In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $ϕ^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $ϕ^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.

arXiv:cond-mat/0612655WORLD SCIENTIFIC eBooksJul 1, 20071 citation

Introduction to the Modern Theory of Phase Transitions

J. Szṅajd

Springer eBooksJan 1, 20031 citation

Revisiting quantum relativistic effects from phase transition by the catastrophe theory

Jiu Hui Wu, Kejiang Zhou, Shuang Yang

Abstract In this paper we start from the Schrödinger equation to revisit some classical quantum mechanics from the perspective of phase transition process. Here the relativistic effect of particles moving at high speed can be regarded as the phase transition process when the velocity variable increases. Considering that the catastrophe theory could describe qualitatively any phase transition process, we adopt the simplest folding catastrophe type as the potential function in the Schrödinger equation to obtain a revised Schrödinger relativistic equation through the dimensionless analysis first, and then further to derive out the steady-state Klein-Gordon equation and Dirac relativistic equation gradually. These results reveal that the quantum relativistic effect could be considered as the phase transition process, which could be described by adopting the catastrophe models as the potential function in the classical Schrödinger equation.Abstract In this paper we start from the Schrödinger equation to revisit some classical quantum mechanics from the perspective of phase transition process. Here the relativistic effect of particles moving at high speed can be regarded as the phase transition process when the velocity variable increases. Considering that the catastrophe theory could describe qualitatively any phase transition process, we adopt the simplest folding catastrophe type as the potential function in the Schrödinger equation to obtain a revised Schrödinger relativistic equation through the dimensionless analysis first, and then further to derive out the steady-state Klein-Gordon equation and Dirac relativistic equation gradually. These results reveal that the quantum relativistic effect could be considered as the phase transition process, which could be described by adopting the catastrophe models as the potential function in the classical Schrödinger equation.

arXiv:2204.03807Europhysics Letters (EPL)Aug 5, 20221 citation

Phase Transitions and Critical Phenomena

David A. Lavis, Roman Frigg

Fundamental theories of physicsJan 1, 202557 citations

Non-Bloch parity-time-symmetric phase transition in quantum walks

Bingzi Huo, Quan Lin, Lei Xiao +3 more

Parity-time ( $\mathcal{P}\mathcal{T}$ ) symmetry is extensively studied within non-Hermitian physical systems as it has the capability to ensure the presence of real eigenenergies of the systems characterized by non-Hermitian Hamiltonians. Previous studies have been mostly based on the Bloch band theory. Recent developments have unveiled intriguing phenomena which can only be explained by non-Bloch band theory. In this paper, we observe the $\mathcal{P}\mathcal{T}$ phase transition in photonic quantum walks, which can only be explained by the non-Bloch band theory, which relies on the open boundary condition. Furthermore, we explore the impact of disorder on $\mathcal{P}\mathcal{T}$ symmetry and demonstrate disorder-induced $\mathcal{P}\mathcal{T}$ phase transitions. Our work provides a valuable perspective for further explorations on non-Hermitian physics.Parity-time ($\mathcal{P}\mathcal{T}$) symmetry is extensively studied within non-Hermitian physical systems as it has the capability to ensure the presence of real eigenenergies of the systems characterized by non-Hermitian Hamiltonians. Previous studies have been mostly based on the Bloch band theory. Recent developments have unveiled intriguing phenomena which can only be explained by non-Bloch band theory. In this paper, we observe the $\mathcal{P}\mathcal{T}$ phase transition in photonic quantum walks, which can only be explained by the non-Bloch band theory, which relies on the open boundary condition. Furthermore, we explore the impact of disorder on $\mathcal{P}\mathcal{T}$ symmetry and demonstrate disorder-induced $\mathcal{P}\mathcal{T}$ phase transitions. Our work provides a valuable perspective for further explorations on non-Hermitian physics.

Physical Review ANov 6, 20244 citations

ϕ^4

theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.

Parity-time (

\mathcal{P}\mathcal{T}

) symmetry is extensively studied within non-Hermitian physical systems as it has the capability to ensure the presence of real eigenenergies of the systems characterized by non-Hermitian Hamiltonians. Previous studies have been mostly based on the Bloch band theory. Recent developments have unveiled intriguing phenomena which can only be explained by non-Bloch band theory. In this paper, we observe the

\mathcal{P}\mathcal{T}

phase transition in photonic quantum walks, which can only be explained by the non-Bloch band theory, which relies on the open boundary condition. Furthermore, we explore the impact of disorder on

\mathcal{P}\mathcal{T}

symmetry and demonstrate disorder-induced

\mathcal{P}\mathcal{T}

phase transitions. Our work provides a valuable perspective for further explorations on non-Hermitian physics.