A complex scalar field theory for charged fluids, superfluids, and fracton fluids

This paper proposes a complex scalar field theory defined on a comoving spacelike hypersurface that unifies the hydrodynamics of charged normal fluids, superfluids, and a novel fracton fluid phase by distinguishing them through specific chemical shift symmetries that govern charge mobility, thereby providing a UV completion to existing effective hydrodynamic theories.

The Gist

Imagine a bustling city where millions of people (particles) are moving around. In physics, we usually study how these crowds move using fluid dynamics. Think of a river flowing, steam rising, or traffic jamming.

This paper proposes a new, very specific way to understand how charged fluids (like a super-hot plasma or a super-cold superfluid) move, using a mathematical "map" that changes depending on how the people in the crowd behave.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Moving Map (The Comoving Hypersurface)

Usually, when we watch a river, we stand on the bank and watch the water flow past us.

• The Paper's Idea: Instead of standing on the bank, imagine you are a leaf floating on the water. You move with the current.
• The Analogy: The author creates a "map" that is glued to the fluid itself. As the fluid moves, the map moves with it. On this moving map, the particles have fixed addresses. This makes it much easier to track what is happening inside the fluid without getting confused by the fluid's overall motion through space.

2. The Three Types of Fluids

The paper uses a single mathematical framework (a complex scalar field, which is like a giant, invisible compass needle) to describe three different "personalities" of fluids.

A. The "Frozen" Crowd (Normal Charged Fluids)

• The Scenario: Imagine a crowded dance floor where everyone is holding hands in a rigid formation. If you try to move, you can't move yourself; you can only move if the whole group shuffles you along.
• The Physics: In a normal charged fluid, the electric charges are "glued" to the fluid parcels. They cannot wiggle around independently on the moving map.
• The "Fracton" Twist: The paper calls this fractonic behavior. It's like the charges are "fractons"—particles that are stuck in place unless the whole system moves them. They can't hop from one spot to another on the map; they are dragged along for the ride.
• The Result: This creates a very restrictive rule (called "chemical shift symmetry") that prevents the charges from rearranging themselves. The fluid behaves like a standard, predictable liquid.

B. The "Free-Range" Crowd (Superfluids)

• The Scenario: Now, imagine the dance floor turns into a zero-friction ice rink. The people (charges) are no longer holding hands. They can slide freely across the ice, rearranging themselves however they want, even while the whole rink is moving.
• The Physics: This is the Superfluid phase. The restrictive "glue" is broken. The charges can now flow independently of the main fluid motion.
• The Result: Because the charges can move freely, a new type of wave appears. In normal fluids, you have sound waves (pressure waves). In superfluids, you get a second type of wave called "Second Sound," which is essentially a wave of heat or entropy moving through the fluid.

C. The "Half-Stuck" Crowd (Fracton Fluids)

• The Scenario: This is the paper's big new idea. Imagine a crowd where people can move, but only in very specific, limited ways. Maybe they can slide left and right, but they can't move forward or backward. Or maybe they can only move in pairs.
• The Physics: The author proposes a middle ground between the "Frozen" and "Free-Range" crowds. They call this a Fracton Fluid.
• The Rule: The charges are constrained by a "linear shift symmetry." Think of it like a rule that says: "You can move, but your movement must be perfectly balanced with your neighbor's."
• The Result: This creates a unique type of wave that behaves like a magnon (a magnetic wave) or a "quadratic" wave. It's slower and moves differently than normal sound. It's a fluid that is neither a solid nor a normal liquid, but something exotic in between.

3. Why Does This Matter? (The "UV Completion")

In physics, we often use "Effective Field Theories" (EFTs).

• The Analogy: An EFT is like a weather forecast. It tells you "it will rain tomorrow" based on current patterns, but it doesn't explain the quantum mechanics of every single water molecule.
• The Problem: Sometimes, these forecasts break down. If you try to predict a storm too far in advance, the math explodes (singularities).
• The Solution: This paper provides the "microscopic blueprint" (the UV Completion). It builds the fluid theory from the ground up using a complex scalar field (the "compass needle").
  • It explains why the normal fluid acts stuck.
  • It explains why the superfluid acts free.
  • It predicts the existence of the new "Fracton Fluid" phase.

Summary

The author has built a universal translator for fluids.

