We construct a novel family of solutions of the holographic superfluid model with the excited states in the probe limit. We observe that the higher excited state or larger superfluid velocity will make the scalar hair more difficult to be developed, and the higher excited state or smaller mass of the scalar field makes it easier for the emergence of translating point from the second-order transition to the first-order one. We note that the difference of the critical chemical potential between the consecutive states increases as the superfluid velocity increases. Interestingly, the “Cave of Winds” phase structure will disappear but the first-order phase transition occurs for the excited states, which is completely different from the holographic superfluid model with the ground state. This means that the excited state will hinder the appearance of the Cave of Winds. Moreover, we find that there exist additional poles in Im[<math><mi>σ</mi><mo>(</mo><mi>ω</mi><mo>)</mo></math>] and delta functions in Re[<math><mi>σ</mi><mo>(</mo><mi>ω</mi><mo>)</mo></math>] arising at low temperature for the excited states, and the higher excited state or larger superfluid velocity results in the larger deviation from the expected relation in the gap frequency.

China

Dong Wang

Qiyuan Pan

Chuyu Lai

Jiliang Jing

Holographic superfluid with excited states

We numerically study the holographic entangle-ment entropy and complexity conjectured with the volume in the holographic superfluid with full backreaction which can realize first and second order phase transitions. Our results show that both the entanglement entropy and complexity exhibit the behaviors characterizing the type of the transition. For the first order phase transition, there is a fast drop of the entanglement entropy and a fast jump of the complexity at the critical temperature, while both of them are continuous but non-differentiable at the second order phase transition point. These suggest that both of holographic entanglement entropy and complexity may be used as a good probe to the type of superfluid phase transition. Moreover, at a fixed temperature in the superfluid phase, we observe that the increasing superfluid velocity increases the entanglement entropy but decreases the complexity. Interestingly, we find that, for the condensation operators  $$\mathcal {O}_{+}$$  <math> <msub> <mi>O</mi> <mo>+</mo> </msub> </math>   and  $$\mathcal {O}_{-}$$  <math> <msub> <mi>O</mi> <mo>-</mo> </msub> </math>  , the dependence of holographic entanglement entropy on the backreaction is inconsistent and so is the dependence of holographic complexity on the superfluid velocity in the normal phase, which indicates that the entanglement entropy and complexity may reflect deep physics about the difference between the two operators in the superfluid dual system.

Entanglement entropy and complexity in the holographic model of superfluid

We investigate the holographic entanglement entropy (HEE) and the holographic subregion complexity (HSC) for holographic superconductors, both in the Einstein and in the Einstein-Gauss-Bonnet gravitational theories. For both ground and excited states, we show that, in the Einstein gravity, the HSC decreases as the temperature increases and the normal phase has a smaller HSC than the superconducting phase, which is opposite to the behavior of the HEE. Moreover, we find out that, for a given temperature T in the superconducting phase, the higher excited state leads to a lager value of the HEE but a smaller value of the HSC. However, the Einstein-Gauss-Bonnet gravity has significantly different effect on the HSC, while the HEE always increases monotonously as the temperature increases and its value in the normal phase always larger than that in the superconducting phase. Our results indicate that the HEE and HSC provide richer physics in phase transitions and condensation of scalar hair for holographic superconductors with excited states.

Xiongying Qiao

Mengjie Wang

Holographic entanglement entropy and subregion complexity for excited states of holographic superconductors

