Dicke Model for Quantum Hall Systems

Quantum Hall (QH) systems consist of many-body electron and nuclear spins. They are coupled so weakly through the hyperfine interaction that normally electron spin dynamics are scarcely affected by the nuclear spins. The dynamics of the QH systems, however, may drastically change when the nuclear spins interact with low-energy collective excitation modes of the electron spins. We theoretically investigate the nuclear-electron spin interaction in the QH systems as hybrid quantum systems driven by the hyperfine interaction. In particular, we study the interaction between the nuclear spins and the Nambu-Goldstone (NG) mode with the linear dispersion relation associated with the U(1) spin rotational symmetry breaking. We show that such an interaction is described as nuclear spins collectively coupled to the NG mode, and can be effectively described by the Dicke model. Based on the model we suggest that various collective spin phenomena realized in quantum optical systems also emerge in the QH systems.


I. INTRODUCTION
Quantum Hall (QH) systems exhibit fascinating macroscopic quantum phenomena [1,2]. For instance, various low-energy electron coherent phenomena in terms of spin and/or pseudospin (layer) internal degrees of freedom are realized. Research interests for the QH physics are not limited to the electron spin physics. However, the electron-nuclear spin dynamics has not yet attracted much attention. For instance, the GaAs semiconductor with the s-type conduction band has a large natural abundance of nuclear spins with the 3/2 nuclear spin angular momentum ( 69 Ga, 71 Ga, and 75 As). Although the nuclear spins interact with the electron spins mainly through the Fermi contact hyperfine interaction, it is usually so weak that electron transport properties are not affected by this interaction. Thus, previous studies of QH physics have mainly been focused on the electron spin physics, whereas the nuclear spins are utilized merely as a tool to investigate the electron magnetic properties [3].
The above situation may change when the nuclear spins interact with low-energy excitation modes of electron spins such as Nambu-Goldstone (NG) modes. These modes appear in the canted antiferromagnetic (CAF) phase [4][5][6][7][8][9][10][11][12], which is the most interesting one among the three phases of the total filling factor ν = 2 bilayer QH systems; the ferromagnetic phase, the spin-singlet phase and the CAF phase. In the CAF phase, the antiferromagnetic correlations between the electron spins in one of the layers (abridged as front layer) and those in the other layer (abridged as back layer) are generated, and the associated linear dispersing NG mode emerges. A new physics is expected to emerge in this phase. In the related context, the nuclear spin relaxation was experimentally estimated by using the resistivity-detected nuclear spin relaxation measurement [10], where the longitudinal resistance R xx is used as a measure of the nuclear-spin polarization. It has been shown that the nuclear-spin relaxation time in the CAF phase is the shortest compared with those in the other two phases (the ferromagnetic phase and the spin-singlet phase). More recent experiment [12], where the spatial nuclear-spin polarization distribution was recorded after the exposure to the CAF phase, showed a sudden change in the nuclear-spin polarization distribution from the initial one. These experimental results suggest a unique many-body interaction between electron and nuclear spin systems.
It is worthwhile to investigate the nuclear spin physics mediated by the hyperfine interaction in the QH systems also from the following three perspectives: First, nonequilibrium phenomena of nuclear spins in the QH systems are still less understood to date. Second, we expect a rich variety of many body effects as well as cooperative phenomena in terms of a nuclear-spin ensemble driven by the magnetic properties of the QH systems, just like the superradiance phenomena in optical cavity quantum electrodynamics systems composed of atomic Bose-Einstein condensate [13] and in quantum dots [14,15]. Third, the electronnuclear spin hybrid system is a candidate for a spin-base quantum information processing and computing with a coherent manipulation [3,16].
