Pseudo Quantum Electrodynamics Research Articles

Pseudo-Quantum Electrodynamics: 30 Years of Reduced QED

Charged quasiparticles, which are constrained to move on a plane, interact by means of electromagnetic (EM) fields which are not subject to this constraint, living, thus, in three-dimensional space. We have, consequently, a hybrid situation where the particles of a given system and the EM fields (through which they interact) live in different dimensions. Pseudo-Quantum Electrodynamics (PQED) is a U(1) gauge field theory that, despite being strictly formulated in two-dimensional space, precisely describes the real EM interaction of charged particles confined to a plane. PQED is completely different from QED(2 + 1), namely, Quantum Electrodynamics of a planar gauge field. It produces, for instance, the correct 1/r Coulomb potential between static charges, whereas QED(2 + 1) produces lnr potential. In spite of possessing a nonlocal Lagrangian, it has been shown that PQED preserves both causality and unitarity, as well as the Huygens principle. PQED has been applied successfully to describe the EM interaction of numerous systems containing charged particles constrained to move on a plane. Among these are p-electrons in graphene, silicene, and transition-metal dichalcogenides; systems exhibiting the Valley Quantum Hall Effect; systems inside cavities; and bosonization in (2 + 1)D. Here, we present a review article on PQED (also known as Reduced Quantum Electrodynamics).

Renormalization group functions in two-dimensional massive Dirac-like systems near an interface

In a recent paper [], pseudo quantum electrodynamics was used to simulate the Coulomb interaction between electrons and determine the mass renormalization in a two-dimensional Dirac-like system. In the present paper, we expand upon these findings by examining this system in a certain distance from a planar interface separating two dielectrics. Using the random-phase approximation, we calculate the renormalization group functions and show how the behaviors of the mass, Fermi velocity, and the anomalous dimension of the fermion field are affected by the presence of this interface. To exemplify an application of our formulas, we calculate the influence of this interface on the renormalized energy band gap in a two-dimensional material. In the appropriate limits, our results recover the corresponding ones reported in the aforementioned publication, as well as in others. Published by the American Physical Society 2024

Effects of the pseudo-Chern-Simons action for strongly correlated electrons in a plane

Chiral symmetry breaking comes from the mass dynamically generated through interaction of Dirac fermions for both quantum electrodynamics in $(2+1)\mathrm{D}$ (QED3) and $(3+1)\mathrm{D}$ (QED4). In QED3, the presence of a Chern-Simons (CS) parameter affects the critical structure of the theory, favoring the symmetric phase where the electron remains massless. Here, we calculate the main effects of a pseudo-Chern-Simons (PCS) parameter $\ensuremath{\theta}$ into the dynamical mass generation of pseudo--quantum electrodynamics (PQED). The $\ensuremath{\theta}$ parameter provides a mass scale for PQED at the classical level and appears as the pole of the gauge-field propagator. After calculating the full electron propagator with the Schwinger-Dyson equation in the quenched-rainbow and large-$N$ approximations, we conclude that $\ensuremath{\theta}$ changes the critical parameters related to the fine-structure constant ${\ensuremath{\alpha}}_{c}(\ensuremath{\theta})$ and the number of copies of the matter field ${N}_{c}(\ensuremath{\theta})$, favoring the symmetric phase. In the continuum limit ($\mathrm{\ensuremath{\Lambda}}\ensuremath{\rightarrow}\ensuremath{\infty}$), nevertheless, the $\ensuremath{\theta}$ parameter does not affect the critical parameters. We also compare our analytical results with numerical findings of the integral equation for the mass function of the electron. In the strong-coupling limit (with a fine-structure constant $\ensuremath{\alpha}\ensuremath{\gg}1$), the PCS mass vanishes and the system presents the same criticality as QED3.

