1 Introduction
In present of the strong interactions, the transport properties behave unconventionally in contrast to normal materials, which is described by Fermi liquid theorem. A prime example is the strange metal phase emerging from the normal states of the high-$T_{c}$ cuprates. Among a number of weird transport properties, two of them are celebrated: one is the linear temperature dependence of the in-plane resistivity $R_{xx}$ and the other is the quadratic temperature dependence of the inverse Hall angle $\cot\Theta_{H}$.[11-15] A comprehensive review on this aspect could be found in Ref. [16]. As early as 90s, some theoretical attempts have been made to explain these behaviors. It is suggested that the scattering of spinons and holons are governed by two different relaxation times in Ref. [17], and the independent behaviors of charge conjugation odd and even quasi-particles are proposed in Ref. [18].
In recent decade, holographic methods become popular to illustrate and realize these anomalous scalings. For instance, the methods of computing transport coefficients by holography have been developed in Refs. [19-20]. The anomalous T-linear in-plane resistivity is solely reproduced in Refs. [21-24] by holography. Particularly in Ref. [21] it coupled D-branes to a Lifshitz background in the bulk, and holographically a T-linear resistivity is observed in appropriate dynamical critical exponent $z=2$. Unfortunately, similar work showed that Lifshitz like scaling fails to reproduce T-quadratic inverse Hall angle.
More holographic models have been set up to try to address the temperature scalings dichotomy between the in-plane resistivity and the inverse Hall angle at once. In Ref. [25], it completely rules out the possibility to realize this dichotomy in the background with scaling symmetry, translational and rotational symmetry, e.g. pure AdS and pure Lifshitz. One way out is to consider more general backgrounds incorporating a non-trivial dynamical critical exponent as well as hyperscaling violating exponent.[24,26-35] While many of these models mainly focus on the IR but are not UV complete, which implies that only the low temperature regime is under consideration.
To extend the analysis to arbitrary finite temperature regime, one approach is to interpolate the scaling geometry to asymptotic AdS or Lifshitz in the UV.[24,26,36] Alternatively, it remains a good choice to start with an analytic background ansatz covering UV and IR simultaneously. Moreover, Refs. [37-38] pointed out Lifshitz spacetimes are unstable using the bottom up approach. So it suggests us to work in analytic background solution approaching asymptotic AdS, like Refs. [39-43]. In Ref. [40], T-linear resistivity and T-quadratic inverse Hall angle can be realized though in different corner of the parameter space. But the gauge field backreacting on the gravity is ignored there. T-linear resistivity is reproduced in Refs. [42-43], which are related to Gubser-Rocha model.[44]
As for the electrodynamics, two classes are extensively investigated: One is Einstein-Maxwell-Dilaton (EMD) type models;[27-32,41] The other is Dirac-Born-Infeld (DBI) type models.[24-27,33-34,39-40] Particularly in Ref. [34], the dichotomy is achieved in probe DBI limit with low charge density and weak magnetic field. Note that all its results are valid at low temperature since only IR geometry is constructed as mentioned above. Furthermore, it argues that the nonlinear dynamics encoded by the DBI interactions supports rich structures and wide spectrum of temperature scalings, making it possible to provide two or more independent quantities with different temperature scales so as to realized the dichotomy.
In present paper we take the holographic model set up in our previous work[45] that generalize the specific electrodynamic to a generic nonlinear electrodynamic (NLED) field. To investigate our NLED holographic model in arbitrary finite temperature, charge density and magnetic field, we solve the background solution in asymptotic AdS and take account for the full backreaction from the gauge field following the methods in Ref. [26]. Moreover, we introduce two axions along the spatial directions to break translational symmetry and generate momentum dissipation as in Ref. [46]. Via gauge/gravity duality, a general expression for the DC conductivities is obtained for any electrodynamics.
Based on our NLED holographic model, we investigate the scalings of temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$ in parameter space for linear Maxwell and nonlinear DBI electrodynamics respectively. Besides, we study if there is overlap between the T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$. To extract the effective scalings of temperature from the complicated expressions of $R_{xx}$ and $\cot\Theta_{H}$, we take advantage of the density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ and ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T$ in parameter space.
The rest of this article is organized as follows. In Sec. 2, we briefly review the holographic model set up in Ref. [45] and give the expressions for in-plane resistivity $R_{xx}$ and inverse Hall angle $\cot\Theta_{H}$. In Sec. 3, we show the density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ and ${\rm d}\log_{10}({\rm d}\cot\Theta_{H} /{\rm d}T)/{\rm d}\log_{10}T$ in various parameter spaces to study the scalings of temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$. We focus on two typical cases: one is Maxwell electrodynamics and the other is nonlinear Born-Infeld electrodynamics, which are discussed in Subsecs. 3.1 and 3.2, respectively. In Sec. 4, we end in short conclusions and discussions.
2 Holographic Setup and DC Conductivity
Consider a holographic model with the action given by
$ S=\int {\rm d}^{4}x\sqrt{-g}\Bigl[R-2\Lambda-\frac{1}{2}\sum_{I=1}^{2} (\partial\psi _{I})^{2}+\mathcal{L}(s,p)\Bigr] , $
where $\Lambda=-3/l^{2}$, and we take $16\pi G=1$ and $l=1$ for simplicity. To break translational symmetry and generate momentum dissipation, we introduce two axions $\psi_{I}$ $(I=1,2)$, which lead to a finite DC conductivity.[46-47] In the action (1), the generic NLED Lagrangian $\mathcal{L}(s,p)$ is a function of $s$ and $p$, where $s\equiv-F^{ab}F_{ab}/4$ and $p\equiv-\epsilon^{abcd}F_{ab}F_{cd}/8$ (the indices $a,b,\ldots$ denote the bulk spacetime $t$, $r$, $x$, and $y$). The two independent nontrivial scalars $s$ and $p$ are built from the electromagnetic field $A_{a}$ using field strength tensor $F_{ab}=\partial _{a}A_{b}-\partial_{b}A_{a}$ and totally anti-symmetric Lorentz tensor $\epsilon^{abcd}$.
