General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions

We examine critically the issue of phase transitions in one-dimensional systems with short range interactions. We begin by reviewing in detail the most famous non-existence result, namely van Hove's theorem, emphasizing its hypothesis and subsequently its limited range of applicability. To further underscore this point, we present several examples of one-dimensional short ranged models that exhibit true, thermodynamic phase transitions, with increasing level of complexity and closeness to reality. Thus having made clear the necessity for a result broader than van Hove's theorem, we set out to prove such a general non-existence theorem, widening largely the class of models known to be free of phase transitions. The theorem is presented from a rigorous mathematical point of view although examples of the framework corresponding to usual physical systems are given along the way. We close the paper with a discussion in more physical terms of the implications of this non-existence theorem.

[1]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[2]  L. Van Hove,et al.  Sur L'intgrale de Configuration Pour Les Systmes De Particules Une Dimension , 1950 .

[3]  L. Landau,et al.  statistical-physics-part-1 , 1958 .

[4]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[5]  D. C. Mattis,et al.  Mathematical physics in one dimension , 1966 .

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  John F. Nagle,et al.  The One-Dimensional KDP Model in Statistical Mechanics , 1968 .

[8]  Freeman J. Dyson,et al.  Existence of a phase-transition in a one-dimensional Ising ferromagnet , 1969 .

[9]  C. Kittel,et al.  Phase Transition of a Molecular Zipper , 1969 .

[10]  H. Stanley,et al.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[11]  D. Thouless.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[12]  C. Thompson.  The Statistical Mechanics of Phase Transitions , 1978 .

[13]  H. D. Watson.  At 14 , 1979 .

[14]  J. Weeks,et al.  Pinning and roughening of one-dimensional models of interfaces and steps , 1981 .

[15]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[16]  Nieuwenhuizen,et al.  Wetting of a disordered substrate: Exact critical behavior in two dimensions. , 1986, Physical review letters.

[17]  M. Plischke,et al.  Equilibrium statistical physics , 1988 .

[18]  Wu,et al.  Absence of localization in a random-dimer model. , 1990, Physical review letters.

[19]  C. Beck,et al.  Thermodynamics of chaotic systems : an introduction , 1993 .

[20]  吉野 崇,et al.  Introduction to operator theory , 1993 .

[21]  Adriaan C. Zaanen,et al.  Introduction to Operator Theory in Riesz Spaces , 1997 .

[22]  F. Moura,et al.  Delocalization in the 1D Anderson Model with Long-Range Correlated Disorder , 1998 .

[23]  M. R. Evans.  Phase transitions in one-dimensional nonequilibrium systems , 2000 .

[24]  V. Baladi.  Positive transfer operators and decay of correlations , 2000 .

[25]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[26]  J. Cuesta,et al.  A theorem on the absence of phase transitions in one-dimensional growth models with on-site periodic potentials , 2001, cond-mat/0112026.

[27]  H. Eugene Stanley,et al.  Metal–insulator transition in chains with correlated disorder , 2002, Nature.

[28]  Apparent phase transitions in finite one-dimensional sine-Gordon lattices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  H. Stanley,et al.  retraction: Metal–insulator transition in chains with correlated disorder , 2003, Nature.