On spinodal points and Lee-Yang edge singularities

Abstract

We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the ${\varphi }^4$ theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid-gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e., $T<T_{\mathrm{c}}$, and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field $H$ for $T>T_{\mathrm{c}}$. The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for $T<T_{\mathrm{c}}$, the Lee-Yang edge singularities are the closest singularities to the real $H$ axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real $H$ axis for $d<4$, in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the $\epsilon =4-d$ expansion, as well as the equation of state of the $\mathrm{O}(N)$-symmetric ${\varphi }^4$ theory at large $N$, are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the $\epsilon$ expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher's ${\varphi }^3$ theory, which remains nonperturbative even for $d\to 4$, where the equation of state of the ${\varphi }^4$ theory is expected to approach the mean-field result. We resolve this paradox by deriving the Ginzburg criterion that determines the size of the region around the Lee-Yang edge singularity where mean-field theory no longer applies.

26 pages, 8 figures; v2: shortened Sec. 4.1 and streamlined arguments/notation in Sec. 4.2, details moved to appendix, added reference 17

References in corpus (5)

• Critical exponents of the 3d Ising and related models from Conformal Bootstrap
• QCD Critical Point and Complex Chemical Potential Singularities
• Density fluctuations in the presence of spinodal instabilities
• Chiral phase transition in the presence of spinodal decomposition
• Lee-Yang model from the functional renormalization group

Cited by in corpus (19)

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• Critical Slowing Down at the Abrupt Mott Transition: When the First-Order Phase Transition Becomes Zeroth-Order and Looks Like Second-Order
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• Universal location of Yang-Lee edge singularity for a one-component field theory in $1\le d\le 4$
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• Locating Critical Points Using Ratios of Lee-Yang Zeros
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