This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.

Reviews of Modern PhysicsMar 1, 1999940 citations

Emergent O( n ) symmetry in a series of three-dimensional Potts models

Chengxiang Ding, Henk W. J. Blöte, Youjin Deng

Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high- ${T}_{c}$ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of $q$ -state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O( $n$ ) symmetry with $n$ = $q$ -1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O( $n$ = $q$ -1) universality classes.Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-${T}_{c}$ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of $q$-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O($n$) symmetry with $n$=$q$-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O($n$=$q$-1) universality classes.

arXiv:1508.04538Physical review. B./Physical review. BSep 1, 201613 citations

Critical Phenomena and the Renormalization Group

WORLD SCIENTIFIC eBooksMar 1, 20080 citations

Extraordinary-Log Surface Phase Transition in the Three-Dimensional XY Model

Minghui Hu, Youjin Deng, Jian-Ping Lv

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance $r$ as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with $d$ the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the $\mathrm{O}(N)$ system. In this logarithmic universality, $g(r)$ decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional $XY$ model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with $L$ as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the $r$ -dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance $r$ as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with $d$ the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the $\mathrm{O}(N)$ system. In this logarithmic universality, $g(r)$ decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional $XY$ model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with $L$ as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the $r$-dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.

arXiv:2104.05152Physical Review LettersSep 16, 202140 citations

Critical phenomena and the renormalization group

Franz Wegner

Springer eBooksApr 6, 20086 citations

{T}_{c}

superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of

q

-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(

n

) symmetry with

n

q

-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(

n

q

-1) universality classes.

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance

r

as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with

d

the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the

\mathrm{O}(N)

system. In this logarithmic universality,

g(r)

decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional

XY

model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of

g(r,L)

has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with

L

as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the

r

-dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.