Fractons (1803.11196v1)

Abstract: We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We introduce the basic phenomena, and discuss the geometric and topological response of fracton phases. We also discuss connections to generalized gauge theories, and explain how gapped fracton phases may be obtained from more familiar theories. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective on fracton phases. We discuss how tensor gauge theory encodes the fracton phenomenon, and how it allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases, and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at non-zero density, before concluding with a brief survey of some open problems.

• The paper provides a comprehensive analysis of gapped and gapless fracton phases, emphasizing their restricted mobility through exactly solvable lattice models like the X-cube and Haah's cubic code.
• The paper employs tensor gauge theories with symmetric tensor fields to derive generalized Gauss’s laws that govern the dipolar conservation laws and restricted excitations in fracton systems.
• The paper draws interdisciplinary connections by relating fracton phenomena to elasticity theory and gravity, highlighting their potential for robust quantum memory and novel material phases.

An Insightful Review of Fractons and Their Significance in Quantum Matter

The reviewed paper, "Fractons," by Nandkishore and Hermele, presents a comprehensive exploration of fracton phases within the domain of quantum matter. Fracton phases are distinguished by quasi-particles that demonstrate restricted mobility, only able to move under certain conditions or in specific directions. This discussion situates fracton phases within the broader context of quantum matter, interacting with a multitude of fields, including topological order, quantum information theory, and gravity.

Key Insights and Contributions

1. Gapped Fracton Phases: The paper provides a detailed exposition on gapped fracton phases, which are most straightforwardly analyzed using exactly solvable lattice models such as the X-cube and Haah's cubic code. These models exhibit novel properties, such as a ground state degeneracy that scales with system size, setting them apart from traditional topologically ordered systems. Notably, the X-cube model introduces immobile fracton excitations and mobile sub-dimensional particles.
2. Tensor Gauge Theory: A significant portion of the paper discusses tensor gauge theories, a crucial tool in understanding fracton phases. By employing symmetric tensor fields, these theories offer novel generalized Gauss's laws, encoding conservation rules leading to the restricted mobility phenomena seen in fractons. This section also bridges the understanding between gapped and gapless phases, elucidating the connections to tensor gauge theories.
3. Gapless Fracton Phases: For gapless fracton phases, the paper illustrates how these phases can be explored through the framework of tensor gauge theories. These phases are represented by gauge theories involving symmetric tensors. The analysis covers how dipolar conservation laws emerge from these theories, leading to electrostatic confinement, where isolated fractons exhibit infinite-energy cost to be unbound.
4. Connections to Other Fields: Fracton phases' interactions with elasticity theory and gravity open up fascinating interdisciplinary approaches to understanding and potentially utilizing these phases in practical applications. The duality between elasticity theory and fractonic phases in two dimensions, for instance, offers novel insights into crystal defects and phonons.
5. Dynamics and Thermodynamics at Finite Density: Exploring the dynamics and thermodynamics of fractonic matter at non-zero densities, the authors reveal the glassy dynamics and many-body localization-like behaviors in certain fracton models. The discussion on dipole creation and motion provides foundational insights into the thermodynamics of fractonic systems, emphasizing potential ground states and phase transitions.

Implications and Future Directions

The theoretical implications of fracton phases are extensive, with potential impacts across condensed matter physics and quantum information science. The unique properties of fracton excitations, such as fractionalized mobility and their implications on system dynamics, open new avenues for exploring matter's quantum states. Furthermore, the potential application of these phases for robust quantum memory due to their stability and complex ground state structure is noteworthy.

The authors conclude with invitations for further research, highlighting the need to discover simpler models, experimental realizations, and broader connections between fracton phases and established theories in physics. The examination of fractons in the field of realistic materials remains a pivotal goal, promising to elevate the understanding and utilisation of these novel quantum states.

This review, through rigorous analysis and broad contextualization, solidifies our understanding of fractons, situating them as an intriguing frontier in the paper of quantum matter. The paper challenges researchers to pursue this promising field with new paradigms, tools, and questions, ensuring a fertile ground for discovery and innovation.