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Phase Transitions

Phase Transitions

Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...

Phase Diagrams

Phase Diagrams

A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent...

Phase Diagram

Phase Diagram

The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).

Phase Transitions: Vaporization and Condensation

Phase Transitions: Vaporization and Condensation

The physical form of a substance changes on changing its temperature. For example, raising the temperature of a liquid causes the liquid to vaporize (convert into vapor). The process is called vaporization—a surface phenomenon. Vaporization occurs when the thermal motion of the molecules overcome the intermolecular forces, and the molecules (at the surface) escape into the gaseous state. When a liquid vaporizes in a closed container, gas molecules cannot escape. As these gas phase...

Phase Transitions: Melting and Freezing

Phase Transitions: Melting and Freezing

Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...

Phase Transitions: Sublimation and Deposition

Phase Transitions: Sublimation and Deposition

Some solids can transition directly into the gaseous state, bypassing the liquid state, via a process known as sublimation. At room temperature and standard pressure, a piece of dry ice (solid CO2) sublimes, appearing to gradually disappear without ever forming any liquid. Snow and ice sublimate at temperatures below the melting point of water, a slow process that may be accelerated by winds and the reduced atmospheric pressures at high altitudes. When solid iodine is warmed, the solid sublimes...

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Related Experiment Video

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Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

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Topological Quantum Criticality from Multiplicative Topological Phases.

R Flores-Calderón^1,2, Elio J König³, Ashley M Cook^1,2

¹Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.

Physical Review Letters

|April 7, 2025

Summary

Multiplicative topological phases offer a new way to understand gapless symmetry-protected topological phases (SPTs). This research connects these phases, enabling concrete models for higher-dimensional, stable, gapless SPTs.

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Area of Science:

Condensed Matter Physics
Topological Matter Physics

Background:

Symmetry-protected topological phases (SPTs) with short-range entanglement are crucial in topological condensed matter physics.
Understanding gapless SPTs is essential for their broader implications.

Purpose of the Study:

To establish a fundamental connection between gapless SPTs and multiplicative topological phases.
To demonstrate multiplicative topological phases as a general approach for realizing gapless SPT models.

Main Methods:

Combining canonical topological insulator and semimetal models.
Utilizing critical gapless models within symmetry-protected tensor product constructions.

Main Results:

Identified a fundamental link between gapless SPTs and multiplicative topological phases.
Demonstrated multiplicative topological phases as an intuitive and general method for realizing gapless SPT models.
Showcased suitability for higher-dimensional, stable, and intrinsically gapless SPTs.

Conclusions:

Multiplicative topological phases provide a powerful framework for studying gapless SPTs.
This work opens new avenues for investigating topology through short-range entanglement.
Facilitates broader and deeper investigations into topological phenomena.