Investigation of transport properties of graphene Dirac fluid by holographic two-current axion coupling model

Abstract

Recently, there has been great interest in the phenomenon of severe violation of the Wiedemann–Franz law in graphene Dirac fluids around 75 K, due to the strong coupling relativistic plasma near the neutral point, where traditional perturbation theory fails. To explain this phenomenon, this article proposes a holographic dual two-current axion coupling model, describing the interaction between electrons and holes in graphene near the charge neutrality point (CNP) and revealing the related physical mechanism. The study shows that the holographic two-current model aligns with experimental results at 100 $\upmu$m$^{-2}$, and correctly predicts conductivity as temperature increases. Additionally, the article explores the behavior of $\beta +\gamma$ and its impact on conductivity and thermal conductivity. The results suggest a frictional effect between electrons and holes. Consequently, this study provides us with a clearer understanding of the properties of graphene Dirac fluids and further confirms the reliability of the holographic duality method.

1 Introduction

In recent years, the study of graphene Dirac fluids has been a hot topic in the field of condensed matter physics. Dirac fluids refer to the relativistic plasma-like state of graphene near charge neutrality, where massless electrons and holes collide at high speeds, giving rise to a series of remarkable phenomena. In 2016, Crossno et al. [1] reported a serious violation of the Wiedemann–Franz law in graphene Dirac fluids near the particle-hole symmetry point, where the thermal power significantly increases and approaches the hydrodynamic limit, especially at $T\sim 75K$. This violation of the Wiedemann–Franz law in graphene was unexpected and not predicted theoretically. To explain this anomaly, Pongsangangan et al. hypothesized in Ref. [2] that the opposite motion of electrons and holes in an electric field suppresses the electrical conductivity due to friction, while the same direction motion in a temperature field leads to infinite thermal conductivity. This suggests a significant difference between the Dirac fluid phenomenon and the Fermi liquid theory.

Nowadays, the hydrodynamic methods promoted by holography have been widely studied [3,4,5]. Yunseok Seo et al. [6] proposed a holographic model of a strongly coupled plasma with two U (1) gauge fields, where one represents electrons and the other represents holes. This the model was used to explain that the existence of two types of currents in graphene may be due to the imbalance between electrons and holes in a linear spectral system, and they found that the transport coefficients in the two-current model match experimental data better than those in the single current model. Subsequently, the holographic two-current model received more attention, as seen in Refs. [7,8,9,10,11,12,13,14,15,16,17].

Afterwards, the two-current model further considered the coupling model of two conserved U (1) currents and found that this model can be applied to the superconducting model. For example, the instability of superconductivity was studied in [17], and a new type of holographic insulator and holographic superconductor was constructed in [15, 16]. Moreover, In [18, 19] not only the interaction between two currents is considered, but also the thermoelectric transport behavior in the vertical magnetic field is studied. In addition, the calculations in [20] indicate that the conductivity is bounded.

Therefore, we will use a coupled model with axion fields and two U(1) gauge fields to investigate the interaction between electrons and holes under varying temperatures and electric fields. We aim to explore to what extent the two-current axion model can simulate experimental results and subsequently investigate the impact of the parameters $\beta$ and $\gamma$ on the electrothermal conductivity. It is important to note that the geometric shape of the system significantly influences the outcomes, requiring careful analysis when comparing with experimental data on graphene.

The paper is organised as follows: In Sect. 2 the holographic two-current model is introduced, and in Sect. 3 the thermoelectric transport properties of the system are analysed and calculated. In Sect. 4, the total electrical and thermal conductivity are discussed. In Sect. 5, the two-current model is compared with experimental data. In Sect. 6, we explore the impact of parameters on the transport properties of graphene. Finally, Sect. 7 summarizes and discusses the current research results and outlines the future research potential in the field of thermal and electrical transport properties of graphene Dirac fluids.

2 Holographic model

After the previous work on the holographic model with gauge-axion coupling [21,22,23,24,25], we consider the following two-current model in 4-dimensional spacetime with a gauge-axion coupling term between the two U(1) gauge fields, denoted by $A_\mu ,\,B_\mu$ respectively, and the axion fields, denoted by $\phi ^I,I=x,y$:

$$\begin{aligned} S{} & {} =\int d^4x\sqrt{-g}(L_0+L_{c}), \end{aligned}$$

(1a)

$$\begin{aligned} L_0{} & {} =R-2\Lambda -\frac{Z_1}{4}F^2-\frac{Z_2}{4}G^2-V(\textrm{Tr}[X]),\end{aligned}$$

(1b)

$$\begin{aligned} L_c{} & {} =\beta _{1} Tr[XF^2]+\beta _{2}Tr[XG^2]+2 \gamma Tr[XFG]. \end{aligned}$$

(1c)

Here, $S_{0}$ is the action of the usual two-current theory, with $\Lambda =-3$ the cosmological constant, $F=dA$ and $G=dB$ the strength of the two gauge fields respectively and $Z_{1}$, $Z_{2}$ are both constants. The axion fields $\phi ^{I}$ are added by a derivative term

$$\begin{aligned} X_{\mu \nu }=\frac{1}{2}\partial _{\mu }\phi ^I\partial _{\nu }\phi ^I \end{aligned}$$

(2)

where $\textrm{Tr}[X]=\frac{1}{2}\partial _{\mu }\phi ^I\partial ^{\mu }\phi ^I$ is the trace of the matrix $X_{\mu \nu }$ and $V(\textrm{Tr}[X])$ means a general function of $\textrm{Tr}[X]$. The gauge-axion coupled term $S_c$ represents a minimum coupling between the two gauge fields $A_\mu ,\,B_\mu$ and the axion fields $\phi ^I$. Explicitly, the trace in $S_c$ means the trace of the matrix product:

$$\begin{aligned} \textrm{Tr}[XF^2]{} & {} =X_{\mu \nu }F^{\nu \alpha }F_\alpha ^{~~\mu }\nonumber \\ \textrm{Tr}[XG^2]{} & {} =X_{\mu \nu }G^{\nu \alpha }G_\alpha ^{~~\mu },\nonumber \\ \textrm{Tr}[XFG]{} & {} =X_{\mu \nu }F^{\nu \alpha }G_\alpha ^{~~\mu }, \end{aligned}$$

(3)

and $\beta _{1},\beta _{2},\gamma$ are coupling constants. The equations of motion for the theory Eq. (1a) are given by Eq. (4a).

$$\begin{aligned}{} & {} R_{\mu \nu }-\frac{R}{2}g_{\mu \nu }+\Bigg (\Lambda +\frac{V}{2}\Bigg )g_{\mu \nu }-V_XX_{\mu \nu }\nonumber \\{} & {} \quad -\frac{Z_1}{2}\Bigg (F_{\mu \alpha }F_\nu ^{~~\alpha }-\frac{F^2}{4}g_{\mu \nu }\Bigg )-\frac{Z_2}{2}(G_{\mu \alpha }G_\nu ^{~~\alpha }-\frac{G^2}{4}g_{\mu \nu })\nonumber \\{} & {} \quad +\frac{\beta _1}{4}\Bigg ((XFF+FFX+FXF)_{\mu \nu }+\frac{\textrm{Tr}(XF^2)}{2}g_{\mu \nu }\Bigg )\nonumber \\{} & {} \quad +\frac{\beta _2}{4}\Bigg ((XGG+GGX+GXG)_{\mu \nu }+\frac{\textrm{Tr}(XG^2)}{2}g_{\mu \nu }\Bigg )\nonumber \\{} & {} \quad +\frac{\gamma }{4}((XFG+XGF+FXG+FGX+GXF+GFX)_{\mu \nu }\nonumber \\{} & {} \quad +\textrm{Tr}(XFG)g_{\mu \nu })=0, \end{aligned}$$

