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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...
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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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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).
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Dynamical phase transitions in the nonreciprocal Ising model.

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Nonreciprocal interactions create stable time-dependent states in 3D systems, exhibiting properties akin to time crystals. These systems transition via critical exponents matching the 3D XY model, revealing broken time translation symmetry.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Complex Systems

Background:

  • Nonreciprocal interactions drive time-dependent states in various natural systems.
  • Understanding the stability and phase transitions of these states is crucial.

Purpose of the Study:

  • Investigate a minimal model with nonreciprocal interactions to understand time-dependent states.
  • Analyze phase transitions and critical behavior in systems with opposing spin goals.

Main Methods:

  • Mean-field analysis to predict system phases.
  • Large-scale numerical simulations in 2D and 3D.
  • Analysis of critical exponents and phase transitions.

Main Results:

  • A minimalistic Ising model with two spin species exhibits disorder, static order, and a time-dependent swap phase.
  • In 3D, the swap phase is stable and displays time crystal properties.
  • The 3D disorder-swap transition shows critical exponents of the 3D XY model, indicating broken time translation symmetry.
  • Defects destabilize the swap phase in 2D; droplet growth destabilizes static order in fully antisymmetric couplings.
  • A droplet-capture mechanism can restore static order in asymmetric couplings.

Conclusions:

  • The study elucidates the phase diagram of systems with nonreciprocal interactions.
  • Identifies conditions for stable time-dependent states and time crystal behavior.
  • Reveals critical phenomena linked to continuous symmetry breaking.