In this paper, for the first step of studying the electron-nuclear spin dynamics in the QH systems as hybrid quantum systems (the schematic illustration is presented in FIG. 1), we theoretically study the interaction between the nuclear spins and the linear dispersing NG mode associated with the U(1) spin rotational symmetry breaking. To make a detailed analysis and clearly understand its physics behind, we focus on the CAF phase in the ν = 2 bilayer QH system. We show that the NG mode couples with collective nuclear spins through the hyperfine interaction in the long-wavelength limit. As a result, such an interaction is effectively described by a Dicke model [17][18][19][20] with a continuous-mode, which has been extensively studied in the quantum optics. It is interesting that the interaction between nuclear spins and the linear dispersing NG mode mediated by the hyperfine interaction in the QH systems can be described by the same model as the two-level atomic system coupled with photonic modes.
Our analysis is not only valid for the nuclear spin-NG mode interaction in the CAF phase, but also for other QH systems where a linear dispersing NG mode is present due to the U(1) spin rotational symmetry breaking. For example, the skyrmion crystal formation in the monolayer QH system in the vicinity of the total filling factor ν = 1 [21] may belong to this category, where a linear dispersing spin wave emerges and is expected to enhance the nuclear spin relaxation rate.
This paper is organized as follows. In Sec. II, we analyze the hyperfine interaction in the QH systems. Based on this analysis, in Sec. III we derive the Dicke model as the effective Hamiltonian describing the interaction between the nuclear spins and the linear dispersing NG mode through the hyperfine interaction. It is the main result of this paper.

II. HYPERFINE INTERACTION IN THE QH SYSTEMS
We first analyze the hyperfine interaction in QH systems to derive the interaction Hamiltonian between the nuclear spins and the NG mode. As shown in FIG. 1, the QH systems consist of many-body nuclei and many-body electrons in the two-dimensional xy plane. We may assume that all electrons are in the s-type conduction band. A high magnetic field is applied perpendicular to the plane, B = (0, 0, −B ⊥ ) with B ⊥ > 0.
The interaction between nuclear and electron spins in the QH state is described by the contact hyperfine interaction [22][23][24], where S j is the electron spin at (x j , z j ), I i the nuclear spin at (X i , Z i ) assuming spin 1/2, and S(X i , Z i ) the three dimensional electron spin density. At sufficiently low temperature electrons are confined within the lowest energy level, and hence the motion of electrons along the z direction is frozen. We may approximate the quantum well by the system where electrons are located at the center of the well, that is, z j = z 0 for all j, and interact with nuclear spins at Z i = z 0 for all i. Then we may set for −L z /2 < Z i < L z /2 with L z the width of quantum well and S(X i ) the two dimensional electron spin density. The quantities γ e (γ n ), µ 0 , u(X i , z 0 ), N , and N e are the gyromagnetic ratio for electron (nucleon), the permeability of vacuum magnetic constant, the Bloch amplitude at (X i , z 0 ), the total number of polarized nuclear spins, and the total electron number, respectively. Since the Bloch amplitude is a periodic function with respect to the nuclear spin separation, we can set |u(X i , z 0 )| 2 = η =const. Then the hyperfine interaction (1) is rewritten as The values of η for Ga and As are given by η Ga = 2.7 × 10 3 and η As = 4.5 × 10 3 , respectively [25]. Here we take η = 10 3 .
The hyperfine coupling is weak compared with the Landau-level energy as well as the thermal energy. It is reasonable to assume that the electronic ground state is unchanged by the hyperfine interaction, and to replace the electron spin density in Eq.

III. DICKE MODEL IN THE QH SYSTEMS
We next derive the interaction Hamiltonian between the nuclear spins and the NG mode. We show that collective nuclear spins couple with the NG mode in the long-wavelength limit, and furthermore that it is effectively described by the Dicke model.