(4a)

$$\begin{aligned}{} & {} \nabla _\mu ( Z_1F^{\mu \nu }+\frac{\beta _1}{2}(XF+FX)^{\mu \nu }{+}\frac{\gamma }{2}(XG+GX)^{\mu \nu }){=}0,\end{aligned}$$

(4b)

$$\begin{aligned}{} & {} \nabla _\mu ( Z_2G^{\mu \nu }+\frac{\beta _2}{2}(XG+GX)^{\mu \nu }{+}\frac{\gamma }{2}(XF+FX)^{\mu \nu }){=}0,\end{aligned}$$

(4c)

$$\begin{aligned}{} & {} \nabla _\mu ((\beta _1F^{\mu \alpha }F_\alpha ^{~~\beta }+\beta _2G^{\mu \alpha }G_\alpha ^{~~\beta }+\gamma F^{\mu \alpha }G_\alpha ^{~~\beta } \nonumber \\{} & {} \quad +\gamma G^{\mu \alpha }F_\alpha ^{~~\beta })\partial _\beta \phi ^I-4V_X\nabla ^\mu \phi ^I)=0. \end{aligned}$$

(4d)

where we write, for instance,$(XFG)_{\mu \nu }=X_{\mu \alpha }F^{\alpha \beta }G_{\beta \nu }$ for short. We take the ansatz as

$$\begin{aligned}&ds^2=\frac{1}{u^2}\left( -f(u)dt^2+\frac{1}{f(u)}du^2+dx^2+dy^2\right) \end{aligned}$$

(5)

$$\begin{aligned}&A=A_{t}(u)dt\,,\qquad B=B_{t}(u)dt\,,\qquad \phi ^I=k \delta ^I_i x^i\,. \end{aligned}$$

(6)

with k the strength constant of the axion fields. The backgrounding solution is as follows:

$$\begin{aligned} A_t{} & {} =\mu _A-\frac{\rho _A}{Z_1}u,\qquad B_t=\mu _B-\frac{\rho _B}{Z_2}u, \end{aligned}$$

(7a)

$$\begin{aligned} f(u){} & {} =1-\left( \frac{u}{u_h}\right) ^3\Bigg (1+\frac{Z_1\mu _A^2u_h^2+Z_2\mu _B^2u_h^2}{4}\Bigg )\nonumber \\{} & {} \quad +u^{3}\Bigg (\int _{u_h}^u \frac{V(k^2 x^2)}{2 x^4}dx\Bigg )+\left( \frac{Z_1\mu _A^2+Z_2\mu _B^2}{4u_h^2}\right) u^4 \end{aligned}$$

(7b)

$$\begin{aligned} f'(u){} & {} =\frac{u^{3}(Z_{1}\mu _{A}^{2}+Z_{2}\mu _{B}^{2})}{4u_{h}^{2}}\!-\!\frac{3}{u}\!+\!\frac{3f(u)}{u}\!+\!\frac{V(k^{2}u^{2})}{2u}\nonumber \\ \end{aligned}$$

(7c)

$u_h$ stands for the horizon. $\mu _A,\mu _B$ are respectively the chemical potential of the gauge field $A_\mu$ and $B_\mu$,and the associated charge density $\rho _A$ and $\rho _B$ are given by

$$\begin{aligned} \rho _A=Z_1\mu _A/u_h,\qquad \rho _B=Z_2\mu _B/u_h . \end{aligned}$$

(8)

Finally, by applying the black hole temperature formula $T=|f'(u_{h})|/(4\pi )$ and Eq. (7c), and considering the boundary conditions $f(u_h) =0$ and $f(0)=1$, we can obtain the expression for the temperature:

$$\begin{aligned} T=\frac{1}{4\pi u_h}\Bigg (3-\frac{V(k^2u_h^2)}{2}-\frac{u_h^2(Z_{1}\mu _{A}^2+Z_{2}\mu _{B}^2)}{4}\bigg ), \end{aligned}$$

(9)

Furthermore, the area A of a black hole is proportional to the entropy S, given by $S=4\pi A$. Here, A denotes the area of a two-dimensional hypersurface at any time, when the event horizon is defined as $u=u_{h}$, and is calculated as $A=\int \sqrt{h}dxdy$, where $h=1/u_{h}^{2}$ represents the metric determinant of the two-dimensional hypersurface with the line element $ds_{2}^{2}=\frac{1}{u_{h}^{2}}(dx^{2}+dy^{2})$. At the same time, $\int dxdy=V$ corresponds to the volume of the dual boundary system. Thus, both the entropy S and the entropy density s can be expressed as:

$$\begin{aligned} \qquad S=\frac{4\pi V}{u_{h}^{2}} \qquad s=\frac{S}{V}=\frac{4\pi }{u_{h}^{2}} \end{aligned}$$

(10)

We should point out that the coupling term $L_c$ does not affect the background solution, which is different from the previously considered coupling term $F_{\mu \nu }G^{\mu \nu }$. Since we attempt to simulate the frictional interaction between electrons and holes that only manifests during transport, and a coupling method that does not affect the background solution seems more reasonable.

3 Thermoelectric transport properties

In this section, we shall calculate the DC thermoelectric conductivities of the dual system, following “membrane paradigm” procedure proposed in [26] (for more details, see [9, 27,28,29,30]). We apply around the background a constant electric field $E_{x}$ and $e_{x}$, associated with the gauge field $A_\mu$ and $B_\mu$ respectively, and a constant temperature gradient $\zeta =-\frac{\nabla T}{T}$. They will generate corresponding charge currents $J_{A}^{x}$ and $J_{B}^{x}$and a heat current $Q^x$. The thermoelectric transport properties in the dual field theory is captured by the generalized Ohm’s law.

$$\begin{aligned} \left( \begin{array}{c} J_{A}^{x} \\ J_{B}^{x}\\ Q^{x} \end{array} \right) = \left( \begin{array}{ccc} \sigma _{AA}&{} \sigma _{AB} &{} \alpha _{A} T\\ \sigma _{BA}&{} \sigma _{BB} &{} \alpha _{B} T\\ \bar{\alpha _{A}}T &{}\bar{\alpha _{B}}T &{} \bar{\kappa }T \end{array} \right) \left( \begin{array}{c} E_{x}\\ e_{x}\\ \zeta \end{array} \right) \end{aligned}$$

(11)

Due to the rotational invariance on the $x-y$ plane, we only consider the transport behavior along the x direction. In order to find the conductivities for the background in question, one takes into account small perturbations around the background solution obtained from Einstein equations of motion. The perturbations imply

$$\begin{aligned}{} & {} \delta A_{x}=(-E_{x}+A_{t}(u)\zeta _{x})t+a_{x}(u) , \end{aligned}$$

(12a)

$$\begin{aligned}{} & {} \delta B_{x}=(-e_{x}+B_{t}(u)\zeta _{x})t+b_{x}(u) ,\end{aligned}$$

(12b)

$$\begin{aligned}{} & {} \delta g_{tx}=\frac{1}{u^2}(-\zeta _{x}f(u)t+h_{tx}(u)),\end{aligned}$$

(12c)

$$\begin{aligned}{} & {} \delta g_{ux}=\frac{1}{u^2}h_{ux}(u),\end{aligned}$$

(12d)

$$\begin{aligned}{} & {} \delta \phi _{x}=\chi (u). \end{aligned}$$

(12e)

