layers further apart should be small. Therefore, we consider that the system with "ve layers reasonably represent the situation of in"nite layers. The results on the Coulomb energy of the three trial states are given in Fig. 1. In the region of large d, the multi-Laughlin state is the lowest. On the other hand the (1 1 1) state is the lowest in the region of small d. There is no region where the itinerant state is the lowest. Hence we "nd that
Fig. 1. Comparison of Coulomb energies per particle for the three states. The result on the (1 1 1) state is shown for a "nite size system, while the other two results are shown for the in"nite size system. The (1 1 1) state has the lowest energy in the region of small inter-layer distance while the multi-Laughlin state becomes stable for large d. The crossing of the two states occurs near d&l . The itinerant state has no region where its Coulomb energy is the lowest.
the transition between the multi-Laughlin state and the (1 1 1) state takes place as long as the tunneling energy is negligible. Taking the tunneling energy into account, the energy of the itinerant state gets lower and the region where it is the lowest appears. Based on the above result a phase diagram on ground states of the present system is proposed as shown in Fig. 2, taking the tunneling energy t and the interlayer distance d as free parameters. It is summarized as follows: In the region d
(7)
where A and are constants. We put appropriate values into A and , namely A"0.2eH/l ,
S. Suzuki, Y. Kuramoto / Physica B 298 (2001) 142}145
Fig. 2. Proposed phase diagram in the plane of t and d. The solid lines are drawn on the basis of the VMC calculation for the (1 1 1) state of the "nite size system. The boundary with the itinerant state is determined from comparison with the Hartree}Fock calculation in the thermodynamic limit. The broken lines indicate formula (7) of t and d with appropriately chosen parameters.
"1.0/l for a case shown as t , and A" 0.3eH/l , "0.6795/l for another case shown as t . The parameters for t with B"10¹ are chosen to reproduce the band width of 4t" 2.5 meV"0.179eH/l at d"0.226 As "2.8l which seems appropriate for a GaAs/AlGaAs superlattice [4]. The change of parameter according to the
145
formula (7) (t or t in Fig. 2) corresponds to applying a uniaxial pressure. Then as shown in Fig. 2 single transition (t ) or double transitions (t ) take place depending on the parameters A and . We should remark whether other states get energetical stability than the three states introduced. For compressible states, inhomogeneous states with electron correlations such as the charge-density wave might have less energy than itinerant state which does not include electron correlations. For incompressible states, it is di$cult to consider other possible states as long as only the two-body correlation is important. In conclusion, we introduced three trial functions including both compressible and incompressible ones for the in"nite multilayer electron system under strong magnetic "eld. By calculating their Coulomb energies, we obtained the phase diagram in the plane of the tunneling energy and the interlayer distance. The quantum phase transitions between the trial states are consequently proposed.
References [1] S. Suzuki, Y. Kuramoto, Phys. Rev. B 62 (2000) 1921. [2] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [3] X. Qiu, R. Joynt, A.H. MacDonald, Phys. Rev. B 40 (1989) 11943. [4] H.L. StoK rmer, J.P. Eisenstein, A.C. Gossard, W. Wiegmann, K. Baldwin, Phys. Rev. Lett. 56 (1986) 85.