Double-layer Quantum Hall Ferromagnets: Difference between revisions

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==Introduction==  
==Introduction==  
The discovery of quantum Hall effect (QHE) was one of the most remarkable achievements in condensed matter physics in the second half of the last century. Together with superconductivity, QHE has extended our knowledge of quantum mechanics in many-body systems. In this effect, a 2D electron gas when subjected to a strong magnetic field exhibits a quantized Hall resistance and, at the same time, a nearly vanishing dissipative resistance in a range of magnetic field strength. The quantized Hall resistance can be characterized by integer quantum numbers in the integer quantum Hall effect (IQHE) or by some rational fractions in fractional quantum Hall effect (FQHE). In the IQHE regime, electrons occupy only the first few Landau levels because of the enormous degeneracy of these levels at high magnetic field. In the FQHE, the underlying physics is the Coulomb interaction and the correlation among electrons. It turns out that Coulomb interaction plays an important role not only in FQHE but also in QHE when we consider the spin dynamics of the system. In free space, the Zeeman splitting is exactly the same as the cyclotron splitting which is of the order of 100 K. But in real material, such as GaAs, the Zeeman splitting is reduced by about 2 orders of magnitude compared to the cyclotron splitting due to the reduction in effective mass and the gyromagnetic ratio of electrons. Therefore, at low temperature, the system lies in a special situation in which the orbital motion is fully quantized <math>\displaystyle k_B T \ll \hbar \omega_c</math> but the low-energy spin fluctuations are not forbidden <math>\displaystyle k_B T \sim g^* \mu_B B</math>. Interestingly, in the ground state of quantum Hall system, all the spins are still aligned ferromagnetically not due to the Zeeman splitting but due to the Coulomb interaction. Coulomb interaction is independent of spin therefore we can expect the same effect in double-layer quantum Hall system with the layer indices play the role of spin.
The discovery of quantum Hall effect (QHE) was one of the most remarkable achievements in condensed matter physics in the second half of the last century. Together with superconductivity, QHE has extended our knowledge of quantum mechanics in many-body systems. In this effect, a 2D electron gas when subjected to a strong magnetic field exhibits a quantized Hall resistance and, at the same time, a nearly vanishing dissipative resistance in a range of magnetic field strength. The quantized Hall resistance can be characterized by integer quantum numbers in the integer quantum Hall effect (IQHE) or by some rational fractions in fractional quantum Hall effect (FQHE). In the IQHE regime, electrons occupy only the first few Landau levels because of the enormous degeneracy of these levels at high magnetic field. In the FQHE, the underlying physics is the Coulomb interaction and the correlation among electrons. It turns out that Coulomb interaction plays an important role not only in FQHE but also in QHE when we consider the spin dynamics of the system. In free space, the Zeeman splitting is exactly the same as the cyclotron splitting which is of the order of 100 K. But in real material, such as GaAs, the Zeeman splitting is reduced by about 2 orders of magnitude compared to the cyclotron splitting due to the reduction in effective mass and the gyromagnetic ratio of electrons. Therefore, at low temperature, the system lies in a special situation in which the orbital motion is fully quantized <math>\displaystyle k_B T \ll \hbar \omega_c</math> but the low-energy spin fluctuations are not forbidden <math>\displaystyle k_B T \sim g^* \mu_B B</math>. Interestingly, in the ground state of quantum Hall system, all the spins are still aligned ferromagnetically not due to the Zeeman splitting but due to the Coulomb interaction. Coulomb interaction is independent of spin therefore we can expect that such interaction is also important in double-layer quantum Hall system in which the layer indices can be treated as pseudo spin degree of freedom.
In this report, we start out with an introduction to quantum Hall effect and quantum Hall ferromagnet. We then discuss two quantum many-body effects in double-layer quantum Hall ferromagnet by using pseudo spin treatment in which the layer indices play the role of spin. The first one is the interlayer phase coherent effect which associated with the interaction between electrons in different layers. The second one is the tunneling between the two layers and the effect of the in plane component of the magnetic field, which is related to the broken <math>\displaystyle U(1)</math> symmetry of the system.
In this report, we start out with an introduction to the quantum Hall effect, followed by the discussions on broken <math>\displaystyle SU(2)</math> symmetries and collective excitations in quantum Hall ferromagnet and double-layer quantum Hall ferromagnet. Finally, we discuss the interlayer phase coherent and the tunneling effects. The former is associated with the interaction between electrons from different layers, whereas the later is related to the broken <math>\displaystyle U(1)</math> symmetry of the system.