The key point of the “membrane paradigm” method is to construct a radially conserved current connecting the horizon and the boundary [28, 31, 32]. This is equivalent to the dual-system current, which is in this model

$$\begin{aligned} J_{A}^{v}{} & {} =\sqrt{-g}(Z_{1}F^{\mu \nu }+\frac{\beta _1}{2}(X_{\alpha }^{~~\mu }F^{\alpha \nu }-X_{\beta }^{~~\nu }F^{\beta \mu })\nonumber \\{} & {} \quad +\frac{\gamma }{2}(X^{\mu }_{~~\sigma }G^{\sigma \nu }-X^{\nu }_{~~\beta }G^{\beta \mu }))=0 \end{aligned}$$

(13)

$$\begin{aligned} J_{B}^{v}{} & {} =\sqrt{-g}(Z_{2}G^{\mu \nu }+\frac{\beta _2}{2}(X_{\sigma }^{~~\mu }G^{\sigma \nu }-X_{\alpha }^{~~\nu }G^{\alpha \mu })\nonumber \\{} & {} \quad +\frac{\gamma }{2}(X_{\alpha }^{~~\mu }F^{\alpha \nu }-X_{\rho }^{~~\nu }F^{\rho \mu }))=0\end{aligned}$$

(14)

$$\begin{aligned} Q^{x}{} & {} =-2\sqrt{-g}\nabla ^{u}k^{x}-A_tJ_{A^{x}}-B_tJ_{B^{x}} \end{aligned}$$

(15)

Subsequently, by employing the expressions in Eq. (11) and Eq. (12), we can work out the DC conductivities. After the calculation (see Appendix B), we obtain the results:

$$\begin{aligned} \sigma _{AA}{} & {} =\frac{J_{A}^{x}(u_h)}{E_{x}}=\frac{\beta _{1}}{4}k^2u_h^2+Z_{1}+\frac{1}{4k^2}\frac{\pi _{A}^2}{\Sigma },\end{aligned}$$

(16a)

$$\begin{aligned} \sigma _{BB}{} & {} =\frac{J_{B}^x(u_h)}{e_{x}}=\frac{\beta _{2}}{4}k^2{u_h}^2+Z_{2}+\frac{1}{4k^2}\frac{\pi _{B}^2}{\Sigma },\end{aligned}$$

(16b)

$$\begin{aligned} \sigma _{AB}{} & {} =\sigma _{BA}=\frac{1}{4}k^2u_h^2\gamma +\frac{1}{4k^2}\frac{\pi _{A}\pi _{B}}{\Sigma },\end{aligned}$$

(16c)

$$\begin{aligned} T\bar{\kappa }{} & {} =\frac{Q^{x}(u_h)}{\zeta _{x}}=\frac{4f'(u_h)^2}{k^2u_h^2\Sigma },\end{aligned}$$

(16d)

$$\begin{aligned} T\alpha _{A}{} & {} =\frac{J_{A}^{x}(u_h)}{\zeta _{x}}=-\frac{\pi _{A}f'(u_h)}{k^2u_h\Sigma },\end{aligned}$$

(16e)

$$\begin{aligned} T\alpha _{B}{} & {} =\frac{J_{B}^{x}(u_h)}{\zeta _{x}}=-\frac{\pi _{B}f'(u_h)}{k^2u_h\Sigma }, \end{aligned}$$

(16f)

The Onsager relation holds $\alpha _{i}=\bar{\alpha _{i}}$. For convenience, we define the following physical quantities:

$$\begin{aligned}{} & {} \Sigma =4V'(k^2u_h^2)\!-u_h^2(\!\beta _1\mu _A^2\!+\beta _2\mu _B^2\!+2\gamma \mu _A\mu _B) ,\end{aligned}$$

(17a)

$$\begin{aligned}{} & {} \pi _{A}=(k^2u_h^2(\beta _1\mu _A+\gamma \mu _B)+4Z_1\mu _A),\end{aligned}$$

(17b)

$$\begin{aligned}{} & {} \pi _{B}=(k^2u_h^2(\beta _2\mu _B+\gamma \mu _A)+4Z_2\mu _B). \end{aligned}$$

(17c)

4 Discussion on total electric thermal conductivity

Transport coefficients of graphene have been experimentally measured and theoretically analyzed in various works, as reviewed in [1, 4, 33,34,35,36]. Additionally, many papers have employed holographic methods, as seen in [18,19,20, 28, 37]. In a recent paper by Seo et al. [6], thermal conductivity was measured and analyzed using a two-current model with holographic methods. The two currents represent the electron and hole currents in the system, with the Fermi energy tuned to coincide with the Dirac point. Notably, our action in Eq. (1a) accounts for the impact of the interaction between them on the thermal transport of graphene near the CNP.

One possible intuitive explanation for the violation of the Wiedemann–Franz law at the charge-neutral point in graphene at 75 K is discussed in Ref. [38]. When electrons and holes move in opposite directions in an electric field, their frictional interaction leads to a finite conductivity. On the other hand, when a temperature gradient is applied, electrons and holes move in the same direction, implying no friction and therefore infinite thermal conductivity in Fig. 1.

Fig. 1

The flow of electrons (red) vs. holes (blue) in the charge neutral Dirac fluid. Both electrons and holes move in the same direction in a temperature gradient, but move in opposite directions in an electric field [38]

To achieve an application and interpretation of Dirac fluids, we shall first define the total electrical current, net charge density, and the corresponding conductivities that can connect directly with experimental quantities. Using the electron–hole picture of Dirac fluids, while gauge field $A_\mu$ and $B_\mu$ will be explained as the electromagnetic field from the electrons and holes respectively. Then $J_A^x,J_B^x$ represent the electron and hole, the charge densities and number densities are related by $\rho _A=-en_{A},\rho _B=en_{B}$. In addition, the electric field $E_x$ and $e_x$ should be identified as $E_x=e_x$, for there is only one voltage driving both the electron and hole current in practice. We thus define the total electrical current $J_{tot}^{x}$, total charge density Q by

$$\begin{aligned} J_{tot}^x=J^x_A+J^x_B,\qquad Q=\rho _A+\rho _B, \end{aligned}$$

(18)

Using the Ohm’s law Eq. (11) with $E_x=e_x$, the observed currents $J_{tot}^{x}$ and heat current $Q_{tot}^{x}$ in Dirac fluids are given by

$$\begin{aligned} J_{tot}^{x}= & {} J_{A}^{x}+J_{B}^{x}\nonumber \\ {}= & {} (\sigma _{AA}+\sigma _{BB}+\!2\sigma _{AB})E_{x}+(\alpha _{A}+\alpha _{B})T\zeta _{x}, \end{aligned}$$

(19a)

$$\begin{aligned} Q_{tot}^{x}= & {} (\alpha _{A}+\alpha _{B})TE_{x}+\bar{\kappa }T\zeta _{x}, \end{aligned}$$

(19b)

and the experimental electrical conductivity $\sigma$ and thermal conductivity $\kappa$ can be extracted from Eq. (19) as follows:

$$\begin{aligned}{} & {} \sigma =\frac{J_{tot}^{x}}{E_x}\Big |_{\zeta _{x}=0}=\sigma _{AA}+\sigma _{BB}+2\sigma _{AB},\end{aligned}$$

