Anomalous Transport and Explicit Symmetry Breaking in Holography

This paper utilizes a five-dimensional holographic Einstein-Maxwell model with Chern-Simons terms to demonstrate that explicit symmetry breaking not only modifies anomaly-induced transport but also extends its influence to the non-anomalous sector, with transport coefficients showing a distinct dependence on the symmetry-breaking mass parameter.

The Gist

Imagine the universe as a giant, complex machine where invisible forces (like electricity and magnetism) flow through it. Physicists have long known that sometimes, the rules of this machine get "glitched" in a very specific way due to quantum mechanics. These glitches are called anomalies. Usually, these glitches cause strange, predictable flows of energy and charge, much like a river always flowing downhill.

This paper explores what happens when you deliberately "break" the symmetry of the machine to see if those glitches change the flow in unexpected places.

Here is a simple breakdown of their study:

1. The Setup: A Holographic Simulation

The authors use a tool called Holography. Think of this like a 3D movie projector. They take a complex, 5-dimensional gravitational world (the "movie") to simulate what happens in a simpler, 4-dimensional world of particles (the "screen"). This allows them to study difficult quantum problems using the easier math of gravity.

In their simulation, they set up three types of "currents" (flows):

• The Vector Flow: A standard flow (like normal electricity).
• The Axial Flow: A flow that is "glitched" by quantum anomalies.
• The Non-Anomalous Flow: A flow that is supposed to be perfectly safe and unaffected by glitches.

2. The Experiment: Breaking the Rules

Usually, if a flow is "non-anomalous" (safe), it ignores the quantum glitches. It's like a car driving on a smooth road that doesn't care about the potholes on the side.

However, the authors introduced a Scalar Field. Imagine this as a heavy weight or a "symmetry-breaking mass" placed on the road. This weight deliberately distorts the road, breaking the perfect symmetry of the system.

3. The Discovery: The "Safe" Flow Gets Infected

The main finding of the paper is surprising. When they added this "weight" (the symmetry breaking):

• The Axial Flow (the one already glitched) behaved as expected, but its behavior changed based on how heavy the weight was.
• Crucially, the Non-Anomalous Flow (the "safe" one) started acting strangely too.

The Analogy: Imagine a group of dancers. One dancer is already tripping over their own feet (the anomaly). The other dancers are moving perfectly in sync (the non-anomalous current).

• Without the weight: The tripping dancer stumbles, but the others keep dancing perfectly.
• With the weight: The authors found that the "weight" caused the perfectly synchronized dancers to start stumbling and moving in weird patterns too, mimicking the tripping dancer.

4. What They Measured

They calculated specific numbers called transport coefficients. Think of these as "sensitivity meters."

• They measured how much the "safe" current moved when they applied magnetic fields or rotation (vorticity).
• They found that the more they increased the "symmetry-breaking mass," the more the "safe" current started to react to these forces, behaving almost like the glitched current.

5. The Conclusion

The paper concludes that explicit symmetry breaking changes the rules of the game. It proves that quantum anomalies aren't just isolated problems affecting only the "glitched" parts of a system. If you break the symmetry of the system (by adding that scalar field mass), the "glitches" can spread and influence the parts of the system that were previously thought to be immune.

In short: You can't just isolate the quantum glitches. If you mess with the symmetry of the whole system, even the "perfect" flows get dragged into the chaos.

Note: The authors mention that while their study is theoretical, it might help us understand materials like "Weyl semi-metals" (a type of crystal), but they do not claim to have tested this on real materials or in a clinical setting. Their work remains a theoretical exploration of how these forces interact.

Technical Summary

1. Problem Statement

The paper addresses the interplay between quantum anomalies and explicit symmetry breaking in the context of relativistic transport phenomena.

• Context: In chiral field theories, anomalies typically induce transport effects (like the Chiral Magnetic Effect and Chiral Vortical Effect) in currents associated with anomalous symmetries.
• Gap: Previous holographic studies (e.g., Ref. [10]) suggested that when symmetries are explicitly broken, anomalies could induce transport in non-anomalous currents (currents with vanishing divergence). However, a comprehensive analysis of how explicit breaking affects the full spectrum of anomalous transport coefficients (including chiral vortical conductivity) and the role of mixed gauge-gravitational anomalies was lacking.
• Goal: To investigate how a scalar field, which explicitly breaks symmetries, modifies the constitutive relations for both anomalous and non-anomalous currents in a strongly coupled holographic system.

2. Methodology

The authors employ the AdS/CFT correspondence (Holography) to model a 5-dimensional strongly coupled quantum field theory.