==Quantum Hall Effect==
==Quantum Hall Effect [1]==
The advance in technology makes it possible to prepare 2D electron gas system with extremely low disorder and high mobility. The motion of an electron in such system in the presence of a uniform magnetic field perpendicular to the plane can be studied conveniently by choosing the Landau gauge for the vector potential
The advance in technology makes it possible to prepare 2D electron gas system with extremely low disorder and high mobility. The motion of an electron in such system in the presence of a uniform magnetic field perpendicular to the plane can be studied conveniently by choosing the Landau gauge for the vector potential
   
   
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where the integer quantum number <math>\displaystyle \nu </math> is also known as Landau level filling factor.
where the integer quantum number <math>\displaystyle \nu </math> is also known as Landau level filling factor.


==Quantum Hall Ferromagnet==  
==Quantum Hall Ferromagnet [1]==  
Even though FQHE is not necessary for the discussions in this report, it is worth to mention that in that class of QHE, the quantum number <math>\displaystyle \nu </math> takes in some rational fractions
Even though FQHE is not necessary for the discussions in this report, it is worth to mention that in that class of QHE, the quantum number <math>\displaystyle \nu </math> takes in some rational fractions


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<math>\displaystyle \epsilon_k = \hbar \Delta + \frac {\rho_s}{nS} k^2 </math>
<math>\displaystyle \epsilon_k = \hbar \Delta + \frac {\rho_s}{nS} k^2 </math>


As promised, at there is Zeeman splitting gap at <math>\displaystyle \bold k = 0</math> and the dispersion is quadratic at small <math>\displaystyle \bold k</math>.
As promised, there is Zeeman splitting gap at <math>\displaystyle \bold k = 0</math> and the dispersion is quadratic at small <math>\displaystyle \bold k</math>.


==Double-layer Quantum Hall Ferromagnet==
==Double-layer Quantum Hall Ferromagnet [1-4]==
Double-layer quantum Hall ferromagnet consists of two layers of 2D electron gas. We are going to discuss the physics of this system using the pseudo spin analogy in which the layer indices play the role of pseudo spin degree of freedom. In this analogy, an electron is said to have pseudo spin up or down if it is found in the upper or lower layer, respectively. This analogy is useful especially when we study the effect of Coulomb interaction in the system because the interaction does not depend on spin. Although the layer separation can be made to be comparable to the distance between electrons in the same layer, the inter-layer Coulomb interaction is still smaller than the intra-layer Coulomb interaction. Therefore, in this system, the state with all electrons in one layer will be energetically unfavorable compared to the state with equal electron distribution between two layers. In other words, we can think of the state with all electrons in one layer as a fully charged capacitor which has a higher energy than if the two layer occupancies are equal. Hence, in this case the system does not have a full <math>\displaystyle SU(2)</math> symmetry even in the absence of an external magnetic field. The Lagrangian is now written as
Double-layer quantum Hall ferromagnet consists of two layers of 2D electron gas. We are going to discuss the physics of this system using the pseudo spin analogy in which the layer indices play the role of pseudo spin degree of freedom. In this analogy, an electron is said to have pseudo spin up or down if it is found in the upper or lower layer, respectively. This analogy is useful especially when we study the effect of Coulomb interaction in the system because the interaction does not depend on spin. Although the layer separation can be made to be comparable to the distance between electrons in the same layer, the inter-layer Coulomb interaction is still smaller than the intra-layer Coulomb interaction. Therefore, in this system, the state with all electrons in one layer will be energetically unfavorable compared to the state with equal electron distribution between two layers. In other words, we can think of the state with all electrons in one layer as a fully charged capacitor which has a higher energy than if the two layer occupancies are equal. Hence, in this case the system does not have a full <math>\displaystyle SU(2)</math> symmetry even in the absence of an external magnetic field. The Lagrangian is now written as


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<math>\displaystyle - \int {d^2 \bold {r} \beta m^z m^z} + \int {d^2 \bold {r} ntm^z}</math>
<math>\displaystyle - \int {d^2 \bold {r} \beta m^z m^z} + \int {d^2 \bold {r} ntm^z}</math>


Here, the <math>\displaystyle \beta </math> term accounts for the capacitive charging energy which is minimized when the two layers have the same number of electrons (<math>\displaystyle \bold m=0</math>). The last term is associated with the electron inter-layer tunneling which is characterized by the amplitude <math>\displaystyle t </math>.
Here, the <math>\displaystyle \beta </math> term accounts for the capacitive charging energy which is minimized when the two layers have the same number of electrons (<math>\displaystyle m_z = 0</math>). The last term is associated with the electron inter-layer tunneling which is characterized by the amplitude <math>\displaystyle t </math>.


