Φ-mapping Theory Research Articles

Topology of Born–Infeld-AdS black hole phase transitions: Bulk and CFT sides

The thermodynamic criticality of the AdS black holes serves as an important structure during the thermal phase transition. This paper discusses about the critical points and their topology during thermal phase transitions of the Born–Infeld AdS black holes. We make such investigations using two different topological approaches, namely, using Duan’s topological current ϕ-mapping theory, and the off-shell free energy. Within Duan’s formalism, we observe that for a given value of the Born–Infeld parameter b, there exists an associated electric charge parameter Q, which is highly sensitive to the topological phase transitions. This way we examine the connections of the first-order phase transition and the topological nature of the critical points. We find that the topological nature has a possible breakdown in certain parametric ranges. In effect, we determine the unconventional and the conventional phase critical points as the creation (topologically vortex) and annihilation (topologically anti-vortex) points (pairs). As the second approach, we call the off-shell free energy to determine the topological classes: of which one corresponds to the AdS-Schwarzschild black hole phases, while the other provides a possible topological phase transition. Here we also reveal a novel phase transition between two unstable phases, namely, the unstable small black hole and the intermediate black holes. For certain parametric values of the Born–Infeld parameter and the pressure, we also study the different topological descriptions that inevitably correspond to the AdS-Reissner–Nordström black hole phases. As a consistency check of the critical points during the topological phase transitions, we study the vortex/anti-vortex annihilation thermodynamics from local as well as global thermodynamic viewpoints. At the end, a special emphasis has been put on phase transition from the conformal field theory perspective revealing that the Born–Infeld-AdS black hole and its dual CFT share identical topology. Consequently, any topological transition that occurs in the bulk corresponds to a parallel transition in the dual CFT.

Thermodynamic topology and photon spheres in the hyperscaling violating black holes

It was shown that a standard ring of light can be imagined outside the event horizon for stationary rotating four-dimensional black holes with axial symmetry using the topological method (Cunha, 2020; Wei and Shao-Wen, 2020). Based on this concept, in this paper, we investigate the topological charge and the conditions of existence of the photon sphere (PS) for a hyperscaling violation (HSV) black hole with various values of the parameters of this model. Then, after carrying out a detailed analysis, we show the conventional topological classes viz Q=−1 for the mentioned black hole and Q=0 for the naked singularities. Also, we propose a new topological class for naked singularities (Q=+1) with respect toz≥1. We also determined thatz≥2, it either shows a naked singularity form with total topological charge+1or has no solution. Therefore, we have the black hole solution only in1≤z<2. Then, we will use two different methods, namely the temperature (Duan’s topological current Φ-mapping theory) and the generalized Helmholtz free energy method, to study the topological classes of our black hole. By considering the black hole mentioned, we discuss the critical and zero points (topological charges and topological numbers) for different parameters of hyperscaling violating black holes, such as (z,θ¯) and other free parameters, and study their thermodynamic topology. We observe that for a given value of the parameters 1≤z<2, θ¯, and other free parameters, there exist two total topological charges (Qtotal=−1,0) with the same phase structure for the T method and total topological numbers (W=+1) for the generalized Helmholtz free energy method. Additionally, we summarize the results for each study as photon sphere, temperature, and generalized Helmholtz free energy in some figures and tables. Finally, we compare our findings with other related studies in the literature.

SELF-DUAL VORTICES IN THE FRACTIONAL QUANTUM HALL SYSTEM

Based on the ϕ-mapping theory, we obtain an exact Bogomol'nyi self-dual equation with a topological term, which is ignored in traditional self-dual equation, in the fractional quantum Hall system. It is revealed that there exist self-dual vortices in the system. We investigate the inner topological structure of the self-dual vortices and show that the topological charges of the vortices are quantized by Hopf indices and Brouwer degrees. Furthermore, we study the branch processes in detail. The vortices are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of the vector field ϕ.

