⚛️ general relativity

Analog Unruh effect of inhomogeneous one-dimensional Dirac fermions

This paper demonstrates that a sudden transition from a homogeneous to an inhomogeneous Dirac velocity in one-dimensional fermions induces an analog Unruh effect characterized by localized particle creation, which persists even for realistic velocity profiles that deviate from the ideal linear Rindler form.

Original authors: Khristian B. Tallent, Daniel E. Sheehy

Published 2026-08-03

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CC BY 4.0

Original authors: Khristian B. Tallent, Daniel E. Sheehy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing on a quiet beach, watching the waves roll in. To you, the ocean looks calm and empty, a perfect vacuum. But now, imagine you hop on a surfboard and start paddling away from the shore, accelerating faster and faster. Suddenly, the water around you doesn't look so empty anymore. To your accelerating eyes, the ocean seems to be bubbling with new waves and particles that weren't there before. This strange phenomenon is called the Unruh effect. It's a mind-bending idea from the world of theoretical physics, suggesting that "empty space" isn't actually empty if you are moving through it with enough acceleration. Instead, you would feel a warm bath of particles, as if the vacuum itself had a temperature.

Why does this matter? Because the Unruh effect is a cousin to the famous Hawking radiation that makes black holes evaporate. If we could create a tiny, controlled version of this effect in a laboratory, we could test the deepest laws of the universe without needing to build a black hole or travel at near-light speeds. Scientists have been trying to build "analog gravity" systems—using things like sound waves in fluids or electrons in special materials—to mimic these cosmic conditions. The big question is: do these lab experiments need to be perfect to work, or can they be a little messy and still show the magic?

This paper by Khristian B. Tallent and Daniel E. Sheehy dives into that exact question using a one-dimensional line of electrons (called Dirac fermions) that behave like tiny, relativistic particles. The researchers set up a thought experiment where they suddenly change the "speed limit" for these electrons. In the beginning, the electrons move at a constant speed everywhere, like cars cruising on a flat, straight highway. Then, in a flash, they switch to a scenario where the speed limit changes depending on where you are: it gets slower and slower the closer you get to the center, and faster as you move away. This changing speed acts like a fake "acceleration," mimicking the conditions of the Unruh effect.

The team first looked at the "perfect" scenario, where the speed limit changes in a perfectly straight line (mathematically, proportional to the distance from the center). As expected, this created a perfect Unruh effect: the electrons suddenly started popping into existence, forming a thermal distribution that looked exactly like a hot gas, even though the system was technically at absolute zero.

But here is the twist: in the real world, it is incredibly hard to build a system where the speed changes perfectly forever. So, the authors asked, "What if the speed limit only changes near the center and then just stays constant far away?" They tested two realistic shapes for this change: one that looks like a smooth "S" curve (sigmoid) and another that looks like a stretched-out "S" (hyperbolic tangent).

Their findings are surprisingly hopeful. They discovered that even with these imperfect, realistic speed profiles, the Unruh effect still happens! The electrons still get excited and create a "bath" of new particles. However, there is a catch: this effect is not spread out everywhere. The new particles are only created in a small, cozy neighborhood right around the center where the speed was changing. Far away from the center, where the speed is constant, the vacuum remains calm and empty.

The paper concludes that while you don't need a perfect, infinite universe to see the Unruh effect, you do need a specific kind of "distortion" in the middle. The resulting particle creation is localized, meaning if you were a tiny observer in this electron line, you would only feel the heat if you were standing right in the middle of the action. This suggests that future lab experiments don't need to be impossible to build; they just need to be clever enough to create a localized "hot spot" where the laws of physics get a little wobbly, allowing us to peek into the secrets of the accelerating universe right here on Earth.

Technical Summary: Analog Unruh Effect of Inhomogeneous One-Dimensional Dirac Fermions

Problem Statement
The Unruh effect predicts that an observer accelerating through a vacuum in Minkowski spacetime perceives a thermal distribution of particles, whereas a stationary observer sees a vacuum. While this is a fundamental prediction of quantum field theory in curved spacetime, direct observation is experimentally infeasible due to the extreme accelerations required. Analog gravity proposes studying systems obeying similar equations to astrophysical or relativistic systems in controlled laboratory settings.

