The thermoelectric properties of inhomogeneous holographic lattices

Abstract: We consider inhomogeneous, periodic, holographic lattices of D=4 Einstein-Maxwell theory. We show that the DC thermoelectric conductivity matrix can be expressed analytically in terms of the horizon data of the corresponding black hole solution. We numerically construct such black hole solutions for lattices consisting of one, two and ten wave-numbers. We numerically determine the AC electric conductivity which reveals Drude physics as well as resonances associated with sound modes. No evidence for an intermediate frequency scaling regime is found. All of the monochromatic lattice black holes that we have constructed exhibit scaling behaviour at low temperatures which is consistent with the appearance of $AdS_2\times\mathbb{R}^2$ in the far IR at T=0.

• The paper presents an analytical derivation of the DC thermoelectric conductivity matrix using black hole horizon data.
• It employs numerical methods to show that lattice black holes exhibit low-temperature scaling behavior consistent with AdS2×ℝ2 geometry.
• The study demonstrates that the figure of merit ZT can become arbitrarily large, suggesting enhanced efficiency for thermoelectric materials.

Overview of the Thermoelectric Properties of Inhomogeneous Holographic Lattices

The paper investigates the thermoelectric properties of inhomogeneous, periodic, holographic lattices within the framework of $D=4$ Einstein-Maxwell theory. The authors provide an analytical expression for the DC thermoelectric conductivity matrix by focusing on black hole horizon data, and they numerically construct pertinent black hole solutions for lattices with various configurations of wave numbers. The approach leverages holographic techniques to link the behavior of strongly coupled systems in real-world materials to theoretical models of black holes and conductivities.

Key Results

1. DC and AC Conductivity:
   • Through analytical methods, the paper demonstrates that the DC thermoelectric conductivity matrix can be described in terms of horizon data from corresponding black hole solutions.
   • The authors showcase the numerical construction of such black hole solutions for lattices with different numbers of wave numbers, specifically one, two, and ten.
   • The AC electric conductivity is calculated numerically, revealing Drude-like behavior as well as resonances which correspond to sound modes.
2. Low Temperature Scaling:
   • The lattice black holes demonstrate scaling behavior at low temperatures, consistent with an $AdS_2\times\mathbb{R}^2$ geometry appearing in the IR limit at zero temperature ($T=0$).
   • The scaling suggests the dominance of irrelevant operators about $AdS_2\times\mathbb{R}^2$ influencing the transport properties in the limit $T\to 0$.
3. Figures of Merit and Efficiency:
   • The research provides expressions for the "figure of merit" $ZT$, which can become arbitrarily large at low temperatures. This offers a measure of efficiency for thermoelectric engines when compared to existing materials.

Implications and Future Directions

The findings from this study hold substantial theoretical implications. The analytical expression derived connects the observable thermoelectric behavior of strongly coupled electronic systems with the underlying holographic principles that describe black hole physics. Practically, this enhances the understanding of phenomena such as the DC resistivity scaling observed in strange metallic phases, offering paths for improved insight on the mechanisms governing such materials.

Future investigations could expand on the current results by exploring the conductivity properties of holographic lattices with greater inhomogeneity, including the dependence of the effective chemical potential on multiple spatial directions. Furthermore, by examining transitions in ground states, researchers may uncover novel properties or stability conditions for holographic lattices and further refine the link between holographic setups and real-world condensed matter systems.

The paper underlines the universality of holographic techniques in tackling diverse issues within condensed matter theory, encompassing the emerging trends in material science and quantum criticality. As such, its implications extend far beyond theoretical physics, providing a foundation upon which new materials, exhibiting unique electronic properties, could be designed and synthesized.