Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap

This paper investigates the equilibrium partition function of non-relativistic conformal field theories in harmonic traps, revealing that the logarithm of the partition function exhibits universal simple poles in the difference between the squared trapping frequency and squared angular velocities, with residues determined by the equation of state in the hydrodynamic regime and by specific thermodynamic variables in the large-angular-momentum limit.

The Gist

Here is an explanation of the paper "Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap" using simple language and creative analogies.

The Big Picture: A Dance in a Bowl

Imagine a giant, invisible bowl (a harmonic trap) sitting on a table. Inside this bowl, you have a swarm of tiny particles (like atoms in a cold gas). These particles are dancing around.

Usually, physicists study how these particles behave when they are just sitting still or moving randomly. But this paper asks a very specific question: What happens if we spin the bowl really, really fast?

The author, Eunwoo Lee, investigates what happens to the "energy bill" (the partition function) of this spinning system when the spin gets so fast that the particles are almost flying out of the bowl, held back only by the bowl's shape.

The Two Main Rules of the Game

The paper looks at two different "modes" of this spinning dance:

1. The "Fluid" Mode (The Smooth Spin)

Imagine the particles are like water in a spinning bucket. If you spin the bucket slowly, the water stays smooth, forming a nice, flat curve.

• The Discovery: In this smooth regime, the math governing the system is surprisingly simple. The "energy bill" depends mostly on how hot the water is and how much "stuff" (particles) is in it.
• The Analogy: It's like a recipe. If you know the temperature and the amount of ingredients, you can predict the taste of the soup perfectly, regardless of whether it's a tomato soup or a potato soup. The paper calls this Universal because the specific type of particle doesn't matter much; the rules of the "fluid" dominate.

2. The "Super-Spin" Mode (The Edge of Chaos)

Now, imagine you spin the bucket so fast that the water is clinging to the very rim, barely staying inside. The centrifugal force (the force pushing things outward) is almost canceling out the gravity (or the trap) pulling things in.

• The Discovery: When the spin gets this extreme, the system enters a "Semi-Universal" state.
• The Analogy: Think of a tightrope walker. In the smooth mode, they walk normally. In the super-spin mode, they are balancing on a razor-thin wire. The way they balance (the math structure) is the same for everyone (a simple pole, or a sharp spike in the math), but how well they balance depends on their specific shoes and training (the specific details of the particles).
• The Result: The math shows a "pole" (a mathematical infinity) that appears when the spin speed hits a critical limit. The height of this spike is determined by a mix of temperature and particle density, but the shape of the spike is the same for many different types of systems.

Why Does This Matter? (The "Twist")

In physics, there's a concept called Twist. Imagine a rubber band. If you stretch it, it has energy. If you twist it, it has "twist" energy.

• In this paper, the "Twist" is the difference between the total energy of the system and the energy of the spinning motion.
• The paper proves that even in these weird, fast-spinning quantum systems, there is a fundamental limit to how much you can twist the system before it breaks or changes phase. It's like a safety valve: the system naturally resists being twisted too far.

Real-World Examples: The Cold Atom Lab

The author doesn't just do math on paper; they check if this works in real experiments.

• Free Particles: Imagine a bunch of independent dancers who don't talk to each other. The math works perfectly.
• Superfluids (The "Super-Dancers"): Imagine a group of dancers holding hands, moving as one giant, frictionless wave (a superfluid). This is what happens in cold atom experiments (like fermions at "unitarity").
  • When these superfluids spin fast, they don't just spin smoothly. They start forming vortices (tiny tornadoes).
  • As the spin increases, these tornadoes multiply until they form a Vortex Lattice (a crystal of tornadoes).
  • The paper shows that even with these complex tornadoes, the "energy bill" still follows the same "Semi-Universal" rule found in the simple fluid mode.

The "Grey Galaxy" Idea (A Glimpse into the Future)

The paper ends with a fascinating speculation about Holography (a theory that says our 3D universe might be a projection of a 2D surface).

• The author suggests that in the "super-spin" limit, the system might behave like a Black Hole that is slowly evaporating into a gas.
• They propose a new phase of matter called a "Grey Galaxy"—a weird middle ground between a dense black hole and a thin gas cloud. It's like a galaxy that is neither fully formed nor fully dissolved, existing in a delicate balance.

Summary in One Sentence

This paper discovers that when you spin a quantum system fast enough, its behavior becomes surprisingly predictable and follows a universal mathematical pattern, whether it's a simple gas or a complex superfluid, revealing a hidden "safety limit" on how much energy and spin the universe can hold before things change fundamentally.

Technical Summary

Here is a detailed technical summary of the paper "Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap" by Eunwoo Lee.

1. Problem Statement

The paper investigates the thermodynamic properties of Non-Relativistic Conformal Field Theories (NRCFTs) governed by the Schrödinger symmetry group, specifically focusing on their equilibrium partition functions in the presence of a harmonic trapping potential and rotation.

While the behavior of relativistic CFTs in rotating frames is well-understood (exhibiting universal scaling at high temperatures and "semi-universal" pole structures near the speed-of-light bound), the analogous behavior for NRCFTs—where a conserved particle number (mass) operator exists—remains less explored. The central problem is to determine the structure of the partition function $\log Z$ in two specific regimes:

1. The Hydrodynamic Regime: Where the system is well-described by fluid dynamics.
2. The Large-Angular-Momentum Regime: Where the angular velocities $\Omega_a$ approach the trapping frequency $\omega$ (the physical bound), a limit where centrifugal forces nearly cancel the trapping potential.

The author seeks to establish whether a "semi-universal" structure exists in NRCFTs similar to that found in relativistic CFTs, where the partition function develops simple poles as $\Omega_a \to \omega$, and to characterize the residues of these poles.

