Anomaly induced transport from symmetry breaking in holography

This paper utilizes a 5-dimensional holographic Einstein-Maxwell model with Chern-Simons terms and a scalar field to demonstrate that explicit symmetry breaking causes anomaly-induced transport phenomena to extend beyond anomalous currents, significantly influencing non-anomalous sectors and making all conductivities sensitive to the symmetry-breaking mass parameter.

The Gist

Imagine you are at a crowded dance party. In physics, this "party" is a fluid made of tiny particles (like electrons in a metal or quarks in a star). Usually, if you push the crowd, they move in a predictable way. But in the quantum world, things get weird because of something called anomalies.

Think of an anomaly as a "glitch in the matrix." It's a rule that says, "Hey, even though we thought this symmetry was perfect, the quantum rules actually break it in a specific way." In our dance party, this glitch causes the crowd to spontaneously start spinning or flowing in a specific direction just because there's a magnetic field or a whirlpool (vorticity) present, without anyone pushing them. This is called Anomaly-Induced Transport.

For a long time, physicists thought these weird flows only happened to the "special" dancers (the anomalous currents). If a dancer didn't have this glitch, they just stood still or moved normally.

The Big Discovery in This Paper
The authors of this paper asked a simple question: What happens if we intentionally break the rules of the dance floor?

In their model, they introduced a "symmetry-breaking" element. Imagine the dance floor has a rule that everyone must hold hands in a perfect circle. The authors then threw a heavy rock (a scalar field) into the middle of the circle, breaking the perfect symmetry. They wanted to see how this "broken" state affected the flow.

Here is what they found, explained simply:

1. The "Contagious" Glitch

Previously, scientists thought the quantum glitch (the anomaly) only affected the specific dancers it was attached to. But this paper shows that the glitch is contagious.

When the symmetry is broken (the rock is thrown in the circle), the weird, glitchy flows start affecting everyone, even the dancers who were supposed to be "normal" (non-anomalous). The "broken" state forces the normal dancers to start flowing in weird directions too, just like the glitchy ones.

2. The Holographic Simulation (The "Shadow" Trick)

How did they figure this out? They didn't build a giant particle collider. Instead, they used a mathematical trick called Holography.

Think of it like this:

• The Real World: A complex, 3D dance party with quantum rules.
• The Shadow: A 2D shadow of that party cast on a wall.

The authors realized that the complex 3D quantum physics is mathematically identical to a simpler 5-dimensional "shadow" world involving gravity and black holes. By solving the equations for the "shadow" (which is actually easier to calculate), they could predict exactly what was happening in the real quantum dance party.

3. The "Mass" Parameter (The Rock in the Circle)

The key variable in their study was a "mass parameter" (let's call it M).

• M = 0: The dance floor is perfect. The rock isn't there. The "normal" dancers stay normal.
• M > 0: The rock is there. The symmetry is broken.

They found that as they increased the size of the rock (M), the behavior of the "normal" dancers changed dramatically. The conductivities (how easily the current flows) became very sensitive to the size of the rock.

4. The Takeaway: Broken Rules Create New Flows

The most exciting part of their discovery is this: You don't need a quantum glitch to get a weird flow if you break the symmetry.

Even if a current (a flow of particles) doesn't have the quantum glitch built-in, if you break the symmetry of the system (like breaking the perfect circle of the dance floor), that current will start behaving as if it has the glitch. It will flow in response to magnetic fields and vortices, just like the "special" dancers.

Why Does This Matter?

• Heavy Ion Collisions: In experiments smashing atoms together (like at the Large Hadron Collider), the symmetry of the universe is temporarily broken. This paper helps explain how energy and charge might flow in those tiny, chaotic explosions.
• New Materials: In materials like Weyl semimetals (a type of super-conductive crystal), electrons behave like these quantum dancers. If we can engineer materials where symmetry is broken, we might be able to create new types of electronic currents that don't lose energy (non-dissipative), leading to super-efficient electronics.

