[Paper Review] Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields

Petro Sarkanych, Yurij Holovatch+1 more|arXiv (Cornell University)|Jun 7, 2018

Theoretical and Computational Physics84 references6 citations

TL;DR

This paper demonstrates that classical phase transitions in one-dimensional short-range spin systems—normally forbidden by no-go theorems—can be induced by either reducing entropy via a negative number of invisible states or by applying complex external fields to those states. Using Lee-Yang zeros analysis, the authors show that both mechanisms shift the phase transition to positive temperatures, bypassing traditional entropic barriers and offering new pathways to criticality in 1D systems.

ABSTRACT

Ising's solution of a classical spin model famously demonstrated the absence of a positive-temperature phase transition in one-dimensional equilibrium systems with short-range interactions. No-go arguments established that the energy cost to insert domain walls in such systems is outweighed by entropy excess so that symmetry cannot be spontaneously broken. An archetypal way around the no-go theorems is to augment interaction energy by increasing the range of interaction. Here we introduce new ways around the no-go theorems by investigating entropy depletion instead. We implement this for the Potts model with invisible states.Because spins in such a state do not interact with their surroundings, they contribute to the entropy but not the interaction energy of the system. Reducing the number of invisible states to a negative value decreases the entropy by an amount sufficient to induce a positive-temperature classical phase transition. This approach is complementary to the long-range interaction mechanism. Alternatively, subjecting positive numbers of invisible states to imaginary or complex fields can trigger such a phase transition. We also discuss potential physical realisability of such systems.

Motivation & Objective

• To overcome the standard no-go theorems that prohibit positive-temperature phase transitions in 1D short-range spin systems.
• To investigate whether entropy depletion—via a negative number of invisible states—can induce a phase transition by reducing entropic excess.
• To explore whether complex external fields acting on invisible states can trigger a positive-temperature phase transition.
• To connect these abstract mechanisms to physical realizability through links to quantum coherence and complex field mappings.
• To analyze the behavior of Lee-Yang zeros and Yang-Lee edges as functions of invisible state count and field parameters.

Proposed method

• Formalizing the (q,r)-state Potts model with r invisible states that contribute to entropy but not interaction energy.
• Introducing a negative number of invisible states as a formal mechanism to deplete entropy, thereby enabling a positive-temperature phase transition.
• Applying complex external fields to invisible states and analyzing their effect on the partition function and phase transition loci.
• Using Lee-Yang zeros analysis to locate critical points and determine the conditions under which phase transitions occur at positive temperatures.
• Mapping complex fields to quantum coherence times via recent theoretical advances, suggesting physical accessibility of complex field parameters.
• Deriving and solving the exact partition function for the 1D (2,3)-state Potts model with complex fields to identify critical temperatures.

Experimental results

Research questions

• RQ1 Can a negative number of invisible states induce a positive-temperature phase transition in a 1D short-range spin system?
• RQ2 How do complex external fields acting on invisible states affect the phase transition behavior in such systems?
• RQ3 What is the role of Lee-Yang zeros and Yang-Lee edges in determining the critical behavior when entropy is depleted or fields are complex?
• RQ4 Are these exotic mechanisms—negative invisible states or complex fields—physically realizable or merely mathematical constructs?
• RQ5 How do these mechanisms bypass the rigorous no-go theorems that forbid phase transitions in 1D short-range systems?

Key findings

• A negative number of invisible states reduces entropy sufficiently to allow a positive-temperature phase transition in the 1D Potts model, violating the standard no-go theorems.
• Complex external fields acting on invisible states induce a phase transition at positive temperature, as confirmed by the locus of Lee-Yang zeros crossing the real axis.
• For the (2,3)-state Potts model, the complex field values $ e^{-\beta h_2} $ form two continuous curves, each corresponding to a physically accessible critical temperature.
• The Yang-Lee edge locus expands with increasing positive r, but only when r is negative or the field is complex does the system cross the real axis at finite temperature.
• The mechanism of entropy depletion via negative r is formally equivalent to a complex chemical potential, suggesting a deep connection between the two bypass mechanisms.
• Complex fields, while initially mathematical, may become physically accessible through their mapping to quantum coherence times in open quantum systems.