Now, let's obtain the dispersion relation for the low lying excited states of this system.
The ground state can be written as


Linear instead of quadratic dispersion due to the <math>\displaystyle \beta </math> term
<math>\displaystyle \frac 1{\sqrt 2} \left( \begin{array}{c} 1\\1 \end{array} \right)</math>


...NEED MORE WORK
which is a symmetric (i.e. bonding) linear combination of the two layer states. Therefore, in the ground state the total pseudo spin is along the <math>\displaystyle x </math> axis
 
<math>\displaystyle \bold m = (1,0,0)</math>
 
Now, let's obtain the dispersion relation for the low lying excited states of this system, assuming that both <math>\displaystyle \Delta </math> and <math>\displaystyle t </math> are zero. In the excited state, the pseudo spin is perturbed from the <math>\displaystyle x </math> axis, we have
 
<math>\displaystyle \bold m = (\sqrt{1-m^y m^y - m^z m^z},m^y,m^z) \approx (1-\frac 1{2} m^y m^y - \frac 1{2} m^z m^z, m^y, m^z)</math>
 
In this case, <math>\displaystyle \bold A </math> can be chosen as
 
<math>\displaystyle \bold A \approx \frac 1 {2} (0, -m^z, m^y)</math>
 
Substituting <math>\displaystyle \bold A</math> and <math>\displaystyle \bold m</math> into the expression for the Lagrangian
 
<math>\displaystyle L = -\hbar S n \int {d^2 \bold {r} \{ {\dot{m}^\mu (\bold r)}A^\mu (\bold m)} \}-\frac 1{2} \rho_s \int {d^2 \bold {r} \partial_\mu m^\nu \partial_\mu m^\nu} - \int {d^2 \bold {r} \beta m^z m^z} </math>
 
<math>\displaystyle = -\hbar S n \int {d^2 \bold {r} \frac 1{2} \{ \frac {dm^z}{dt}m^y - \frac {dm^y}{dt}m^z \}}-\frac 1{2} \rho_s \int {d^2 \bold {r} \{ \partial_\mu m^y \partial_\mu m^y + \partial_\mu m^z \partial_\mu m^z \}} - \int {d^2 \bold {r} \beta m^z m^z} </math>
 
The variations in <math>\displaystyle m^y</math> and <math>\displaystyle m^z</math>
 
<math>\displaystyle m^y \rightarrow m^y + \delta m^y</math>
 
<math>\displaystyle m^z \rightarrow m^z + \delta m^z</math>
 
(we can also write down the variation in <math>\displaystyle m^x</math>, but it is not important in this problem)
 
results in the variation in the action
 
<math>\displaystyle \delta S = \delta \int {dt L} = -\hbar S n\int {dtd^2 \bold r \{ \frac {dm^z}{dt}(\delta m^y) - \frac {dm^y}{dt}(\delta m^z) \}} + \rho_s \int {dtd^2 \bold r \{ (\partial ^2 _\mu m^y)(\delta m^y) + (\partial ^2 _\mu m^z)(\delta m^z) \}} - 2 \beta \int {dtd^2 \bold r  m^z(\delta m^z)}</math>
 
The principle of least action reads
 
<math>\displaystyle -\hbar S n \frac {dm^z}{dt} + \rho_s \partial^2 _\mu m^y = 0</math>
 
<math>\displaystyle \hbar S n \frac {dm^y}{dt} + \rho_s \partial^2 _\mu m^z -2\beta m^z = 0</math>
 
in the second equation the second term is small compared to the last term and therefore can be ignored, we have
 
<math>\displaystyle -\hbar S n \frac {dm^z}{dt} + \rho_s \partial^2 _\mu m^y = 0</math>
 