In a Nutshell:
The paper shows that in the quantum world, breaking the rules doesn't just stop the game; it changes the rules for everyone. Even the "boring" parts of the system start acting weird and interesting when the symmetry is broken, creating new ways for energy and charge to move that we didn't expect.

Technical Summary

1. Problem Statement

The paper investigates the transport properties of relativistic fluids in the presence of quantum anomalies (specifically chiral and gravitational anomalies) when explicit symmetry breaking is introduced.

• Context: Anomaly-induced transport phenomena, such as the Chiral Magnetic Effect (CME) and Chiral Vortical Effect (CVE), are well-understood in systems with exact symmetries. These effects generate non-dissipative currents (e.g., electric currents parallel to magnetic fields or vorticity) driven by quantum corrections.
• The Gap: Previous studies (e.g., Ref. [20]) showed that in the presence of explicit symmetry breaking, anomalies could induce transport in non-anomalous currents (currents with vanishing divergence). However, these studies were limited to specific sectors (primarily CME) and often neglected the full backreaction of gauge fields on the spacetime metric or the mixed gauge-gravitational anomaly.
• Objective: The authors aim to extend this analysis to:
  1. Include the Chiral Vortical Effect (CVE) and energy transport sectors.
  2. Incorporate the mixed gauge-gravitational anomaly.
  3. Account for the full backreaction of gauge fields on the metric tensor.
  4. Determine how explicit symmetry breaking (controlled by a scalar field mass parameter $M$) modifies conductivities in both anomalous and non-anomalous sectors.

2. Methodology

The authors employ the AdS/CFT correspondence (Holography) to model a strongly coupled relativistic fluid.

• Holographic Model:

  • Action: A 5-dimensional Einstein-Maxwell theory with a negative cosmological constant ($\Lambda = -6/\ell^2$).
  • Fields:
    • Three $U(1)$ gauge fields: Vector ($V$), Axial ($A$), and an additional non-anomalous symmetry ($W$).
    • A scalar field ($\phi$) charged under both $A$ and $W$, responsible for explicit symmetry breaking.
  • Chern-Simons (CS) Terms: The action includes:
    • Pure gauge CS terms ($\kappa$) mimicking the axial anomaly.
    • Mixed gauge-gravitational CS terms ($\lambda$) mimicking the gravitational contribution to the anomaly.
  • Symmetry Breaking: The scalar field potential includes a mass term $m^2 = -3/\ell^2$ (corresponding to an operator dimension $\Delta=3$). The boundary value $M$ of the scalar field acts as the explicit symmetry breaking parameter.

• Background Solution:

  • The authors solve the coupled Einstein-Maxwell-scalar equations of motion numerically using the shooting method.
  • They consider a black brane background with chemical potentials $\mu_v, \mu_a, \mu_w$ and temperature $T$.
  • Crucially, they include the full backreaction of the gauge fields on the metric, which is essential for capturing the CVE and energy transport correctly.

• Fluctuations and Kubo Formulae:

  • Linear perturbations are applied to the background metric and gauge fields.
  • The Kubo formulae are used to extract transport coefficients (conductivities) from the retarded Green's functions of the currents ($J_v, J_a, J_w$) and the energy-momentum tensor ($T^{0i}$).
  • The study focuses on the zero-frequency ($\omega \to 0$) and zero-momentum ($p \to 0$) limits to isolate the anomalous transport coefficients ($\sigma^B$ for magnetic response, $\sigma^V$ for vortical response).

• Numerical Technique:

  • The fluctuation equations are solved using the pseudo-spectral method (Chebyshev polynomials) on a Gauss-Lobatto grid.
  • Results are compared against analytical limits: the massless case ($M=0$) and the large mass limit ($M \to \infty$).