A. Electron Spin Configuration and Effective Hamiltonian for the NG mode
When the U(1) spin rotational symmetry is spontaneously broken around the z-axis, the in-plane component of the classical electron spin density is expressed as where S is a constant in the range 0 < |S| < 1. In the ground state the electron spins are in a spatially homogeneous configuration with a fixed orientation angle ϑ 0 of the in-plane spin component. The fluctuation field δϑ(x) is the associated NG mode. For the z-component, it is assumed that the fluctuation of the electron spin density is negligible, because it is gapped. We expand the above spin densities in terms of δϑ(x) up to the first order and substitute it to the hyperfine interaction (3). The zeroth order terms with respect to δϑ(x) are where S g z is the ground-state expectation value of the z component of the normalized electron spin density satisfying 0 < |S g z | < 1. The first term describes the in-plane magnetic field induced by the hyperfine interaction, which nuclear spins experience, while the second term generates the Knight shift K s ω s . These two terms have the same order of magnitude as the coupling g/ ∼ 100 rad·kHz/T, which was mentioned in the previous section, i.e., K s ω s ∼ 100 rad·kHz/T. Here we note that this value is comparable with the experimental result reported in [10], i.e., (K s ω s ) Exp ∼ 10 rad·kHz. We can drop these terms because they are negligible compared with the nuclear-spin Larmor frequency, which is ω s ∼ 10 rad·MHz/T.
The first-order terms in δϑ(x) describe the interaction between the nuclear spins and the NG mode, To understand the situation more clearly, we present an example, the CAF phase in the ν = 2 bilayer QH state. As presented in FIG. 2, electron spins have ferromagnetic correlations in each layer, whereas they have antiferromagnetic correlations between the two layers. The electron spin configuration in the front (back) layer S f(b) a (a = x, y, z) is described as where S f(b) a (a = x, y) and S f(b) z are the in-plane and z components of the electron spin in the front (back) layer, respectively. Here S and S caf z satisfy the conditions 0 < |S| < 1 and 0 < |S caf z | < 1. As a result, spins are canted coherently. By focusing on the in-plane component, electron spins orient homogeneously characterized by an angle ϑ 0 . Although the ground state energy does not depend on ϑ 0 , the ground state itself does, reflecting that the CAF state is the U(1) spin rotational symmetry broken state around z axis. The small fluctuation mode δϑ is the corresponding NG mode. We present the explicit formula of (7) for the case of the CAF phase in Appendix A.
The intriguing feature of the NG mode is that it has a linear dispersion in the CAF phase. We denote the Fourier transform of δϑ(x) as δϑ k , and introduce the canonical conjugate variable δσ k satisfying [δσ k , δϑ † k ′ ] = iδ(k − k ′ ). By making the momentum expansion, the effective Hamiltonian for the NG mode has the following form, where a, b and c are positive constants. This Hamiltonian is diagonalized by introducing another set of canonical variables, Now the effective Hamiltonian (9) is diagonalized as with E k a linear dispersion relation for the NG mode. We present the explicit formulas of Eqs.

B. Dicke Model
We now describe the hyperfine interaction Hamiltonian (6) in terms of the NG mode r k , r † k and the nuclear spins I x,y i . Using Eqs. (8) and (10), we obtain whereĨ ± = e ∓i(ϑ0+π/2) I ± (I ± = I x ± iI y ) is the rotated in-plane nuclear spin. In the rest of this paper we just writeĨ ± as I ± . To derive (13), we used the relations G k = G −k , δσ † k = δσ −k and ϑ † k = ϑ −k . They follow from the fact that G k are even function of k, and δσ(x) = (d 2 k/2π)e −ikx δσ † k and δϑ(x) are real fields. Using the rotating-wave approximation for (13), we obtain Furthermore, the i dependence disappears from R i and R † i in the long wave-length limit e ikXi ≈ 1. We may rewrite (14) as Indeed, as the NG mode has a long wavelength, the approximation e ikXi ≈ 1 is valid in this system. For instance, in the case of the CAF phase the wavelength of the NG mode becomes λ s ∼ 10 7 Å for E k = ω s with ω s ∼ 10 rad·MHz/T, where E k = γ|k| (γ = 2 √ ac > 0) is the linear dispersion relation for the NG mode. The value of λ s is about the same as the sample size L ∼ 100µm (see Eq. (B11)). Thus it is a good approximation at this energy scale. If the dispersion for the NG mode were quadratic with the same coefficient as the linear one, the wavelength at E k = ω s would be around λ s ∼ 10 3 Å, which is much smaller than the sample size. Such a case, the long-wavelength limit is not a good approximation so that the interaction Hamiltonian H SR cannot be expressed in terms of the NG mode and the collective spin.