(20a)

$$\begin{aligned}{} & {} \kappa =\frac{Q_{tot}^{x}}{\zeta _x}\Big |_{J=0}=\bar{\kappa }-\frac{(\alpha _{A}+\alpha _{B})^2T}{\sigma _{AA}+\sigma _{BB}+2\sigma _{AB}}. \end{aligned}$$

(20b)

The underlying theory behind the thermal conductivity expression is based on Ref. [38], which suggests that both electrons and holes are driven by the same electric field. Additionally, The total charge density is $Q=\rho _{A}+\rho _{B}$, and the net charge density is $Q_{n}=\rho _{A}-\rho _{B}$. There exists a proportional relationship between the total charge density and net charge density, which can be expressed as follows:

$$\begin{aligned} Q_{n}=g_{n}Q. \end{aligned}$$

(21)

Building on the above,we assume $\beta _1=\beta _2=\beta \ne \gamma ,Z_1=Z_2=Z$ is substituted into the action. In this case, the expression for electrical conductivity can be obtained through Ohm’s law and the ingoing boundary condition:

$$\begin{aligned}{} & {} \sigma _{0}\!=2Z+\frac{k^2}{2}u_h^2(\beta +\gamma ),\end{aligned}$$

(22a)

$$\begin{aligned}{} & {} \sigma \!=\sigma _{0}\Bigg (1-\frac{Q^2u_h^2(\!4Z\!+\! k^{2}u_h^{2}(\!\beta +\!\gamma \!)) }{k^2(u_h^4Q^2(\beta (1\!+\!g_n^2)+\!\gamma (\!1-\!g_n^2\!)\!)\!-\!8Z^2V')}\!\Bigg ),\end{aligned}$$

(22b)

$$\begin{aligned}{} & {} \bar{\kappa }\!=-\frac{128\pi ^2TZ^2}{k^2u_h^2(Q^2u_h^4(\beta (1+\!g_n^2)+\gamma (1-g_n^2)) -8Z^2V')},\end{aligned}$$

(22c)

$$\begin{aligned}{} & {} \kappa \!=\!-\frac{128\pi ^2T Z^{2}}{Q^2u_h^4(g_n^2k^2u_h^2(\beta -\!\gamma )-4Z)-\!8k^2u_h^2Z^2V'},\nonumber \\ \end{aligned}$$

(22d)

Further simplification can be achieved by replacing $s=4\pi /{u_h^2}$:

$$\begin{aligned}{} & {} \sigma _{0}=2(Z+\frac{k^2\pi (\beta +\gamma )}{s}),\end{aligned}$$

(23)

$$\begin{aligned}{} & {} \sigma =\!\sigma _{0}\Bigg (1-\!\frac{\pi Q^2 s \sigma _{0}}{k^2(2\pi ^2Q^2(\beta (\!1+\!g_n^2)+\!\gamma (\!1-\!g_n^2))-\!s^2Z^2}\!\Bigg ),\end{aligned}$$

(24)

$$\begin{aligned}{} & {} \bar{\kappa }=\!-\frac{4\pi s^3T Z^2}{k^2(2(\!1+\!g_n^2)\beta \pi ^2Q^2\!-\!s^2Z^2-\!2(g_n^2-\!1)\pi ^2Q^2\gamma )},\end{aligned}$$

(25)

$$\begin{aligned}{} & {} \kappa =-\frac{4\pi s^3 T Z^2}{2\pi Q^2(g_n^2k^2\pi (\beta -\gamma )-sZ)-k^2s^2Z^2}. \end{aligned}$$

(26)

5 Comparing with experiments

Previous to this, we obtained the expression for the total thermal conductivity of graphene Dirac fluid. In this section, we will mainly explore the anomalous behavior of graphene Dirac fluid at 75K, which manifests as a significant increase in thermal conductivity near the charge neutrality point and a serious violation of the Wiedemann–Franz law.

Before comparing with experiments, it was found that entropy density s not only depends on T but also varies with n. Since the transport coefficients of graphene are closely related to s, caution is required in the selection and handling of entropy density. As shown in Fig. 2, larger values of $g_{n}$ result in a more apparent change of entropy density concerning number density. To simplify the analysis, we select the case $v_{F}=6*10^6$ m/s, $g_{n}=2$, where the change in entropy density is relatively small. In this case, we approximate the entropy density as a constant, denoted by $s=3538 \,\, \upmu$m$^{-2}$.

Fig. 2

a Entropy density s as a function of temperature T at the charge neutrality point $(n=0)$. b Entropy density s as a function of charge density n at $T=75K$. The plot shows the regulatory effect of charge fitting value g on s, and as g increases, the rate of change of s with n becomes faster

By observing the experimental data [1], we obtained $\sigma$ and $\kappa$ values for CNP at $T=75K$:

$$\begin{aligned}{} & {} \kappa _{0}=\frac{4\pi sT k_{B}^2}{k^2\hbar }=7.7 \,\, {\text {nW/k}}, \end{aligned}$$

(27a)

$$\begin{aligned}{} & {} \sigma _{0}=2\frac{e^2}{\hbar }\Bigg (Z+\frac{k^2\pi (\beta +\gamma )}{s}\Bigg )=0.338\,\, {\text {k}}\Omega ^{-1}. \end{aligned}$$

(27b)

In parameter selection, Z is set to 1 based on [21,22,23, 25, 29, 37, 39, 40] practices for consistency and comparability. Subsequently, $k^2$ is determined as $1000\mu m^{-2}$ using the temperature expression Eq. (A.5), and $s=3538\mu m^{-2}$. These key parameters reveal a constant difference between $\beta$ and $\gamma$ at a fixed temperature. Solving the conductivity equation yields $\sigma _{0}=0.338 k\Omega ^{-1}= 2e^2(1+\pi k^2(\beta +\gamma )/s)/\hbar$, resulting in $\beta +\gamma =-0.342$. In the thermal conductivity fitting process, it is established that $\gamma \ge 0$. With other parameters fixed, careful adjustments to $\gamma$ and $\beta$ led to the optimal fit: $\gamma =8$ and $\beta =-8.342$. This systematic approach ensures rational and reliable parameter selection for a more accurate fitting of experimental observations.

In Appendix C, we will work in units where $\hbar =k_{B}=V_{F}=e=1$. It is straightforward to restore these units with dimensional analysis. Following the previous settings, the relationship between the electrical and thermal conductivity and the number density is obtained as follows:

$$\begin{aligned}{} & {} \kappa (n)\!=\frac{\kappa _{0}}{1-\frac{2\pi (g_n^2k^2\pi (\beta -\gamma )-sZ)n^2}{k^2s^2Z^2}},\end{aligned}$$

(28a)

$$\begin{aligned}{} & {} \sigma (n)\!=\sigma _{0}\Bigg (\!1-\!\frac{n^2}{\frac{2\pi n^2k^2}{s\sigma _{0}}(\beta (\!1+\!g_n^2)+\!\gamma (\!1-\!g_n^2))\!-\!\frac{k^2 s Z^2}{\pi \sigma _{0}}}\Bigg ).\nonumber \\ \end{aligned}$$

(28b)

Fig. 3

Theory vs. data: a the measured conductivity $\sigma$ as functions of number density n. b The measured thermal conductivity $\kappa$ as functions of charge density n. Blue lines is the analytic Eq. (28) and the red dot are for experimentally data used in [1, 5]

Our model demonstrates strong agreement with experimental data across the range of $-100\, \upmu$m$^{-2} \le n \le 100\, \upmu$m$^{-2}$, as shown in Fig. 3a. In the absence of disorder, conductivity remains finite due to the establishment of electron–hole equilibrium at specific temperatures. This equilibrium nullifies the total momentum induced by an electric field, obviating the need for contaminants to relax momentum. Instead, electrons and holes are bound together through the drag mechanism [2]. As shown in Eq. (27), it is evident that $\beta +\gamma$ exclusively influences the electrical conductivity $\sigma _{0}$, while the thermal conductivity $\kappa _{0}$ is solely dependent on entropy density and temperature. We hypothesize that at the charge neutrality point (CNP), $\beta$ and $\gamma$ exhibit behavior reminiscent of the Drag effect.

