JHEP06(2020)013 Published for SISSA by Springer Received: November 15, 2019 Revised: April 6, 2020 Accepted: May 12, 2020 Published: June 1, 2020 Excited states of holographic superconductors Yong-Qiang Wang, 1 Tong-Tong Hu, Yu-Xiao Liu, Jie Yang and Li Zhao Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, People’s Republic of China E-mail: [email protected], [email protected], [email protected], [email protected], [email protected]Abstract: In this paper we re-investigate the model of the anti-de Sitter gravity coupled to Maxwell and charged scalar fields, which has been studied as the gravitational dual to a superconductor for a long time since the famous work [Phys. Rev. Lett. 101 (2008) 031601]. By numerical method, we present a novel family of solutions of holographical superconductor with excited states, and find there exists a lower critical temperature in the corresponding excited state. Moreover, we study the condensate and conductivity in the excited states. It is very interesting that the conductivity σ of each excited state has an additional pole in Im[σ] and a delta function in Re[σ] arising at the low temperature inside the gap, which is just the evidence of the existence of excited states. Keywords: Holography and condensed matter physics (AdS/CMT), AdS-CFT Corre- spondence ArXiv ePrint: 1910.07734 1 Corresponding author. Open Access,c The Authors. Article funded by SCOAP 3 . https://doi.org/10.1007/JHEP06(2020)013
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JHEP06(2020)013
Published for SISSA by Springer
Received: November 15, 2019
Revised: April 6, 2020
Accepted: May 12, 2020
Published: June 1, 2020
Excited states of holographic superconductors
Yong-Qiang Wang,1 Tong-Tong Hu, Yu-Xiao Liu, Jie Yang and Li Zhao
Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University,
In condensed matter physics, there is still no consensus on the mechanism of high tem-
perature superconductivity. Over the past decade, the AdS/CFT correspondence [1–3] is
one of the most important results from string theory, which establishes the relationship
between the strongly correlated field on the boundary and a weak gravity theory in one
higher-dimensional bulk spacetime. In the past ten years, the AdS/CFT correspondence
has provided a novel way to study condensed matter theory, and received a great deal
of attention. In the seminal papers [4–6], the authors investigated the model of a com-
plex scalar field coupled to a U(1) gauge field in a (3 + 1)-dimensional Schwarzschild-AdS
black hole and found that due to the U(1) symmetry breaking below the critical temper-
ature Tc, the condensate of the scalar field could be interpreted as the Cooper pair-like
superconductor condensate. Moreover, there exists a gap in the optical conductivity of
the superconducting state, and the value of the gap is close to the value of a high tem-
perature superconductor. Thus, this model can be regarded as the dual explanation to
the high temperature superconductor. When replacing the scalar field with other matter
fields, one can also obtain the condensate of matter fields corresponding to various kinds
of holographic superconductors. For example, holographic d-wave model was constructed
by introducing a symmetric, traceless second-rank tensor coupled to a U(1) gauge field in
the bulk [7–9]. With the SU(2) Yang-Mills field coupled to gravity, the holographic p-wave
model has also been discussed in [10]. Two alternative holographic realizations of p-wave
superconductivity could arise from the condensate of a two-form field [11] and a complex,
massive vector field with U(1) charge [12, 13], respectively. The model of holographic su-
perconductor can also be extended to study the holographic Josephson junction [14–26],
which is made up of two superconductor materials with weak link barrier [27]. A top-down
construction of holographic superconductor from superstring theory was discussed in [28],
and a similar construction using an M-theory truncation was proposed in [29, 30]. For
reviews of holographic superconductors, see [31–34].
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JHEP06(2020)013
Until now, lots of work of holographic superconductor were investigated in the ground
state,1 that is, the scalar field can keep sign along the radial direction. It is well-known that
the excited states could have some nodes along the radial direction, where the value of the
scalar field could change the sign. On the other hand, it is easy to see in quantum theory
that the bound states with given angular momentum and other quantum numbers similarly
could form towers of states known as radial excitations with otherwise identical quantum
numbers. Especially, holographic phenomenological models [37, 38] have the potential
to provide a better understanding of strongly interacting systems of quarks and gluons,
including the excited states of hadrons. These models are presently known as AdS/QCD.
Furthermore, in the background of Schwarzschild black hole, holographic model of QCD
could be extended to the finite temperature system [39], There are also many new progresses
on the excited states of hadrons from holographic QCD, such as [40–42].
So, it is interesting to see whether there exist the solutions of holographic supercon-
ductor with excited states of the scalar field. In the present paper, we would like to numer-
ically solve the Maxwell-Klein-Gordon system with the background of a four-dimensional
Schwarzschild-AdS black hole and give a family of excited states of holographic supercon-
ductors with different critical temperatures. Moreover, we will also study the condensate
and optical conductivity in the excited states of holographic superconductors.
The paper is organized as follows: in section 2, we review the model of a U(1) gauge
field coupled with a charged scalar field in (3 + 1)-dimensional AdS spacetime and show a
gravity dual model of holographic superconductor. We show the numerical results of the
excited states and study the characteristics of the condensate and conductivity in section 3.
The conclusion and discussion are given in the last section.
2 Review of holographic superconductors
Let us introduce the model of a Maxwell field and a charged complex scalar field in the
four-dimensional Einstein gravity spacetime with a negative cosmological constant. The
bulk action reads
S =1
16πG
∫d4x
[R+
6
`2− 1
4FµνFµν − (Dµψ)(Dµψ)∗ −m2ψψ∗
], (2.1)
where Fµν = ∂µAν−∂νAµ is the field strength of the U(1) gauge field, and Dµ = ∇µ−iqAµψis the gauge covariant derivative with respect to Aµ. The constant ` is the AdS length
scale, m and q are the mass and charge of the complex scalar field ψ, respectively. Due
to the existence of Maxwell and complex scalar field, the strength of the backreaction of
the matter fields on the spacetime metric could be tuned by the charge q. In order to
see that how the effect of backreaction varies with the charge q, we could introduce the
scaling transformations A → A/q and ψ → ψ/q, and the Lagrangian density in eq. (2.1)
changes into
16πGL = R+6
`2+ κ
(−1
4FµνFµν − |∇ψ − iAψ|2 −m2ψψ∗
), (2.2)
1In [35, 36] the ground state represents the zero-temperature limit of holographic superconductor while
in this paper the ground state refers to the scalar field without nodes.
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JHEP06(2020)013
with the constant κ ≡ 1/q2. From the above, we can see that the value of the parameter
κ will decrease when q increases, and the backreaction of matter fields on the spacetime
metric could be ignored in the large q → ∞ limit, which is also called as the probe limit.
Here we adopt to the probe approximation. Thus, the following equations of the scalar
and Maxwell fields can be derived from the Lagrangian density (2.2):