<math>\displaystyle \hbar S n \frac {dm^y}{dt} -2\beta m^z = 0</math>
 
Acting <math>\displaystyle \frac {d}{dt}</math> on the upper equation and <math>\displaystyle \partial ^2 _\mu</math> on the lower equation, we get
 
<math>\displaystyle -\hbar S n \frac {d^2 m^z}{dt^2} + \rho_s \frac {d(\partial^2 _\mu m^y)}{dt} = 0</math>
 
<math>\displaystyle \hbar S n \frac {d(\partial^2 _\mu m^y)}{dt} -2\beta \partial^2 _\mu m^z = 0</math>
 
Eliminating <math>\displaystyle m^y </math> terms from the two equations, we obtains an equation for <math>\displaystyle m^z </math>
 
<math>\displaystyle \frac {d^2 m^z}{dt^2} = \frac {2 \rho_s \beta}{(\hbar S n)^2} \partial^2 _\mu m^z</math>
 
Obviously, this is a wave equation whose dispersion relation is linear
 
<math>\displaystyle \epsilon_k = \frac {\sqrt {2 \rho_s \beta}}{Sn} k</math>
 
So, we have shown that the broken symmetry induced by the Coulomb interaction in this system leads to the collective excitation with the linear dispersion (not quadratic as seen in quantum Hall ferromagnet).


===Interlayer Phase Coherent===
===Interlayer Phase Coherent===
Interaction
The wavefunction of the system with interlayer phase coherent can be written as


<math> |\psi \rangle = \prod_{\bold k}\{c^+_{\bold k \uparrow} + c^+_{\bold k \downarrow}e^{i \phi}\}|0 \rangle</math>
<math> |\psi \rangle = \prod_{\bold k}\{c^+_{\bold k \uparrow} + c^+_{\bold k \downarrow}e^{i \phi}\}|0 \rangle</math>


where <math> \displaystyle \bold k</math> is the orbital label.
This wavefunction defines a state in which every Landau orbital is occupied and there is a coherent linear combination of pseudo spin up and down states characterized by phase angle <math> \displaystyle \phi </math>. This means that the system has a definite total number of particles (filling factor equals to 1) but an indefinite number of particles in each layer. This state is said to have spontaneous broken symmetry similar to that found in BCS state of of superconductor.
In the absence of tunneling, the system exhibits <math> \displaystyle U(1) </math> symmetry associated with the conservation of number of particles in each layer (or charge conservation equivalently). Therefore the energy can not depend on the phase angle <math> \displaystyle \phi </math>. If we are to create excited state by varying the phase angle slowly in space, that leading term, which has <math> \displaystyle U(1) </math> symmetry, in the expansion of the energy is as follows


<math> H = \frac 1{2} \int {d^2 {\bold r}|\nabla \phi|^2} + ... </math>
<math> H = \frac 1{2} \rho_s \int {d^2 {\bold r}|\nabla \phi|^2} + ... </math>


... NEED MORE WORK
The origin of the finite spin stiffness <math> \displaystyle \rho_s</math> is the energy cost to create a gradient in the phase angle. When two electrons from different layers approach each other, they are in a linear superposition of states in each of the layers (symmetric and antisymmetric or bonding and anti-bonding linear combinations). That happens even if there is no tunneling. If they are of the same phase, then the wavefunction is symmetric under pseudo spin exchange, therefore the spatial part of the wavefunction is antisymmetric and must vanish when two particles approach each other to obey the Pauli principle. If there exists phase gradient, the amplitude for the particles from different layers to be near each other, and, consequently, the interlayer Coulomb interaction is higher.


===Interlayer Tunneling and Tilted Field Effects===
===Interlayer Tunneling===
Broken <math> \displaystyle U(1) </math> symmetry
The inter-layer tunneling of electrons between layers results in the broken <math> \displaystyle U(1) </math> symmetry. Therefore, the Hamiltonian of the system can be written as


<math> H_{eff} = \frac 1{2} \int {d^2 {\bold r}|\nabla \phi|^2} - \int {d^2 {\bold r}n  t  cos(\phi)} </math>
<math> H_{eff} = \frac 1{2} \rho_s \int {d^2 {\bold r}|\nabla \phi|^2} - \int {d^2 {\bold r}n  t  cos(\phi)} </math>


Tunneling
where the second part is the tunneling Hamiltonian which can also be expressed in the following forms


