Quantum critical theories in a periodic potential: strange metallic thermoelectric and magnetotransport

This paper utilizes holographic models of near-extremal AdS black holes to demonstrate that 2D quantum critical theories with strong translational symmetry breaking and zero average chemical potential exhibit enhanced conductivity, bad-metal electrical transport with non-Drude thermal behavior, and approximately linear-in-field magnetoresistance.

The Gist

The Big Picture: The "Strange Metal" Mystery

Imagine you are trying to understand how electricity flows through a super-conducting material (like those used in high-temperature superconductors). Physicists call these materials "strange metals." They are weird because they don't follow the standard rules of how electrons usually behave (like cars driving on a highway).

In normal metals, electricity flows smoothly. In strange metals, it's chaotic, and the heat and electricity seem to get mixed up in confusing ways. One of the biggest puzzles is that when you put a magnet near them, the resistance to electricity doesn't behave the way standard physics predicts.

This paper asks: What happens if we take these strange materials and put them in a "labyrinth" of varying energy? Specifically, what if the "chemical potential" (the energy pushing the electrons) changes back and forth like a wave, but averages out to zero?

The authors used a powerful mathematical tool called Holography (which treats a 3D problem like a 2D shadow) to simulate this. They looked at two scenarios: a 1D lattice (a single line of obstacles) and a 2D lattice (a checkerboard of obstacles).

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The Setup: The "Zero-Average" Labyrinth

Imagine a river (the flow of electrons) trying to move through a landscape.

• The Normal Case: Usually, the river has a steady current, and the landscape has small bumps.
• This Paper's Case: The river has no average current (it's "charge neutral"). However, the landscape is a wild rollercoaster. Sometimes the ground is high (pushing electrons one way), and sometimes it's low (pulling them the other way). Crucially, the average height of the ground is exactly zero.

The researchers asked: "If we make this landscape very bumpy (strong lattice), how does the river flow?"

Key Finding 1: The "Detour" Effect (Better Conductors)

The Analogy: Imagine a crowd of people trying to walk through a hallway filled with pillars.

• Standard Logic: You'd think pillars would block the path, making it harder to walk (higher resistance).
• The Paper's Discovery: Surprisingly, the crowd walks faster when the pillars are there!

Why? In this quantum world, the electrons are like water. When they hit a "high energy" wall, they don't stop; they flow around it. In a 2D grid (a checkerboard), the water finds clever, winding paths to bypass the obstacles. It's like a river finding a meandering path around rocks; the total flow actually increases because the water is forced to find the "path of least resistance" through the gaps.

• 1D vs. 2D: In a single line (1D), the electrons just get stuck or flow in a straight line. But in a 2D grid, they can weave around obstacles, making the material a better conductor as the obstacles get bigger.

Key Finding 2: The "Ghost" Heat and Electricity

The Analogy: Imagine a busy train station.

• Normal Metals: The trains (electrons) and the heat (passengers) move together in a single, coordinated line. If you stop the trains, you stop the heat.
• This Paper's Discovery: The trains and the passengers have split up.
  • Electricity becomes "incoherent." It's like a chaotic swarm of bees. There is no single "train" moving; it's just a jumble of particles bouncing around.
  • Heat remains "coherent." It's like a smooth, flowing river of passengers.

The paper found that even though the electricity is a mess, the heat flows very smoothly (like a Drude peak, which is a fancy way of saying "smooth flow"). This separation is a signature of "strange metals."

Key Finding 3: The Magnetic Mystery (The "B-Linear" Effect)

The Analogy: Imagine spinning a coin on a table.

• Normal Metals: If you add a magnetic field, the coin spins in a tight circle and eventually stops. The resistance goes up in a predictable curve (like a parabola).
• This Paper's Discovery: When they added a magnetic field to their 2D lattice, the resistance didn't curve. It went up in a straight line forever.

Why? This is the "Holy Grail" of strange metal physics. In real experiments (like with cuprate superconductors), scientists see this straight-line resistance, but no one knows why.
The paper suggests that because the electrons are weaving through a chaotic, heterogeneous landscape (like water flowing through a complex maze of rocks), the magnetic field stretches these paths out linearly. It's not a simple spin; it's a complex navigation through a maze that scales perfectly with the strength of the magnet.

The "Effective Medium" Secret

The authors realized that their complex quantum system behaves exactly like a theory called Effective Medium Theory (EMT).

