2+1维量子电动力学中带电费米子的辐射质量效应

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"这篇论文详细探讨了2+1维量子电动力学（QED _{2+1}）框架下，恒定均质磁场中平面带电费米子的辐射动力学质量。研究内容包括对有质量和无质量费米子的有效拉格朗日算子和质量算子的计算，以及这些量在无质量情况下的具体分析。作者V.R.Khalilova在文中指出，当带电的无质量费米子处于最低Landau能级时，对于纯QED _{2+1}和简化QED _{3+1}（2-brane）的辐射校正进行了研究。此外，还讨论了在简化的QED中，即使裸质量极小（$$ m_0 $$）或为零的情况下，如何在2+1维的外部磁场中动态生成费米子质量，这在单层石墨烯等材料中可能有所体现。" 这篇论文的主要知识点包括： 1. **量子电动力学（QED）**：这是物理学中的一个理论，描述了光子、电子和其他基本粒子之间的相互作用。在这里，研究集中在2+1维的量子电动力学，即QED _{2+1}。 2. **恒定均质磁场**：这是一种理想化的磁场模型，其中磁场强度在空间中是常数且均匀分布的。在这种条件下，费米子的行为可以被精确分析。 3. **平面带电费米子**：费米子是物质的基本构成单元，如电子和夸克。在这个研究中，它们带有电荷并且位于二维平面上。 4. **有效拉格朗日算子和质量算子**：这些是量子场论中的重要概念，用于描述物理系统的动态行为和粒子的质量。在近似计算中，它们可以帮助理解费米子的性质和相互作用。 5. **Landau能级**：在磁场中，电子的能级会形成一系列离散的态，称为Landau能级。最低Landau能级是电子在磁场中可能的最低能量状态。 6. **辐射校正**：这是量子电动力学中的一个概念，指的是由于电磁辐射的存在，粒子的实际属性（如质量）与经典预期有所不同。 7. **动态质量生成**：在某些条件下，费米子的质量可以由其与环境的相互作用动态产生，即使其初始裸质量为零或非常小。 8. **2-brane和QED _{3+1}**：这是一个将QED应用于3+1维时空（常规四维宇宙）的概念，但通过某种方式简化，以便与2+1维的情况进行比较。 9. **单层石墨烯**：一种二维碳材料，其电子行为受到磁场影响时，可能会表现出动态质量生成的现象。通过这项工作，研究人员不仅深化了对2+1维量子电动力学的理解，也提供了关于带电费米子在特定条件下质量行为的新见解，这对于理解基本粒子的性质以及开发新型材料具有潜在的意义。

Eur. Phys. J. C (2019) 79:196

https://doi.org/10.1140/epjc/s10052-019-6717-4

Regular Article - Theoretical Physics

Radiative dynamical mass of planar charged fermion in a constant

homogeneous magnetic ﬁeld

V.R. Khalilov

Faculty of Physics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia

Received: 29 November 2018 / Accepted: 22 February 2019 / Published online: 5 March 2019

Abstract The effective Lagrangian and mass operator are

calculated for planar charged massive and massless fermions

in a constant external homogeneous magnetic ﬁeld in the

one-loop approximation of the 2+1 dimensional quantum

electrodynamics (QED

2+1

). We obtain the renormalizable

effective Lagrangian and the fermion mass operator for a

charged fermion of mass m and then calculate these quan-

tities for the massless case. The radiative corrections to the

mass of charged massless fermion when it occupies the low-

est Landau level are found for the cases of the pure QED

2+1

as well as the so-called reduced QED

3+1

on a 2-brane. The

fermion masses were found can be generated dynamically in

an external magnetic ﬁeld in the pure QED

2+1

if the charged

fermion has small bare mass m

and in the reduced QED

3+1

on a 2-brane even at m

= 0. The dynamical mass seems to

be likely to be revealed in monolayer graphene in the pres-

ence of constant homogeneous magnetic ﬁeld (normal to the

graphene sample).

1 Introduction

Quantum systems of planar charged fermions in external

electromagnetic ﬁelds are interesting in view of possible

applications of the corresponding ﬁeld-theory models to a

number of condensed-matter quantum effects such as, for

example, the quantum Hall effect [1] and high-temperature

superconductivity [2] as well as in connection with problems

of graphene (see, [3–6]). In graphene, the electron dynam-

ics near the Fermi surface can be described by the Dirac

equation in 2+1 dimensions for a zero-mass charged fermion

[4] though the case of massive charged fermions is also of

interest [7]. The ﬁeld-theory models applied for study are the

pure 2+1 dimensional quantum electrodynamics (QED

2+1

)

as well as the so-called reduced QED

3+1

on a 2-brane. In

the latter model, fermions are conﬁned to a plane, neverthe-

e-mail: khalilov@phys.msu.ru

less the electromagnetic interaction between them is three-

dimensional [8,9].

The radiative one-loop shift of an electron energy in the

ground state in a constant homogeneous magnetic ﬁeld in

QED

2+1

was calculated in [10] and the one-loop electron

self-energy in the topologically massive QED

2+1

at ﬁnite

temperature and density was obtained in [11]. The effective

Lagrangian, the electron mass operator and the density of

vacuum electrons (induced by the background ﬁeld) in an

external constant homogeneous magnetic ﬁeld were derived

in the one-loop QED

2+1

approximation in [12].

Since the effective ﬁne structure constant in graphene

is large, the QED

2+1

effects can be signiﬁcant already in

the one-loop approximation. The polarization operator in

graphene in a strong constant homogeneous magnetic ﬁeld

perpendicular to the graphene membrane has been obtained

in the one-loop approximation of the QED

2+1

in [9,13–15].

The effective potential and vacuum current in graphene in

a superposition of a constant homogeneous magnetic ﬁeld

and an Aharonov–Bohm vortex was studied in [16]. We

also note that the induced vacuum current in the ﬁeld of a

solenoid perpendicular to the graphene sample was investi-

gated in [17], and the vacuum polarization of massive and

massless fermions in an Aharonov–Bohm vortex in the one-

loop approximation of the QED

2+1

was studied in [18]. The

one-loop self-energy of a Dirac electron of mass m in a thin

medium simulating graphene in the presence of external mag-

netic ﬁeld was investigated in the reduced QED

3+1

on a 2-

brane in [19], in which it was shown that the radiative mass

correction in the lowest Landau level does not vanish at the

limit m → 0.

In this work, we calculate the effective Lagrangian and

the mass operator of planar charged fermions in the pres-

ence of an external constant homogeneous magnetic ﬁeld in

the one-loop approximation of the QED

2+1

. We also calcu-

late the radiative corrections to the mass of charged fermion

when it occupies the lowest Landau level for the cases of the

123

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