Along the lines of Ref. [45], we take the following ansatz to construct a black brane solution with asymptotic AdS spacetime:
$ {\rm d}s^{2} =-f(r){\rm d}t^{2}+\frac{{\rm d}r^{2}}{f(r)}+r^{2}({\rm d}x^{2}+{\rm d}y^{2}) , \\ {A} =A_{t}(r){\rm d}t+\frac{h}{2}(x{\rm d}y-y{\rm d}x) , \\ \psi_{1} =\alpha x , \quad\quad \psi_{2} =\alpha y , $
where $h$ denotes the magnetic field and $\alpha$ denotes the strength ofmomentum dissipation. Plugging the ansatz into the action (1)and varying it with respect to $g_{ab}$, $A_{a}$, and $\psi_{I}$, we obtain theequations of motions:
$ f(r)-3r^{2}+rf^{^{\prime}}(r) =-\frac{\alpha^{2}}{2}+\frac{r^{2}}{2}[\mathcal{L}(s,p)+A_{t}^{^{\prime}}(r)G^{rt}] ,$
$ 2f^{^{\prime}}(r)-6r+rf^{^{\prime\prime}}(r) =r[\mathcal{L}(s,p)+hG^{xy}] ,$
$ [r^{2}G^{rt}]^{^{\prime}}=0 ,$
where the prime denotes the derivative to radial direction, and $G$ is defined as $G^{ab}\equiv-\partial\mathcal{L}(s,p)/\partial F_{ab}$. Equation(5) leads to $G^{rt}=-\rho/r^{2},$ where $\rho$ could beinterpreted as the charge density of the dual field theory.[45] The horizon located at $r_{h}$ is determined by $f(r_{h})=0$, and theHawking temperature is given by $T=f^{^{\prime}}(r_{h})/4\pi$. Therefore, Eq.(3) gives
$ -3r_{h}^{2}+4\pi r_{h}T=-\frac{\alpha^{2}}{2}+\frac{r_{h}^{2}}{2} [\mathcal{L}(s_{h},p_{h})+A_{t}^{^{\prime}}(r_{h}) G_{h}^{rt}] ,$
with
$ s_{h} =\frac{A_{t}^{^{\prime}2}(r_{h})}{2}-\frac{h^{2}}{2r_{h}^{4} } , \quad p_{h} =-\frac{hA_{t}^{^{\prime}}(r_{h})}{r_{h}^{2}} , \\ G_{h}^{rt} =-\frac{\rho}{r_{h}^{2}}=-\mathcal{L}^{(1,0)}(s_{h},p_{h} )A_{t}^{^{\prime}}(r_{h}) +\mathcal{L}^{(0,1)}(s_{h},p_{h}) \frac{h}{r_{h}^{2} } ,$
where the superscripts $(1,0)$ and $(0,1)$ denote the partial derivative of $\mathcal{L}(s,p)$ with respect to $s$ and $p$, respectively.
Via gauge/gravity duality, the electromagnetic field $A_{a}$ living in thebulk would be dual to a conserved current $\mathcal{J}^{i}$ (the indices$i,j,\ldots$ denote the co-dimensional boundary spacetime $t$, $x$, and $y$)living in the boundary. As a consequence, the DC conductivities for$\mathcal{J}^{i}$ can be derived using the method developed in Refs. [41,48]. The detailed calculation was carried out in Ref. [45], and here we only display the final expressions for DCconductivities $\sigma$:
$ \sigma_{xx} =\frac{\alpha^{2}r_{h}^{2}\left[ h^{2}+{\alpha^{2} r_{h}^{2}}/{\mathcal{L}^{(1,0)}(s_{h},p_{h})}+A_{t}^{^{\prime}2}(r_{h} )r_{h}^{4}\right] }{\left[ h^{2}+{\alpha^{2}r_{h}^{2}}/{\mathcal{L} ^{(1,0)}(s_{h},p_{h})}\right] ^{2}+h^{2}A_{t}^{^{\prime}2}(r_{h})r_{h}^{4} } , \\ \sigma_{xy} =\frac{hA_{t}^{^{\prime}}(r_{h})r_{h}^{2}\left[ 2\alpha ^{2}r_{h}^{2}+\mathcal{L}^{(1,0)}(s_{h},p_{h})(h^{2}+A_{t}^{^{\prime}2} (r_{h})r_{h}^{4})\right] }{\left[ h^{2}+{\alpha^{2}r_{h}^{2} }/{\mathcal{L}^{(1,0)}(s_{h},p_{h})}\right] ^{2}+h^{2}A_{t}^{^{\prime}2} (r_{h})r_{h}^{4}} -\mathcal{L}^{(0,1)}(s_{h},p_{h}) . $
To express DC conductivities in terms of the temperature $T$, the chargedensity $\rho,$ the magnetic field $h$, the strength of momentum dissipation$\alpha$, we need to solve Eqs. (3) and(5) for $r_{h}$ and $A_{t}^{^{\prime}}(r_{h})$ and plugthem into Eqs. (8). The in-plane resistivity$R_{xx}$ and inverse Hall angle $\cot\Theta_{H}$ are defined as
$ R_{xx}=\frac{\sigma_{xx}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}},\quad \cot\Theta_{H}=\frac{\sigma_{xx}}{\sigma_{xy}} . $
Notice that $R_{xx}$ and $\cot\Theta_{H}$ remain invariant under the scaling transformation
$ T\rightarrow\lambda T,\quad\alpha\rightarrow\lambda\alpha,\quad h\rightarrow\lambda ^{2}h,\quad\rho\rightarrow\lambda^{2}\rho , (10) $
for some positive constant $\lambda.$ From now on we will rescale $T$, $h$,$\rho$, and $\alpha$ to $T/\alpha$, $h/\alpha^{2}$, $\rho$/$\alpha^{2}$ and $1$by taking the scaling factor $\lambda=1/\alpha$.
