Quantum turbulence in the many-body regime

Technical Summary: Quantum Turbulence in the Many-Body Regime

Problem Statement
The paper addresses a gap in the understanding of turbulent hydrodynamics in quantum fluids. While the phenomenology of turbulence in superfluids is well-established within the mean-field approximation (governed by the Gross-Pitaevskii or GP equation), this regime assumes that quantum fluctuations are negligible compared to the mean boson density. In typical superfluids deep within the ordered phase, local number fluctuations are small, allowing the system to be described by classical fields and classical hydrodynamic equations (specifically, a compressible Euler fluid with a quantum potential).

However, the authors argue that there exists a distinct "many-body regime" of quantum turbulence where quantum fluctuations are strong and play a crucial role in determining macroscopic observables. This regime occurs near zero-temperature quantum phase transitions, such as the superfluid-insulator transition, where the order parameter expectation value vanishes ($\langle \hat{\psi} \rangle = 0$) and fluctuations occur at all length and time scales. In these conditions, the mean-field GP equation breaks down, and the hydrodynamic behavior may exhibit signatures of genuine many-body quantum effects that are not captured by classical turbulence theories or standard mean-field quantum hydrodynamics.

Methodology and Theoretical Framework
The paper employs a phenomenological and theoretical approach to synthesize existing knowledge of quantum turbulence with modern many-body physics concepts. The methodology involves:

1. Review of Mean-Field Turbulence: The authors first review the established phenomenology of turbulence in superfluids described by the GP equation. They detail the hierarchy of length scales ($L \gg \lambda \gg \xi$, where $L$ is system size, $\lambda$ is inter-vortex spacing, and $\xi$ is the healing length) and the resulting two inertial ranges:

   • Inertial Range I ($L > \ell > \lambda$): Exhibits a Kolmogorov-like $k^{-5/3}$ energy spectrum, interpreted as a classical Richardson cascade of coarse-grained vortices.
   • Inertial Range II ($\lambda > \ell > \xi$): Dominated by Kelvin wave cascades on vortex lines, also yielding specific power laws.
   • The authors note that even at $T=0$, the GP fluid behaves largely classically at large scales because the quantum potential term in the Madelung transformation becomes negligible compared to interaction energy (Thomas-Fermi limit).

2. Extension to Strong Fluctuations: The paper proposes moving beyond the mean-field approximation by treating the boson field as a quantum operator $\hat{\psi}$ rather than a classical wavefunction. This involves considering systems where the root-mean-square fluctuation in boson number ($\delta n$) is comparable to or larger than the average density ($\langle n \rangle$).

3. Identification of Candidate Systems: The authors identify specific physical settings where these strong fluctuations are realized:

   • Lattice Bosons near Criticality: Systems described by the Bose-Hubbard model near the superfluid-Mott insulator transition. Here, the system is tuned to a quantum critical point (e.g., the Wilson-Fisher fixed point in 2D).
   • Low-Dimensional Systems: Specifically one-dimensional (1D) and two-dimensional (2D) systems. In 1D, the lack of long-range order (Mermin-Wagner theorem) and the presence of Luttinger liquid physics imply that the GP equation is insufficient. The relevant excitations are phase slips rather than vortices.

4. Hypothesis on Scaling and Hydrodynamics: The authors hypothesize that in these regimes:

   • The conservation of vortex number (a key feature in GP turbulence) may be lost, potentially leading to a single scaling regime rather than the dual regimes seen in mean-field turbulence.
   • Quantum fluctuations may renormalize the scaling dimensions of hydrodynamic operators (such as the velocity field), leading to new turbulent power laws distinct from both classical fluids and weakly interacting superfluids.
   • The "Kolmogorov constant" (a dimensionless prefactor) might become tunable, depending on system parameters like the ratio of boson mass to interaction strength.

Key Contributions and Results

• Definition of a New Frontier: The paper formally defines the "many-body regime" of quantum turbulence, distinguishing it from the well-studied mean-field regime. It posits that turbulence in fluids with strong quantum fluctuations represents a new frontier for non-equilibrium quantum dynamics.
• Theoretical Motivation for New Scaling Laws: The authors provide a theoretical argument that near quantum critical points (e.g., the Wilson-Fisher fixed point), the spontaneous production and annihilation of vortices from the vacuum, combined with emergent Lorentz invariance, could lead to distinct turbulent scaling structures.
• Identification of Experimental Platforms: The paper highlights that modern experimental platforms, specifically ultracold atom setups and quantum computing platforms capable of realizing the Bose-Hubbard model, are now accessible enough to probe these regimes. It also suggests clean superconducting thin films and structured light interactions ("quantum printing") as potential solid-state platforms.
• Proposed Theoretical Tools: The authors suggest that studying these systems requires tools beyond standard hydrodynamics, including:
  • Quantum two-fluid models (coupling superfluid and vortex fluids).
  • Boson-vortex duality.
  • Non-thermal (dynamical) fixed points for classifying non-equilibrium scaling.
  • Numerical techniques suitable for low dimensions, such as exact diagonalization and Density Matrix Renormalization Group (DMRG).

Significance and Claims
The paper claims to synthesize the know-how of turbulence phenomenology and quantum many-body theory to set the stage for future research. Its primary significance lies in:

1. Motivating a Shift in Perspective: It argues that the study of turbulence in quantum fluids must expand beyond the mean-field GP equation to include regimes where quantum fluctuations are dominant.
2. Highlighting Open Questions: It explicitly lists open questions, such as the fate of the Kolmogorov constant in the many-body regime, the specific power laws in 1D and 2D critical fluids, and the nature of hydrodynamic variables near quantum critical points.
3. Feasibility: The authors assert that with current advancements in cold-atom experiments and quantum simulators, the proposed questions regarding many-body quantum turbulence are experimentally answerable.

The paper does not present new numerical simulations or experimental data but serves as a roadmap and theoretical proposal for investigating how many-body quantum effects manifest in turbulent hydrodynamics, particularly near quantum critical points and in low-dimensional systems.