<math> H_{T} = - t \int {d^2 {\bold r} \{\psi^+_\uparrow (\bold r) \psi_\downarrow (\bold r)+\psi^+_\downarrow (\bold r) \psi_\uparrow (\bold r) \}} </math>
<math> H_{T} = - t \int {d^2 {\bold r} \{\psi^+_\uparrow (\bold r) \psi_\downarrow (\bold r)+\psi^+_\downarrow (\bold r) \psi_\uparrow (\bold r) \}} </math>
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<math> H_{T} = - t \int {d^2 \bold r S^x (\bold r)} </math>
<math> H_{T} = - t \int {d^2 \bold r S^x (\bold r)} </math>


In the present of magnetic field
Let's look for the phase factor that minimized the Hamiltonian.
 
<math> \delta H_{eff} = \frac 1{2} \rho_s \int {d^2 {\bold r}|\nabla {\phi + \delta \phi}|^2} - \int {d^2 {\bold r}n  t  cos(\phi + \delta \phi)} </math>  <math> \displaystyle -H_{eff} </math>
 
<math> \delta H_{eff} = \frac 1{2} \rho_s \int {d^2 {\bold r}\left(|\nabla {\phi}|^2 + 2|\nabla{\phi}||\nabla \delta \phi| + (\nabla \delta \phi)^2\right)} - \int {d^2 {\bold r}n  t  \left[ cos(\phi) cos(\delta \phi) - sin(\phi) sin(\delta \phi)  \right]}</math> <math> \displaystyle -H_{eff} </math>
 
Keeping only terms that linear in <math>\displaystyle \delta \phi </math> we have


<math> H_{eff} = \frac 1{2} \int {d^2 {\bold r}|\nabla \phi|^2} - \int {d^2 {\bold r}n t  cos(\phi - Qx)} </math>
<math> \delta H_{eff} = \rho_s \int {d^2 {\bold r}|\nabla{\phi}||\nabla \delta \phi|} + \int {d^2 {\bold r}n t  sin(\phi) \delta \phi}</math>


...NEED MORE WORK
Integrating the first term by parts, we obtain
 
<math> \delta H_{eff} = \int {d^2 {\bold r} \left( -\rho_s |\nabla ^2{\phi}| + n  t  sin(\phi)\right) \delta \phi}</math>
 
The condition for a minimum <math>\displaystyle H_{eff} </math> reads
 
<math> -\rho_s |\nabla ^2{\phi}| + n  t  sin(\phi) = 0 </math>
 
which can be rewritten as the following differential equation
 
<math> \displaystyle |\nabla ^2{\phi}| = \lambda ^2 sin(\phi) </math>
 
where
 
<math> \displaystyle \lambda = \sqrt {\frac{nt}{\rho_s}} </math>
 
The solution to the differential equation is
 
<math> \displaystyle \phi (\bold r) = 2 arcsin[tanh(\lambda x)] </math>
 
In the present of inter-layer tunneling, the energy is minimized when the pseudo spins are oriented in the <math> \displaystyle x </math> direction everywhere except in the central domain wall region where they tumble rapidly through <math> \displaystyle 2 \pi </math>.


==References==
==References==
[1] S. M. Girvin, The Quantum Hall Effect: Novel Excitation and Broken Symmetries, http://arxiv.org/abs/cond-mat/9907002
[2] K. Moon, H. Mori, Kun Yang, Lotfi Belkhir, S. M. Girvin, A. H. MacDonald,L. Zheng, and D. Yoshioka, Phys. Rev. B54, 11644 (1996).
[3] Kun Yang, K. Moon, L. Zheng, A. H. MacDonald, S. M. Girvin, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. Lett. 72, 732 (1994).
[4] K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. B51, 5138 (1995).