As a result, the interaction between the nuclear spins and the linear dispersing NG mode is described as an interaction between the collective nuclear spin J = i I i and the NG mode R with the coupling constant g in the long-wavelength limit.
The other relevant terms are the nuclear-spin Hamiltonian H S describing the Larmor precision, The total effective Hamiltonian H = H S + H R + H SR is given by Consequently, the interaction between the nuclear spins and the linear dispersing NG mode mediated by the hyperfine interaction is described effectively by the Dicke model [17][18][19][20] with a continuous-mode, where the collective spin operator J a (a = x, y, z) with its magnitude N/2 interacts with the NG mode with the coupling g. It is interesting that the nuclear spin-NG mode interaction mediated by the hyperfine interaction in the QH systems can be described by the analogue model as the two-level atomic systems surrounded by the electromagnetic field in the vacuum: Nuclear-spins 1/2 correspond to the two-level atoms, whereas the NG mode corresponds to the electromagnetic field in the vacuum [20]. For an explicit derivation of this Dicke model in the case of the CAF phase, see Appendix C.

IV. CONCLUSION
In this paper, we have presented the theoretical studies on the interaction between the nuclear spins and the linear dispersing NG mode due to the spin U(1) rotational symmetry breaking in the QH systems mediated by the hyperfine interaction. Since the NG mode has a long wavelength, the nuclear spins couple collectively with the NG mode. Consequently, the nuclear spin-NG mode interaction can be described effectively by the Dicke model with a continuous-mode. This physics could be captured by regarding the QH systems as hybrid quantum systems comprise of electron and nuclear spins. In this paper, we focused on the CAF phase in ν = 2 bilayer QH systems and demonstrated that the interaction between the nuclear spins and the NG mode is described by the above Dicke model. This Dicke model must be also applied to QH systems with the linear dispersing NG mode due to the spin U(1) rotational symmetry breaking in general, for instance, the nuclear spin-NG mode interaction in the vicinity of ν = 1 monolayer QH systems in the presence of the skyrmion crystal formation.
We would like to emphasize that the Dicke model derived in this paper can lead to new directions for the study of QH physics, which cannot be obtained solely from the perspectives of the solid state physics. Combining the perspectives of solid state physics with these from quantum optics, we may reveal rich and new phenomena in the QH systems. For instance, the realization of phenomena similar to those in the quantum optical systems, for example, superradiance in terms of nuclear spins in the QH systems might be one possibility. The study of the spin-boson dynamics inherent to the QH system might be another interesting direction.

Appendix A: Order Parameters in the CAF phase
We present the derivation of the Dicke model by constructing the Hamiltonian H R for the NG mode and the interaction Hamiltonian H SR between the nuclear spin and the NG mode in the case of the CAF phase. To derive the Dicke model, we make a concise review of the effective Hamiltonian density for the ground state and the NG modes in bilayer QH systems [2,8,11]. We start with the discussion on the phase structure as well as the spin density configuration, and the associated NG modes at ν = 2. Then we present the effective Hamiltonian density for the linear dispersing NG mode in the CAF phase.