5.1 Discussion on $\sigma (T)$and $\kappa (T)$

In this section, we will concentrate on examining the characteristics of thermoelectric transport coefficients at various temperatures. Through experimental observations, it is noted that, at zero charge density, the conductivity $\sigma _{0}$ exhibits a monotonic increase with temperature. This observed behavior aligns with our intuitive understanding of electrical conductivity in everyday scenarios.

Subsequently, substituting s(T) into the expressions for the conductivity $\sigma _{0}(T)$ and thermal conductivity $\kappa _{0}(T)$ at the neutrality point.

$$\begin{aligned}{} & {} \sigma _{0}(T,n)|_{n=0}=2\left( Z+\frac{k^2\pi (\beta +\gamma )}{s(T)}\right) , \end{aligned}$$

(29a)

$$\begin{aligned}{} & {} \kappa _{0}(T,n)|_{n=0}=\frac{4\pi s(T) T}{k^2}. \end{aligned}$$

(29b)

Fig. 4

Theory vs. data: a The measured conductivity $\sigma _{0}$ as functions of temperature T at the charge neutrality point $(n=0)$. b The measured thermal conductivity $\kappa _{0}$ as functions of temperature T at the charge neutrality point $(n=0)$. Blue lines is the analytic Eq. (29) and the red dot are for experimentally data used in [1, 5]

Our theory not only provides quantitative fits for $\kappa _{0}(T)$ and $\sigma _{0}(T)$ but also faithfully reproduces existing research results as much as possible, which is shown in Fig. 4. Particularly noteworthy is that our theory captures the phenomenon of increasing $\sigma _{0}(T)$ with rising temperature, offering valuable insights into the momentum relaxation mechanism.

In comparison to the work by Yunseok Seo et al. [6, 41, 42], $\sigma _{0}(T)$ is no longer a constant; our model based on Eq. (4a) results in an additional term $\pi k^2(\beta +\gamma )/s(T)$. The inclusion of this term enhances the precision of our simulation results, providing a more accurate reflection of the trend in electrical conductivity with increasing temperature. However, the correspondence between $\kappa _{0}(T)$ and experimental results is not ideal, and we speculate that this discrepancy may be due to the temperature-dependent relationship of entropy density.

5.2 The Wiedemann–Franz ratio

Fermi liquid has a noteworthy property [43]: the Lorenz ratio of thermal conductivity $\kappa$ to electric conductivity $\sigma$ remains constant at low temperatures. This property is known as the Wiedemann–Franz (WF) law [44]:

$$\begin{aligned} L=\frac{\kappa }{\sigma T}=\frac{\pi ^2}{3}\frac{(k_{B})^2}{e^2}=L_{0} \end{aligned}$$

(30)

However, the WF law is fragile. This law has been found to be broken in the strongly interacting non-Fermi liquids [45, 46]. Because of the inelastic scattering between charged and neutral degrees of freedom, it can be ascribed to heat and charge transport in different ways [45].

For the charge neutral point region of graphene’s Dirac fluid, it has been experimentally observed that the WF law is strongly violated at around 75K. By substituting Eq. (28) into Eq. (30), we obtained expressions for the Lorenz number L(n) based on the transport coefficient of our model.

$$\begin{aligned}{} & {} L(n)|_{T=75K}=\frac{\kappa _{0}(T)}{(1-\frac{2\pi (g_n^2k^2\pi (Z-\gamma )-s(T)w)n^2}{k^2s(T)^2w^2})\Big (\sigma _{0}(T)(1-\frac{n^2}{\frac{2\pi n^2k^2}{s(T)\sigma _{0}(T)}(Z(1+g_n^2)+\gamma (1-g_n^2))-\frac{k^2 s(T) w^2}{\pi \sigma _{0}(T)}})\Big )T} \end{aligned}$$

(31)

Fig. 5

a The measured Lorenz ratio L/L0 at T = 60 K as function of number density n. The red solid circle in the figure represents the experimental data results at 60K [1], the blue solid line is the image taken by our model at 60 K. b The measured Lorenz ratio as function of temperatures T, and the Lorentz ratio L/L0 here is represented by substituting s(T) into the Eq. (31)

According to Fig. 5a, we can see a significant peak of the Lorentz ratio L/L0 near the charge neutral point, while it approaches zero in other regions. This indicates that the L/L0 in graphene is no longer constant, accurately reflecting changes in the Dirac fluid state near the neutral point. Our model has effectively captured this characteristic. To put it simply, the unusual behavior of graphene near the neutral point is due to the uniform movement of electrons and holes in the same direction in the temperature field, without any frictional effects or loss of momentum. The lack of frictional effects and momentum loss significantly enhances the thermal conductivity of graphene, and the impact on its conductivity can be almost negligible.

At high temperatures (above 100K), experimental data shows that the heat transfer from electrons to phonons is equivalent to the degree of electron diffusion [47,48,49]. However, this electron–phonon scattering leads to the degradation of Dirac fluids, resulting in a decrease in the Lorentz number L at a temperature of approximately 100K. Our theory has shortcomings at this point, partly because our system did not take into account the scattering of phonons and electrons. On the other hand, our theoretical entropy density is linearly related to temperature, as shown in Fig. 2.

Fig. 6

a  Measured thermoelectric coefficient $\alpha$ as a function of density n for various temperatures. As we increase T, the maximum value of $\alpha$ shifts towards high concentration areas. b   The measured thermoelectric coefficient $\alpha$ as a function of temperatures T for various density n. As we increase number density n, the peak value of $\alpha$ will shift towards the high temperature region

Fig. 7

a Measured Seebeck coefficient S as a function of density n for various temperatures. As we increase T, the maximum value of S shifts and increases towards high concentration areas. b The measured Seebeck coefficient S as a function of temperatures for various density n. As we increase number density n, the rate of change of S with temperature decreases, but it still maintains an upward trend

5.3 Calculating the Seeback coefficients

This section will explore thermoelectric coefficient $\alpha$ properties of graphene and the Seebeck coefficient S. The distinctive feature of the Seebeck coefficient is its zero value at the neutral point, as electrons and holes of equal concentration move in the same direction in the temperature gradient field. Consequently, the current $J=0$, leading to $\alpha =0$.

