We invastigate numerically the holographic entanglement entropy (HEE) and holographic subregion complexity (HSC) for a p-wave superconductor with backreaction in the framework of dRGT massive gravity. We calculate the HEE and HSC as functions of subregion strip-width or temperature by following the RT formula and CV conjecture. It is shown that both the HEE and HSC exhibit a discontinuity in their slope at the critical temperature, hence the two physical quantities can be able to probe the p-wave superconducting phase transition. Both the HEE and HSC increase linearly at the large strip-widths, which is consistent with the “area law” of entanglement entropy and the definition of complexity in quantum information. We notice that the HEE in the superconducting phase is always lower than that in the normal phase. In contrast, the behavior of HSC shows a distinct and intriguing dependence on the strip-width. Through comparing the values of HEE and HSC for different massive coupling constants or backreaction, we find that increasing backreaction or introducing massive gravity term results in a notable increase in both HEE and HSC.

China

Yiliang Hu

Yabo Wu

Junwang Lu

Cheng-Yuan Zhang

Jianan Chi

Wenzhong Liu

Numerical study of holographic entanglement entropy and subregion complexity for a p-wave superconductor in massive gravity

Based on the holographic duality, we construct the new <italic>p</italic>-wave metal/superconductor phase transitions triggered by the electric and magnetic fields. Especially, using the axion field, we introduce simultaneously the spontaneous (SBT) and explicit (EBT) breaking of translation and then investigate how the SBT and EBT corrections influence on the critical temperature, vector condensate and the optical conductivity. Concretely, for the case with pure electric field, both the EBT correction with weak SBT correction and the SBT correction lower the critical temperature <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> and soften the energy gap. Interestingly, for the strong enough SBT correction, the increasing EBT correction enhances firstly and then inhibits the phase transition. Meanwhile, the increasing EBT correction suppresses the SBT effect on <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math>. At the lowest Landau energy level, the stronger SBT correction suppresses the phase transition for the weak magnetic field, whose inhibiting effect is finally eliminated by the strong enough magnetic field. The monotonic dependence of <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> on the EBT correction only exists for the weak SBT correction. For the Landau energy level in the excited state, the monotonic dependence of <math altimg='si1.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> on the EBT correction vanishes for the intermediate SBT correction with large magnetic field.

Bao-Ping Dong

Jun-Wang Lu

Ya-Bo Wu

Juan-Juan Luo

Jian-Xin Cheng

Hao Liao

Axion effects on new <italic>p</italic>-wave superconductors induced by electromagnetic fields

At the probe limit, the effects of three typical nonlinear corrections of Born-Infeld electrodynamics (BINE), logarithmic Lagrangian (LNE) and exponential Lagrangian (ENE) on the holographic ferromagnetic model under the four-dimensional Gauss-Bonnet gravity background are studied by numerical calculation. For comparison, the Gauss-Bonnet parameter <italic>&#945;</italic> is fixed in the article. In the absence of an external magnetic field, three different nonlinear electrodynamic corrections reduce the critical temperature respectively and make the spontaneous magnetic moment more difficult to form. In the case of opening the external magnetic field, the magnetic susceptibility density always satisfies the Curie-Weiss law, and the increase of the nonlinear parameters under the three corrections will prolong the period of the external magnetic field. Comparing the three corrections horizontally, the exponential form of nonlinear electrodynamics correction has a more significant effect on the parameters.

Ruo-Tong Wei

Te Huang

Jun-Ping Wang

Holographic ferromagnetic phase transitions with three typical nonlinear electrodynamics in Gauss-Bonnet gravity

At the probe level, we investigate the holographic p-wave conductor/superconductor phase transition by both numerical and analytical methods. Especially, due to the nonlinear axion correction, the translational symmetry in superconductor is broken spontaneously. We then study the effects of the axion correction (or disorder) and the mass of vector field on the superconductor phase transition. Concretely, as the disorder parameter and the scaling dimension of vector operator increase, the critical temperature decreases and the stable value of condensate at the low temperature as well as the energy gap increase, which suggests that the larger mass of vector field and the disorder correction inhibit the phase transition, and increase the strength of interaction in superconductor. Near the critical point, the system always undergoes a second-order phase transition, which is independent of the disorder strength and the mass of vector field. According to the behaviors of grand potential and conductivity, the hairy state is verified to be thermodynamical favored and indeed superconducting state. Meanwhile, the analytical results uphold the numerical ones in terms of the critical behavior of condensate.