In the bilayer QH systems electrons possess the four internal degrees of freedom, the spin and the layer (pseudospin). We denote the two layers as the "front" and "back" layers. The electron field operator in the bilayer QH systems is represented as Ψ(x) = (ψ ↑f (x), ψ ↓f (x), ψ ↑b (x), ψ ↓b (x)). The physical operators in this system are expressed in terms of the following sixteen operators, the density operator and SU(4) isospin operators. In terms of the electron field operator Ψ(x), they are given by where a, b = x, y, z and τ spin a = τ a 0 0 τ a , τ ppin with τ a denoting the Pauli matrices. The operator ρ(x) is the density operator, while the SU(4) isospin density operator S a (x), P a (x) and R ab (x) represent the spin density operator, the pseudospin density operator, and the R-spin density operator, respectively. On the other hand, the total Hamiltonian in this system is given by [2], H = H K + H C + H Z + H PZ , where H K is the kinetic term which generates the Landau level, H C the Coulomb interaction term, H Z the Zeeman interaction term, and H PZ the pseudo-Zeeman term composed of the tunnelling interaction term and the bias term, which represents the creation of the densityimbalanced configuration between the two layers. The Coulomb interaction is decomposed into the form represents the SU(4) invariant (non-invariant) term.
We analyze the classical Hamiltonian density of H = d 2 xH(x) by performing the lowest Landau level projection. We denote the generic lowest Landau level state as |S . We derive the form of H cl (x) = S|H(x)|S , which are represented by the classical density operators ρ cl (x) = S|ρ(x)|S , S cl a (x) = S|S a (x)|S , P cl a (x) = S|P a (x)|S , and R cl ab (x) = S|R ab (x)|S . Actually we express H cl in terms of the normalized SU(4) operators defined by S cl a (x) = ρ Φ S cl a (x), P cl a (x) = ρ Φ P cl a (x), and R cl ab (x) = ρ Φ R cl ab (x), by setting ρ cl (x) = ρ 0 due to the incompressibility of the QH state. In the ν = 2 bilayer QH system the SU(4) order parameters, which are the expectation values of the normalized SU(4) operators in the ground state, are given by [8] with where ∆ Z is the Zeeman gap and ∆ SAS the tunneling gap; α, β (|α|, |β| ≤ 1) and ω are real parameters. The quantity σ 0 is the imbalanced parameter, representing the density difference between the front and back layers, and defined by 0 the electron density in the front (back) layer. Here the parameters α and β are determined by minimizing the classical Hamiltonian given by (B1), where the normalized isospin densities are in spatially homogeneous configurations. They satisfy the condition (S 0 a ) 2 + (P 0 a ) 2 + (R 0 ba ) 2 = 1. We demonstrate later that the parameter ω is associated with the NG mode in the CAF phase: See (B3).
It was shown [8] that for α = 0, the ground state is the ferromagnetic phase, where only the spin is polarized as S 0 z = 1. On the other hand, for α = 1 the ground state is the spin-singlet phase, where only the pseudospin is polarized as P 0 z = σ 0 and P 0 x = 1 − σ 2 0 . The CAF phase is realized for 0 < α < 1. From 2S f a = S 0 a + R 0 az and 2S b a = S 0 a − R 0 az , the relations between the spin densities in the front and back layers are From the above equation, we see that in the CAF phase the antiferromagnetic correlation is built up between the two layers.
Here the angle ω describes the orientation angle of the in-plane spin component. The order parameters (A3) are obtained from the ones with ω = 0 by the spin rotation exp[iT z0 ω]. Furthermore, the Hamiltonian density (B1) is invariant under this rotation. Thus, the CAF phase is the U Tz0 (1) spin rotational symmetry broken state. At ν = 2, four complex NG modes emerge due to the symmetry breaking pattern SU(4)→U(1)×SU(2)×SU (2). All the NG modes get gapped due to the presence of the Coulomb interactions, the Zeeman and pseudo-Zeeman terms. We analyze the system in the limit ∆ SAS → 0, where only one NG mode responsible to the interlayer phase coherence becomes gapless. In this limit the order parameters (A3) are reduced to with the relations which are obtained in the limit ∆ SAS → 0. We consider the case σ 0 > 0 explicitly, while the case σ 0 < 0 is similarly discussed. The order parameters (A6) imply that the SU(4) isospins are given by with all the others being zero, where σ(x) and ϑ(x) are the canonical set of the NG mode. The ground-state expectation values of these fields must be σ(x) = σ 0 and ϑ(x) = ϑ 0 = −ω. Since ρ Φ σ(x) represents the density while ϑ(y) the angle variable, the following canonical commutation relation holds, We refer the detailed derivation of the SU(4) isospins (A8) and the canonical commutation relation (A9) to Ref. [11].