• EMT Analogy: Imagine a patchwork quilt made of two different fabrics: one that conducts electricity perfectly and one that blocks it completely.
• Instead of calculating every single thread, you just look at the "average" flow. The current doesn't go through the whole quilt evenly; it snakes through the conductive patches, avoiding the bad ones.
• The paper shows that this "snaking" behavior explains why the resistance is linear with the magnetic field and why the material conducts better with more obstacles.

Conclusion: Why This Matters

This paper solves a few pieces of the "Strange Metal" puzzle:

1. Dimension Matters: A 1D line of obstacles acts differently than a 2D grid. The 2D grid allows for "detours" that improve conductivity.
2. Separation of Powers: Electricity and heat can flow independently in these extreme conditions.
3. The Magnetic Clue: The strange, straight-line resistance seen in real superconductors might just be the result of electrons navigating a chaotic, zero-average energy landscape, behaving like water flowing through a complex maze.

In short: When you shake up the energy landscape enough, electrons stop acting like individual cars and start acting like a fluid navigating a maze, leading to some very strange and useful behaviors.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the transport properties (DC and AC, thermoelectric, and magnetotransport) of 2D quantum critical theories subjected to strong translational symmetry breaking via a spatially varying chemical potential lattice with zero average ($\bar{\mu} = 0$).

• Context: Quantum critical points (QCPs) usually require fine-tuning. However, in real materials like high-$T_c$ cuprates and graphene at charge neutrality, strong lattice effects and disorder are prevalent.
• The Regime: The authors focus on the "strong lattice" regime where the variance of the chemical potential ($\delta\mu$) far exceeds the average ($\bar{\mu}$), specifically the extremal limit $\bar{\mu} = 0$. This eliminates background scales, potentially revealing universal transport regimes.
• The Puzzle:
  • Standard hydrodynamics predicts a single momentum relaxation rate (Drude peak) governing both longitudinal and Hall transport.
  • Experimental observations in "strange metals" (e.g., cuprates) show a discrepancy: longitudinal resistivity scales linearly with $T$ (or $B$), while the Hall angle scales as $T^2$. This implies two distinct relaxation timescales.
  • Previous weak-coupling models for 2D periodic potentials predicted no change in electrical transport, while strong-coupling 1D models showed distinct behavior. The authors aim to resolve the 2D strong-coupling case to see if it explains the "two-time-scale" puzzle and the observed unsaturating linear magnetoresistance.

2. Methodology

The authors employ Holographic Duality (AdS/CFT) to model these strongly correlated systems.

• Holographic Setup:
  • Bulk Theory: 4D Einstein-Maxwell gravity in asymptotically Anti-de Sitter (AdS$_4$) space.
  • Action: $S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g} [R - 2\Lambda - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}]$.
  • Boundary Conditions: The chemical potential on the boundary is modulated as a lattice:
    • 1D: $\mu(x) = \delta\mu \cos(Gx)$
    • 2D: $\mu(x,y) = \frac{\delta\mu}{2} [\cos(Gx) + \cos(Gy)]$
    • Crucially, the average $\bar{\mu} = 0$.
  • Magnetic Field: A background magnetic field $B$ is included for magnetotransport studies.
• Numerical Techniques:
  • Background Solution: Solved using the de Turck trick to make Einstein's equations elliptic. The equations are solved numerically on a grid (Chebyshev-Lobatto in radial $z$, finite difference in transverse $x,y$) using the PETSc library.
  • DC Transport: Extracted by solving Stokes flow equations on the black hole horizon. This avoids the need for $\omega \to 0$ limits of AC data.
  • AC Transport: Computed via linear perturbations of the background geometry and gauge fields, extracting retarded Green's functions ($G^R_{JJ}, G^R_{JQ}, G^R_{QQ}$).
  • Pole Analysis: The authors analyze the pole structure in the complex frequency plane to identify Drude modes, Umklapp modes, and their collisions.

3. Key Contributions and Results

A. 1D Charge-Neutral Lattices (Review and Extension)

• DC Electrical Conductivity ($\sigma_{DC}$): Increases with lattice strength ($\delta\mu$) at low temperatures. This is explained by an emergent anisotropy in the near-horizon geometry (AdS$_4$) which, upon coordinate rescaling, enhances conductivity. The scaling follows $\sigma_{DC} \approx 1 + \frac{1}{2}(\frac{\delta\mu}{4G})^2$.
• DC Thermal Conductivity ($\kappa_{DC}$): Exhibits a crossover. At high $T$, it behaves like a Drude fluid. At low $T$, it becomes a "strong thermal conductor," increasing exponentially as the lattice becomes irrelevant to the IR fixed point.
• AC Transport & Pole Collisions:
  • Electrical: Dominated by an Umklapp sound mode at $\omega \approx G/\sqrt{2}$.
  • Thermal: Contrary to simple hydrodynamics, the thermal response is not a single Drude peak. Instead, the Drude pole collides with an Umklapp charge diffusion pole, splitting into two symmetric poles slightly offset from the imaginary axis ($\omega = \pm \omega_{peak}$). This creates a mid-IR peak.