Simplifying according to Eq. (16), we obtain the thermoelectric coefficients $\alpha _{A}$ and $\alpha _{B}$ separately:

$$\begin{aligned}&T\alpha _{A}=\frac{2\pi Q s T Z((1+g_{n})\beta k^2\pi +(1+g_{n})s Z+(1-g_{n})k^2\pi \gamma ))}{k^2(2(1+g_{n}^2)\beta \pi ^2Q^2-s^2Z^2+2(1-g_{n}^2)\pi ^2Q^2\gamma )}\nonumber \\&T\alpha _{B}=\frac{2\pi Q sTZ((1-g_{n})\beta k^2\pi +(1-g_{n})sZ+(1+g_{n})k^2\pi \gamma )}{k^2(2(1+g_{n}^2)\beta \pi ^2Q^2-s^2Z^2+2(1-g_{n}^2)\pi ^2Q^2\gamma )} \end{aligned}$$

(32)

The definition for the total thermoelectric coefficient $\alpha$ and Seebeck coefficient S are given as:

$$\begin{aligned} \alpha =\alpha _{A}+\alpha _{B}\qquad S=\frac{\alpha }{\sigma } \end{aligned}$$

(33)

Using Eqs. (33) and (27), the expressions for the $\alpha$ and S are derived as follows:

$$\begin{aligned}{} & {} \alpha =\frac{k_{B}|e|}{\hbar }\frac{4\pi s Q Z( \pi k^2 (\beta +\gamma )+sZ)}{k^2\Theta } \end{aligned}$$

(34)

$$\begin{aligned}{} & {} S=\frac{k_{B}}{|e|}\frac{\kappa _{0}Z Q}{2T\Theta (1-\frac{\pi Q^2 s\sigma _{0}}{k^2\Theta })}\end{aligned}$$

(35)

$$\begin{aligned}{} & {} \Theta =2\pi ^{2}Q^{2}(\beta (1+g_{n}^{2})+\gamma (1-g_{n}^2))-s^{2}Z^{2} \end{aligned}$$

(36)

Figures 6 and 7 show the relationship between the thermoelectric coefficient and the Seebeck coefficient as a function of temperature and carrier density. As the system crosses the Fermi liquid region at high carrier densities, the enhancement decreases as disorder and scattering increase

Fig. 8

a DC electric conductivity $\sigma$ as a function of n with different values of $\gamma$ and $\beta$ at $T=75K$. b DC thermal conductivity $\kappa$ as the function of n with different $\gamma$ and $\beta$ at $T=75K$. Due to the relationship between $\beta + \gamma$ and Z, it is necessary to ensure that their sum remains constant even when increasing $\beta$ and $\gamma$

Furthermore, near the neutrality point, the Seebeck coefficient increases with temperature. We also simulate the conductivity behavior with temperature in different carrier density regimes, showing that the rate of change of Seebeck coefficient with temperature is smaller near charge neutrality [35]. These results provide a valuable direction for future in-depth studies, as the variation of the Seebeck coefficient in graphene is influenced by multiple factors, including the transport properties of electrons and holes, the position of the Fermi level, and the temperature gradient.

6 Parameter impact on transport coefficients

In this section we discuss the influence of the parameter $\beta +\gamma$ on $\sigma (n,T)$ and $\kappa (n,T)$. Under isothermal conditions, the electrical conductivity maintains a minimum near CNP. However, at higher densities, the increase of $\beta$ and $\gamma$ further suppresses $\sigma (n,T)$ and diminishes the characteristic features of graphene conductivity near the CNP, as shown in Fig. 8a. This indicates a significant influence of $\beta$ and $\gamma$ parameters on the electrical conductivity under high density conditions.

According to $\sigma _{0}=0$ and $\kappa =\kappa _{0}$, the following relationships among $\beta$, $\gamma$, and Z can be obtained:

$$\begin{aligned} \beta -\gamma =\frac{sZ}{g_{n}^2k^2\pi }\qquad \beta +\gamma =-\frac{sZ}{k^2\pi } \end{aligned}$$

(37)

At the same time, $\kappa (n,T)$ is also affected by these parameters, although to a lesser extent. While our model does not explicitly delineate the relative influence of $\beta$ and $\gamma$ on electrothermal transport, it reveals the potential for opposite motion of electrons and holes in an electric field. This hypothesis offers an intriguing perspective for interpreting the electrothermal transport properties of graphene, although more in-depth investigation is required to quantify the specific degree of influence.

7 Summary and conclusion

This paper firstly applies the holographic two-current-axion coupling model to investigate the transport properties of graphene Dirac fluids. Compared with previous study in Refs. [6, 19, 41], the interaction between eletrons and holes is considered. with the interaction term introduced in Eq. (1c), we found that $\sigma _{0}(n=0,T)$ is no longer a constant. Instead, it now includes an additional term $k^2\pi (\beta +\gamma )/s(T)$. we further consider the relationship between entropy density s, temperature T, and number density n, and emphasize that entropy density is not an independent free parameter as done in Ref. [41]. Due to the introduction of the extra term and the consideration of s(T) as a function of T at the CNP, $\sigma _{0}(T)$ performs better in fitting the experimental data, compared with previous results in Refs. [2, 5, 6, 41], and successfully reflects the fact that the conductivity $\sigma _{0}(n,T)$ with the temperature is approximately a linearly function of T, as shown in Fig. 4a. This confirms the rationality of our consideration of s(n, T).

On the other hand, this article considers the influence of coupling parameters $\beta$ and $\gamma$ on DC conductivities. We observed that as the coupling parameters increased, the conductivity $\sigma (n)$was suppressed faster in the high concentration region, while the thermal conductivity $\kappa (n)$ was less affected. This indicates that coupling parameters will suppress the scattering of electrons and holes under the influence of an exerting electric field (see Fig. 1). Unfortunately, we are currently unable to fully distinguish the proportion of $\beta$ and $\gamma$ in electron–hole interactions, which is also a direction that can be further explored in the future.

In addition, we have also investigated the Lorentz ratio $L/L_{0}$, the electric thermal coefficient $\alpha$, and the Seebeck coefficient S of the graphene Dirac fluid as a function of temperature. Our results are consistent with the hydrodynamic mechanics method [2, 5], providing important references for a better understanding of the transport properties in graphene Dirac fluids.

In summary, this study preliminarily reveals the influence of electron and phonon heat transfer on thermal conductivity during electron and hole scattering at high temperatures. Future research can explore more directions, such as exploring different sources of momentum relaxation, interactions in magnetic transport, and considering electron hole plasma situations in this model. In the future, we anticipate holographic theories to provide a more accurate description of graphene’s thermoelectric properties under high-temperature conditions.

Appendices

Appendix A: Discussing the relationship between s, T, and n

In this appendix, we will discuss the relationship between entropy density s, temperature T, and number density n. Previously, we obtained expressions for temperature Eq. (9) and entropy density Eq. (10). Since both T and s contain $u_h$, here we substitute $u_h = \sqrt{4\pi /s}$ into the temperature expression in Eq. (9). Utilizing the relationship between chemical potential and charge density, $\rho _A = Z_1\mu _A/u_h$ and $\rho _B = Z_2\mu _B/u_h$, where the total charge density is $Q = \rho _A + \rho _B$, and the net charge density is $Q_n = \rho _A - \rho _B$. We assume a relationship between total charge density and net charge density as $Q_n = g_n Q$, leading to simplification.

$$\begin{aligned} T=\frac{\sqrt{s}}{\left( 4\pi \right) ^{3/2}}\Bigg (3-\frac{2\pi \ k^2}{s}-\frac{2\pi ^2(1+g_n^2)n^2}{Z\ s^2}\Bigg ) \end{aligned}$$

(A.1)

After restoring the units according to the Appendix C,

$$\begin{aligned} T=\frac{\hbar v_F}{k_B}\frac{\sqrt{s}}{\left( 4\pi \right) ^{3/2}}\Bigg (3-\frac{2\pi \ k^2}{s}-\frac{2\pi ^2(1+g_n^2)n^2}{Z\ s^2}\Bigg ) \end{aligned}$$

(A.2)

