Metallic and Insulating Phases in Two-Dimensional Strongly-Coupled Field Theories With a Gravitational Dua

1. Introduction

We examined electrical transport in strongly coupled holographic quantum field theories at zero charge density, constructing perfect metals amidst disorder. Our findings have implications for realistic models of disordered strange metals.

2. Conductivity

Consider a static, asymptotically anti-de Sitter space with a black hole horizon sourced entirely by uncharged bulk matter and a dynamical metric. We can choose the bulk metric using diffeomorphism invariance.

$$\mathrm{d}{s}^2=L^2\left[P\mathrm{d}{r}^2-Q\mathrm{d}{t}^2+G_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j\right].$$

(1)

$i,j$ indices represent the spatial boundary directions, while $M,N$ represent all dimensions, and L is AdS radius. All functions in the metric are functions of r and $\mathbf{x}$. We further choose bulk coordinate $0<r<\infty$, with $r=0$ black hole horizon, and $r=\infty$ AdS boundary. Uncharged matter not required, energy conditions obeyed.

We add a U(1) gauge field to the bulk, so the action of our theory is

$$S=\int {\mathrm{d}}^{d+2}x\sqrt{-g}\left({\mathcal{L}}_{\mathrm{uncharged}}-\frac{Z}{4}{F}^2\right).$$

(2)

Function Z is a parameter of (uncharged) scalar matter, but for our purposes it’s an arbitrary function of r and $\mathbf{x}$. Gauge field’s two-point functions correspond to current-current correlation functions in the boundary theory, including electrical conductivity matrix ${\sigma }^{ij}$. The conductivity may be related, via membrane paradigm [1], to data on the horizon of the black hole alone. The expected value of the boundary current is given by

$$J^i={\sigma }^{ij}{E}_j=\mathbb{E}\left[Z\sqrt{\gamma }{\gamma }^{ij}\left(E_j+{\partial }_j\alpha \right)\right],$$

(3)

where $E_j$ is the applied electric field, $\mathbb{E}[\cdots ]$ denotes a uniform spatial average, ${\gamma }_{ij}=G_{ij}(r=0)$ is the induced metric on the horizon, and $\alpha$ is the unique function which obeys equation

$$0={\partial }_i\left(Z\sqrt{\gamma }{\gamma }^{ij}\left(E_j+{\partial }_j\alpha \right)\right)$$

(4)

with appropriate boundary conditions (for example, periodicity in compact boundary spatial directions). The membrane paradigm was used in holographic systems in [2], and similar computations appear in [3,4,5] for black holes with translational symmetry broken only in one direction. These results are special cases of this general formula. This formula may break down if black hole horizon fragments and becomes disconnected, as was considered in [6,7].

We can interpret (4) as a hydrostatic equation enforcing local charge conservation in an emergent horizon fluid. This is subtle – the local “electric current" in (4) is not the same as the expected value of the local current in the dual theory; only their spatial averages are equal. A powerful set of techniques have been developed to understand the qualitative behavior of transport in such fluids [8]; for example, it immediately follows from (4) that ${\sigma }_{ij}={\sigma }_{ji}$.

In particular, if $\sigma [Z;{\gamma }_{ij}]$ is the conductivity matrix with given Z and ${\gamma }_{ij}$:

$$det\left(\sigma [Z;{\gamma }_{ij}]\right)det\left(\sigma \left[\frac{1}{Z};{\gamma }_{ij}\right]\right)=\frac{1}{e^8}.$$

(5)

If we set $Z=1$, (5) gives

$$det\left(\sigma \right)=\frac{1}{e^4}.$$

(6)

If we expect that on average for a disordered sample, the conductivity matrix is isotropic (${\sigma }^{ij}=\sigma {\delta }^{ij}$), that fixes conductivity to be $\sigma =1/e^2$, exactly the clean result!

A simple way to understand this result: suppose that in local coordinates, the metric is given by

$${\gamma }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j\approx a_x^2\mathrm{d}{x}^2+a_y^2\mathrm{d}{y}^2.$$

(7)

Then we expect “locally" ${\sigma }_{xx}\sim a_y/a_x$ and ${\sigma }_{yy}\sim a_x/a_y$[9]. On average $a_y$ and $a_x$ should have identical distributions, so we expect that ${\sigma }_{xx}$ and $1/{\sigma }_{xx}$ have the same distributions. This implies $\sigma =1/e^2$; analogous statements are known for random resistor lattices in $d=2$ with analogous (e.g., log-normal) resistance distributions. And more generally, if $logZ$ is symmetrically distributed about 0, then in an isotropic theory, $\sigma =1/e^2$ follows from (5) in the thermodynamic limit.