1. Normal Fluid: Charges are glued to the fluid. (Like a rigid dance).
2. Superfluid: Charges are free to roam. (Like an ice rink).
3. Fracton Fluid: Charges are partially stuck, moving only in specific patterns. (Like a dance where you can only step in time with a partner).

By using a "moving map" (comoving hypersurface) and a single mathematical field, the paper shows how all these different behaviors emerge from the same underlying rules, offering a deeper, more complete understanding of how matter flows in the universe, from the smallest quantum particles to the largest cosmic plasmas.

Technical Summary

1. Problem Statement

Hydrodynamics is traditionally formulated as an effective field theory (EFT) based on collective variables (density, velocity, temperature) and phenomenological transport coefficients. While successful, this approach often lacks a microscopic "UV completion," leading to pathologies such as caustic singularities in nonlinear regimes and an inability to fully capture the microscopic origin of transport coefficients.

Specifically, there is a need for a unified field-theoretic framework that:

1. Describes charged relativistic fluids (normal phase) and finite-temperature superfluids (two-fluid model) from first principles.
2. Explains the emergence of fractonic behavior (restricted mobility of charges) in the normal phase.
3. Provides a UV completion for existing EFTs of hydrodynamics that are formulated solely in terms of Goldstone bosons.
4. Interpolates between normal fluids and superfluids via intermediate phases with partially restricted charge mobility.

2. Methodology

The author proposes a comoving hypersurface formulation of fluid dynamics. Instead of defining fields on spacetime directly, the theory is constructed on a spacelike hypersurface that moves with the fluid.

• Geometric Setup: The fluid is described by scalar fields $\phi^I$ ($I=1,2$) acting as comoving coordinates. These define a foliation of spacetime. The geometry of this comoving space is encoded in a metric $B_{IJ}$ and a volume form.
• Field Content: A complex scalar field $\Phi$ is defined on this comoving hypersurface. This field transforms linearly under the global $U(1)_Q$ charge symmetry.
• Symmetry Structure: The core innovation lies in the realization of symmetries:
  • Normal Phase: The theory possesses a chemical shift symmetry where the phase of $\Phi$ can shift by an arbitrary function of the comoving coordinates, $\Phi \to e^{i\Lambda(\phi^I)}\Phi$. This implies the conservation of all multipole moments of the charge density in the comoving frame, effectively freezing charge motion relative to the fluid elements (fractonic behavior).
  • Superfluid Phase: The symmetry is relaxed to a constant shift $\Phi \to e^{i\lambda}\Phi$, allowing free redistribution of charge on the hypersurface.
  • Fracton Fluid Phase: An intermediate phase is proposed where only a subset of the shift symmetries (e.g., linear shifts) is preserved, restricting mobility to conserve specific multipole moments (e.g., dipole moment).
• Action Principle: The dynamics are derived from a relativistic action $S = \int d^3x \sqrt{-g} \mathcal{L}$, where the Lagrangian is constructed to be invariant under Poincaré transformations, Area-Preserving Diffeomorphisms (APDs) of the comoving coordinates, and the specific charge symmetries.

3. Key Contributions

A. UV Completion of Charged Fluids and Superfluids

The paper constructs explicit actions that serve as UV completions for the EFTs of charged fluids [6] and finite-temperature superfluids [7].

• Normal Fluid Action:
  $$S = \int d^3x \sqrt{-g} \left[ \frac{i}{2}(\Phi^\dagger D_0 \Phi - \Phi D_0 \Phi^\dagger) - E(|\Phi|, b) \right]$$
  Here, $D_0 = u^\mu(\partial_\mu - iA_\mu)$ is the covariant derivative along the fluid velocity. The restrictive chemical shift symmetry ensures that charges are "locked" to fluid elements, reproducing the standard relativistic Euler equations.
• Superfluid Action:
  The superfluid action includes a spatial gradient term on the comoving hypersurface (via the projector $\Pi^{\mu\nu}$), allowing charge transport transverse to the fluid velocity:
  $$S_{SF} = \int d^3x \sqrt{-g} \left[ \frac{i}{2}(\Phi^\dagger D_0 \Phi - \Phi D_0 \Phi^\dagger) - \frac{1}{2m} \Pi^{\mu\nu} D_\mu \Phi^\dagger D_\nu \Phi - V(|\Phi|) + F(b) \right]$$
  This reproduces the Landau two-fluid model, including the coupling between normal and superfluid components.

B. Emergence of Fractonic Phenomenology

The paper demonstrates that the chemical shift symmetry in the normal phase naturally leads to fractonic immobility.