3 Temperature Dependence of $R_{xx}$ and $\cot\Theta_{H}$
In this section we will discuss the scalings of the temperature dependence of the in-plane resistivity $R_{xx}$ and inverse Hall angle $\cot\Theta_{H}$ in parameter space spanned by $T/\alpha$, $h/\alpha^{2}$, $\rho/\alpha^{2}$ and some possible parameters from $\mathcal{L}(s,p)$. However, the temperaturedependence of $R_{xx}$ and $\cot\Theta_{H}$ is highly nontrivial. To comparewith the results from experiments, we can fit $R_{xx}$ and $\cot\Theta_{H}$with some power of $T/\alpha$:
$ R_{xx}\sim A+B\left( T/\alpha\right)^{n} , \quad \cot\Theta_{H}\sim C+D\left( T/\alpha\right)^{m} , $
where the terms $A$, $B$, $C$, and $D$ can depend on $h/\alpha^{2}$ and $\rho/\alpha^{2}$. The effective power factors $n$ and $m$ are usually focused in experiments since they mainly govern the temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$. To extract the effective power factors $n$ and $m$ from Eq. (11), we display the density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ and ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T$ in the parameter space. For latter convenience, we introduce $N$ and $M$ as
$ \frac{{\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)}{{\rm d}\log_{10}T}\equiv N \Longrightarrow R_{xx}\sim(T/\alpha)^{N+1} , $
$ \frac{{\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)}{{\rm d}\log_{10}T}\equiv M \Longrightarrow \cot\Theta_{H}\sim(T/\alpha)^{M+1} , $
where $N=0$ and $M=1$ correspond to the linear temperature dependence ofin-plane resistivity $R_{xx}$ and the quadratic temperature dependence of inverse Hall angle $\cot\Theta_{H}$, respectively. In the following, we focus on Maxwell electrodynamics in Subsec. 3.1 and Born-Infeld electrodynamics in Subsec. 3.2.
3.1 Maxwell Electrodynamics
The Lagrangian for Maxwell Electrodynamics reads
$ \mathcal{L}(s,p)=s\text . (14) $
Combining Eqs. (6), (7), (8), (9), and (14), one can obtain
$ R_{xx} =\frac{r_{h}^{2}\alpha^{2}(h^{2}+\rho^{2}+r_{h}^{2}\alpha^{2} )}{r_{h}^{4}\alpha^{4}+2r_{h}^{2}\alpha^{2}\rho^{2}+h^{2}\rho^{2}+\rho^{4} } , (15)$
$ \cot\Theta_{H} =\frac{r_{h}^{2}\alpha^{2}(h^{2}+\rho^{2}+r_{h}^{2} \alpha^{2})}{h\rho(h^{2}+\rho^{2}+2r_{h}^{2} \alpha^{2})} , (16) $
with the horizon $r_{h}$ satisfying
$ -12r_{h}^{4}+16\pi Tr_{h}^{3}+2\alpha^{2}r_{h}^{2}+h^{2}+\rho^{2}=0 . (17) $
In this paper we focus on $h/\alpha^{2}\geq0$ and $\rho/\alpha^{2}\geq0$, thus $R_{xx}$ and $\cot\Theta_{H}$ are non-negative in Maxwell electrodynamics. We numerically solve Eq. (17) for $r_{h}$ and then use Eqs. (15) and (16) to study the scalings of temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$, respectively.
(i) In-plane Resistivity
We depict the density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/ $ $ {\rm d}\log_{10}T$ at some fixed values of $h/\alpha^{2}$ or $\rho/\alpha^{2}$ in Fig. 1. The vertical axis is labeled by $\log_{10}(T/\alpha)$, and the range of the temperature $T/\alpha$ varies from 0.01 to 100.
Fig. 1 (Color online) The temperature dependence of $R_{xx}$ in the Maxwell case. Upper row: Density plots of ${\rm d}\log _{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $\rho/\alpha^{2}$ and $\log _{10}(T/\alpha)$ at fixed $h/\alpha^{2}=0$, $1$ and $10$ from left to right. Lower row: Density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $h/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $\rho/\alpha^{2}=0$, $1$, and $10$ from left to right. |
From all figures in Fig. 1, we find two common features. First, one finds that $N\sim-3$ at $T/\alpha\gtrsim10$. Actually, in the high temperature limit $T/\alpha\gg(h/\alpha^{2},\rho/\alpha^{2})$, Eq. (17) reduces to $T\propto r_{h}$, which leads to
$ R_{xx}\sim\text{constant}+(h^{2}-\rho^{2})(\alpha T)^{-2}+\mathcal{O}(T^{-4}) . $
More interestingly, we find that the resistivity varies linearly in temperature at $T/\alpha\lesssim0.1$.