The robustness of $\sigma$ in these strongly disordered $d=2$ models is remarkable, and deserves further comments. In models where momentum dissipation is introduced through massive gravity [10] or “Q-lattice" axions [11], one finds the hydrodynamic result [12]

$$\sigma ={\sigma }_q+\frac{{\mathcal{Q}}^2\tau }{ϵ+P},$$

(8)

where $\mathcal{Q}$ is charge density, $ϵ$ energy density, P pressure, ${\sigma }_q$ dissipative “quantum critical" conductivity without disorder, and $\tau$ a “momentum relaxation time", inversely related to graviton mass. Before now, it was unclear whether the fact that (8) holds beyond the hydrodynamic limit was an unrealistic feature of massive gravity or similar theories. Our work confirms this is a sensible prediction of massive gravity for many systems at $\mathcal{Q}=0$. (8) further implies another mechanism, $\tau \to 0$, by which the conductivity can reach its lower bound, ${\sigma }_q$. The conductivity saturating this lower bound, at least qualitatively, is likely to occur at strong disorder [8]. Confirmation that strongly-disordered charged holographic models (with $Z=1$) have a conductivity no smaller than $1/e^2$ in $d=2$ would be a further non-trivial test of predictions of simple mean-field physics.

In $d\ne 2$, and/or if Z is distributed more generically, it’s valuable to employ insight gained from equivalence between Markov chains on lattices and resistance of a resistor lattice [13]. For arbitrary Z, this analogy can be leveraged to find lower and upper bounds to $\sigma$, for a self-averaging disordered sample: [8]

$$\frac{L^{d-2}}{e^2}\mathbb{E}{\left[\frac{{\gamma }_{ii}}{dZ\sqrt{\gamma }}\right]}^{-1}\le \sigma \le \frac{L^{d-2}}{e^2}\frac{\mathbb{E}\left[Z\sqrt{\gamma }{\gamma }^{ii}\right]}{d}.$$

(9)

It is straightforward to test these results and bounds by numerically solving (4) for various disorder realizations. Good agreement with our exact analytic results and consistency with our bounds is obtained.

3. Conductor-Insulator Transition

(9) constrains $\sigma$ to deviate from the clean result by the strength of fluctuations in Z and ${\gamma }_{ij}$. It’s evident from (9) that if ${\gamma }_{ij}$ and Z are finite at all points on the horizon, then the black hole necessarily conducts electrical current, no matter how strong the disorder. This is a remarkable result. In contrast, in non-interacting quantum field theory, a conductor-insulator transition occurs at a finite disorder strength [14] in $d>2$, and at arbitrarily small disorder in $d\le 2$[15]. This transition relates to the destructive interference of matter waves scattering off of the disorder. Apparently, bulk fluctuations of the gauge field in holographic theories do not suffer from such interference. While it’s known [16,17] that metal-insulator transitions occur at a finite disorder strength in an interacting quantum system, even such systems ultimately succumb to (many-body) localization at strong disorder. Perhaps holographic models have taken the “coupling $\to \infty$" limit first, rendering such a transition impossible.

Realizing a holographic conductor-insulator transition takes more care. A “helical lattice" approach has generated such a transition in [18,19], but there is no satisfying physical interpretation. However, even in these papers, the conductivity in the insulating phase only decays as algebraically in T as $T\to 0$, in contrast to canonical insulators.

Assuming $d=2$ and a probe limit with AdS-Schwarzschild geometry, we need a large $\mathbb{E}[1/Z]$ for $\mathcal{C}=0$, requiring percolating $Z\to 0$ bubbles across the horizon. When these finite-Z regions disconnect, charge transport is halted, causing a disorder-driven holographic metal-insulator transition, similar to random resistor lattices [20].

Numerically compute conductivity for Z ansatz with "bubbles" where $Z\to 0$ percolate across horizon to test proposal. Numerics support this; see Figure 1 and Figure 2.

