arXiv:1004.4983 · Physical Review B

Universal amplitude ratios in the three-dimensional Ising universality class

Martin Hasenbusch B

Published Nov 23, 2010• 46 citations

Abstract

We compute a number of universal amplitude ratios in the three-dimensional Ising universality class. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. For example, we obtain ${A}_{+}/{A}_{\ensuremath{-}}=0.536(2)$ and ${C}_{+}/{C}_{\ensuremath{-}}=4.713(7)$, where ${A}_{\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts $+$ and $\ensuremath{-}$ indicate the high- and the low-temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high- and low-temperature series expansions, field theoretic methods, and experiments.

DOI: 10.1103/physrevb.82.174434

Related papers

A Monte Carlo study of the three-dimensionalXYuniversality class: universal amplitude ratios

Martin Hasenbusch

arXiv:0810.2716Journal of Statistical Mechanics Theory and ExperimentDec 12, 200823 citations

Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

Martin Hasenbusch B

arXiv:1711.10946Physical review. EJan 16, 201814 citations

Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

arXiv:1004.4486Physical Review BNov 23, 2010333 citations

Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model

Martin Hasenbusch B

arXiv:1908.01702Physical review. EFeb 24, 202039 citations

The three-dimensionalXYuniversality class: a high precision Monte Carlo estimate of the universal amplitude ratioA+/A−

Martin Hasenbusch B

arXiv:cond-mat/0607189Journal of Statistical Mechanics Theory and ExperimentAug 25, 200612 citations

Critical exponents and equation of state of the three-dimensional Heisenberg universality class

Massimo Campostrini B, Martin Hasenbusch B, Andrea Pelissetto C +2 more

arXiv:cond-mat/0110336Physical review. B, Condensed matterApr 2, 2002482 citations

arXiv:1004.4983 · Physical Review B

Universal amplitude ratios in the three-dimensional Ising universality class

Martin Hasenbusch B

Published Nov 23, 2010• 46 citations

arXiv HTML PDF

Abstract

{A}_{+}/{A}_{\ensuremath{-}}=0.536(2)

and

{C}_{+}/{C}_{\ensuremath{-}}=4.713(7)

, where

{A}_{\ifmmode\pm\else\textpm\fi{}}

and

{C}_{\ifmmode\pm\else\textpm\fi{}}

are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts

+

and

\ensuremath{-}

indicate the high- and the low-temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high- and low-temperature series expansions, field theoretic methods, and experiments.

DOI: 10.1103/physrevb.82.174434

Related papers

A Monte Carlo study of the three-dimensionalXYuniversality class: universal amplitude ratios

Martin Hasenbusch

We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. In addition to the XY model, we study the three-dimensional two-component ϕ 4 model on the simple cubic lattice. The parameter of the ϕ 4 model is chosen such that leading corrections to scaling are small. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R ϒ = 0.411(2), R B = 2.83(1), R ξ + = 0.3562(10) and R ξ − = 0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the λ-transition of 4 He.We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. In addition to the XY model, we study the three-dimensional two-component ϕ 4 model on the simple cubic lattice. The parameter of the ϕ 4 model is chosen such that leading corrections to scaling are small. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R ϒ = 0.411(2), R B = 2.83(1), R ξ + = 0.3562(10) and R ξ − = 0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the λ-transition of 4 He.

arXiv:0810.2716Journal of Statistical Mechanics Theory and ExperimentDec 12, 200823 citations

Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

Martin Hasenbusch B

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$ . As a check, also simulations of the spin- $\frac{1}{2}$ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$ -point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where $L$ is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading ${Z}_{2}$ -even scalar $\ensuremath{\epsilon}$.We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$. As a check, also simulations of the spin-$\frac{1}{2}$ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where $L$ is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading ${Z}_{2}$-even scalar $\ensuremath{\epsilon}$.

arXiv:1711.10946Physical review. EJan 16, 201814 citations

Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$ , where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

arXiv:1004.4486Physical Review BNov 23, 2010333 citations

Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model

Martin Hasenbusch B

We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.

arXiv:1908.01702Physical review. EFeb 24, 202039 citations

The three-dimensionalXYuniversality class: a high precision Monte Carlo estimate of the universal amplitude ratioA+/A−

Martin Hasenbusch B

We simulate the improved three-dimensional two-component ϕ 4 model on the simple cubic lattice in the low and the high temperature phase for reduced temperatures down to | T − T c |/ T c ≈ 0.0017 on lattices of size up to 350 3 . Our new results for the internal energy and the specific heat are combined with the accurate estimates of β c and data for the internal energy and the specific heat at β c recently obtained in Campostrini et al (2006 Preprint cond-mat/0605083 ). We find R α = (1 − A + / A − )/α = 4.01(5), where α is the critical exponent of the specific heat and A ± is the amplitude of the specific heat in the high and the low temperature phase, respectively.We simulate the improved three-dimensional two-component ϕ 4 model on the simple cubic lattice in the low and the high temperature phase for reduced temperatures down to | T − T c |/ T c ≈ 0.0017 on lattices of size up to 350 3 . Our new results for the internal energy and the specific heat are combined with the accurate estimates of β c and data for the internal energy and the specific heat at β c recently obtained in Campostrini et al (2006 Preprint cond-mat/0605083 ). We find R α = (1 − A + / A − )/α = 4.01(5), where α is the critical exponent of the specific heat and A ± is the amplitude of the specific heat in the high and the low temperature phase, respectively.

arXiv:cond-mat/0607189Journal of Statistical Mechanics Theory and ExperimentAug 25, 200612 citations

Critical exponents and equation of state of the three-dimensional Heisenberg universality class

Massimo Campostrini B, Martin Hasenbusch B, Andrea Pelissetto C +2 more

We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.

arXiv:cond-mat/0110336Physical review. B, Condensed matterApr 2, 2002482 citations

{A}_{+}/{A}_{\ensuremath{-}}=0.536(2)

and

{C}_{+}/{C}_{\ensuremath{-}}=4.713(7)

, where

{A}_{\ifmmode\pm\else\textpm\fi{}}

and

{C}_{\ifmmode\pm\else\textpm\fi{}}

are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts

+

and

\ensuremath{-}

L=1600

. As a check, also simulations of the spin-

\frac{1}{2}

Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of

N

-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where

L

is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading

{Z}_{2}

-even scalar $\ensuremath{\epsilon}$.

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter

D

on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter

D

, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for

D=0.641

and 0.655 on lattices of a linear size up to

L=360

we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.