The density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $\rho/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at $h/\alpha^{2}=0$, $1$ and $10$ are displayed in the upper row of Fig. 1. At vanishing magnetic field, as one increases the temperature, the corresponding $N$ monotonically decreases from $0$ to $-3$ at $\rho/\alpha^{2}\lesssim1$. However at $\rho/\alpha^{2}\gtrsim1$, $N$ first increases from $0$ to a maximum value and then decreases to $-3$. In the $h/\alpha^{2}=1$ case, the scalings behavior at $\rho/\alpha\gtrsim1$ is similar to that in the previous case while at $\rho/\alpha\lesssim1$ new behavior appears. One significant character of this new behavior is the discontinuity between the purple region and the red one as shown in the upper middle panel. The line separating these two regions, we call it \textquotedblleft extremum line\textquotedblright, is determined by ${\rm d}R_{xx}/{\rm d}T=0$ and thus the value of ${\rm d}\log_{10}({\rm d}R_{xx} /{\rm d}T)/{\rm d}\log_{10}T$ diverges on this line resulting in the discontinuity. Furthermore, ${\rm d}R_{xx}/{\rm d}T=0$ on the extremum line implies an extreme value of $R_{xx}$ locally, indicating a transition between metal and insulator, which is consistent with Ref. [45]. The presence of discontinuity provides richer behavior and supports a wider spectrum of temperature scalings. Increasing the magnetic field to $h/\alpha^{2}=10$, the extremum line stretches across nearly all the values of $\rho/\alpha^{2}$ in the upper right panel. The behavior below the extremum line is similar to the $h/\alpha^{2}=0$ case, but with much lower scalings exhibits. Due to the discontinuity, a narrow strip-like region of T-linear resistivity survives into $T/\alpha\gtrsim1$ above the extremum line.
The density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $h/\alpha ^{2}$ and $\log_{10}(T/\alpha)$ at $\rho/\alpha^{2}=0$, $1$ and $10$ are displayed in the lower row of Fig. 1. At vanishing charge density, $N$ decreases monotonically from about $0$ to $-3$ as the temperature increases. In the $\rho/\alpha^{2}=1$ case, the metal-insulator transition appears and a narrow stripe of T-linear resistivity presents at $T/\alpha\sim1$. For the case with $\rho/\alpha^{2}=10$, $N$ first increases from $0$ to $1$ and then decreases to $-3$ with the increasing temperature.
To summarize, T-linear resistivity dominates in the low temperature regime with $T/\alpha \lesssim0.1$ for almost all the range of $\rho/\alpha^{2}$ and $h/\alpha^{2}$ in Fig. 1 and survives into higher temperatures in a narrow strip-like manner.
(ii) Inverse Hall Angle
We display the density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}$ $\log_{10}T$ at some fixed values of $h/\alpha^{2}$ in Fig. 2. Note that $\cot\Theta_{H}$ remains invariant under the interchange between $h/\alpha^{2}$ and $\rho/\alpha^{2}$ from Eqs. (16) and (17). Equation (16) shows that $\cot\Theta_{H}$ diverges at $h/\alpha^{2}=0$, so we take a small but non-vanishing magnetic field $h/\alpha^{2}=0.01$.
Fig. 2 (Color online) The temperature dependence of $\cot\Theta_{H}$ in the Maxwell case. Density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/$ ${\rm d}\log_{10}T$ versus $\rho/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $h/\alpha^{2}=0.01$, $1$, and $10$ from left to right. |
From Fig. 2, one can see that $M\sim1$ at $T/\alpha\gtrsim10$, indicating the T-quadratic $\cot\Theta_{H}$ in the high temperature regime. This is easy to understand from the high temperature limit of Eq. (16). At $T/\alpha\lesssim0.1$, we find that $M\sim0$. Similar to $R_{xx}$, the inverse Hall angle $\cot \Theta_{H}$ behaves linearly in temperature at low temperatures. The two cases with fixed $\rho/\alpha^{2}=0.01$ and $\rho/\alpha^{2}=1$ are similar. At $h/\alpha^{2}\gtrsim4$ the scalings of these two cases both have a non-monotonic behavior. As temperature increases, $M$ first increases from $0$ to about $1$, then decreases to about $0.7$, and then again increases to $1$. In the $h/\alpha^{2}=10$ case, the pattern of the right panel of Fig. 2 is similar to those at $\rho/\alpha^{2}\gtrsim4$ of two previous cases.
To summarize, T-quadratic $\cot\Theta_{H}$ not only dominates in the high temperature regime but also extends to much lower temperatures, even reaches $T/\alpha\sim0.1$ for small magnetic field and charge density.
(iii) Overlap
We check if there is overlap between the regions of T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$. It seems unpractical to take exact $N=0$ as the linear temperature dependence of in-plane resistivity and $M=1$ as the quadratic temperature dependence of inverse Hall angle both in the Mathematica programming and real experiments. So we approximately take $-0.2<N<0.2$ as the T-linear $R_{xx}$ and $0.8<M<1.2$ as the T-quadratic $\cot\Theta_{H}$, respectively. These criteria are just in some arbitrariness and approximation since one can imagine that if taking $-0.1<N<0.1$ and $0.9<M<1.1$, then the related figures would be much similar to Figs. 3 and 9 except for the smaller area of the overlap. In Fig. 3, we show the regions of T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$ in yellow and green at some fixed $h/\alpha^{2}$ or $\rho/\alpha^{2}$, respectively.