B. 2D Charge-Neutral Lattices (Novel Findings)

• DC Electrical Conductivity:
  • Behavior: Unlike 1D, $\sigma_{DC}$ increases with temperature at high $T$ before saturating to the conformal field theory (CFT) bound ($\sigma_\infty = 1$) at low $T$.
  • Mechanism: The current flows around "obstacles" (regions of high chemical potential) via curved paths. This is analogous to Effective Medium Theory (EMT). The variance in the current flow ($J_y$) correlates with the enhanced conductivity.
  • RG Flow: The lattice modulation acts as an irrelevant operator in $d=2$ (unlike marginal in $d=1$), explaining the flow toward the homogeneous CFT value at low $T$.
• AC Transport:
  • Electrical: Displays "bad metal" behavior with a weak zero-frequency diffusive peak in addition to the Umklapp sound peak. This new peak arises from mesoscopic Effective Medium effects (remnants of local Drude modes in particle/hole regions) that do not cancel out in 2D as they do in 1D.
  • Thermal: Similar to 1D, it features the mid-IR peak from pole collisions. However, it also exhibits a new zero-frequency diffusive peak distinct from the electrical one.
  • Decoupling: Crucially, the thermoelectric conductivity $\alpha$ remains zero, confirming that electrical and thermal transport mechanisms remain distinct despite the complex flow patterns.
• Magnetotransport (2D):
  • Linear Magnetoresistance: The longitudinal magnetoresistance $\rho_{xx}(B)$ exhibits an approximately linear, unsaturating dependence on the magnetic field ($B$) at large fields.
  • Scaling: This matches the form predicted by Effective Medium Theory for systems with large spatial fluctuations: $\rho(B) \propto \sqrt{(\alpha T)^2 + (\gamma B)^2}$.
  • Nernst Effect: The Nernst coefficient shows distinct scaling regimes, transitioning at the onset of the linear magnetoresistance regime.

4. Significance and Implications

1. Resolution of the "Two-Time-Scale" Puzzle: The paper provides a holographic mechanism where longitudinal and transverse (Hall) transport are governed by different physics. In 2D, the longitudinal transport is dominated by Effective Medium-like flow (multiple local relaxation rates), while the Hall response (in a charged system) would be governed by global hydrodynamics. This naturally separates the timescales without ad-hoc assumptions.
2. Connection to Strange Metals: The results offer a potential explanation for the phenomenology of high-$T_c$ cuprates:
   • Bad Metal Behavior: The coexistence of Drude-like AC thermal transport and incoherent/bad-metal electrical transport.
   • Linear Magnetoresistance: The emergence of unsaturating $B$-linear magnetoresistance purely from strong translational symmetry breaking in a quantum critical system, matching experimental observations in cuprates and pnictides.
3. Dimensionality Matters: The study highlights a fundamental difference between 1D and 2D lattices. In 1D, currents are constrained, leading to emergent symmetries. In 2D, currents can meander around potential barriers, leading to EMT-like behavior and new diffusive modes.
4. Limitations and Outlook: The current model is charge-neutral ($\bar{\mu}=0$), so the Hall effect vanishes. The authors suggest that adding finite density ($\bar{\mu} \neq 0$) while maintaining strong lattice fluctuations ($\delta\mu \gg \bar{\mu}$) could yield a finite Hall effect while preserving the EMT signatures in longitudinal transport, potentially fully resolving the strange metal puzzle.

Conclusion

The paper demonstrates that strong translational symmetry breaking in quantum critical systems leads to non-Drude, multi-scale transport signatures. In 2D, this manifests as Effective Medium Theory behavior, where currents navigate a heterogeneous landscape of particle and hole regions. This results in enhanced electrical conductivity at high temperatures, distinct AC pole structures, and unsaturating linear magnetoresistance, offering a robust holographic framework for understanding the anomalous transport properties of strange metals.