3.1. Holographic Realizations

We now ask whether the percolation mechanism proposed above for a disorder-driven metal-insulator transition can occur in a “realistic" holographic model: a bottom-up Einstein-Maxwell-dilaton ($\Phi$)-axion ($\alpha$) theory with action

$$S=\int {\mathrm{d}}^{d+2}x\sqrt{-g}\left(\frac{R-2\Lambda }{16\pi G}-{\mathcal{M}}^d\left[\frac{1}{2}{(\partial \Phi )}^2+\frac{{\mathrm{e}}^{-\Phi }}{2}{(\partial \alpha )}^2-\frac{V\left(\alpha \right)+U\left(\Phi \right)}{L^2}\right]-\frac{Z\left(\alpha \right)}{4e^2}{F}^2\right).$$

(10)

Here $\mathcal{M}$ is a mass scale, whose precise value is unimportant – we choose it so that $\Phi$ is strictly dimensionless, for simplicity, and

$$\Lambda =-\frac{d(d+1)}{2L^2}.$$

(11)

At $G\to 0$, generalizing choices yields similar results, but (10) with axio-dilaton scalar kinetic terms is essential. $Z\left(\alpha \right)$’s cosine potentials may suit our needs, and arise due to instanton effects in effective actions (as in QCD). In our holographic model, $Z\left(\alpha \right)$ isn’t suppressed by $G/L^d$ (the scale of bulk’s quantum corrections).

For conductor-insulator transitions, $V\left(\alpha \right)$ must have at least two minima, ${\alpha }_{\mathrm{c}}$ and ${\alpha }_{\mathrm{i}}$, with $Z\left({\alpha }_{\mathrm{c}}\right)>0$ and $Z\left({\alpha }_{\mathrm{i}}\right)=0$. $\alpha$ drives the transition and $\Phi$ stabilizes it, although theories with finite Lifshitz or hyperscaling-violating exponents may also work [21]. Insulators form when bubbles of $\alpha ={\alpha }_{\mathrm{i}}$ percolate across the horizon; we aim to demonstrate how to create and maintain these bubbles at low temperatures. [21] is a citation. For this purpose, a simple choice of potentials, though certainly not the only, is

$$\begin{array}{cc}\hfill U\left(\Phi \right)& =\frac{7{\lambda }^2}{2}-3\lambda \Phi -4{\lambda }^2{\mathrm{e}}^{-\Phi /\lambda }+\frac{{\lambda }^2}{2}{\mathrm{e}}^{-2\Phi /\lambda },\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill V\left(\alpha \right)& =-{\alpha }^2+\frac{{\alpha }^4}{2{\alpha }_0^2},\hfill \end{array}$$

(12b)

$$\begin{array}{cc}\hfill Z\left(\alpha \right)& ={\left(1-\frac{\alpha }{{\alpha }_0}\right)}^2.\hfill \end{array}$$

(12c)

Using Z in [22], we set ${\alpha }_0\to 0$, $\lambda >2$, and $V\left(\alpha \right)$ marginal to avoid axion backreaction on the dilaton. The Harris criterion [21] implies inability to source disordered modes of all wavelengths without UV geometry backreaction.

Let us begin by sourcing the dilaton with (positive) $\delta$-like sources on the AdS boundary – analogous to point-like impurities in the dual theory. Each impurity produces an expanding bubble which becomes insulating; width of the “bubbles" of $\alpha$ is $\sim 1/T$. If density of the impurities is n, then the bubbles percolate across the horizon when $T\lesssim \sqrt{n}$. Within each bubble, $\alpha \to {\alpha }_0$, and thus at low temperatures we obtain an insulator

A second mechanism for obtaining the transition is as follows: suppose $\alpha$.

As $T\to 0$ in the insulating phase, we predict:

$$\sigma \left(T\right)\sim exp\left[-\frac{8}{\lambda }{\left(\frac{\zeta }{T}\right)}^{\lambda /2}\right]$$

(13)

4. Outlook

Recent models [23,24,25,26] propose momentum non-conservation in (quasi-2d) strange metals. We constructed perfect conductors in strong disorder and predict finite charge density will not decrease conductivity. We encourage extending holographic approach to charged black holes and finding non-holographic field theories with disorder-resistant ${\sigma }_q$.