2. Methodology

The paper employs a multi-faceted approach combining hydrodynamics, effective field theory (EFT), and explicit microscopic calculations:

• Hydrodynamic Derivative Expansion:

  • The author constructs stationary solutions for non-relativistic fluids in a harmonic trap.
  • By imposing shearless and divergence-free conditions, the fluid velocity is identified as a Killing vector field corresponding to rigid rotations with angular velocities $\Omega_a$.
  • The equilibrium partition function is derived by integrating the pressure profile over the spatial volume, utilizing the local density approximation.

• Newton-Cartan Geometry and Effective Action:

  • To go beyond the strict fluid regime, the partition function is constructed as a functional of background geometric data using Newton-Cartan geometry (the non-relativistic limit of spacetime geometry).
  • A derivative expansion of the generating functional $W$ is performed, constrained by Schrödinger symmetry, Galilean invariance, and Weyl invariance.
  • The analysis focuses on the behavior of terms in the expansion as $\Omega_a \to \omega$, demonstrating that higher-order derivative terms share the same divergence structure as the leading order.

• Microscopic Verification:

  • Free Theories: The exact partition functions for free bosons and fermions in a harmonic trap are calculated using the grand canonical ensemble. The large-spin limit ($\beta(\omega - \Omega) \ll 1$) is explicitly expanded to verify the predicted pole structure.
  • Superfluids (Cold Atoms): The paper analyzes strongly interacting systems, specifically fermions at unitarity, using the Large-Charge Effective Field Theory. It examines the transition from phonon modes to vortex formation and finally to a vortex lattice as angular momentum increases.

3. Key Contributions and Results

A. Universal Structure in the Hydrodynamic Regime

In the fluid-dynamical regime (valid when the Knudsen number is small), the logarithm of the partition function takes the form:
$$\log Z \approx \frac{\mu^d F(\mu/T)}{\prod_{a=1}^r (\omega^2 - \Omega_a^2)} \times \begin{cases} 1/\omega & d=2r+1 \\ 1 & d=2r \end{cases}$$

• Result: The partition function exhibits simple poles at $\Omega_a = \omega$.
• Universality: The residue is determined by a single-variable function $F(\mu/T)$, making it highly universal across different NRCFTs with the same equation of state.

B. Semi-Universality in the Large-Angular-Momentum Limit

In the limit where $\Omega_a \to \omega$ (large spin), the system may exit the strict hydrodynamic regime. However, the paper argues that the pole structure persists:
$$\log Z \approx \frac{\mu^d F(\mu/T, \omega/T)}{\prod_{a=1}^r (\omega^2 - \Omega_a^2)}$$

• Key Distinction: Unlike the hydrodynamic regime, the residue function $F$ now depends on two dimensionless parameters: $\mu/T$ and $\omega/T$.
• Significance: This dependence on $\omega/T$ introduces theory-specific dynamical information, rendering the result "semi-universal." The functional form (the pole structure) is universal, but the coefficient is not.

C. Microcanonical Entropy and Twist

By performing a Legendre transform, the author derives the microcanonical entropy $S$ in the limit of large angular momenta $J_a$:
$$S(\tau, Q, J_a) \approx \left( \prod_{a=1}^r J_a \right)^{\frac{1}{r+1}} s_{int} \left( \frac{\tau}{(\prod J_a)^{\frac{1}{r+1}}}, \frac{Q}{(\prod J_a)^{\frac{1}{r+1}}} \right)$$

• Here, $\tau = E - \omega \sum J_a$ is the non-relativistic analogue of the "twist."
• The term $(\prod J_a)^{\frac{1}{r+1}}$ acts as an effective spatial volume.
• The paper discusses the positivity of the twist ($\tau \geq 0$), noting that while it holds for fluids and free theories, it is not guaranteed by unitarity bounds alone for general NRCFTs.

D. Verification via Examples

• Free Bosons/Fermions: Explicit calculation confirms the pole structure $\sim (\omega^2 - \Omega^2)^{-1}$ and the specific form of the residue functions $\tilde{h}_{bos/ferm}$.
• Cold Atom Superfluids:
  • At low angular momentum, the system is described by phonons.
  • At intermediate angular momentum, the system forms a vortex lattice. The effective volume of the cloud diverges as $V \sim [\mu/(\omega^2 - \Omega^2)]^{d/2}$, reproducing the semi-universal scaling.
  • At extremely high angular momentum (near the Regge limit), the system transitions to a Fractional Quantum Hall (FQH) phase or a dilute gas, yet the semi-universal scaling behavior is argued to persist.

4. Significance and Implications

• Extension of Relativistic Results: The paper successfully extends the concept of "semi-universality" from relativistic CFTs (studied in works like [5]) to the non-relativistic domain, establishing a robust framework for NRCFT thermodynamics.
• Holographic Applications: The results provide a concrete prediction for the thermodynamics of rotating black holes in Schrödinger spacetimes (the holographic duals of NRCFTs). The author speculates on the existence of a "Grey Galaxy" phase (analogous to the relativistic case) interpolating between black holes and thermal gas, governed by the derived scaling laws.
• Experimental Relevance: The analysis of cold atoms at unitarity provides a direct link to current experimental setups in ultracold quantum gases, offering theoretical predictions for the behavior of rotating superfluids in harmonic traps.
• Theoretical Insight: The derivation clarifies the role of the conserved particle number in non-relativistic conformal symmetry and how it modifies the scaling of thermodynamic potentials compared to the relativistic case.

In summary, Eunwoo Lee demonstrates that despite the lack of a full gravitational dual for many NRCFTs, the interplay between conformal symmetry, harmonic trapping, and rotation leads to a robust, semi-universal thermodynamic structure characterized by simple poles in the partition function, with residues encoding the specific dynamics of the theory.