From Eq. (A.2) can be seen that when the temperature is 75K, the entropy density is still related to the charge density. Near the charge-neutral point, because our entropy density value is very large, the last term can be ignored and the entropy density can be considered as a function only of temperature, where $v_F=6*10^6m/s,g_n=2,n=100(\mu m)^{-2}$, and other parameters are obtained according to the values in the previous section, as shown below.

$$\begin{aligned}{} & {} T=\frac{\hbar v_F}{k_B}\frac{\sqrt{s}}{(4\pi )^{3/2}}\Bigg (3-\frac{2\pi k^2}{s}-\frac{2\pi ^2(1+g_n^2)n^2}{Z s^2}\Bigg ) \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} s={3584}{(\mu m)^{-2}} \end{aligned}$$

(A.4)

At the charge-neutral point $n=0$:

$$\begin{aligned}{} & {} T=\frac{\hbar v_F}{k_B}\frac{\sqrt{s}}{(4\pi )^{3/2}}\Bigg (3-\frac{2\pi k^2}{s}\Bigg )\end{aligned}$$

(A.5)

$$\begin{aligned}{} & {} s={3538}{(\mu m)^{-2}} \end{aligned}$$

(A.6)

Comparing Eqs. (A.4) and (A.6), it was found that the difference between s at $n=100(\upmu {\text {m}})^{-2}$ and s at CNP is very small. Therefore, this article believes that entropy density can be regarded as a function solely dependent on temperature. Note that if this range is exceeded, entropy density cannot be considered a constant.

Appendix B: Derivation of DC conductivity

In this appendix, we illustrate the process of calculating DC conductivity using the approach outlined in [27]. The central concept of this method involves establishing a radially-conserved current linking the horizon and the boundary. As a result, the DC conductivity of the dual boundary system can be directly deduced from the data at the horizon.

The consistent perturbations around the background are given by

$$\begin{aligned}{} & {} \delta A_{x}=(-E_{x}+A_{t}(u)\zeta _{x})t+a_{x}(u) , \end{aligned}$$

(B.7a)

$$\begin{aligned}{} & {} \delta B_{x}=(-e_{x}+B_{t}(u)\zeta _{x})t+b_{x}(u) ,\end{aligned}$$

(B.7b)

$$\begin{aligned}{} & {} \delta g_{tx}=\frac{1}{u^2}(-\zeta _{x}f(u)t+h_{tx}(u)),\end{aligned}$$

(B.7c)

$$\begin{aligned}{} & {} \delta g_{ux}=\frac{1}{u^2}h_{ux}(u),\end{aligned}$$

(B.7d)

$$\begin{aligned}{} & {} \delta \phi _{x}=\chi (u). \end{aligned}$$

(B.7e)

From the Maxwell Eqs. (4c), (4d), and Einstein Eq. (4b), one can construct radially-conserved currents in the bulk, which have the following forms:

$$\begin{aligned} J_{A}^{v}{} & {} =\sqrt{-g}(Z_{1}F^{\mu \nu }+\frac{\beta _1}{2}(X_{\alpha }^{~~\mu }F^{\alpha \nu }-X_{\beta }^{~~\nu }F^{\beta \mu })\nonumber \\{} & {} \quad +\frac{\gamma }{2}(X^{\mu }_{~~\sigma }G^{\sigma \nu }-X^{\nu }_{~~\beta }G^{\beta \mu }))=0 \end{aligned}$$

(B.8)

$$\begin{aligned} J_{B}^{v}{} & {} =\sqrt{-g}(Z_{2}G^{\mu \nu }+\frac{\beta _2}{2}(X_{\sigma }^{~~\mu }G^{\sigma \nu }-X_{\alpha }^{~~\nu }G^{\alpha \mu })\nonumber \\{} & {} \quad +\frac{\gamma }{2}(X_{\alpha }^{~~\mu }F^{\alpha \nu }-X_{\rho }^{~~\nu }F^{\rho \mu }))=0\end{aligned}$$

(B.9)

$$\begin{aligned} Q^{x}{} & {} =-2\sqrt{-g}\nabla ^{u}k^{x}-A_tJ_{A^{x}}-B_tJ_{B^{x}} \end{aligned}$$

(B.10)

Furthermore, the explicit expressions for the electric and heat currents can be provided as follows:

$$\begin{aligned} J_{A}^{t}{} & {} =-Z_{1}A_{t}'(u)=\rho _{A}\quad J_{B}^{t}=-Z_{2}B_{t}'(u)=\rho _{B} \end{aligned}$$

(B.11a)

$$\begin{aligned} J_{A}^{x}{} & {} =\frac{1}{4}[h_{tx}(u)((4Z_1 +k^2 u^2 \beta _{1})A_{t}'(u)+k^2 u^2\gamma B_{t}'(u))\nonumber \\{} & {} \quad + f(u)((4Z_{1}+k^{2}u^{2}\beta _{1})a_{x}'(u)+k^2 u^2\gamma b_{x}'(u))]\end{aligned}$$

(B.11b)

$$\begin{aligned} J_{B}^{x}{} & {} =\frac{1}{4}[h_{tx}(u)(k^2 u^2 \gamma A_{t}'(u)+(4Z_{2}+k^{2}u^2 \beta _{2})B_{t}'(u))\nonumber \\{} & {} \quad + f(u)(k^2 u^2 \gamma a_{x}'(u)+(4Z_{2}+k^2 u^2\beta _{2})b_{x}'(u))]\end{aligned}$$

(B.11c)

$$\begin{aligned} Q^{x}{} & {} =\frac{1}{u^2}[h_{tx}(u)f'(u)-f(u)h_{tx}'(u)]\nonumber \\{} & {} \quad - A_{t}(u)J_{A}^{x}-B_{t}(u)J_{B}^{x} \end{aligned}$$

(B.11d)

In addition, these currents are conserved along radial direction u. Therefore their boundary values are related to that of horizon data, which can be determined from the regularity at the horizon [9, 31, 39, 50]:

$$\begin{aligned}{} & {} a_{x}'(u)|_{u=u_{h}}=\frac{E_{x}}{f(u)} \qquad b_{x}'(u)|_{u=u_{h}}=\frac{e_{x}}{f(u)} \end{aligned}$$

(B.12)

$$\begin{aligned}{} & {} h_{tx}(u)|_{u=u_h}=-f(u)h_{ux}(u) \end{aligned}$$

(B.13)

According to the regular boundary condition near the horizon and the following constraint relation derived from the linearized Einstein equation,

$$\begin{aligned} h_{ux}(u){} & {} =-\frac{(E_{x}M_{A}\!+\!e_{x}M_{B})}{\Gamma }-\frac{u^{3}(Z_{1}\mu _A^{2}\!+\!Z_{2}\mu _{B}^{2})(3u-4u_h)}{\Gamma }\nonumber \\{} & {} \quad -\frac{k^{2}u^{5}W(u-u_h)}{\Gamma }\!+\!\frac{u_h^{2}(4\Lambda +\!12f(u)+\!2V)}{\Gamma }\!+\!\frac{\chi '(u)}{k}\nonumber \\ \end{aligned}$$

(B.14)

In order to facilitate exposition, the following physical quantities have been defined.

$$\begin{aligned} W{} & {} =\beta _1 \mu _{A}^{2}+\mu _{B}(2\gamma \mu _{A}+\beta _{2}\mu _{B}) \end{aligned}$$

(B.15a)

$$\begin{aligned} \Gamma{} & {} =k^2 u f(u)[u^4 W-4u_h^{2}V'(k^2 u_h^{2})]\end{aligned}$$

