⚛️ high-energy theory

Emergent Carroll symmetry at phase separation in one-dimensional lattice systems

This paper establishes that an emergent Carrollian symmetry governs the physics of phase separation in one-dimensional lattice systems, demonstrating that Carroll Conformal Field Theory provides a successful analytic and numerical framework for describing density-density correlations where standard relativistic CFT techniques fail.

Original authors: Sourav Biswas, Ashutosh Dubey, Saikat Mondal, Aritra Banerjee, Arijit Kundu, Arjun Bagchi

Published 2026-07-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sourav Biswas, Ashutosh Dubey, Saikat Mondal, Aritra Banerjee, Arijit Kundu, Arjun Bagchi

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 the universe as a giant, bustling dance floor. In most places, the dancers (particles) move with a certain rhythm, bumping into each other, flowing like a liquid. Physicists have a fantastic set of tools to describe this dancing, called "Relativistic Conformal Field Theory." Think of it as the ultimate choreography guide that works perfectly when the dancers are moving at a steady, universal speed limit, much like how light travels. This guide helps us understand how materials behave when they are on the edge of changing their state, like water freezing into ice.

But what happens when the music stops, or the speed limit suddenly drops to zero? Imagine a dance floor where the dancers can't move forward or backward at all; they can only wiggle in place, instantly reacting to their neighbors without any travel time. This is a strange, frozen kind of physics. For a long time, scientists thought their usual choreography guides broke down completely in these "frozen" zones. They called this breakdown a "Phase Separation," where the liquid suddenly splits into clumps of dancers and empty spaces, and the old math just couldn't explain what was happening. It was a mystery corner of the science world where the rules seemed to vanish.

Now, a team of researchers has stepped in with a new, surprising idea. They discovered that when the dance floor freezes, it doesn't just break the old rules; it switches to a completely different set of rules called "Carrollian symmetry." If the old rules were like a fast-paced highway, these new rules are like a system where time moves, but space is stuck. The paper shows that this "frozen" state isn't a glitch, but a new kind of order governed by this ultra-local symmetry. They didn't just guess this; they used powerful computer simulations to watch the particles behave and found that their movements matched the predictions of this new "Carroll" dance perfectly.

The Paper's Discovery: When the Dance Floor Freezes

The paper focuses on a specific type of one-dimensional system, which you can imagine as a single, long line of particles (like beads on a string) that can hop around and push against each other. Usually, when these particles interact, they form a "Luttinger Liquid," a state that behaves like a fluid and is well-understood by the standard relativistic rules. However, the researchers were interested in what happens when you crank up the interaction strength between the particles to a critical point.

At this specific point, the particles stop flowing like a liquid and instead "phase separate." They clump together into dense islands, leaving empty gaps between them. In the past, physicists struggled to describe this transition because the standard tools failed. The authors of this paper propose that the solution lies in an "emergent Carroll symmetry."

To understand this, think of the speed of sound in the material. In a normal liquid, sound waves travel at a certain speed. As the system approaches the phase separation point, this speed of sound slows down and eventually hits zero. The paper argues that when the speed of sound (or any characteristic velocity) hits zero, the system doesn't just stop; it transforms. It becomes "Carrollian." In this Carrollian world, the system becomes "ultralocal," meaning what happens at one spot is instantly connected to its immediate neighbor, but there is no propagation of information across space in the usual way. It's as if the universe at that point has turned off the "travel" setting and only kept the "instant reaction" setting.

What They Found: The Math Matches the Simulation

The team didn't just write down equations; they tested them. They used a method called Density Matrix Renormalization Group (DMRG), which is like a super-advanced computer simulation, to model a lattice of spinless fermions (particles that can't occupy the same spot). They watched what happened as they tuned the interaction strength to the critical point where phase separation occurs.

They looked at how the particles correlated with each other—essentially, how the density of particles at one point related to the density at another point.

Before the transition: When the system was a normal liquid, the correlation followed a straight line (linear) when plotted against momentum. This matched the old, standard physics.
At the transition: As they hit the phase separation point, the correlation suddenly changed its shape. It stopped being a straight line and became a curve that grew much faster (quadratic).

The paper shows that this specific change—from a linear relationship to a quadratic one—is exactly what the Carrollian symmetry predicts. The math of the "Carroll CFT" (Conformal Field Theory) predicted this quadratic behavior, and the computer simulation confirmed it.

Furthermore, the paper highlights two other weird features that appear at this point, which the Carroll framework explains naturally:

Infinite Degeneracy: The system has a massive number of different ways to arrange itself that all have the exact same energy. It's like having a million different dance formations that all look equally "frozen" and cost the same amount of energy to hold. The Carroll symmetry, with its "supertranslation" properties, explains why this happens.
Divergent Compressibility: The material becomes incredibly easy to squeeze. The paper shows that as you approach this point, the compressibility (how much the volume changes when you push) shoots up to infinity. This is a classic sign of phase separation, and the Carroll model ties this directly to the vanishing speed of sound.

Beyond the Line: What It Means

The authors are careful to note that while they have found a perfect match for this specific model (the $t-V$ model of spinless fermions) and another model (the $t-J$ model), they aren't claiming this is the only way phase separation happens everywhere. However, they suggest a powerful new way of thinking: whenever a system transitions from a flowing state to a clumped state because the "speed limit" drops to zero, it likely enters a Carrollian phase.

They even speculate that this idea might reach far beyond tiny particles in a lab. They mention that in the early universe, during a period called inflation, or even in the extreme physics of string theory near a "Hagedorn temperature," similar "frozen" conditions might arise where Carroll symmetry could be the key to understanding what's happening.