We systematically study the complexity of a strip-shaped subregion in a fully backreacted holographic model of a superconductor with the nonlinear electrodynamics by the “complexity=volume” (CV) conjecture, and compare it with the holographic entanglement entropy. We consider three types of typical nonlinear electrodynamics and find that the holographic complexity can be utilized as a good probe of the superconductor phase transition in the nonlinear electrodynamics like the holographic entanglement entropy does. For the operator <math><msub><mrow><mi>O</mi></mrow><mrow><mo>−</mo></mrow></msub></math>, the complexity decreases (or increases) monotonically as the absolute value of the nonlinear parameter <math><mo>|</mo><mi>b</mi><mo>|</mo></math> grows in the superconducting (or normal) phase, which is the opposite of the behavior of the holographic entanglement entropy, and this property holds for various types of the nonlinear electrodynamics. For the operator <math><msub><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msub></math>, in the superconducting phase, it is interesting to note that the complexity is a monotonic decreasing function of <math><mo>|</mo><mi>b</mi><mo>|</mo></math> for the Logarithmic nonlinear electrodynamics (LNE), but in systems with the Born-Infeld nonlinear electrodynamics (BINE) and Exponential nonlinear electrodynamics (ENE), as the parameter <math><mo>|</mo><mi>b</mi><mo>|</mo></math> increases, the complexity first decreases and arrives at its minimum at some threshold, then increases monotonously. Whereas the non-monotonic variation of the holographic entanglement entropy can be seen in all the three types of the nonlinear electrodynamics, concretely, it first rises, then descends with larger <math><mo>|</mo><mi>b</mi><mo>|</mo></math>, and has a peak at the inflection point. Furthermore, comparing with the BINE and LNE, we find that the ENE has stronger effect on the condensation formation, the subregion complexity and the entanglement entropy of the holographic superconductors with backreaction.

Complexity for holographic superconductors with the nonlinear electrodynamics

In Gauss–Bonnet gravity, we analytically investigate the p-wave superfluid models in five dimensional AdS soliton and AdS black hole in order to explore the influences of the higher curvature correction on the holographic superfluid phase transition. We observe that the analytical findings are in good agreement with the numerical computations. Our results show that the critical chemical potential of the system increases with the increase of the Gauss–Bonnet parameter in AdS soliton background, while the critical temperature decreases as the Gauss–Bonnet factor grows if the phase transition of the system is of the second order in AdS black hole background, both of which indicate that the higher curvature correction hinders the formation of the condensation of the vector operator. Moreover, the critical exponent of the system takes the mean-field value 1/2, which is independent of the Gauss–Bonnet parameter and the spatial component of the gauge field.

Tangmei He

Analytical study of holographic p-wave superfluid models in Gauss–Bonnet gravity

We analytically study the holographic superfluid phase transition in the AdS soliton background by using the variational method for the Sturm–Liouville eigenvalue problem. By investigating the holographic s-wave and p-wave superfluid models in the probe limit, we observe that the spatial component of the gauge field will hinder the phase transition. Moreover, we note that, different from the AdS black hole spacetime, in the AdS soliton background the holographic superfluid phase transition always belongs to the second order and the critical exponent of the system takes the mean-field value in both s-wave and p-wave models. Our analytical results are found to be in good agreement with the numerical findings.

Brazil

Yongjiu Wang

Analytical study on holographic superfluid in AdS soliton background

Based on the Sturm–Liouville eigenvalue problem, Banerjee et al. proposed a perturbative approach to analytically investigate the properties of the ( 2+1 )-dimensional superconductor with Born–Infeld electrodynamics (Banerjee et al., 2013) [29] . By introducing an iterative procedure, we will further improve the analytical results and the consistency with the numerical findings, and can easily extend the analytical study to the higher-dimensional superconductor with Born–Infeld electrodynamics. We observe that the higher Born–Infeld corrections make it harder for the condensation to form but do not affect the critical phenomena of the system. Our analytical results can be used to back up the numerical computations for the holographic superconductors with various condensates in Born–Infeld electrodynamics.

On analytical study of holographic superconductors with Born–Infeld electrodynamics

We present an explicit construction of a family of steady state density matrices for an open integrable spin-1 chain with bilinear and biquadratic interactions, also known as the Lai–Sutherland model, driven far from equilibrium by means of two oppositely polarizing Markovian dissipation channels localized at the boundary. The steady state solution exhibits n+1 fold degeneracy, for a chain of length n , due to existence of (strong) Liouvillian U(1) symmetry. The latter can be exploited to introduce a chemical potential and define a grand canonical nonequilibrium steady state ensemble. The matrix product form of the solution entails an infinitely-dimensional representation of a non-trivial Lie algebra (semidirect product of sl2 and a non-nilpotent radical) and hints to a novel Yang–Baxter integrability structure.