Global phase diagram of $\ensuremath{\nu}=2$ quantum Hall bilayers in tilted magnetic fields

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 – Published 24 September 2004

Abstract

We consider a bilayer quantum Hall system at total filling fraction $ν = 2$ in tilted magnetic field allowing for charge imbalance as well as tunneling between the two layers. Using an “unrestricted Hartree Fock,” previously discussed by Burkov and MacDonald [Phys. Rev. B 66, 115323 (2002)], we examine the zero-temperature global phase diagrams that would be accessed experimentally by changing the in-plane field and the bias voltage between the layers while keeping the tunneling between the two layers fixed. In accordance with previous work, we find symmetric and ferromagnetic phases as well as a first-order transition between two canted phases with spontaneously broken U(1) symmetry. We find that these two canted phases are topologically connected in the phase diagram and, reminiscent of a first-order liquid-gas transition, the first-order transition line between these two phases ends in a quantum critical point. We develop a physical picture of these two phases and describe in detail the physics of the transition.

Received 7 January 2004

DOI:https://doi.org/10.1103/PhysRevB.70.115325

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

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 (2004)

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Vol. 70, Iss. 11 — 15 September 2004

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Figure 1
Global phase diagram for a $ν = 2$ bilayer sample in perpendicular magnetic field. The Zeeman energy is $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ and the interlayer spacing is $d = l$ . The tunneling strength $Δ_{SAS}^{0}$ and the bias voltage $Δ_{V}$ are given in units of $(e^{2} ∕ ε l)$ . The phase $S$ is the spin-singlet phase, $F$ is the ferromagnetic phase, and $C$ is the canted phase. The thick line along $Δ_{SAS}^{0} = 0$ represents the $I$ phase.Reuse & Permissions
Figure 2
Energy profile of charge-balanced $ν = 2$ bilayers with Zeeman energy $Δ_{Z} = 0.05 (e^{2} ∕ ε l)$ , interlayer spacing $d = l$ , and a range of tunneling amplitudes. The dashed line represents the energy of the ferromagnetic state. The long dashed line is the energy of the spin-singlet state. In the absence of interactions, there would be a first order phase transition at the intersection of the two dashed lines. The interactions effectively “smooth out” the profile by stabilizing the canted phase.Reuse & Permissions
Figure 3
Global phase diagrams for $ν = 2$ bilayer samples of different tunneling strengths in tilted magnetic field. The tunneling amplitudes $Δ_{SAS}^{0}$ are given in the lower right corner of each panel. The Zeeman energy $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ and the distance between the layers $d = l$ , are the same for all the phase diagrams. The solid lines indicate second-order phase transitions and the dashed lines indicate first order. The phases are $S$ : spin-singlet, $F$ : ferromagnetic, $I$ : incommensurate, SC: spin-singlet commensurate, $C 1$ : canted commensurate, and $C 2$ : canted spin-isospin commensurate.Reuse & Permissions
Figure 4
Evolution of the spin-wave wave vector $Q_{S}$ (solid line) as a function of the in-plane field wave vector $Q_{‖}$ (given in units of $1 ∕ l$ ). The dashed line is $Q_{I} = Q_{‖}$ , given for comparison. The wave vector $Q_{S}$ is small in the phase $C 1$ (low $Q_{‖}$ ) and abruptly jumps to $Q_{S} \approx Q_{‖}$ at the phase transition to $C 2$ . The Zeeman energy in this figure is $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ , the interlayer spacing is $d = l$ , the tunneling constant is $Δ_{SAS}^{0} = 0.06 (e^{2} ∕ ε l)$ , and the external bias voltage is $Δ_{V} = 0.8 (e^{2} ∕ ε l)$ .Reuse & Permissions
Figure 5
Expectation values $⟨ I_{Q}^{x} ⟩$ (solid lines), $⟨ I^{z} ⟩$ (dashed lines), and $⟨ I^{z} + S^{z} ⟩$ (dash-dotted lines), per flux quantum, across $C 1 − C 2$ (thick lines) and the naive commensurate-incommensurate (thin lines) phase transitions. The Zeeman energy is $Δ_{Z}^{0} = 0.01 (e^{2} ∕ ε l)$ , the interlayer spacing is $d = l$ , the tunneling constant is $Δ_{SAS}^{0} = 0.06 (e^{2} ∕ ε l)$ , and the external bias voltage is $Δ_{V} = 0.8 (e^{2} ∕ ε l)$ .Reuse & Permissions
Figure 6
Schmatic representation of the comensurate $C 1$ and $C 2$ states. In the left figure, an $I$ state is represented. The $I$ state is shown in Sec. 4D to be an eigenstate of the operator $I^{z} + S^{z}$ . Thus, the $I$ state is represented as a vector pointing in the $I^{z} + S^{z}$ direction. In the presence of tunneling, the system can satisfy the Aharonov-Bohm phases in the tunneling term by winding around either the $I^{z} + S^{z}$ axis, yielding the $C 2$ -commensurate state shown above, or by winding around the $I^{z}$ axis yielding the $C 1$ -commensurate state. The $C 2$ state asymptotically approaches the $I$ state in large in-plane field where the winding becomes very fast and therefore must be very tight (due to “spin stiffness”).Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

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 – Published 24 September 2004

Abstract

Authors & Affiliations

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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 (2004)

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Physical Review B

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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 – Published 24 September 2004

Abstract

Authors & Affiliations

See Also

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 (2004)

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

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