• The symmetry $\Phi \to e^{i\Lambda(\phi^I)}\Phi$ implies the conservation of all comoving multipole moments $Q_{I_1...I_N} = \int \rho \phi^{I_1}...\phi^{I_N}$.
• Consequently, the Goldstone mode associated with the phase $\psi$ (where $\Phi = \sqrt{\rho}e^{i\psi}$) lacks a kinetic term in the comoving frame.
• Result: In the normal phase, the phase mode is non-propagating (fractonic), and the only propagating mode is the standard longitudinal sound wave.

C. Proposal of Fracton Fluids

The author introduces a new phase of matter: Fracton Fluids.

• Symmetry: These fluids preserve only a finite subgroup of the chemical shift symmetry (e.g., linear shifts $\Lambda(\phi) = \Lambda_0 + \Lambda_I \phi^I$).
• Consequence: This requires restricting the APD symmetry group from $SDiff(\mathbb{R}^2)$ to the affine subgroup $SL(2,\mathbb{R}) \ltimes \mathbb{R}^2$.
• Dynamics: The resulting theory supports a propagating "second sound" mode with a quadratic dispersion relation ($\omega \sim k^2$), characteristic of magnons in fracton systems, alongside the standard linear sound mode.

4. Key Results

1. Recovery of Hydrodynamic Equations:

   • The equations of motion derived from the proposed actions exactly match the relativistic Euler equations for charged fluids and the Landau two-fluid equations for superfluids.
   • Explicit constitutive relations for the stress-energy tensor $T^{\mu\nu}$ and charge current $J^\mu$ are derived, mapping the microscopic fields ($\Phi, \phi^I$) to hydrodynamic variables ($\rho, u^\mu, T, \mu$).

2. Symmetry Breaking Patterns:

   • Normal Phase: $ISO(2,1) \times (SDiff \ltimes CShift(\infty)) \to (SO(2) \ltimes \mathbb{R}^2) \times \mathbb{R}$.
   • Superfluid Phase: The infinite chemical shift symmetry breaks down to constant $U(1)$, allowing the phase mode to acquire a kinetic term.
   • Fracton Fluid Phase: The symmetry breaks to a subgroup preserving dipole moments, leading to a specific pattern of gapless excitations.

3. Excitation Spectra:

   • Normal Fluid: One linear sound mode ($\omega = c_s k$). The phase mode is gapped/non-propagating due to the restrictive symmetry.
   • Superfluid: Two linear sound modes (First and Second sound). The phase mode becomes propagating.
   • Fracton Fluid: One linear sound mode and one quadratic mode ($\omega \sim k^2$). This quadratic mode arises from the comoving dipole symmetry and represents a novel excitation not found in ordinary fluids.

4. Thermodynamic Consistency:
   The models satisfy standard thermodynamic relations (e.g., $\epsilon + p = Ts + \mu \rho$) and correctly identify temperature and entropy density in terms of the field variables.

5. Significance and Impact

• Unification: The work provides a single, coherent field-theoretic framework that unifies the description of normal charged fluids, superfluids, and exotic fracton fluids.
• Microscopic Origin of Hydrodynamics: By providing a UV completion via a complex scalar field, the paper resolves the issue of caustic singularities found in pure Goldstone EFTs and offers a microscopic derivation of transport coefficients and constitutive relations.
• New Phase of Matter: The proposal of "Fracton Fluids" as an interpolating phase between normal and superfluid states is a significant theoretical advancement. It suggests that fractonic behavior (restricted mobility) is not limited to crystalline solids or specific lattice models but can emerge naturally in relativistic fluid dynamics through symmetry constraints.
• Bridge to Gravity and Geometry: The comoving hypersurface approach aligns fluid dynamics with geometric formulations of elasticity and gravity, offering new tools to study fluids in curved spacetime and potentially in the context of holography.
• Future Directions: The paper outlines paths toward incorporating dissipation (via Schwinger-Keldysh formalism), spinful fluids (relevant for heavy-ion collisions), and incompressible quantum Hall fluids, suggesting a broad applicability of this framework.

In summary, Głodkowski's paper establishes a rigorous field-theoretic foundation for relativistic hydrodynamics, revealing that the mobility of charges in fluids is fundamentally governed by the structure of shift symmetries on the comoving hypersurface, thereby bridging the gap between standard hydrodynamics and the emerging physics of fractons.