SKYRMION EXCITATIONS OF SPIN HALL EFFECT IN ATOMS

By making use of ϕ-mapping topological current theory and the decomposition of gauge potential theory, we investigate the skyrmion excitations of spin Hall effect induced by optical fields in neutral atomic system. It is shown that the topological structures of these skyrmions are characterized by ϕ-mapping topological numbers: Hopf indices and Brouwer degrees. Based on the bifurcation theory of ϕ-mapping theory, it is found that the skyrmions can be generated or annihilated at the limit point and they encounter, split or merge at the bifurcation point.

Topological Vortices in a Spin-Triplet Superconductor

Based on the complex three-component order parameter model of a spin-triplet superconductor, by using the ϕ-mapping theory, we derive a new equation describing the distribution of the magnetic field for vortices, which can be reduced to the modified London equation in the case of | ψ2|2 = |ψ3|2 = 0 and Wl1 = 1. A magnetic flux quantization condition for vortices in a spin-triplet superconductor is also derived, which is topological-invariant. Furthermore, the branch processes during the evolution of the vortices in a spin-triplet superconductor are discussed. We also point out that the sum of the magnetic flux quantization that those vortices carried is 2nΦ0 (Φ0 is the unit magnetic flux), that is to say, the sum of winding number is even, which needs to be proved by experiment.

Topological Aspects of Superconductors at Dual Point

We study the properties of the Ginzburg–Landau model at the dual point for the superconductors. By making use of the U(1) gauge potential decomposition and the ϕ-mapping theory, we investigate the topological inner structure of the Bogomol'nyi equations and deduce a modified decoupled Bogomol'nyi equation with a nontrivial topological term, which is ignored in conventional model. We find that the nontrivial topological term is closely related to the N-vortex, which arises from the zero points of the complex scalar field. Furthermore, we establish a relationship between Ginzburg–Landau free energy and the winding number.

Evolution of Nielsen–Olesen's String from Chern–Simons FieldTheory

We study the topology of Nielsen–Olesen's local field theory ofsingle dual string. Based on the Chern–Simons field theory in three dimensons,we find many strings that can form world sheets in four dimensions. Thesestrings have important relation to the zero point of the complex scalarfield. These world sheets of strings can be expressed by the topologicalinvariant number, Hopf index, and Brower degree. Nambu–Goto's action isobtained from the Nielsen's action definitely by using ϕ-mapping theory.

Topological Aspects of TripletSuperconductors

In this paper, using the ϕ-mapping theory, it is shown thattwo kinds of topological defects, i.e., the vortex lines and the monopolesexist in the helical configuration of magnetic field in tripletsuperconductors. And the inner topological structure of these defects isstudied. Because the knot solitons in the triplet superconductors arecharacterized by the Hopf invariant, we also establish a relationshipbetween the Hopf invariant and the linking number of knots family, andreveal the inner topological structure of the Hopf invariant.

TOPOLOGICAL STRUCTURE AND EVOLUTION OF SPACE–TIME DISLOCATIONS AND DISCLINATIONS

By making use of the gauge potential decomposition theory and ϕ-mapping theory, the topological structure and the topological quantization of dislocations and disclinations are studied in the framework of Riemann–Cartan space–time manifold. The evolution of dislocation strings and disclination points is also studied from the topological properties of the order parameter field. The dislocations and disclinations are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of the order parameter field.

MAGNETIC BRANES FROM GENERALIZED 't HOOFT TENSOR

't Hooft–Polykov magnetic monopole regularly realizes the Dirac magnetic monopole in terms of a two-rank tensor, the so-called 't Hooft tensor in 3D space. Based on the Chern kernel method, we propose the arbitrary rank 't Hooft tensors, which universally determine the quantized low energy boundaries of generalized Georgi–Glashow models under asymptotic conditions. Furthermore, the dual magnetic branes theory is built up in terms of ϕ-mapping theory.

Regular Magnetic Monopole from Generalized 't Hooft Tensor

It is well known that 't Hooft–Polykov magnetic monopole regularlyrealizes the Dirac magnetic monopole in terms of a two-rank tensor, i.e.the so-called 't Hooft tensor in three-dimensional space, which has beengeneralized to all even rank ones. We propose an arbitrary odd rank 'tHooft tensor, which universally determines the quantized low-energyboundaries of the even dimensional Georgi–Glashow models underasymptotic conditions. Furthermore, the dual magnetic monopole theory isbuilt up in terms of the ϕ-mapping theory.