Fig. 3 (Color online) The overlap between T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$ in the Maxwell case. Upper Row: Region plots of $-0.2<{\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log _{10}T<0.2$ and $0.8<{\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T<1.2$ versus $\rho/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $h/\alpha^{2}=0.01$, $1$ and $10$ from left to right. Lower Row: Region plots of $-0.2<{\rm d}\log _{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T<0.2$ and $0.8<{\rm d}\log_{10}({\rm d}\cot\Theta _{H}/{\rm d}T)/{\rm d}\log_{10}T<1.2$ versus $h/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $\rho/\alpha^{2}=0.01$, $1$ and $10$ from left to right. The regions in yellow and green correspond to the T-linear $R_{xx}$ and the T-quadratic $\cot\Theta_{H}$, respectively. |
Generally speaking, the T-linear $R_{xx}$ dominates in the low temperature regime with $T/\alpha\lesssim0.1$ and may survive into higher temperatures in a narrow strip-like way, while T-quadratic $\cot\Theta_{H}$ dominates in the high temperature regime with $T/\alpha\gtrsim10$ and can extend to lower temperatures. The overlap does not occur in the cases with fixed $\rho/\alpha^{2}=0.01$ and $10$. However as shown in Fig. 3, there exists the overlap in other cases, which occurs at $0.1\lesssim T/\alpha\lesssim1$.
Several EMD-like theories[27-32,41] are mentioned in the Introduction. In Ref. [41], it claimed that charge-conjugation symmetric conductivity contributes to $R_{xx}$ but not to $\cot\Theta_{H}$. It was argued in Ref. [31] that the irrelevance of the charge density operator is crucial to the dichotomy. And a massive Einstein gravity holographic model is studied in Ref. [29]. Particularly in Ref. [29], an RN-AdS geometry case shows that in the high temperature region $R_{xx}$ and $\cot\Theta_{H}$ are both T-quadratic, making it suspense to reproduce the dichotomy. Actually, almost all EMD-like models are constructed in general backgrounds with non-trivial dynamical critical exponent $z$ and hyperscaling violating exponent $\theta$. For instance, the realization of dichotomy imposes $z=6/5$ and $\theta=8/5$ in Ref. [29] and $z=1$ and $\theta=1$ in Ref. [32] etc. However, even in trivial $z=1$ and $\theta=0$ in our model, the expressions of $R_{xx}$ and $\cot\Theta_{H}$ look extremely complicated from Eqs. (15), (16), and (17). So whether dichotomy exists or not need detailed analysis. Using our method, we extract and show the concerning temperature scalings explicitly in Fig. 3. One can see that in some certain region of the parameter space, the dichotomy does exist. And note that the overlap may happen at strong magnetic field and charge density where the backreaction is necessary indeed.
3.2 Born-Infeld Electrodynamics
Born-Infeld theory is a typical nonlinear realization of electrodynamics arising from the effective string theory at low energy scale with the Lagrange given by
$ \mathcal{L}(s,p)=\frac{1}{a}(1-\sqrt{1-2as-a^{2}p^{2}}) , $
where the coupling parameter $a$ is related to the string tension $\alpha^{\prime}$ as $a=(2\pi\alpha^{\prime})^{2}$. To get $R_{xx}$ and $\cot\Theta_{H}$, we should first obtain $A_{t}^{^{\prime}}(r)$ in terms of $r_{h}$, $a$, $h$, and $\rho$ by Eqs. (7) and (19):
$ A_{t}^{^{\prime}}(r_{h})=\frac{\rho}{\sqrt{r_{h}^{4}+a(h^{2}+\rho^{2})} } . $
The reality of $A_{t}^{^{\prime}}(r_{h})$ gives a constraint
$ r_{h}^{4}+a(h^{2}+\rho^{2})>0.$
For $a>0$, the above constraint holds automatically, while for $a<0$ it puts an upper bound on the charge density and magnetic field. The physical interpretation is that the singularity of the black brane needs to hide behind the horizon. After some arrangements of Eqs. (6), (7), (8), (9), (19) and (20), the resistivity $R_{xx}$ and inverse Hall angle $\cot \Theta_{H}$ read
$ R_{xx} =\frac{\alpha^{2}r_{h}^{2}( h^{2}+\rho^{2}+\alpha^{2} \sqrt{r_{h}^{4}+a(h^{2}+\rho^{2})}) }{\alpha^{4}r_{h}^{4}+\rho ^{2}( a\alpha^{4}+h^{2}+\rho^{2}+2\alpha^{2} \sqrt{r_{h}^{4}+a(h^{2} +\rho^{2})}) } ,$
$ \cot\Theta_{H}=\frac{\alpha^{2}r_{h}^{2}( h^{2}+\rho^{2}+\alpha ^{2}\sqrt{r_{h}^{4}+a(h^{2}+\rho^{2})}) }{h\rho( a\alpha ^{4}+h^{2}+\rho^{2}+2\alpha^{2}\sqrt{r_{h}^{4}+ a(h^{2}+\rho^{2})})} ,$
with $r_{h}$ satisfying
$ -(1+6a)r_{h}^{2}+8\pi aTr_{h}+a\alpha^{2}+\sqrt{r_{h}^{4}+a(h^{2}+\rho^{2} )}=0.$
For $|a|\ll1$, Maxwell electrodynamics is recovered as expected. Note that $\cot\Theta_{H}$ still possesses the symmetry between $h/\alpha^{2}$ and $\rho/\alpha^{2}$.