Yu Tang

Hui Zhu

Holographic -wave Superconductor With Spontaneously Broken Translation

In this work, we construct a holographic antiferromagnetism model by introducing two real antisymmetric tensor fields coupling to the background gauge field strength and interacting with each other in the four-dimensional Lifshitz black hole, and investigate the effects of Lifshitz dynamical exponent z on the paramagnetism-antiferromagnetism phase transition in detail. Using the numerical shooting method, we find that there is still a phase transition in asymptotically Lifshitz space-time in the probe limit. In the absence of an external magnetic field and in low temperature, the magnetic moments condense spontaneously in antiparallel manner with the same magnitude. But the critical temperature and the staggered magnetization decrease with increasing z, which makes paramagnetism-antiferromagnetism phase transition harder to happen. In the presence of the weak external magnetic field, the magnetic susceptibility has a peak at the critical temperature and increasing values of z makes the peak higher. In the case of the strong external magnetic field, there exists a critical magnetic field <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub></math> in the antiferromagnetic phase. When the magnetic field reaches <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub></math>, the system turns into the paramagnetic phase. Besides, the increasing of z results in increasing of <math><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub></math>.

Ting Zhang

Guang-Zhi Xu

Lifshitz scaling effects on the holographic model for paramagnetism-antiferromagnetism phase transition

With the numerical method, we research the properties of the holographic paramagnetism/ferromagnetism phase transition for the Power - Maxwell field in the background of a Schwarzschild-AdS black hole. When there is no external magnetic field, we work out the values of the critical temperature and the magnetic moment with the different power parameter q for the Power - Maxwell field. It can be seen that the larger power parameter q reduces the critical temperature and makes it harder for the magnetic moment to be formed. What is more, in the presence of an external magnetic field, we get the values of the magnetic susceptibility density with the different q, which satisfies the Cure-Weiss law. When the external magnetic field changes, the hysteresis loop appears for different q.

The influence of Power-Maxwell field on holographic paramagnetism model

We extend on-shell bootstrap methods from spacetime amplitudes to the world sheet objects of the Cachazo-He-Yuan formalism. We observe that the integrands corresponding to tree-level nonlinear sigma model, Yang-Mills and <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory are determined by demanding enhanced UV scaling under Britto-Cachazo-Feng-Witten shifts. Back in spacetime, we also find that <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory is fixed by gauge invariance/UV scaling and simple locality assumptions.

France

Laurentiu Rodina

UV consistency conditions for Cachazo-He-Yuan integrands

The Cachazo-He-Yuan formula was originally proposed to describe on-shell scattering of particles from a single massless field. We present a method to modify it to include several interacting scalar fields, all possessing different masses and possibly off-shell momenta. The method is applied to Yukawa interactions between a number of scalar nucleons and pions, and to the <math><msub><mi>ϕ</mi><mn>1</mn></msub><msub><mi>ϕ</mi><mn>2</mn></msub><msub><mi>ϕ</mi><mn>3</mn></msub></math> coupling of three different scalar fields. Composite models constructed from existing theories can be used to broaden the scope of the method. The modification is applied to describe Compton scattering from a massive particle and to photon bremsstrahlung. It is also employed to generalize the disk function <math><mi>Z</mi></math> and the sphere function <math><mi>J</mi></math>.

Canada

C. S. Lam

Cachazo-He-Yuan theory for several fields

Möbius invariance is used to construct gluon tree amplitudes in the Cachazo, He, and Yuan (CHY) formalism. If it is equally effective in steering the construction of off-shell tree amplitudes, then the S-matrix CHY theory can be used to replace the Lagrangian Yang-Mills theory. Unfortunately that is not possible. We find that the CHY formula can indeed be modified to obtain a Möbius-invariant off-shell amplitude <math><msub><mi>M</mi><mi>P</mi></msub></math>, but this modified amplitude lacks local gauge invariance, which can be restored to give the correct Yang-Mills amplitude only by the addition of a complementary amplitude <math><msub><mi>M</mi><mi>Q</mi></msub></math>. Although neither <math><msub><mi>M</mi><mi>P</mi></msub></math> nor <math><msub><mi>M</mi><mi>Q</mi></msub></math> is fully gauge invariant, both are partially gauge invariant in a sense to be explained.

Off-shell Yang-Mills amplitude in the Cachazo-He-Yuan formalism

In the probe limit, we investigate the effect of the quartic field strength correction to the usual Maxwell field on the holographic paramagnetism/ferromagnetism phase transition in the background of a 4-dimensional Schwarzschild-AdS black hole spacetime. We find that the presence of the higher correction parameter a to the Maxwell field decreases the critical temperature and makes the condensation harder to form in the case without external magnetic field. Furthermore, the increase of a will result in shortening the period of external magnetic field.