Bosonization, mass generation, and the pseudo-Chern-Simons action

We discuss several aspects of a generalization of the Chern-Simons action containing the pseudo-differential operator$\sqrt{-\Box}$, which we shall call pseudo Chern-Simons (PCS). Firstly, we derive the PCS from the bosonization of free massive Dirac particles in (2+1)D in the limit when $m^2\ll p^2$, where $m$ is the fermion mass and $p$ is its momentum. In this regime, the whole bosonized action also has a modified Maxwell term, involving the same pseudo-differential operator. Furthermore, the large-mass $m^2\gg p^2$ regime is also considered. We also investigate the main effects of the PCS term into the Pseudo quantum electrodynamics (PQED), which describes the electromagnetic interactions between charged particles in (2+1)D. We show that the massless gauge field of PQED becomes massive in the presence of a PCS term, without the need of a Higgs mechanism. In the nonrelativistic limit, we show that the static potential has a repulsive term (given by the Coulomb potential) and an attractive part (given by a sum of special functions), whose competition generates bound states of particles with the same charge. Having in mind two-dimensional materials, we also conclude that the presence of a PCS term does not affect the renormalization either of the Fermi velocity and of the band gap in a Dirac-like material.

Influence of the four-fermion interactions in a (2+1)D massive electron system

The description of the electromagnetic interaction in two-dimensional Dirac materials, such as graphene and transition-metal dichalcogenides, in which electrons move in the plane and interact via virtual photons in 3d, leads naturally to the emergence of a projected non-local theory, called pseudo-quantum electrodynamics (PQED), as an effective model suitable for describing electromagnetic interaction in these systems. In this work, we investigate the role of a complete set of four-fermion interactions in the renormalization group functions when we coupled it with the anisotropic version of massive PQED, where we take into account the fact that the Fermi velocity is not equal to the light velocity. We calculate the electron self-energy in the dominant order in the $1/N$ expansion in the regime where $m ^ 2 \ll p ^ 2$. We show that the Fermi velocity renormalization is insensitive to the presence of quartic fermionic interactions, whereas the renormalized mass may have two different asymptotic behaviors at the high-density limit, which means a high-energy scale.

Dynamical mass generation in pseudoquantum electrodynamics with Gross-Neveu interaction at finite temperature

We describe the dynamical mass generation in pseudoquantum electrodynamics (PQED) coupled to the Gross-Neveu (GN) interaction, in ($2+1$) dimensions, at both zero and finite temperatures. We start with a gapless model and show that, under particular conditions, a dynamically generated mass emerges. In order to do so, we use a truncated Schwinger-Dyson equation, at the large-$N$ approximation, in the imaginary-time formalism. In the instantaneous-exchange approximation (the static regime), we obtain two critical parameters, namely, the critical number of fermions ${N}_{c}(T)$ and the critical coupling constant ${\ensuremath{\alpha}}_{c}(T)$ as a function of temperature and of the finite cutoff $\mathrm{\ensuremath{\Lambda}}$, which must be provided by experiments. In the dynamical regime, we find an analytical solution for the mass function $\mathrm{\ensuremath{\Sigma}}(p,T)$ as well as a zero-external momentum solution for $p=0$. In the continuum theory $\mathrm{\ensuremath{\Lambda}}=\ensuremath{\infty}$, where scale-invariance is respected, it is shown that the model has a dynamically generated mass for any value of the coupling constant $\ensuremath{\alpha}$. Furthermore, after calculating the effective potential for PQED, we prove that the dynamically generated mass is an energetically favorable solution in comparison to the massless phase. We compare our analytical results with numerical tests and a good agreement is found.

Two-loop photon self-energy in pseudoquantum electrodynamics in the presence of a conducting surface

In the present paper, we investigate the influence of a grounded perfectly conducting surface on the photon self-energy of a $2+1\mathrm{D}$ system of massless Dirac fermions, whose electron interaction is described by pseudoquantum electrodynamics. We calculate the temporal component of the polarization tensor up to 2-loop perturbation order in the presence of the conducting surface and, applying the Kubo formula, we obtain the longitudinal and optical conductivities of such system. When the distance between the system and the plate tends to infinity, we recover the correspondent results found in the literature. Since pseudoquantum electrodynamics proved to give a good description of the transport properties of graphene, our results can be useful as an alternative way to control the longitudinal and optical conductivities of this material.