In short, the paper solves a long-standing puzzle about how particles behave when they stop flowing and start clumping. It reveals that the "broken" state isn't a failure of physics, but a switch to a new, ultra-local kind of physics called Carrollian symmetry. By matching computer simulations with this new theory, they've shown that when the speed of sound hits zero, the universe doesn't go silent; it just starts dancing to a different, ultra-local beat.

Technical Summary: Emergent Carroll Symmetry at Phase Separation in One-Dimensional Lattice Systems

Problem Statement
The theory of phase transitions in one-dimensional (1D) condensed matter systems is traditionally governed by relativistic Conformal Field Theories (CFT), specifically the Tomonaga-Luttinger Liquid (LL) framework. However, this paradigm encounters a fundamental breakdown at the point of Phase Separation (PS). In the LL parameter space, the PS region corresponds to a limit where the effective velocity $u \to 0$ while the Luttinger parameter $K \to \infty$ (specifically $uK \to 1$ but $u/K \to 0$ ). At this point, standard relativistic CFT techniques fail to describe the physics, leading to an instability that has historically been difficult to analyze analytically. The authors identify a gap in understanding the continuum description of this specific regime where conventional scaling symmetries associated with Poincaré invariance are no longer applicable.

Methodology
The paper employs a dual approach combining continuum field theory analysis with numerical lattice simulations:

Continuum Analysis: The authors analyze the bosonized Luttinger Hamiltonian in the limit $u \to 0$ . They demonstrate that this limit reduces the system to an "Electric" Carrollian scalar field theory. They utilize the known symmetry generators, Ward identities, and correlation functions of Conformal Carroll (CC) CFTs to derive analytical predictions for physical observables, specifically focusing on the static density-density structure factor and the nature of the ground state degeneracy.
Numerical Simulation: To test these continuum predictions, the authors perform Density Matrix Renormalization Group (DMRG) calculations on two distinct lattice models:
- The spinless fermionic $t-V$ model (nearest-neighbor hopping and interaction).
- The spinful fermionic $t-J$ model.
  They compute the density-density structure factor $S(q)$ , the compressibility $\kappa$ , and the scaling exponent $\epsilon$ (defined as $d \log S(q) / d \log q$ ) across the phase separation boundary ( $V=2$ in the $t-V$ model).

Key Contributions and Results

Emergence of Carroll Symmetry: The central finding is that the approach to the Phase Separation point is governed by an emergent Carrollian symmetry. Unlike Galilean symmetry (obtained as $c \to \infty$ ), Carroll symmetry arises when the characteristic velocity scale (here, the effective Fermi velocity) goes to zero ( $c \to 0$ ). The authors show that the PS region is naturally described by an Electric Carroll scalar field theory.
Scaling of Density-Density Correlations:
- In the standard LL regime ( $V < 2$ ), the density-density correlator in momentum space scales linearly with momentum ( $S(q) \sim |q|$ ), consistent with relativistic CFT.
- At the PS point and beyond ( $V \ge 2$ ), the authors predict and numerically verify a transition to a quadratic dependence ( $S(q) \sim q^2$ ). This corresponds to a scaling exponent $\epsilon$ shifting from 1 to 2.
- This quadratic scaling is derived analytically from the Carroll two-point function $\langle \rho(q)\rho(-q) \rangle \sim -q^2$ , which reflects the ultralocal nature of Carroll theories.
Ground State Degeneracy and Supertranslations: The Carroll CFT framework predicts an infinite-dimensional supertranslation symmetry. In the lattice models, this manifests as an extensive low-energy manifold of degenerate ground states. The energy of these states depends on the number of domain walls (separating particle-rich and hole-rich regions) rather than their positions, leading to a massive degeneracy observed in the numerical spectra.
Divergent Compressibility: The study confirms that the compressibility $\kappa$ diverges at the PS point ( $V=2$ ). This is consistent with the $K \to \infty$ limit of the LL theory and is interpreted within the Carroll framework as a consequence of the vanishing sound velocity and the specific structure of the Carroll Hamiltonian.
Universality Across Models: The transition from $\epsilon \sim 1$ to $\epsilon \sim 2$ and the associated features (divergent compressibility, domain wall segregation) are observed in both the spinless $t-V$ model and the spinful $t-J$ model, suggesting that the Carrollian description is robust across different microscopic contents.

Significance and Claims
The paper posits that the physics of Phase Separation in Luttinger Liquids, where conventional relativistic methods fail, is fundamentally captured by an emergent Carrollian description. The authors claim that this framework provides a "single structural origin" for three simultaneous phenomena observed at the transition:

The $q^2$ scaling of the static structure factor.
The divergence of compressibility.
The extensive ground-state degeneracy associated with supertranslations.

The authors are explicit about the scope of their claims:

They establish that the PS boundary is governed by a Carroll CFT, supported by matching lattice observations with continuum predictions.
They suggest, but do not prove, that the gapped phase beyond the PS point ( $V > 2$ ) may be described by a generic Carroll Quantum Field Theory (QFT), noting that this is a "suggestive correspondence" rather than an established equivalence, as the gapped phase can also be described via standard Ising ferromagnet domain walls.
They propose a broader conjecture: whenever a phase transition involves a gapless phase governed by a relativistic CFT where the effective velocity is driven to zero, the transition point is naturally captured by a Carroll CFT.

The work adds to the growing body of literature exploring the relevance of Carroll physics in condensed matter systems, offering a new analytic paradigm for studying instabilities in 1D systems that were previously inaccessible to standard CFT techniques.

Technical Summary: Emergent Carroll Symmetry at Phase Separation in One-Dimensional Lattice Systems

More like this