From (A9), we obtain the commutaion relations By using (B6), the Hamiltonian density (B4) in the momentum space is diagonalized as where E k is given by Hence, the NG mode has the linear dispersion relation. The wave length of the NG mode at ω s ∼ 10 rad·MHz/T is estimated as where k s = ω s γ −1 , and we have set l B = d = 230.967 Å, or equivalently, ρ 0 = 0.59669 × 10 −5 Å −2 , while σ 0 = 0.3676, as typical values for QH samples, and k B = 1.3807 × 10 −23 J/K. The wavelength (B11) is about the same size as the sample size, L ∼ 100µm. Thus the long-wavelength approximation is valid in this case.
Sinceσ(x) and δϑ(x) are real fields and G k are even function of k, we obtain the relations G k = G −k ,σ † k =σ −k and δϑ † k = δϑ −k . Thus, by using them the phase field δϑ(x) is described in terms of (B6) as In the long wave-length limit, we have e ikx → 1, and there is no x dependence in κ x,k . In this limit we just write it as κ k .

Appendix C: Dicke Model in the CAF phase
We proceed to derive the interaction Hamiltonian between the nuclear spins and the NG mode in the CAF phase based on (3). We assume that only the nuclear spins in one of the layers are dynamically polarized. This is indeed the case in the experiment [12]. Thus we consider the interaction between nuclear spins in the front layer and the NG mode δϑ. From now on, we omit the pseudospin index in the spin density. First from (A8), we see that S z is dynamically frozen, because the imbalanced field σ has a gap much larger than the thermal energy, and therefore, its excitation is suppressed. Thus the interaction between the nuclear spins and the NG mode is solely expressed by the spin-spin interaction for the in-plane component. For simplicity, we start from Eq. (A5) where any limits are not taken. By setting −ω → ϑ(X i ) = ϑ 0 + δϑ(X i ), and expanding S a (X i ) in terms of δϑ up to the linear order, we have S x (X i ) = S (cos ϑ 0 − sin ϑ 0 · δϑ(X i )) + O(δϑ 2 ), S y (X i ) = S (sin ϑ 0 + cos ϑ 0 · δϑ(X i )) + O(δϑ 2 ), where S = ∆ SAS α √ 1 − α 2 1 − β 2 /2∆ 0 with 0 ≤ S < 1. Then, from (3) the hyperfine interaction for the in-plane component becomes gδϑ(X i )(− sin ϑ 0 I x i + cos ϑ 0 I y i ), where g =gS. The first term in (C2) represents the in-plane Knight shift term and is much smaller compared with the Larmor frequency ω s (see the discussion in Sec. II). With the same reason, we can neglect the interaction term between the z component of nuclear spins and that of electron spins. Hence we only retain the second term, representing the interaction between the nuclear spins and the NG mode. By introducing I ± = I x ± iI y and using (B12), we have whereĨ ± = e ∓i(ϑ0+π/2) I ± is the rotated in-plane nuclear spin. We just write them as I ± in the rest of this Appendix. By using the rotating-wave approximation, we obtain In the long-wavelength limit the i dependence disappears from R i and R † i . Thus we obtain the interaction Hamiltonian for the nuclear spins and the NG mode in the CAF phase, By combining the Larmor-precision Hamiltonian (16), the effective Hamiltonian for the NG mode (B9), and the interaction Hamiltonian (C5), we have the Dicke model in the CAF phase.