A non-Hermitian form of QED is presented which describes interacting Dirac monopoles. The theory is related by a canonical transformation to a model proposed by Milton. As in Hermitian QED an abelian gauge potential is coupled to a four-component fermion. Under proper Lorentz transformations and time-reversal the fermion field transforms like a Dirac spinor but has a non-standard parity transformation. This implements the property that magnetic charge, unlike electric charge, is parity-odd. A consequence of the non-Hermiticity is that there is an attractive force between identical charged particles, at least in the weakly coupled regime. This effect can be understood even at the classical level; a simple calculation of the force between classical Dirac monopoles is presented which shows that like charge monopoles attract and opposite charges repel.
Monopoles, duality, and large charge in Chern-Simons QED$_3$
Abdus Salam Centre for Theoretical Physics, Imperial College London（阿卜杜斯·萨拉姆理论物理中心，伦敦帝国理工学院）；Departement de physique, Universite de Montreal（蒙特利尔大学物理系）
We initiate the conformal bootstrap study of Quantum Electrodynamics in $2+1$ space-time dimensions (QED$_{3}$) with $N$ flavors of charged fermions by focusing on the 4-point function of four monopole operators with the lowest unit of topological charge. We obtain upper bounds on the scaling dimension of the doubly-charged monopole operator, with and without assuming other gaps in the operator spectrum. Intriguingly, we find a (gap-dependent) kink in these bounds that comes reasonably close to the large $N$ extrapolation of the scaling dimensions of the singly-charged and doubly-charged monop…
We study monopole operators with the lowest possible topological charge $q=1/2$ at the infrared fixed point of scalar electrodynamics in $2+1$ dimension (scalar QED$_3$) with $N$ complex scalars and Chern-Simons coupling $|k|=N$. In the large $N$ expansion, monopole operators in this theory with spins $\ell<O(\sqrt{N})$ and associated flavor representations are expected to have the same scaling dimension to sub-leading order in $1/N$. We use the state-operator correspondence to calculate the scaling dimension to sub-leading order with the result $N-0.2789+O(1/N)$, which improves on existing le…
We study the quantum phase transition from a Dirac spin liquid to an antiferromagnet driven by condensing monopoles with spin quantum numbers. We describe the transition in field theory by tuning a fermion interaction to condense a spin-Hall mass, which in turn allows the appropriate monopole operators to proliferate and confine the fermions. We compute various critical exponents at the quantum critical point (QCP), including the scaling dimensions of monopole operators by using the state-operator correspondence of conformal field theory. We find that the degeneracy of monopoles in QED3 is lif…
Monopole operators are studied in a large family of quantum critical points between Dirac and topological quantum spin liquids (QSLs): chiral and Z$_{2}$ QSLs. These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2+1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED$_{3}$-GN), while for the transitions to Z$_{2}$ QSLs it is a superconducting pairing term with general spin/valley structure (generalized QED$_{…
We present a direct Monte-Carlo determination of the scaling dimension of a topological defect operator in the infrared fixed point of a three-dimensional interacting quantum field theory. For this, we compute the free energy to introduce the background gauge field of the $Q=1$ monopole-antimonopole pair in three-dimensional non-compact QED with $N=2,4$ and $12$ flavors of massless two-component fermions, and study its asymptotic logarithmic dependence on the monopole-antimonopole separation. We estimate the scaling dimension in the $N=12$ case to be consistent with the large-$N$ (free fermion…
We discuss how the topological defects in the dark sector affect the Standard Model sector when the dark photon has a kinetic mixing with the QED photon. In particular, we consider the dark photon appearing in the successive gauge symmetry breaking, $\mathrm{SU}(2)\to \mathrm{U}(1) \to \mathbb{Z}_2$, where the remaining $\mathbb{Z}_2$ is the center of $\mathrm{SU(2)}$. In this model, the monopole is trapped into the cosmic strings and forms the so-called bead solution. As we will discuss, the dark cosmic string induces the QED magnetic flux inside the dark string through the kinetic mixing. Th…
Confining Properties of Abelian(-Projected) Theories
导出了阿贝尔投影SU（2）-和SU（3）-胶子动力学的磁流表示。除二次项外，所得到的有效作用量还包含了这些电流与四维电弦世界面或三维电涡旋线的相互作用。其次，我们证明了3D紧致QED是具有外磁单极子的3D Abel Higgs模型的小规范玻色子质量极限，并给出了约束弦论作为量子密度积分的物理解释。最后，我们导出了三维紧致QED弱场极限下的双局域场强度关联式，结果与随机真空模型预言的结果一致。
We study monopole operators at the infrared fixed points of $U(1)$ Chern-Simons-matter theories (QED$_3$, scalar QED$_3$, ${\cal N} =1$ SQED$_3$, and ${\cal N} = 2$ SQED$_3$) with $N$ matter flavors and Chern-Simons level $k$. We work in the limit where both $N$ and $k$ are taken to be large with $κ= k/N$ fixed. In this limit, we extract information about the low-lying spectrum of monopole operators from evaluating the $S^2 \times S^1$ partition function in the sector where the $S^2$ is threaded by magnetic flux $4 πq$. At leading order in $N$, we find a large number of monopole operators with…
Nonlinear $σ$ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or $\text{QED}$ theory. In this paper, we propose a dimensional reduction scheme to derive the $\text{U}(1)$ gauged topological NLSM in $n$-dimensional spacetim…
Quantum spin liquids host novel emergent excitations, such as monopoles of an emergent gauge field. Here, we study the hierarchy of monopole operators that emerges at quantum critical points (QCPs) between a two-dimensional Dirac spin liquid and various ordered phases. This is described by a confinement transition of quantum electrodynamics in two spatial dimensions (QED3 Gross-Neveu theories). Focusing on a spin ordering transition, we get the scaling dimension of monopoles at leading order in a large-N expansion, where 2N is the number of Dirac fermions, as a function of the monopole's total…
