双层石墨烯电场磁场相图研究：量子霍尔新现象

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"双层石墨烯在电场和平行磁场空间的相图研究，通过非微扰Schwinger-Dyson方程分析量子霍尔相，考虑动力学极化函数和四费米子相互作用。" 双层石墨烯是一种由两层单层石墨烯堆叠而成的二维材料，其独特的电子结构使其在量子霍尔效应（Quantum Hall Effect, QHE）领域表现出特殊的性质。QHE是物理学中的一个重要现象，它发生在强磁场和低温度条件下，表现为电阻呈现精确的量子化台阶。在双层石墨烯系统中，由于层间相互作用，可以观察到更为丰富的量子霍尔相。本文的研究由Junji Jia、E. V. Gorbar、V. P. Gusynin和V. A. Miransky等人进行，他们采用非微扰Schwinger-Dyson方程来研究双层石墨烯在电场和两种磁场（平行及弱垂直）下的量子霍尔相。Schwinger-Dyson方程是量子场论中处理粒子自能和格林函数的重要工具，它可以用来求解系统中的非线性和非微扰问题。研究者在计算中考虑了动力学极化函数，这是描述电子在材料中集体行为的一个关键因素，它影响着电子的散射和能带结构。此外，还引入了四费米子相互作用，这涉及到电子之间的直接相互作用，对双层石墨烯的电子性质有显著影响。在没有外部场的情况下，双层石墨烯系统被认为处于层反铁磁相，这是一种两层间磁矩相互抵消的有序状态。然而，随着电场的增大，系统会发生相变，转变为量子谷霍尔相（Quantum Valley Hall State, QVHS）。这种相变是由于电场引起的能谷极化，导致谷极化的电子在边界形成导电通道。在电场-平行磁场空间，反铁磁相和量子谷霍尔相之间存在一个相界，可以表示为E = E01 + c1B²∥。这个边界表明电场E和平行磁场B的关系，并且c1是一个与材料参数相关的系数。同样，在电场-垂直磁场空间，也存在一个相界，用E = E02 - c2B²⊥表示，其中c2是另一个系数，反映了垂直磁场的影响。实验结果与理论计算的相界进行了比较，两者在定性上表现出良好的一致性。这些发现对于理解双层石墨烯的量子霍尔效应以及可能的应用，如高精度电阻标准和量子计算等，具有重要的科学价值。关键词：双层石墨烯、量子霍尔相、极化函数、Schwinger-Dyson方程中图分类号：O469（物理学）、O413（凝聚态物理学）这项研究进一步揭示了双层石墨烯的丰富物理现象，为探索二维材料的新奇量子态提供了理论基础，对于未来的量子器件设计和新型量子技术的发展具有重要意义。

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The Coulomb interaction term V (r) in H

int

is the bare intralayer potential whose Fourier

transform is given by V (p) = 2πe

/(κp), where κ is the dielectric constant. The Fourier

transform of the interaction V

(r) equals V

(p) = 2πe

−pd

−1)/(κp). The interaction V

(r)

describes the d dependent part of the interlayer electron interactions and unlike the Coulomb

interaction is not invariant with respect to the spin-valley SU(4) symmetry. Since d is small

in bilayer graphene, this interaction is weak. The two-dimensional charge densities in the two

layers are (the total charge density ρ = ρ

+ ρ

)

(r) = Ψ

†

(r)P

Ψ(r) , ρ

(r) = Ψ

†

(r)P

Ψ(r) , (3)

where P

= (1 + η

)/2 and P

= (1 −η

)/2 are projectors on the states in the layers 1 and

2, respectively. When the dynamical screening eﬀects are taken into account, the potentials

V (r) and V

(r) are replaced by eﬀective interactions V

eff

(t, r) and V

eff

(t, r) which are no longer

instantaneous.

If external electric and magnetic ﬁelds are absent and the interaction V

is neglected, then

the Hamiltonian H = H

+ H

int

, with H

and H

int

in Eqs.(1) and (2), possesses the spin-valley

SU(4) symmetry and all the QAH, QSH, LAF, and QVH states discussed in Introduction are

degenerate in energy in bilayer graphene at the neutral point. Taking into account the V

interaction makes the layer polarized QVH state to have larger energy than other three QAH,

QSH, and LAF states which remain degenerate in energy.

For a suﬃciently large electric ﬁeld E

⊥

perpendicular to the planes of graphene, the layer-

polarized QVH state is realized. The experiments performed in Refs. [15, 16] demonstrated

that as the value E

⊥

decreases there is a phase transition to another state. This eliminates the

QVH state as a possible candidate for the ground state of bilayer graphene at the neutral point

in the absence of external ﬁelds. On the other hand, the recent experiment [42] observed a

quantum phase transition at large in-plane magnetic ﬁeld to a state with conductance of order

/~ consistent with the QSH state that also excludes the QSH state as the ground state.

Since the QAH state has four topologically protected edge states, hence nonzero conductance,

the experimental data in Ref. [42] single out the insulating LAF phase as the ground state of

bilayer graphene in the absence of external electric and magnetic ﬁelds. According to Ref. [26],

the LAF state transforms in out-of-plane magnetic ﬁeld into the CAF state once the Zeeman

coupling is taken into account. Moreover, for B

⊥

= 0, the CAF state continuously crossovers

into the QSH state as in-plane magnetic ﬁeld increases.

Since the QAH, QSH, and LAF states are degenerate in energy for the Hamiltonian given by

Eqs.(1) and (2), new terms breaking the SU(4) symmetry should be added to the Hamiltonian in

order to ensure that the LAF is the ground state of bilayer graphene in the absence of external

ﬁelds. To provide this, the following local valley-asymmetric four-fermion interaction terms

were added to the Hamiltonian of the model in Refs. [26] (we retain the simplest interaction

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双层石墨烯电场磁场相图研究：量子霍尔新现象

Methods for PHASE DIAGRAM DETERMINATION.pdf

Open the first file and push data in numpy.array for plotting -ray diffraction diagram.

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