Renormalization of the band gap in 2D materials through the competition between electromagnetic and four-fermion interactions in largeNexpansion

Recently the renormalization of the band gap $m$, in both WSe$_2$ and MoS$_2$, has been experimentally measured as a function of the carrier concentration $n$. The main result establishes a decreasing of hundreds of meV, in comparison with the bare band gap, as the carrier concentration increases. These materials are known as transition metal dichalcogenides and their low-energy excitations are, approximately, described by the massive Dirac equation. Using Pseudo Quantum Electrodynamics (PQED) to describe the electromagnetic interaction between these quasiparticles and from renormalization group analysis, we obtain that the renormalized mass describes the band gap renormalization with a function given by $m(n)/m_0=(n/n_0)^{C_\lambda/2}$, where $m_0=m(n_0)$ and $C_\lambda$ is a function of the coupling constant $\lambda$. We compare our theoretical results with the experimental findings for WSe$_2$ and MoS$_2$, and we conclude that our approach is in agreement with these experimental results for reasonable values of $\lambda$. In addition we introduced a Gross-Neveu (GN) interaction which could simulate an disorder/impurity-like microscopic interaction. In this case, we show that there exists a critical coupling constant, namely, $\lambda_c \approx 0,66$ in which the beta function of the mass vanishes, providing a stable fixed point in the ultraviolet limit. For $\lambda>\lambda_c$, the renormalized mass decreases while for $\lambda<\lambda_c$ it increases with the carrier concentration.

Pseudo quantum electrodynamics and Chern-Simons theory coupled to two-dimensional electrons

We study a nonlocal theory that combines both the Pseudo quantum electrodynamics (PQED) and Chern-Simons actions among two-dimensional electrons. In the static limit, we conclude that the competition of these two interactions yields a Coulomb potential with a screened electric charge given by $e^2/(1+\theta^2)$, where $\theta$ is the dimensionless Chern-Simons parameter. This could be useful for describing the substrate interaction with two-dimensional materials and the doping dependence of the dielectric constant in graphene. In the dynamical limit, we calculate the effective current-current action of the model considering Dirac electrons. We show that this resembles the electromagnetic and statistical interactions, but with two different overall constants, given by $e^2/(1+\theta^2)$ and $e^2\theta/(1+\theta^2)$. Therefore, the $\theta$-parameter does not provide a topological mass for the Gauge field in PQED, which is a relevant difference in comparison with quantum electrodynamics. Thereafter, we apply the one-loop perturbation theory in our model. Within this approach, we calculate the electron self-energy, the electron renormalized mass, the corrected gauge-field propagator, and the renormalized Fermi velocity for both high- and low-speed limits, using the renormalization group. In particular, we obtain a maximum value of the renormalized mass for $\theta\approx 0.36$. This behavior is an important signature of the model and relations with doping control of band gap size are also discussed in the conclusions.

Bosonic interactions in a nonlocal theory in (2+1) dimensions

Pseudo-Quantum Electrodynamics (PQED) provides an excellent description of the interaction between charged particles confined to a plane. When we couple pseudo-gauge field with a bosonic matter field, we obtain the so-called Scalar Pseudo-Quantum Electrodynamics (SPQED). In this work, we make a perturbative analysis of SPQED via Feynman diagrams. We compute the one loop Green functions: bosonic field self-energy, electromagnetic field self-energy, and vertex corrections. Finally, we consider the non-relativistic interaction potential between two bosonic particles. We compute the radiative corrections to the usual Coulomb potential and comment on the analogies and the differences with the fermionic case.