At $a>0$, we find that the temperature dependence of $R_{xx}$ and $\cot \Theta_{H}$ are quite similar to those of Maxwell electrodynamics so we only show some examples in Fig. 4. One could find that the behavior in Fig. 4 is similar to that in Figs. 1 and 2. The slight difference between them is the range of $N$ and $M$. For instance, at $\rho/\alpha^{2}=0$, the minimum of $N$ is $-3$ in the Maxwell case shown in Fig. 1 while it becomes $-3.5$ in the Born-Infeld case shown in the lower left panel of Fig. 4. The similarity among the Maxwell case and Born-Infeld case with $a>0$ was also noticed in Refs. [26,45].
Fig. 4 The temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$ in the Born-Infeld case with $a>0$. Density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ and ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T$ for various fixed values of $a$, $\rho /\alpha^{2}$ and $h/\alpha^{2}$. |
Things become quite different in the $a<0$ case. In the following, we will discuss the scalings of temperature dependence of $R_{xx}$ and $\cot\Theta _{H}$ in various parameter spaces for $a<0$.
(i) In-plane Resistivity
In Fig. 5, we depict the density plots of ${\rm d}\log_{10}$ $({\rm d}R_{xx}/{\rm d}T)/$ ${\rm d}\log_{10}T$ at fixed $a=-1$ and some values of $h/\alpha^{2}$ or $\rho/\alpha^{2}$. We would only focus on $a=-1$ since the cases with other fixed negative value of $a$ are much alike.
Compared to the case with $a>0$, the most obvious distinction is the appearance of the white region due to the constraint (21). The discontinuity usually occurs in the region close to the white region, which implies the mental-insulator transition. The behavior at $T/\alpha\gtrsim1$ in Fig. 5 is reminiscent of that in Maxwell case. Though the new behavior appears and the range of temperature scalings generally becomes broader for $a<0$, the region of the T-linear resistivity becomes very smaller than that in the $a>0$ case.
Fig. 5 The temperature dependence of $R_{xx}$ in the Born-Infeld case with $a=-1$. Upper Row: Density plots of ${\rm d}\log_{10} ({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $\rho/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $h/\alpha^{2}=0$, $1$ and $10$ from left to right. Lower Row: Density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $h/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $\rho/\alpha^{2}=0$, $1$ and $10$ from left to right. |
In the following, we study the temperature dependence of $R_{xx}$ with respect to the parameter $a$ since it is a quantity characterizing the coupling of the Born-Infeld electrodynamics. We show the density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ as a function of $a$ and $\log_{10}$ $(T/\alpha)$ at various fixed values of $h/\alpha^{2}$ and $\rho/\alpha^{2}$ in Fig. 6. Note that we choose a small but non-vanishing charge density $\rho/\alpha^{2}=0.01$ in the upper row due to the triviality of constant resistivity $R_{xx}=1$ for both strictly vanishing magnetic filed and charge density.
Fig. 6 The temperature dependence of $R_{xx}$ in the Born-Infeld case. Density plots of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ versus $a$ and $\log_{10}(T/\alpha)$ at various fixed values of $h/\alpha^{2}$ and $\rho/\alpha^{2}$. The fixed $\rho/\alpha^{2}$ for each row, from upper to lower, are set as $0.01$, $1$ and $10$. And the fixed $h/\alpha^{2}$ for each column, from left to right, are set as $0$, $1$ and $10$. |
We first focus on the upper row with fixed $\rho/\alpha^{2}=0.01$. In the $h/\alpha^{2}=0$ case, there is no white region since $a$ is not large enough to violate the constraint (21). At $a<0$, as the temperature increases, $N$ first decreases from $0$ to $-\infty$, then directly jumps to $+\infty$ on the extremum line and finally decreases to $-3$ at high temperatures as expected. And at $a>0$, as the temperature increases, $N$ first decreases from $0$ to a minimum around $-3.5$ and then increases to $-3$. The $h/\alpha^{2}=1$ and $10$ cases are quite similar to the $h/\alpha^{2}=0$ case except the appearance of the white region.
We then consider the middle row with fixed $\rho/\alpha^{2}=1$. For vanishing magnetic field, at $a>0$, $N$ first increases from $0$ to a maximum of around $0.5$ and then decreases to $-3$ with the increasing temperature. For the $h/\alpha^{2}=1$ case, $N\sim-5$ at sufficiently high temperatures rather than the usual scaling $-3$. The reason is that in high temperature limit, the $(T/\alpha)^{-2}$ term in Eq. (18) vanishes due to $h/\alpha^{2}=\rho/\alpha^{2}$, and thus the leading term dependent on the temperature is at the order of $(T/\alpha)^{-4}$. Moreover, there are two extremum lines at $a<0$, indicating two metal-insulator transitions at $T/\alpha\sim1$. In the $h/\alpha^{2}=10$ case, in addition to the expected discontinuity located at $a<0$, a new one appears at $a>0$.
We finally study the lower row with fixed $\rho/\alpha^{2}=10$. In both $h/\alpha^{2}=0$ and $h/\alpha^{2}=1$ cases, at $a>0$, $N$ first increases from $0$ to a maximum of around $1$ and then decreases to $-3$ as the temperature increases. In the case with $h/\alpha^{2}=10$, the reason for the unusual scaling at high temperatures is same as that in the case with $h/\alpha^{2}=1$ and $\rho/\alpha^{2}=1$.
Generally speaking, the regions of T-linear $R_{xx}$ in the Born-Infeld case with $a>0$ is similar to those in the Maxwell case. For the Born-Infeld case with $a<0$, they are strips at $T/\alpha\sim1$.