3. Key Contributions

1. Inclusion of Mixed Gauge-Gravitational Anomaly: The model explicitly incorporates the $\lambda$ term, allowing for the study of gravitational contributions to transport, which are vital for the CVE and energy currents.
2. Full Backreaction: Unlike previous probe-limit studies, this work solves the full coupled system where gauge fields modify the geometry, ensuring the results for energy transport and CVE are physically consistent.
3. Non-Anomalous Current Activation: The paper demonstrates that explicit symmetry breaking ($M \neq 0$) "activates" transport in the non-anomalous sector ($J_w$). Even though the $W$ symmetry is non-anomalous (no CS term directly couples to it), the mixing with the anomalous sector via the scalar field induces non-zero conductivities ($\sigma^B_{wv}, \sigma^V_w$, etc.).
4. Analytical Large-M Limit: The authors derive analytical expressions for conductivities in the limit $M \to \infty$, showing a redistribution of anomaly coefficients where the non-anomalous sector effectively inherits half the anomaly strength of the broken sector.

4. Key Results

• Massless Limit ($M=0$): The numerical results perfectly reproduce known analytical formulas for CME and CVE in the symmetric phase. All conductivities related to the non-anomalous symmetry $W$ vanish.
• Finite Mass ($M > 0$):
  • Sensitivity to $M$: All conductivities exhibit a distinct sensitivity to the symmetry breaking parameter $M$.
  • Non-Anomalous Currents: The non-anomalous current $\vec{J}_w$ acquires non-zero responses to magnetic fields ($\vec{B}_v, \vec{B}_a$) and vorticity ($\vec{\Omega}$).
    • $\sigma^B_{wv}$ grows linearly with $\mu_v$ and its slope increases with $M$.
    • $\sigma^V_w$ shows a quadratic dependence on $\mu_v$ with increasing curvature as $M$ increases.
  • Redistribution of Response: As $M$ increases, the magnetic response is redistributed. Conductivities inducing the axial current ($\vec{J}_a$) are suppressed, while those inducing the non-anomalous current ($\vec{J}_w$) are enhanced.
  • Zero Chemical Potentials: Even when $\mu_v = \mu_a = \mu_w = 0$, certain transport coefficients (specifically $\sigma^V_a, \sigma^V_w, \sigma^B_{\epsilon a}, \sigma^B_{\epsilon w}$) remain non-zero at finite temperature. This implies that symmetry breaking alone is sufficient to induce anomalous responses in the absence of chemical potentials.
• Large Mass Limit ($M \to \infty$):
  • The system approaches a new fixed point where the broken symmetry ($A_-$) decouples, and the remaining symmetry ($A_+$) behaves as if its chemical potential is renormalized ($\mu_+ = \mu_a + \mu_w$).
  • A specific relation emerges: $\sigma^B_{av}(\infty) = \sigma^B_{wv}(\infty) = \frac{1}{2}\sigma^B_{+v}(\infty)$.
  • In the special case where $\mu_a = \mu_w$, four specific conductivities ($\sigma^B_{vv}, \sigma^B_{\epsilon v}, \sigma^V_v, \sigma^V_\epsilon$) become independent of $M$ entirely.

5. Significance

• Theoretical Insight: The work establishes that anomaly-induced transport is not confined to strictly anomalous currents. Explicit symmetry breaking acts as a bridge, allowing quantum anomalies to influence "safe" (non-anomalous) sectors of the theory. This challenges the intuition that non-anomalous currents are immune to anomaly-driven effects.
• Strongly Coupled Systems: The findings are relevant for understanding strongly coupled systems where symmetries are not exact, such as in the quark-gluon plasma (heavy ion collisions) or condensed matter systems.
• Condensed Matter Applications: The authors suggest a potential application to Weyl and Dirac semimetals. In these materials, multiple Weyl nodes (valleys) exist. If valley symmetries are broken (e.g., by strain or disorder), the paper predicts that "valley currents" (non-anomalous in the strict sense) could exhibit CME-like or CVE-like behaviors, offering a new mechanism for valleytronics.
• Methodological Robustness: By including full backreaction and mixed anomalies, the paper provides a more complete holographic framework for studying transport in realistic, symmetry-broken environments.

In summary, the paper demonstrates that explicit symmetry breaking fundamentally alters the transport landscape, enabling anomaly-induced currents to permeate non-anomalous sectors, with implications for both high-energy physics and condensed matter phenomenology.