Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry

Abstract: We discuss a realization of the nonrelativistic conformal group (the Schroedinger group) as the symmetry of a spacetime. We write down a toy model in which this geometry is a solution to field equations. We discuss various issues related to nonrelativistic holography. In particular, we argue that free fermions and fermions at unitarity correspond to the same bulk theory with different choices for the near-boundary asymptotics corresponding to the source and the expectation value of one operator. We describe an extended version of nonrelativistic general coordinate invariance which is realized holographically.

• The paper introduces a novel geometric realization of Schrödinger symmetry by deforming the AdS metric for nonrelativistic systems.
• It employs a toy gravity dual from Einstein-Proca equations to connect nonrelativistic fermions with holographic models.
• The study outlines an operator-field correspondence that opens new avenues for exploring nonrelativistic holography in cold atom physics.

An Overview of "Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry" by D. T. Son

This paper explores the intriguing possibility of extending the holographic AdS/CFT correspondence to nonrelativistic systems, focusing on fermions at unitarity, which share similarities with strongly coupled gauge theories like the N=4 supersymmetric Yang-Mills theory. The author embarks on constructing a geometric structure with Schrödinger symmetry, which is pivotal in understanding the nonrelativistic conformal symmetry analogous to relativistic conformal symmetry.

Key Contributions and Theoretical Insights

1. Schrödinger Symmetry in Nonrelativistic Systems: The paper elucidates how the Schrödinger symmetry group, inherent to systems like fermions at unitarity, can be embedded within the larger framework of relativistic conformal symmetry, specifically the conformal algebra in a higher-dimensional spacetime, O(d+2, 2). This embedding is pivotal for realizing Schrödinger symmetry geometrically.
2. Geometric Realization: A significant contribution is the construction of a geometric model that respects Schrödinger symmetry. This involves deforming the AdS metric to maintain the invariance of the Schrödinger algebra while reducing symmetry from the full conformal group to the Schrödinger group. The resulting metric is shown to exhibit full Schrödinger symmetry, providing a plausible "spacetime" where the duality could manifest.
3. Gravity Duals and Nonrelativistic Holography: By positing a toy model where the constructed metric results from coupled Einstein and Proca equations, the author takes a preliminary step towards a gravity dual for the unitarity Fermi gas. Although a comprehensive dual theory is not claimed, the work sets the stage for further exploration into nonrelativistic holography, which might offer insights akin to those seen in AdS/CFT for relativistic theories.
4. Operator-Field Correspondence: The essay explores the operator-state correspondence in this nonrelativistic holographic setup. It explores how operators in the boundary theory relate to fields in the bulk, akin to the AdS/CFT dictionary. Particularly interesting is the discussion on scaling dimensions and the spectrum of primary operators, highlighting cases where free fermions and unitarity fermions offer different realizations within the same theoretical framework.

Numerical Results and Theoretical Implications

The paper does not numerically quantify results, given its focus on theoretical development. However, it highlights crucial theoretical implications, such as the universal thermodynamic relation between energy and pressure in nonrelativistic conformal field theories and the potential lower bound on operator dimensions. These insights can play a significant role in understanding how properties of nonrelativistic systems are encoded in higher-dimensional geometries.

Future Directions

Several intriguing paths for future research are hinted at. Extending the gravity dual concept to finite density states or finite-temperature scenarios could unveil rich phenomenological analogs to strongly coupled condensed matter systems. The exploration of nonrelativistic hydrodynamics and superfluidity in this geometric framework could also merge insights from condensed matter physics with high energy theoretical models.

In summary, this paper serves as a foundational reference for researchers examining the AdS/CFT analogy in nonrelativistic realms, providing a geometric scaffold that aligns with the principles of holography. While it leaves open many questions concerning the practicalities and specifics of the gravity dual for unitarity Fermi gases, it importantly establishes a cornerstone for continued theoretical endeavors in connecting cold atom physics with high-energy theoretical constructs.