Topological vortices in a two-gap superconductor

Based on the ϕ-mapping theory, we derive a new rigorous equation describing the distribution of the magnetic field for vortices in a two-gap superconductor, of which the so-called modified London equation is just a special case in a one-flavor limit. We explicitly investigate the London penetration depth, the Meissner and mixed states and Josephson effect. A magnetic flux quantization condition for vortices in a two-gap superconductor is also derived, from which it follows that in a two-gap superconductor there exist vortices which carry an arbitrary fraction of magnetic flux quantum. The branch processes during the evolution of the vortices in a two-gap superconductor are discussed.

Self-Dual Chern–Simons Vortices in Higgs Field

By decomposing the Bogomol'nyi self-dual equation in the AbelianChern–Simons Higgs model, we find a self-dual topological term that wasignored all the time in the Bogomol'nyi self-duality equation due to theimproper decomposition of the complex Higgs field. We also present a newself-dual equation that includes the topological term. It is shown thatthe self-dual vortex just arises from the symmetric phase of the Higgsfield ϕ = 0. Using our ϕ-mapping theory, the inner topologicalstructure of the vortex and double vortex is given.

KNOTS IN THE NON-ABELIAN CHERN–SIMONS THEORY

We employ a spinor decomposition of SU (2) connection to study the Chern–Simons form. Our results show that the SU (2) Chern–Simons action yields the link invariants. Then the structure of the knots is discussed with the ϕ-mapping theory.

Knot-like Mermin-Ho vortices in the superfluid [formula omitted]He–A

We study the topological properties of knots in superfluid 3 He–A. The decomposition of SO(3) connection and its Abelian projection is presented. The topological structure of Mermin-Ho vortices is obtained with the Abelian projection from the viewpoint of φ-mapping theory. Furthermore, it is shown that the linking number of the knot-like Mermin-Ho vortices can be described with the Hopf invariant. At last, the evolution of the knots is discussed.

ϕ-MAPPING TOPOLOGICAL THEORY IN TWO-GAP SUPERCONDUCTOR

In this paper, the topological structure of the two-gap superconductor is discussed in detail based on the ϕ-mapping theory. The expression of the vorticity ∇×V of the composite vortex is given and the relation between the vorticity and the magnetic field which is carried by the composite vortex is discussed. The curl of velocity ∇×V has important relation to δ2(ϕ) or we can say that ∇×V has an important relation to the zero points of ϕ. The inner structure of the topological current is characterized by the ϕ-mapping topological numbers Hopf-index and Brouwer degrees.

Velocity field vortices and Mermin-Ho vortices in two-component spinor BEC

In light of the φ-mapping topological current theory, two important vortex structures in two-component spinor BEC—the velocity field vortices and the Mermin-Ho vortices are discussed. It is revealed that these two different kinds of vortices are created respectively from the zero points of two different order parameter configurations in the condensates, and both their topological charges, locations and motions can be determined by the φ-mapping theory.

INNER STRUCTURE OF GAUSS–BONNET–CHERN THEOREM AND THE MORSE THEORY

We define a new one-form HA based on the second fundamental tensor [Formula: see text], the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.

Topological tensor current of p̃-branes in the φ-mapping theory

We present a new topological tensor current of p̃-branes by making use of the φ-mapping theory. It is shown that the current is identically conserved and behaves as δ(φ⃗), and every isolated zero of the vector field φ⃗(x) corresponds to a “magnetic” p̃-brane. Using this topological current, the generalized Nambu action for multi p̃-branes is given, and the field strength F corresponding to this topological tensor current is obtained. It is also shown that the magnetic charges carried by p̃-branes are topologically quantized and labeled by Hopf index and Brouwer degree, the winding number of the φ mapping.

Topological Structure in the O(n) Symmetric Time-Dependent-Ginzburg-Landau Model

A new topological current in the O(n) symmetrical time-dependent-Ginzburg-Landau model is discussed. Using ϕ-mapping theory, this topological current is studied. It is shown that the inner structure of topological defects can be classified by Brouwer degrees and Hopf indices of the ϕ-mapping.