(B.15b)

$$\begin{aligned} M_{A}{} & {} =u^3 u_h[4Z_{1}\mu _{A}+k^2 u^2(\gamma \mu _B+\beta _{1}\mu _{A})]\end{aligned}$$

(B.15c)

$$\begin{aligned} M_{B}{} & {} =u^3 u_h[4Z_{2}\mu _{B}+k^2u^2(\gamma \mu _A+\beta _{2}\mu _B)] \end{aligned}$$

(B.15d)

Finally, replacing Eqs. (B.14) and (B.12) with the conserved current expression in Eq. (B.11), one can establish the relationship between the conserved currents $J_{A}^{x}$, $J_{B}^{x}$ and $Q^{x}$ and the perturbations in the electric field $E_{x}$, $e_{x}$ and the temperature gradient $\zeta _{x}$. The DC conductivities can be determined by further application of Ohm’s law from the Eq. (11).

$$\begin{aligned}{} & {} \sigma _{AA}=\frac{J_{A}^{x}(u_h)}{E_{x}}=\frac{\beta _{1}}{4}k^2u_h^2+Z_{1}+\frac{1}{4k^2}\frac{\pi _{A}^2}{\Sigma }, \end{aligned}$$

(B.16a)

$$\begin{aligned}{} & {} \sigma _{BB}=\frac{J_{B}^x(u_h)}{e_{x}}=\frac{\beta _{2}}{4}k^2{u_h}^2+Z_{2}+\frac{1}{4k^2}\frac{\pi _{B}^2}{\Sigma },\end{aligned}$$

(B.16b)

$$\begin{aligned}{} & {} \sigma _{AB}=\sigma _{BA}=\frac{1}{4}k^2u_h^2\gamma +\frac{1}{4k^2}\frac{\pi _{A}\pi _{B}}{\Sigma },\end{aligned}$$

(B.16c)

$$\begin{aligned}{} & {} T\bar{\kappa }=\frac{Q^{x}(u_h)}{\zeta _{x}}=\frac{4f'(u_h)^2}{k^2u_h^2\Sigma },\end{aligned}$$

(B.16d)

$$\begin{aligned}{} & {} T\alpha _{A}=\frac{J_{A}^{x}(u_h)}{\zeta _{x}}=-\frac{\pi _{A}f'(u_h)}{k^2u_h\Sigma },\end{aligned}$$

(B.16e)

$$\begin{aligned}{} & {} T\alpha _{B}=\frac{J_{B}^{x}(u_h)}{\zeta _{x}}=-\frac{\pi _{B}f'(u_h)}{k^2u_h\Sigma }, \end{aligned}$$

(B.16f)

The following physical quantities are defined for ease of giving by

$$\begin{aligned}{} & {} \Sigma =4V'(k^2u_h^2)\!-u_h^2(\!\beta _1\mu _A^2\!+\beta _2\mu _B^2\!+2\gamma \mu _A\mu _B) ,\end{aligned}$$

(B.17a)

$$\begin{aligned}{} & {} \pi _{A}=(k^2u_h^2(\beta _1\mu _A+\gamma \mu _B)+4Z_1\mu _A),\end{aligned}$$

(B.17b)

$$\begin{aligned}{} & {} \pi _{B}=(k^2u_h^2(\beta _2\mu _B+\gamma \mu _A)+4Z_2\mu _B). \end{aligned}$$

(B.17c)

Appendix C: Dimensional analysis

We can restore units through dimensional analysis and set $\hbar =k_{B} =e=v_{F} =T=1$, with the following settings before restoring the unit: $1/t\sim \omega ,\hbar \omega =k_{B}T$. For thermal conductivity, there are:

$$\begin{aligned} \kappa =\frac{J^{Q}}{\nabla T}\sim \frac{\frac{\hbar \omega }{t}}{\frac{\hbar \omega }{k_{B}}}\sim \frac{\hbar \omega ^2}{\frac{\hbar \omega }{k_{B}}}\sim k_{B}\omega \sim \frac{k_{B}^2T}{\hbar } \end{aligned}$$

(C.18)

Therefore, the thermal conductivity at CNP before restoring the unit is: $\kappa _{0}=\frac{4\pi sT}{k^2}$, the thermal conductivity after reduction is:

$$\begin{aligned} \kappa _{0}^{*}=\frac{k_{B}^2 4\pi s T}{k^2\hbar }\sim \frac{k_{B}^2T}{\hbar }\sim \frac{k_{B}^2}{\hbar }\kappa _{0} \end{aligned}$$

(C.19)

For conductivity, there are:

$$\begin{aligned}{} & {} R=\frac{U}{I}=\frac{eU}{eI}\sim \frac{\hbar \omega }{\frac{e^2}{t}}\sim \frac{\hbar }{e^2} \end{aligned}$$

(C.20)

$$\begin{aligned}{} & {} \sigma =\frac{1}{R}\sim \frac{e^2}{\hbar } \end{aligned}$$

(C.21)

Before reducing the conductivity of the unit CNP:

$$\begin{aligned} \sigma _{0}=2\left( Z+\frac{k^2\pi (\beta +\gamma )}{s}\right) \end{aligned}$$

(C.22)

After restoration, there will be:

$$\begin{aligned} \sigma _{0}^{*}=\frac{e^2}{\hbar }2\left( Z+\frac{k^2\pi (\beta +\gamma )}{s}\right) =\frac{e^2}{\hbar }\sigma _{0} \end{aligned}$$

(C.23)

The results of reducing the unit of conductivity and thermal conductivity when not at the neutral point of charge:

$$\begin{aligned}{} & {} \kappa ^{*}=\frac{\kappa _{0}^{*}}{1-\frac{2\pi Q^2 (g_{n}^2 k^2 \pi (\beta -\gamma )^{*})-sZ}{k^2 (s^2) (Z^2)}}, \end{aligned}$$

(C.24)

$$\begin{aligned}{} & {} \sigma ^{*}=\sigma _{0}^{*}\left( \!1-\!\frac{n^2}{\frac{2\pi n^2k^2}{s\sigma _{0}}(\beta (\!1+\!g_n^2)+\!\gamma (\!1-\!g_n^2))\!-\!\frac{k^2 s Z^2}{\pi \sigma _{0}}}\right) . \end{aligned}$$

(C.25)

Finally, let’s address the matter of restoring the temperature unit:

$$\begin{aligned}{} & {} T=\frac{\hbar \omega }{k_{B}}\sim \frac{J s s^{-1}}{JK}\sim K, \end{aligned}$$

(C.26)

$$\begin{aligned}{} & {} T=\frac{1}{4\pi u_h}\left( 3-\frac{V(k^2u_h^2)}{2}-\frac{u_h^2(Z_{1}\mu _{A}^2+Z_{2}\mu _{B}^2)}{4}\right) \sim L^{-1}.\nonumber \\ \end{aligned}$$

(C.27)

To restore the temperature unit to Kelvin (K), it is necessary to multiply by a factor of m on the basis of $T=\hbar \omega /{k_{B}}$. This is equivalent to multiplying the temperature by an additional factor of $(\hbar v_{F})/(k_{B})$.

$$\begin{aligned}{} & {} T^{*}=\frac{\hbar v_F}{k_B}\frac{\sqrt{s}}{(4\pi )^{3/2}}\left( 3-\frac{2\pi k^2}{s}-\frac{2\pi ^2(1+g_n^2)n^2}{Z s^2}\right) ,\nonumber \\{} & {} T^{*}=\frac{\hbar v_F}{k_B}T\sim K. \end{aligned}$$

(C.28)