A specific challenge in realizing an analog Unruh effect using Dirac fermions is the requirement for a spatially varying Dirac velocity $v(x)$ that mimics the Rindler metric. Ideally, this requires a profile where $v(x) \propto |x|$ , corresponding to a uniformly accelerating observer. However, realizing a velocity profile that scales linearly with position for all $x$ is experimentally difficult, as it implies a highly non-perturbative change in local Dirac velocity. The central problem addressed by this paper is to determine the extent to which an analog Unruh effect can be realized in a 1D Dirac fermion system using more realistic, localized velocity profiles (e.g., $v(x) \propto \tanh(|x|/\lambda)$ or sigmoid functions) that are linear only for small $|x|$ and constant for large $|x|$ .

Methodology
The authors analyze a 1D system of Dirac fermions governed by the Hamiltonian:
$H = \sqrt{v(x)} \sigma_x p_x \sqrt{v(x)}$
where $p_x = -i\hbar \partial_x$ and $\sigma_x$ is a Pauli matrix. The study employs a "sudden quench" protocol:

Initial State: The system is prepared in the ground state (Minkowski vacuum) of a homogeneous velocity profile $v(x) = v$ . In this state, negative energy states are fully occupied, and positive energy states are empty.
Quench: The velocity profile is suddenly switched to an inhomogeneous profile $v(x)$ .
Analysis: Within the sudden approximation, the expectation values of observables in the final system are computed with respect to the initial vacuum state. This involves calculating the Bogoliubov transformation between the initial mode operators ( $c_{p\alpha}$ ) and the final mode operators ( $d_{E\alpha}$ ) associated with the eigenfunctions of the inhomogeneous Hamiltonian.

The authors first analyze the ideal case where $v(x) \propto |x|$ , deriving the exact Unruh effect. They then generalize this to two realistic nonlinear profiles:

Sigmoid: $v(x) = v(1 - e^{-|x|/\lambda})$
Hyperbolic Tangent: $v(x) = v \tanh(|x|/\lambda)$

For these nonlinear cases, the authors utilize integral representations of the mode transformation functions and evaluate the resulting integrals using distribution theory (Sokhotski-Plemelj formula) and asymptotic approximations to determine the final particle occupation numbers.

Key Contributions and Results

Recovery of the Ideal Unruh Effect:
For the linear profile $v(x) \propto |x|$ , the authors rigorously derive that the final state exhibits a Fermi-Dirac distribution of particles with a temperature:
$T_U = \frac{v\hbar}{2\pi \lambda k_B}$
This confirms that the 1D Dirac system with a linear velocity gradient exactly reproduces the Unruh effect, where the "acceleration" is effectively determined by the ratio $v/\lambda$ .
Approximate Unruh Effect in Realistic Profiles:
The paper demonstrates that for realistic profiles (sigmoid and tanh) which are linear only for $|x| \ll \lambda$ and constant for $|x| \gg \lambda$ , an approximate Unruh effect still emerges.
- The final occupation number of positive-energy states follows a Fermi distribution characterized by the same Unruh temperature $T_U$ as the ideal linear case.
- While the effect is not exact (due to the deviation from linearity at large $|x|$ ), the dominant term in the occupation probability is proportional to an energy delta function, yielding a thermal-like distribution.
- The authors show that the deviation from a perfect thermal state is captured by sub-leading terms involving principal value integrals, but the thermal signature remains robust.
Spatial Localization of Created Particles:
A significant finding is the spatial distribution of the created positive-energy fermions. The local density of these particles, $\Delta n(x)$ , is found to be proportional to $1/v(x)$ .
- Since $v(x) \to 0$ as $x \to 0$ for the studied profiles, the created particles are spatially localized near $x=0$ .
- Conversely, there is a depletion of negative-energy states (creation of holes) in the same region.
- The total local charge density remains unchanged, but the energy-dependent density creates a distinct signature that could be probed by energy-sensitive techniques.

Significance
The paper claims that the physics of the Unruh effect is not restricted to idealized, globally linear velocity profiles but can emerge in condensed matter systems that only approximately realize the necessary conditions. By showing that realistic, experimentally feasible velocity profiles (such as those achievable via strain engineering or light fields in graphene-like systems) still yield a thermal distribution of particles with the correct Unruh temperature, the work broadens the scope of analog gravity simulations.

The authors emphasize that this is not a true thermal state (the system remains in a zero-temperature vacuum state globally), but the observables mimic a thermal distribution. The spatial localization of the effect near the region of maximum velocity gradient ( $x=0$ ) provides a specific, testable signature for future experimental verification in 1D Dirac materials. The study suggests that the Unruh effect is a robust phenomenon in analog systems, persisting even when the mapping to a uniformly accelerating observer is only local rather than global.

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