We study a class of defect quantum field theories where the quantum field theory in the 3+1-dimensional bulk is a free photon and charged matter and the interactions of the photons with the charges occur entirely on a 2+1-dimensional defect. We observe that at the fully quantum level, the effective action of such a theory is still a defect field theory with free photons propagating in the bulk and the nonlinearities in the quantum corrections to the Maxwell equations confined to the defect. We use this observation to show that the defect field theory has interesting electromagnetic properties.…
The Maxwell equations imply that, under the background of non-zero $\boldsymbol{B}$, varying $θ$ term produces $\boldsymbol{E} \cdot \boldsymbol{B}$. An interesting example is the Witten effect where a magnetic monopole becomes a dyon which, however, should disappear in the exact massless limit of the fermion. Underlying mechanism of this phenomenon has been understood by Callan by the presence of an effective axion-like degree of freedom around the monopole, which is roughly the phase of the fermions. The configuration of this axion cancels the effect of the $θ$ term. Now, the chiral anomaly …
Symmetry Breaking from Monopole Condensation in QED$_3$
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校理论物理曼尼·L·巴奥米克研究所，物理与天文学系）
In this work, taking the QED effect into account, we investigate the shadows of the static black hole with magnetic monopoles and neutral black holes in magnetic fields through the numerical backward ray-tracing method. For a static black holes with magnetic monopole, we obtain the relation between the shadow radius and the coupling constant. For neutral black holes in the uniform magnetic fields, we find that the shadow curves deviate very small from the ellipses for equatorial observer, and we read the linear relation between the eccentricity and the coupling constant. For $θ_o\neqπ/2$, we f…
We present the results of a conformal bootstrap study of the presumed unitary IR fixed point of quantum electrodynamics in three dimensions (QED$_3$) coupled to $N_f=4$ two-component Dirac fermions. Specifically, we study the four-point correlators of the $SU(4)$ adjoint fermion bilinear $r$ and the monopole of lowest topological charge $\mathcal{M}_{1/2}$. Most notably, the scaling dimensions of the fermion bilinear $r$ and the monopole $\mathcal{M}_{1/2}$ are found to be constrained into a closed island with a combination of spectrum assumptions inspired by the $1/N_f$ perturbative results a…
We present an alternative description of magnetic monopoles by lifting quantum mechanics from 3-dimensional space into a one with 2 complex dimensions. Magnetic monopoles are realized as a generalization of the considered states. Usual algebraic relations and magnetic fields describing monopoles are reproduced, with the Dirac quantization condition satisfied naturally.
Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital $SU(4)$ symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by $(2+1)d$ quantum electrodynamics (QED$_3$) with $N_f=8$ Dirac fermions. We generalize methods previously developed for $SU(2)$ spin systems to analyze the symmetry pr…
Despite decades-long efforts, magnetic monopoles were never found as elementary particles. Monopoles and associated currents were directly measured in experiments and identified as topological quasiparticle excitations in emergent condensed matter systems. These monopoles and the related electric-magnetic symmetry were restricted to classical electrodynamics, with monopoles behaving as classical particles. Here we show that the electric-magnetic symmetry is most fundamental and extends to full quantum behavior. We demonstrate that at low temperatures magnetic monopoles can form a quantum Bose …
We describe the internal composition of a topologically stable monopole carrying a magnetic charge of $6π/e$ that arises from the spontaneous breaking of the trinification symmetry $SU(3)_c\times SU(3)_L\times SU(3)_R$ ($G$). Since this monopole carries no color magnetic charge, a charge of $6π/e$ is required by the Dirac quantization condition. The breaking of $G$ to the Standard Model occurs in a number of steps and yields the desired topologically stable monopole ("magnetic baryon"), consisting of three confined monopoles. The confined monopoles ("magnetic quarks") each carry a combination …
We consider a Kaluza-Klein vacuum solution which is closely related to the Gross-Perry-Sorkin (GPS) magnetic monopole. The solution can be obtained from the Euclidean Taub-NUT solution with an extra compact fifth spatial dimension within the formalism of Kaluza-Klein reduction. We study its physical properties as appearing in $(3+1)$ spacetime dimensions, which turns out to be a static magnetic monopole. We then boost the GPS magnetic monopole along the extra dimension, and perform the Kaluza-Klein reduction. The resulting four-dimensional spacetime is a rotating stationary system, with both e…
It is proposed that Magnetic Monopoles (MMs) could originate from a new $\mbox{U(1)}_{M}$ symmetry. Such an abelian symmetry is then assumed to be related to the conservation of a magnetic number $M$. This magnetic number is associated with massive MMs from an expected high scale breaking of this magnetic symmetry. The involved scales are approached and the properties of such MMs are investigated along with the prospects for their detection.
(also inside: this manuscript introduces the reader to the argument against the existence of magnetic monopoles, which forms an essential part of Staruszkiewicz's Quantum Mechanics of the Electric Charge) Ultra-high energy (UHE) photons with energies exceeding $10^{18}\eV$ can potentially be observed. They are produced in various processes involving electrically charged particles. However, more exotic scenarios are also possible. UHE photons could be emitted in encounters of massive magnetically charged monopole--antimonopole pairs or in processes associated with monopoles accelerated to high …
We consider the creation of 't Hooft-Polyakov magnetic monopoles by scattering classical wave packets of gauge fields. An example with eight clearly separated magnetic poles created with parity violating helical initial conditions is shown. No clear separation of topological charge is observed with corresponding parity symmetric initial conditions.