(ii) Inverse Hall Angle
We now consider the scalings of temperature dependence of the inverse Hall angle. We show the density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T$ at fixed $a=-1$ and some values of $h/\alpha^{2}$ in Fig. 7. The cases at other fixed negative $a$ are in quite similarity.
Fig. 7 (Color online) The temperature dependence of $\cot\Theta_{H}$ in the Born-Infeld case with $a=-1$. Density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/$ ${\rm d}T)/{\rm d}\log_{10}T$ versus $\rho/\alpha^{2}$ and $\log_{10}(T/\alpha)$ at fixed $h/\alpha^{2}=0.01$, $1$ and $10$ from left to right. |
The behavior in the upper plane of the three figures in Fig. 7 is similar to that in the Maxwell case, which is shown in Fig. 2. As before, the white region appears due to the constraint (21). The presence of the discontinuity largely widens the spectrum of the temperature scalings of $\cot\Theta_{H}$. As a side note, there are two possibilities for the origins of the discontinuity here. One possibility, in similar to the discontinuity appearing in the temperature dependence of $R_{xx}$, is the vanishing of ${\rm d}\cot\Theta_{H}/{\rm d}T$ that causes the divergence of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T$. The other possibility arises from the fact that $\cot\Theta_{H}$ could be infinite if the parameter $a$ takes a proper negative value making the denominator of Eq. (23) vanish. Note that the latter possibility would not happen in the Maxwell case. While which possibility causing the discontinuity in Fig. 7 could not be read from the figures directly, and it might be the case that both possibilities exist in one figure simultaneously. Actually one needs detailed calculations of $\cot\Theta_{H}$ and ${\rm d}\cot\Theta_{H}/{\rm d}T$ to find out the origin. Similar to the Maxwell and Born-Infeld with $a>0$ cases, the T-quadratic $\cot\Theta_{H}$ dominates in the temperature regime with $T/\alpha\gtrsim1$. In the case with $h/\alpha^{2}=0.01$, we find that T-quadratic $\cot\Theta _{H}$ is observed at low temperatures with $T/\alpha\sim0.05$ for weak charge density $\rho/\alpha^{2}\lesssim0.25$.
We display the density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/$ ${\rm d}T)/{\rm d}\log_{10}T$ against $a$ and $\log_{10}(T/\alpha)$ at various fixed values of $h/\alpha ^{2}$ and $\rho/\alpha^{2}$ in Fig. 8. For small but non-vanishing magnetic filed and charge density as shown in the upper left panel, no white region exhibits as expected. As the temperature increases, $M$ increases monotonically from $0.4$ to $2$ in the region of $a<0$ and $T/\alpha\lesssim0.1$. At $a>0$, $M$ first increases from $0.2$ to a maximum and then decreases to $1$. T-quadratic $\cot\Theta_{H}$ dominates in $T/\alpha\gtrsim1$ for all values of $a$, and also presents at $T/\alpha \sim0.05$ for $a<0$, which is consistent with the $h/\alpha^{2}=0.01$ case in Fig. 7. For the cases in the upper middle and upper right panels, both are similar to the previous case except the presence of the white region. The three cases in lower row are all similar. At $a>0$, with the increasing temperature, $M$ first increases from $0$ to around $0.9$, then decreases to around $0.6$ and finally increases to $1$. The region of T-quadratic $\cot\Theta_{H}$ presents in $T/\alpha\gtrsim1.5$ and $0.25\lesssim T/\alpha\lesssim 1$.
(iii) Overlap
We end this section by discussing the overlap between T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$ for Born-Infeld electrodynamics. For the $a>0$ case, the overlaps plotted in the $h/\alpha^{2}$ $(\rho/\alpha^{2})$% -$\log_{10}\left( T/\alpha\right) $ plane with fixed $a$ are similar to those in the Maxwell cases in Fig. 3. For the $a<0$ case, the region of T-linear $R_{xx}$ in the $h/\alpha^{2}$ $(\rho/\alpha^{2})$-$\log_{10}\left( T/\alpha\right) $ plane with fixed $a$ is very small as shown in Figs. 5. In Fig. 9, we display the region plots of T-linear $R_{xx}$ in yellow and T-quadratic $\cot\Theta_{H}$ in green as a function of $a$ and $\log_{10}(T/\alpha)$ at several fixed values of $h/\alpha^{2}$ and $\rho/\alpha^{2}$.
Generally speaking, T-linear $R_{xx}$ mainly lives in low temperatures with $T/\alpha\lesssim0.1$ for $a>0$ and survives in some strip-like regions at $T/\alpha\gtrsim0.1$ for all values of $a$. And T-quadratic $\cot\Theta_{H}$ dominates at high temperatures for all range of $a$ and extends down to low temperatures. In the upper row, no overlap exhibits. In the middle row with fixed $\rho/\alpha^{2}=1$, the overlap would occur at $a>0$ and $0.1\lesssim T/\alpha\lesssim1$, which is reminiscent of the Maxwell case in Fig. 3. In the third row with $\rho/\alpha^{2}=10$, a narrow strip-like overlap presents at $a\lesssim-5$ and $T/\alpha\sim1$, distinguishing from the Maxwell case.