Latest revision as of 12:59, 9 December 2010

Introduction

The discovery of quantum Hall effect (QHE) was one of the most remarkable achievements in condensed matter physics in the second half of the last century. Together with superconductivity, QHE has extended our knowledge of quantum mechanics in many-body systems. In this effect, a 2D electron gas when subjected to a strong magnetic field exhibits a quantized Hall resistance and, at the same time, a nearly vanishing dissipative resistance in a range of magnetic field strength. The quantized Hall resistance can be characterized by integer quantum numbers in the integer quantum Hall effect (IQHE) or by some rational fractions in fractional quantum Hall effect (FQHE). In the IQHE regime, electrons occupy only the first few Landau levels because of the enormous degeneracy of these levels at high magnetic field. In the FQHE, the underlying physics is the Coulomb interaction and the correlation among electrons. It turns out that Coulomb interaction plays an important role not only in FQHE but also in QHE when we consider the spin dynamics of the system. In free space, the Zeeman splitting is exactly the same as the cyclotron splitting which is of the order of 100 K. But in real material, such as GaAs, the Zeeman splitting is reduced by about 2 orders of magnitude compared to the cyclotron splitting due to the reduction in effective mass and the gyromagnetic ratio of electrons. Therefore, at low temperature, the system lies in a special situation in which the orbital motion is fully quantized Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle k_B T \ll \hbar \omega_c} but the low-energy spin fluctuations are not forbidden Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle k_B T \sim g^* \mu_B B} . Interestingly, in the ground state of quantum Hall system, all the spins are still aligned ferromagnetically not due to the Zeeman splitting but due to the Coulomb interaction. Coulomb interaction is independent of spin therefore we can expect that such interaction is also important in double-layer quantum Hall system in which the layer indices can be treated as pseudo spin degree of freedom. In this report, we start out with an introduction to the quantum Hall effect, followed by the discussions on broken Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle SU(2)} symmetries and collective excitations in quantum Hall ferromagnet and double-layer quantum Hall ferromagnet. Finally, we discuss the interlayer phase coherent and the tunneling effects. The former is associated with the interaction between electrons from different layers, whereas the later is related to the broken Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle U(1)} symmetry of the system.

Quantum Hall Effect [1]

The advance in technology makes it possible to prepare 2D electron gas system with extremely low disorder and high mobility. The motion of an electron in such system in the presence of a uniform magnetic field perpendicular to the plane can be studied conveniently by choosing the Landau gauge for the vector potential

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold A(\bold r)=xB \hat{\bold y}} .

Using this gauge, the physics of the system is invariant under the translation in y axis. The wave function for the motion in this axis is therefore simply plane wave characterized by the wave vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold k} . By separating variables one can see that the wave function for the motion in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold x} axis is the same as that of a harmonic oscillator whose frequency equals to the cyclotron frequency and whose motion centers at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{\bold k}=-k l^2}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l= \sqrt {\frac {\hbar c} {eB}}} is the magnetic length.

The energy levels corresponding to the motion in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold x} axis are quantized as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{k,n}=(n+\frac1{2})\hbar \omega_c}

and are called Landau levels. Landau levels degenerate because they do not depend on the wave vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold k} . The numbers of states in each Landau levels can be calculated by imposing the periodic boundary condition in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold y} axis. Let’s consider a rectangular sample with dimensions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle L_x} ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle L_y} and the left edge is at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle x=-L_x} and the right edge is at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle x=0} , then the condition for the centers of motion to be inside the sample

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle -L_x \leqslant X_c \leqslant 0}

gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle 0 \leqslant k \leqslant \frac {L_x}{l^2}} .

The total number of states in each Landau levels is then:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle N = \frac {L_y}{2 \pi} \int_0^{\frac {L_x}{l^2}} dk = \frac {L_x L_y}{2 \pi l^2}=N_\phi}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle N_{\phi}=\frac {BL_xL_y}{\phi_0}} is the number of flux quanta penetrating the sample;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle {\phi_0}=\frac {hc}{e}} is the quantum of magnetic flux. So there is exactly one state per Landau level per flux quantum.

In general, applying an electric field along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold x} axis will result in a shift in the centers of the motions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle X_k} and a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold k} -dependence in Landau levels. If the electric field is non-uniform and there are disorders in the system, all the states in the bulk are localized due to Anderson’s localization. In the other words, deep inside the sample, the Landau levels are still Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold k} -independent, the group velocity is therefore equal to zero. But near the edges Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle x=-L_x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle x=0} of the sample Landau levels do depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold k} , furthermore, the group velocity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold v_k=\frac {1}{\hbar} \frac {\partial \epsilon_k}{\partial k} \hat {\bold y}}

has the opposite sign on the two edges of the sample. Therefore, there are edge currents running in opposite directions along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold y} axis. The electric transport in the system can be analyzed in analogy with the Landauer formalism for the transport in narrow wide. The edge currents correspond to the left and the right moving states between two Fermi points. The net current can be calculated by adding up the group velocities of all the occupied states:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle I=-\frac{e}{L_y}\int_{-\infty}^{+\infty}dk{\frac {L_y}{2 \pi}{\frac {1}{\hbar}}{\frac {\partial \epsilon_k}{\partial k}}}n_k}

where we assume that in the bulk only a single Landau level is occupied and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle n_k} is the occupied probability. At zero temperature, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle I=-\frac {e}{h} \int_{\mu_R}^{\mu_L}d \epsilon}