A. The Holographic Model

• Action: The model is defined by a 5D Einstein-Maxwell action augmented with:
  • Pure Gauge Chern-Simons (CS) terms: Coupling the axial gauge field ($A$) and vector gauge field ($V$).
  • Mixed Gauge-Gravitational CS terms: Coupling the gauge fields to the curvature tensor ($R$).
  • Scalar Field ($\phi$): A charged scalar field introduced to explicitly break the symmetries. It carries charge under both the axial ($A$) and a third non-anomalous ($W$) symmetry.
• Symmetries:
  • $U(1)_V$: Vector symmetry (anomalous).
  • $U(1)_A$: Axial symmetry (anomalous).
  • $U(1)_W$: Non-anomalous symmetry.
• Symmetry Breaking Mechanism: The scalar field $\phi$ acquires a non-zero boundary value $M$, which acts as the explicit symmetry-breaking mass parameter. The covariant derivative is defined as $D_M \phi = [\partial_M - i(A_M - W_M)]\phi$.

B. Background Solution

• Ansatz: The authors assume a static, translationally invariant black brane metric and gauge fields depending only on the radial coordinate $r$ (or the dimensionless variable $u = r_h^2/r^2$).
• Equations of Motion: A system of six coupled non-linear differential equations is derived for the metric functions ($f, \chi$), gauge potentials ($V_t, A_{t\pm}$), and the scalar field ($\phi$).
• Numerical Solution: The system is solved numerically using the shooting method, integrating from the horizon ($u=1$) to the boundary ($u=0$) with appropriate boundary conditions (regularity at the horizon and fixed chemical potentials/mass at the boundary).

C. Linear Perturbations and Transport Coefficients

• Fluctuations: The authors introduce linear perturbations to the metric and gauge fields, decomposing them into Fourier modes with momentum $p$ along the $z$-axis.
• Kubo Formulae: Transport coefficients are computed using the Kubo formula, which relates them to the retarded Green's functions (correlators) of the charge and energy currents at zero frequency ($\omega \to 0$) and first order in momentum ($p$).
• Numerical Technique: The fluctuation equations are solved using the pseudo-spectral method, expanding the fluctuations in Chebyshev polynomials on a Gauss-Lobatto grid.

3. Key Contributions

1. Extension to Chiral Vortical Conductivity: Unlike previous works focusing primarily on magnetic effects, this study explicitly computes the chiral vortical conductivity in the presence of explicit symmetry breaking.
2. Role of Mixed Anomalies: The model incorporates mixed gauge-gravitational anomalies, allowing the authors to isolate their specific contribution to the generation of non-anomalous currents.
3. Full Backreaction: The analysis includes the full backreaction of the scalar field and gauge fields on the spacetime metric, ensuring a consistent holographic description of the symmetry-breaking phase.
4. Non-Anomalous Current Response: The paper provides concrete evidence that explicit symmetry breaking allows anomalies to "leak" into the non-anomalous sector, inducing currents in $J_W$ that mimic chiral magnetic and vortical effects.

4. Results

The study analyzes transport coefficients as functions of chemical potentials ($\mu_v, \mu_a, \mu_w$) and the symmetry-breaking parameter $M$.

A. Conductivities vs. Chemical Potentials (Finite $M$)

• The authors computed the conductivities ($\sigma^B_{wv}, \sigma^B_{wa}, \sigma^B_{ww}, \sigma^V_w$) that induce the non-anomalous current $J_W$.
• Finding: Even though $U(1)_W$ is non-anomalous, the presence of the scalar field ($M \neq 0$) generates non-zero conductivities. This confirms that a Chiral Magnetic Effect (CME) and Chiral Vortical Effect (CVE) can exist for non-anomalous currents when symmetry is explicitly broken.
• The results show a clear dependence on the chemical potentials, deviating from the analytic large-$M$ limit predictions as $M$ decreases.

B. Conductivities vs. Symmetry Breaking Parameter ($M$)

• Axial Current ($J_A$): The conductivities inducing the axial current are monotonically suppressed as the symmetry-breaking parameter $M$ increases.
• Non-Anomalous Current ($J_W$): Conversely, the conductivities inducing the non-anomalous current $J_W$ are enhanced as $M$ increases.
• Interplay: This demonstrates a distinct competition: explicit symmetry breaking damps the anomalous response of the axial sector while simultaneously activating anomalous transport mechanisms in the non-anomalous sector.

5. Significance and Implications

• Theoretical Insight: The work clarifies that the distinction between "anomalous" and "non-anomalous" transport is not absolute in the presence of explicit symmetry breaking. Anomalies can fundamentally alter the transport properties of conserved currents if the underlying symmetries are broken by scalar condensates.
• Phenomenological Relevance: The authors suggest potential applications in Weyl semi-metals. In these materials, multiple Weyl nodes (valleys) exist. While the total system may conserve valley symmetry, individual valley symmetries might be broken. The results imply that "valley currents" (non-anomalous in the total sense) could exhibit chiral magnetic/vortical effects similar to those observed in anomalous currents, provided valley symmetries are explicitly broken.
• Future Directions: The paper highlights the need to compare these strong-coupling holographic results with weak-coupling field theory calculations to determine which features are universal consequences of quantum anomalies versus artifacts of the holographic regime.

In summary, this paper establishes that explicit symmetry breaking acts as a bridge, allowing quantum anomalies to induce transport phenomena in sectors of the theory that would otherwise be inert, thereby enriching the phenomenology of chiral fluids and condensed matter systems.