Two-dimensional Yukawa interactions from nonlocal Proca quantum electrodynamics

We derive two versions of an effective model to describe dynamical effects of the Yukawa interaction among Dirac electrons in the plane. Such short-range interaction is obtained by introducing a mass term for the intermediate particle, which may be either scalar or an abelian gauge field, both of them in ($3+1$) dimensions. Thereafter, we consider that the fermionic matter field propagates only in ($2+1$) dimensions, whereas the bosonic field is free to propagate out of the plane. Within these assumptions, we apply a mechanism for dimensional reduction, which yields an effective model in ($2+1$) dimensions. In particular, for the gauge-field case, we use the Stueckelberg mechanism in order to preserve gauge invariance. We refer to this version as nonlocal-Proca quantum electrodynamics (NPQED). For both scalar and gauge cases, the effective models reproduce the usual Yukawa interaction in the static limit. By means of perturbation theory at one loop, we calculate the mass renormalization of the Dirac field. Our model is a generalization of Pseudo quantum electrodynamics (PQED), which is a gauge-field model that provides a Coulomb interaction for two-dimensional electrons. Possibilities of application to Fermi-Bose mixtures in mixed dimensions, using cold atoms, are briefly discussed.

Dynamical mass generation in pseudoquantum electrodynamics with four-fermion interactions

We describe dynamical symmetry breaking in a system of massless Dirac fermions with both electromagnetic and four-fermion interactions in (2+1) dimensions. The former is described by the Pseudo Quantum Electrodynamics (PQED) and the latter is given by the so-called Gross-Neveu action. We apply the Hubbard-Stratonovich transformation and the large$-N_f$ expansion in our model to obtain a Yukawa action. Thereafter, the presence of a symmetry broken phase is inferred from the non-perturbative Schwinger-Dyson equation for the electron propagator. This is the physical solution whenever the fine-structure constant is larger than a critical value $\alpha_c(D N_f)$. In particular, we obtain the critical coupling constant $\alpha_c\approx 0.36$ for $D N_f=8$., where $D=2,4$ corresponds to the SU(2) and SU(4) cases, respectively, and $N_f$ is the flavor number. Our results show a decreasing of the critical coupling constant in comparison with the case of pure electromagnetic interaction, thus yielding a more favorable scenario for the occurrence of dynamical symmetry breaking. For two-dimensional materials,in application in condensed matter systems, it implies an energy gap at the Dirac points or valleys of the honeycomb lattice.

Inhibition of the Fermi velocity renormalization in a graphene sheet by the presence of a conducting plate

We investigate the renormalization of the Fermi velocity in a plane graphene sheet in the presence of a parallel conducting plate. We use the pseudo-quantum electrodynamics to describe the Coulombian interaction between the electrons, but taking into account that this interaction is changed by the conducting plate. Incorporating the influence of the plate into the gauge field, we obtain the correspondent photon propagator and electron self-energy, showing that the logarithmic renormalization of the Fermi velocity is inhibited by the presence of the plate. Our result may be useful as an alternative way to control the electronic properties of graphene.

Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions

A brief introduction to topological phases is provided, considering several two-band Hamiltonians in one- and two-dimensions. Relevant concepts of the topological insulator theory, such as: Berry phase, Chern number, and the quantum adiabatic theorem, are reviewed in a basic framework, which is meant to be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ models. The role of the electromagnetic interaction in the topological insulator theory is addressed in the light of the Pseudo quantum electrodynamics (PQED). The well known parity anomaly for massless Dirac particle is reviewed in terms of the Chern number. Within the continuum limit, a half-quantized Hall conductivity is obtained. Thereafter, by considering the lattice regularization of the Dirac theory, we show how one may obtain the well known quantum Hall conductivity for a single Dirac cone. The renormalization of the electron energy spectrum, for both small and large coupling regime, is derived. In particular, it is shown that massless Dirac particles may, only in the strong correlated limit, break either chiral or parity symmetries. For graphene, this implies the generation of Landau-like energy levels and the quantum valley Hall effect.