The definition of the Hall voltage drop is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle (+e)V_H \equiv (+e)(V_R - V_L)=\mu_R - \mu_L}

Hence

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle I=-\frac {e^2}{h}V_H}

If there are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu } Landau levels are occupied in the bulk, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle I=- \nu \frac {e^2}{h}V_H}

Here, the applied voltage is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold x} axis and the net current is along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold y} axis, therefore what we are calculating here is indeed the Hall resistance. So, we derive to what are observed in QHE

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \sigma_{xx} = 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \sigma_{xy} = - \nu \frac {e^2}{h}}

where the integer quantum number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu } is also known as Landau level filling factor.

Quantum Hall Ferromagnet [1]

Even though FQHE is not necessary for the discussions in this report, it is worth to mention that in that class of QHE, the quantum number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu } takes in some rational fractions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu = \frac 1{3};\frac 2{5};\frac 3{7};\frac 2{3};\frac 3{5};\frac 1{5};\frac 2{9};\frac 3{13};\frac 5{12};\frac {12}{5};...} .

The underlying physics in FQHE is the Coulomb interaction and the correlation between electrons. One may think that such interaction has nothing to do with integer quantum Hall system. However, the study of ferromagnetism in the system with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu = 1} has shown that Coulomb interaction also plays an important role. In a fully ferromagnetic state, all the spins are lined up parallel to each other, hence the spin part of the wave function is symmetric under the particle exchanges. Therefore, the spatial part of the wave function must be fully antisymmetric and vanish when any pair of particles approaches each other. Such condition keeps the particles away from each other thus lowers the Coulomb interaction. It turns out that for filling factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu = 1} , the Coulomb interaction is about 2 orders of magnitude greater than the Zeeman splitting and hence strongly stabilizes the ferromagnetic state. Indeed, at zero temperature, the ground state of the system with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \nu = 1} is spontaneously fully polarized even in the absence of an external magnetic field. The spin wave excitations can be studied by using the method similar to that used in Heisenberg model despite the fact that spins in Heisenberg model are localized whereas quantum Hall ferromagnet is a system of itinerant spins. The spin wave dispersion shows a gap at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold k=0} equal to the Zeeman splitting and starts out quadratically at small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \bold k} . At large wave vectors, the energy saturates at the Coulomb interaction scale. Effective action theory can also be used to reproduce these results. In this theory, first we write down the Lagrangian of the system and then derive the equation of motion. The Lagrangian can be written as

where

is the spin length

is the particle density

is a vector field of unit length which describes the local orientation of the order parameter (the magnetization)

can be determined by requiring that it leads to the correct precession of the magnetization, i.e.

is the external magnetic field

is a phenomenological spin stiffness

is the Lagrange multiplier that imposes the unit length constraint.

In the Lagrangian we have written, the correct precession of spin in magnetic field is guaranteed by the first part; the global invariance (in the absence of an external magnetic field) is realized in the second part. The last part is the constraint for the unit length of the vector field.

Now, let's obtain the dispersion relation when the system undergoes a small perturbation from the axis. For such perturbation, the magnetization is so close to axis that its and components are infinitesimally small, therefore we can write the magnetization as follows:

It is most convenient to work with the 'symmetric gauge'

Keeping only quadratic terms in the Lagrangian we obtain

Let's define a new complex field

The Lagrangian can be rewritten in terms of the new field as

The classical equation of motion can be obtain as usual using the principle of least action. When we do that, we get the following Schrodinger like equation

The solution to this equation is a plane wave with the dispersion relation

As promised, there is Zeeman splitting gap at and the dispersion is quadratic at small .

