According to renormalization theory, Ising systems above their upper critical dimensionality ${d}_{u}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ have classical critical behavior and the ratio of magnetization moments $Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}〈{m}^{2}{〉}^{2}/〈{m}^{4}〉$ has the universal value $0.456947\dots{}$ . However, Monte Carlo simulations of $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$ Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which ${d}_{u}<4$ , and a careful analysis of the corrections to scaling. Our results are in perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.According to renormalization theory, Ising systems above their upper critical dimensionality ${d}_{u}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ have classical critical behavior and the ratio of magnetization moments $Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}〈{m}^{2}{〉}^{2}/〈{m}^{4}〉$ has the universal value $0.456947\dots{}$. However, Monte Carlo simulations of $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$ Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which ${d}_{u}<4$, and a careful analysis of the corrections to scaling. Our results are in perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.

arXiv:cond-mat/9601130Physical Review LettersMar 4, 199698 citations

Experimental evidence for classical critical behavior

IMPERIAL COLLEGE PRESS eBooksMay 1, 20000 citations

Protected percolation: a new universality class pertaining to quantum critical systems

Sean Fayfar

We describe a new universality class - dubbed protected percolation - that we show to be relevant to quantum critical systems. Percolation theory describes phase transitions where long-range order is lost when parts of a system become disconnected from other parts; in the vicinity of the transition, critical behavior is observed, captured by universal power laws. Protected percolation has the added restriction that only sites from the system spanning connection can be removed. We developed a new technique to simulate protected percolation, and we used it to determine the critical exponents of this new universality class in 2, 3, and 4 dimensions. We relate the exponents analytically to those of standard percolation. The Harris criterion predicts whether a phase transition is stable against impurities. We prove that protected percolation violates this criterion in 3 dimensions and higher, implying that impurities result in the loss of universal behavior in systems governed by protected percolation. We investigated the change in critical exponents for various types of impurities, focusing on the case for three dimensions where protected percolation models quantum critical systems. We detail how our simulations can be used for direct comparison to experimental results on such quantum critical systems.We describe a new universality class - dubbed protected percolation - that we show to be relevant to quantum critical systems. Percolation theory describes phase transitions where long-range order is lost when parts of a system become disconnected from other parts; in the vicinity of the transition, critical behavior is observed, captured by universal power laws. Protected percolation has the added restriction that only sites from the system spanning connection can be removed. We developed a new technique to simulate protected percolation, and we used it to determine the critical exponents of this new universality class in 2, 3, and 4 dimensions. We relate the exponents analytically to those of standard percolation. The Harris criterion predicts whether a phase transition is stable against impurities. We prove that protected percolation violates this criterion in 3 dimensions and higher, implying that impurities result in the loss of universal behavior in systems governed by protected percolation. We investigated the change in critical exponents for various types of impurities, focusing on the case for three dimensions where protected percolation models quantum critical systems. We detail how our simulations can be used for direct comparison to experimental results on such quantum critical systems.

Dec 1, 20210 citations

Critical phenomena and scale invariance

Franz Wegner

Springer eBooksDec 11, 20055 citations

Critical behavior of an impurity at the boson superfluid–Mott-insulator transition

Seth Whitsitt, Subir Sachdev A

We present a universal theory for the critical behavior of an impurity at the two-dimensional superfluid--Mott-insulator transition. Our analysis is motivated by a numerical study of the Bose-Hubbard model with an impurity site by Huang et al. [Phys. Rev. B 94, 220502 (2016)]; they found an impurity phase transition as a function of the trapping potential, while the bulk was critical. The bulk theory is described by the $\text{O}(2)$ symmetric Wilson-Fisher conformal field theory, and we model the impurity by a localized spin-1/2 degree of freedom. We also consider a generalized model by considering an $\text{O}(N)$ symmetric bulk theory coupled to a spin- $S$ degree of freedom. We study this field theory using the $\ensuremath{\epsilon}=3\ensuremath{-}d$ expansion, where the impurity-bulk interaction flows to an infrared stable fixed point at the critical trapping potential. We determine the scaling dimensions of the impurity degree of freedom and the associated critical exponents near the critical point. We also determine the universal contribution of the impurity to the finite-temperature compressibility of the system at criticality. Our results are compared with numerical simulations.We present a universal theory for the critical behavior of an impurity at the two-dimensional superfluid--Mott-insulator transition. Our analysis is motivated by a numerical study of the Bose-Hubbard model with an impurity site by Huang et al. [Phys. Rev. B 94, 220502 (2016)]; they found an impurity phase transition as a function of the trapping potential, while the bulk was critical. The bulk theory is described by the $\text{O}(2)$ symmetric Wilson-Fisher conformal field theory, and we model the impurity by a localized spin-1/2 degree of freedom. We also consider a generalized model by considering an $\text{O}(N)$ symmetric bulk theory coupled to a spin-$S$ degree of freedom. We study this field theory using the $\ensuremath{\epsilon}=3\ensuremath{-}d$ expansion, where the impurity-bulk interaction flows to an infrared stable fixed point at the critical trapping potential. We determine the scaling dimensions of the impurity degree of freedom and the associated critical exponents near the critical point. We also determine the universal contribution of the impurity to the finite-temperature compressibility of the system at criticality. Our results are compared with numerical simulations.

arXiv:1709.04919Physical Review ANov 15, 20176 citations

According to renormalization theory, Ising systems above their upper critical dimensionality

{d}_{u}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4

have classical critical behavior and the ratio of magnetization moments

Q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}〈{m}^{2}{〉}^{2}/〈{m}^{4}〉

has the universal value

0.456947\dots{}

. However, Monte Carlo simulations of

d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5

Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which

{d}_{u}<4

, and a careful analysis of the corrections to scaling. Our results are in perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.

\text{O}(2)

symmetric Wilson-Fisher conformal field theory, and we model the impurity by a localized spin-1/2 degree of freedom. We also consider a generalized model by considering an

\text{O}(N)

symmetric bulk theory coupled to a spin-

S

degree of freedom. We study this field theory using the $\ensuremath{\epsilon}=3\ensuremath{-}d$ expansion, where the impurity-bulk interaction flows to an infrared stable fixed point at the critical trapping potential. We determine the scaling dimensions of the impurity degree of freedom and the associated critical exponents near the critical point. We also determine the universal contribution of the impurity to the finite-temperature compressibility of the system at criticality. Our results are compared with numerical simulations.