The influence of a weak magnetic field in the Renormalization-Group functions of (2 + 1)-dimensional Dirac systems

The experimental observation of the renormalization of the Fermi velocity $v_{F}$ as a function of doping has been a landmark for confirming the importance of electronic interactions in graphene. Although the experiments were performed in the presence of a perpendicular magnetic field $B$, the measurements are well described by a renormalization-group (RG) theory that did not include it. Here we clarify this issue, for both massive and massless Dirac systems, and show that for the weak magnetic fields at which the experiments are performed, there is no change in the renormalization-group functions. Our calculations are carried out in the framework of the Pseudo-quantum electrodynamics (PQED) formalism, which accounts for dynamical interactions. We include only the linear dependence in $B$, and solve the problem using two different parametrizations, the Feynman and the Schwinger one. We confirm the results obtained earlier within the RG procedure and show that, within linear order in the magnetic field, the only contribution to the renormalization of the Fermi velocity arises due to interactions. In addition, for gapped systems, we observe a running of the mass parameter.

Chiral-symmetry breaking in pseudoquantum electrodynamics at finite temperature

We use the Schwinger-Dyson equations in the presence of a thermal bath, in order to study chiral symmetry breaking in a system of massless Dirac fermions interacting through pseudo quantum electrodynamics (PQED3), in (2+1) dimensions. We show that there is a critical temperature $T_c$, below which chiral symmetry is broken, and a corresponding mass gap is dynamically generated, provided the coupling is above a certain, temperature dependent, critical value $\alpha_c$. The ratio between the energy gap and the critical temperature for this model is estimated to be $2 \pi$. These results are confirmed by analytical and numerical investigations of the Schwinger-Dyson equation for the electron. In addition, we calculate the first finite-temperature corrections to the static Coulomb interaction. The relevance of this result in the realm of condensed matter systems, like graphene, is briefly discussed.

Interaction Induced Quantum Valley Hall Effect in Graphene

We use pseudo-quantum electrodynamics in order to describe the full electromagnetic interaction of the p electrons in graphene in a consistent 2D formulation. We first consider the effect of this interaction in the vacuum polarization tensor or, equivalently, in the current correlator. This allows us to obtain the T→0 conductivity after a smooth zero-frequency limit is taken in Kubo’s formula. Thereby, we obtain the usual expression for the minimal conductivity plus corrections due to the interaction that bring it closer to the experimental value. We then predict the onset of an interaction-driven spontaneous quantum valley Hall effect below an activation temperature of the order of 2 K. The transverse (Hall) valley conductivity is evaluated exactly and shown to coincide with the one in the usual quantum Hall effect. Finally, by considering the effects of pseudo-quantum electrodynamics, we show that the electron self-energy is such that a set of P- and T-symmetric gapped electron energy eigenstates are dynamically generated, in association with the quantum valley Hall effect.Received 8 October 2013DOI:https://doi.org/10.1103/PhysRevX.5.011040This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

Unitarity of theories containing fractional powers of the d’Alembertian operator

We examine the unitarity of a class of generalized Maxwell U(1) gauge theories in (2+1) D containing the pseudodifferential operator $\Box^{1-\alpha}$, for $\alpha \in [0,1)$. We show that only Quantum Electrodynamics (QED$_3$) and its generalization known as Pseudo Quantum Electrodynamics (PQED), for which $\alpha =0$ and $\alpha = 1/2$, respectively, satisfy unitarity. The latter plays an important role in the description of the electromagnetic interactions of charged particles confined to a plane, such as in graphene or in hetero-junctions displaying the quantum Hall effect.

Chiral symmetry breaking in the pseudo-quantum electrodynamics

In this work, we study the chiral symmetry breaking in pseudo-quantum electrodynamics in ($2+1$) dimensions, which is designed to reproduce a Coulomb potential for charged particles on a plane interacting via photons propagating in ($3+1$) space-time dimensions and would be relevant for applications to condensed-matter systems. Using an ultraviolet cutoff in the momentum integrals, we show that there is a critical dimensionless coupling ${\ensuremath{\alpha}}_{c}=\ensuremath{\pi}/4$ above which there is chiral symmetry breaking. In the case of the theory with $N$ massless fermions, we obtain a critical value of the number of fermion flavors, ${N}_{c}$, below which the chiral symmetry breaking occurs. Finally, we discuss the relevance of our results to graphene in the ultimate deep infrared regime where the Fermi velocity of the Dirac fermions approaches the velocity of light.