Collective modes of $\ensuremath{\nu}=2$ quantum Hall bilayers in tilted magnetic fields

Collective modes of $ν = 2$ quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115326 – Published 24 September 2004

Abstract

We use the time-dependent Hartree Fock approximation to study the collective-mode spectra of $ν = 2$ quantum Hall bilayers in tilted magnetic fields, allowing for charge imbalance as well as tunneling between the two layers. In a previous companion paper to this work, we studied the zero-temperature global phase diagram of this system, which was found to include symmetric and ferromagnetic phases as well as a first-order transition between two canted phases with spontaneously broken U(1) symmetry. We further found that this first-order transition line ends in a quantum critical point within the canted region. In the current work, we study the excitation spectra of all of these phases and pay particular attention to the behavior of the collective modes near the phase transitions. We find, most interestingly, that the first-order transition between the two canted phases is signaled by a near softening of a magnetoroton minimum. Many of the collective-mode features explored here should be accessible experimentally in light-scattering experiments.

Received 7 January 2004

DOI:https://doi.org/10.1103/PhysRevB.70.115326

Authors & Affiliations

Anna Lopatnikova¹, Steven H. Simon², and Eugene Demler¹

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²Lucent Technologies Bell Labs, Murray Hill, New Jersey 07974, USA

Global phase diagram of $ν = 2$ quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115325 (2004)

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Vol. 70, Iss. 11 — 15 September 2004

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Figure 1
Global phase diagram of charge-unbalanced $ν = 2$ bilayers in tilted magnetic field. The phase diagram is calculated for bare tunneling gap $Δ_{SAS}^{0} = 0.06 e^{2} ∕ (ε l)$ , Zeeman energy $Δ_{Z}^{0} = 0.01 e^{2} ∕ (ε l)$ , and the distance between the layers $d = l$ . The axes are the amplitude of the in-plane field wave vector ${\vec{Q}}_{‖} = \hat{z} \times {\vec{B}}_{‖} ∕ (B_{⊥} l^{2} ∕ d)$ and the external bias voltage $Δ_{V}$ . This choice of axes is particularly suitable, since current experimental techniques allow to vary the bias voltage and the in-plane field in situ over a wide range of values. SC is the spin-singlet commensurate phase, $C 1$ is the simple, isospin-commensurate phase, and $C 2$ is the spin-isospin commensurate phase. Solid lines represent the second-order quantum phase transitions and the dashed line represents the first-order $C 1 − C 2$ transition.Reuse & Permissions
Figure 2
The collective-mode dispersions of the charge-unbalanced $ν = 2$ bilayers in perpendicular field (ferromagnetic phase). The dispersions in the left column are given for a system with $Δ_{SAS}^{0} = 0.0 (e^{2} ∕ ε l)$ ; in the right column, for a system with $Δ_{SAS}^{0} = 0.02 (e^{2} ∕ ε l)$ ; the Zeeman energy in all panels is $Δ_{Z} = 0.01 (e^{2} ∕ ε l)$ ; the bias voltage is given in the upper right corner of each panel in units of $(e^{2} ∕ ε l)$ . Some collective-mode dispersions are degenerate. The low-energy, low-wave-vector region is shown in the insets.Reuse & Permissions
Figure 3
The collective-mode dispersions of the charge-unbalanced $ν = 2$ bilayers in perpendicular field (many-body phase). The dispersions in the left column are given for a system with $Δ_{SAS}^{0} = 0.0 (e^{2} ∕ ε l)$ ; in the right column, for a system with $Δ_{SAS}^{0} = 0.02 (e^{2} ∕ ε l)$ ; the Zeeman energy in all panels is $Δ_{Z} = 0.01 (e^{2} ∕ ε l)$ ; the bias voltage is given in the upper right corner of each panel in units of $(e^{2} ∕ ε l)$ . Some collective-mode dispersions are degenerate. The low-energy, low-wave-vector region is shown in the insets.Reuse & Permissions
Figure 4
The collective-mode dispersions of the charge-unbalanced $ν = 2$ bilayers in perpendicular field (spin-singlet phase). The dispersions in the left column are given for a system with $Δ_{SAS}^{0} = 0.0 (e^{2} ∕ ε l)$ ; in the right column, for a system with $Δ_{SAS}^{0} = 0.02 (e^{2} ∕ ε l)$ ; the Zeeman energy in all panels is $Δ_{Z} = 0.01 (e^{2} ∕ ε l)$ ; the bias voltage is given in the upper right corner of each panel in units of $(e^{2} ∕ ε l)$ . Some collective-mode dispersions are degenerate. The low-energy, low-wave-vector region is shown in the insets.Reuse & Permissions
Figure 5
The collective-mode dispersions of the charge-unbalanced $ν = 2$ bilayers in tilted field—evolution across the $C 1 − C 2$ phase transition. The dispersions are given for a system with $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ , $Δ_{SAS}^{0} = 0.06 (e^{2} ∕ ε l)$ , and $Δ_{V} = 0.8 (e^{2} ∕ ε l)$ . The strength of the in-plane component of the tilted magnetic field is given by the wave vector $Q_{‖}$ , which, for every set of dispersion curves, is represented by a number in the units of $1 ∕ l$ . When $Q_{‖} l ⩽ 0.96$ , the system is in the $C 1$ phase; when $Q_{‖} l > 0.96$ , it is in the $C 2$ phase. In the bottom left panel, the dotted line is for comparison of the dispersion curves in the $C 2$ phase to those in the $I$ phase.Reuse & Permissions
Figure 6
The collective-mode dispersions of the charge-unbalanced $ν = 2$ bilayers in tilted field at a critical point terminating the $C 1 − C 2$ transition. The dispersions are given for a system with $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ and $Δ_{SAS}^{0} = 0.06 (e^{2} ∕ ε l)$ . The critical point for this system occurs at $Δ_{V} = 0.5328$ and $Q_{‖} = 0.9362$ ; $Q_{S} \approx 0.474$ for these parameters. The top plot zooms in on the long-wavelength region, to demonstrate the $q_{⊥}^{3}$ dependence of the Goldstone-mode dispersions in the positive $q_{⊥}$ direction.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Collective modes of $ν = 2$ quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115326 – Published 24 September 2004

Abstract

Authors & Affiliations

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Global phase diagram of $ν = 2$ quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115325 (2004)

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Physical Review B

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Collective modes of ν=2 quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115326 – Published 24 September 2004

Abstract

Authors & Affiliations

See Also

Global phase diagram of ν=2 quantum Hall bilayers in tilted magnetic fields

Anna Lopatnikova, Steven H. Simon, and Eugene Demler

Phys. Rev. B 70, 115325 (2004)

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Collective modes of $ν = 2$ quantum Hall bilayers in tilted magnetic fields

Global phase diagram of $ν = 2$ quantum Hall bilayers in tilted magnetic fields