Fig. 8 (Color online) The temperature dependence of $\cot\Theta_{H}$ in the Born-Infeld case. Density plots of ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/$ ${\rm d}T)/{\rm d}\log_{10}T$ versus $a$ and $\log_{10}(T/\alpha)$ at various fixed values of $h/\alpha^{2}$ and $\rho/\alpha^{2}$ after taking into account the symmetry between $h/\alpha^{2}$ and $\rho/\alpha^{2}$. |
4 Discussion and Conclusion
In this paper, we investigated the temperature dependence of the in-plane resistivity $R_{xx}$ and inverse Hall angle $\cot\Theta_{H}$ for the NLED holographic model developed in our previous work.[45] To extract the effective scalings of temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$, we took the advantage of the density plot of ${\rm d}\log _{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ and ${\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log _{10}T$ in parameter space. In Sec. 3, we focused on two specific cases in the model: one is Maxwell electrodynamics and the other is the nonlinear Born-Infeld electrodynamics.
For Maxwell electrodynamics, a wide spectrum of the scalings of the temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$ can be observed. In general, the in-plane resistivity has been shown to vary as $R_{xx}\sim T$ at low temperatures and $R_{xx}\sim T^{-2}$ at high temperatures. And the inverse Hall angle varies as $\cot\Theta_{H}\sim T$ at low temperatures and $\cot\Theta_{H}\sim T^{2}$ at high temperatures. Moreover, the presence of discontinuity in the density plot of ${\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T$ implies the metal-insulator transition. In general, the T-liner $R_{xx}$ dominates at low temperatures with $T/\alpha\lesssim0.1$ and might survive into higher temperatures in a narrow strip-like manner. And the T-quadratic $\cot\Theta_{H}$ dominates at high temperatures with $T/\alpha\gtrsim10$ and extends down to lower temperatures, even to $T/\alpha\sim0.1$ at small magnetic field and charge density. The overlap, if occurs, generally locates in the intermediate temperate regime within $0.1\lesssim T/\alpha\lesssim1$ as shown in Fig. 3.
For nonlinear Born-Infeld electrodynamics with $a>0$, the temperature dependence of $R_{xx}$ and $\cot\Theta_{H}$ and their overlap are quite similar to Maxwell case. While at $a<0$, the constraint (21) generally results in white region at low temperatures, which provides richer behavior and broader range of scalings than Maxwell case. At $a<0$, the T-linear $R_{xx}$ presents in a narrow strip-like region at $T/\alpha\sim1$ and T-quadratic $\cot\Theta_{H}$ still dominates at high temperatures. And the overlap in the $a<0$ case could occur at strong charge density as shown in Fig. 9.
Fig. 9 (Color online) The overlap between T-linear $R_{xx}$ and T-quadratic $\cot\Theta_{H}$ in the Born-Infeld case. Region plots of $-0.2<{\rm d}\log_{10}({\rm d}R_{xx}/{\rm d}T)/{\rm d}\log_{10}T<0.2$ and $0.8<{\rm d}\log_{10}({\rm d}\cot\Theta_{H}/{\rm d}T)/{\rm d}\log_{10}T<1.2$ versus $a$ and $\log _{10}(T/\alpha)$ for several fixed values of $h/\alpha^{2}$ and $\rho /\alpha^{2}$. The fixed $\rho/\alpha^{2}$ for each row, from upper to lower, are set as $0.01$, $1$ and $10$. And the fixed $h/\alpha^{2}$ for each column, from left to right, are set as $0.01$, $1$ and $10$. The regions in yellow and green correspond to the T-linear $R_{xx}$ and the T-quadratic $\cot\Theta_{H} $, respectively. |
We would like to address more on the parameter $a$. From Refs. [45,49], we see that $a$ is a quantity measuring the correction to the coulomb force between two electrons due to the nonlinearity encoded by Born-Infeld electrodynamics. More precisely, the strength of repulsive force between the electrons tends to decrease for $a>0$ and increase for $a<0$. Particulary for negative $a$, the stronger force between the electrons tends to cause the electronic traffic jam resulting in the Mott-like behavior, namely strong enough $e$-$e$ interactions prevent the available mobile charge carriers to efficiently transport charges. This mechanism provides us a natural physical interpretation about the appearance of white region in Figs. 5 to 8: the temperature is ought to be high enough to overcome the electronic traffic jam. The complicated physical process deduced from the nonlinearity in the microscope generates rich structures and wide spectrum of temperature scalings in the macroscope, which increases the chance to provide two or more macroscopic independent quantities with different temperature scales so as to realized the dichotomy. Similar arguments also appears in Ref. [34].
Notice that we used rescaled quantities, e.g., the rescaled temperature $T/\alpha$, all the time. However, it is quite unknown how the value of $\alpha$ is related to experiments. In Ref. [50], it is argued that the universality of T-linear $R_{xx}$ may appear in incoherent regime (the regime of strong momentum relaxation). And Ref. [42] showed that a strong value of $\alpha$ would allow T-linear $R_{xx}$ survives in the high temperature regime, even up to room temperature as detected by experiments. Since the overlap in Figs. 3 and 9 is depicted by rescaled temperature $T/\alpha$, the range of the real temperature $T$ would be large for strong $\alpha$. This indicates that though the overlap seems small in Maxwell and Born-Infeld cases, our model remains a chance to achieve the dichotomy in a wide range of realistic temperature for a proper value of $\alpha$. Furthermore, increasing the degrees of freedom in holographic models could enlarge the overlap. For instance, one could introduce the dilaton field which is widely investigated in literatures, e.g., Refs. [26,34,41].
Finally, we found that in the Maxwell case with vanishing magnetic field, see Fig. 1, $N$ ranges from $0$ to $1$ at $T/\alpha\lesssim1$. It might imply a combination dependence of temperature $T+T^{2}$ on $R_{xx}$, which could help to explain the unconventional behaviors observed at low temperatures of two prototypical copper oxide superconductors LSCO and TBCO.[51] It deserves future study to make concrete comparisons with experiments.