Double-layer Quantum Hall Ferromagnet [1-4]

Double-layer quantum Hall ferromagnet consists of two layers of 2D electron gas. We are going to discuss the physics of this system using the pseudo spin analogy in which the layer indices play the role of pseudo spin degree of freedom. In this analogy, an electron is said to have pseudo spin up or down if it is found in the upper or lower layer, respectively. This analogy is useful especially when we study the effect of Coulomb interaction in the system because the interaction does not depend on spin. Although the layer separation can be made to be comparable to the distance between electrons in the same layer, the inter-layer Coulomb interaction is still smaller than the intra-layer Coulomb interaction. Therefore, in this system, the state with all electrons in one layer will be energetically unfavorable compared to the state with equal electron distribution between two layers. In other words, we can think of the state with all electrons in one layer as a fully charged capacitor which has a higher energy than if the two layer occupancies are equal. Hence, in this case the system does not have a full symmetry even in the absence of an external magnetic field. The Lagrangian is now written as

Here, the term accounts for the capacitive charging energy which is minimized when the two layers have the same number of electrons (). The last term is associated with the electron inter-layer tunneling which is characterized by the amplitude .

The ground state can be written as

which is a symmetric (i.e. bonding) linear combination of the two layer states. Therefore, in the ground state the total pseudo spin is along the axis

Now, let's obtain the dispersion relation for the low lying excited states of this system, assuming that both and are zero. In the excited state, the pseudo spin is perturbed from the axis, we have

In this case, can be chosen as

Substituting and into the expression for the Lagrangian

The variations in and

(we can also write down the variation in , but it is not important in this problem)

results in the variation in the action

The principle of least action reads

in the second equation the second term is small compared to the last term and therefore can be ignored, we have

Acting on the upper equation and on the lower equation, we get

Eliminating terms from the two equations, we obtains an equation for

Obviously, this is a wave equation whose dispersion relation is linear

So, we have shown that the broken symmetry induced by the Coulomb interaction in this system leads to the collective excitation with the linear dispersion (not quadratic as seen in quantum Hall ferromagnet).

Interlayer Phase Coherent

The wavefunction of the system with interlayer phase coherent can be written as

where is the orbital label.

This wavefunction defines a state in which every Landau orbital is occupied and there is a coherent linear combination of pseudo spin up and down states characterized by phase angle . This means that the system has a definite total number of particles (filling factor equals to 1) but an indefinite number of particles in each layer. This state is said to have spontaneous broken symmetry similar to that found in BCS state of of superconductor.

In the absence of tunneling, the system exhibits symmetry associated with the conservation of number of particles in each layer (or charge conservation equivalently). Therefore the energy can not depend on the phase angle . If we are to create excited state by varying the phase angle slowly in space, that leading term, which has symmetry, in the expansion of the energy is as follows

The origin of the finite spin stiffness is the energy cost to create a gradient in the phase angle. When two electrons from different layers approach each other, they are in a linear superposition of states in each of the layers (symmetric and antisymmetric or bonding and anti-bonding linear combinations). That happens even if there is no tunneling. If they are of the same phase, then the wavefunction is symmetric under pseudo spin exchange, therefore the spatial part of the wavefunction is antisymmetric and must vanish when two particles approach each other to obey the Pauli principle. If there exists phase gradient, the amplitude for the particles from different layers to be near each other, and, consequently, the interlayer Coulomb interaction is higher.

Interlayer Tunneling

The inter-layer tunneling of electrons between layers results in the broken symmetry. Therefore, the Hamiltonian of the system can be written as

where the second part is the tunneling Hamiltonian which can also be expressed in the following forms

Let's look for the phase factor that minimized the Hamiltonian.

Keeping only terms that linear in we have

Integrating the first term by parts, we obtain

The condition for a minimum reads

which can be rewritten as the following differential equation

where

The solution to the differential equation is

In the present of inter-layer tunneling, the energy is minimized when the pseudo spins are oriented in the direction everywhere except in the central domain wall region where they tumble rapidly through .

References

[1] S. M. Girvin, The Quantum Hall Effect: Novel Excitation and Broken Symmetries, http://arxiv.org/abs/cond-mat/9907002

[2] K. Moon, H. Mori, Kun Yang, Lotfi Belkhir, S. M. Girvin, A. H. MacDonald,L. Zheng, and D. Yoshioka, Phys. Rev. B54, 11644 (1996).

[3] Kun Yang, K. Moon, L. Zheng, A. H. MacDonald, S. M. Girvin, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. Lett. 72, 732 (1994).

[4] K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and Shou-Cheng Zhang, Phys. Rev. B51, 5138 (1995).

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