Universal amplitude ratios of the Ising universality class

edit · history · discussion · files · short url · statistical mechanics physics

Numbers

ratio

universal amplitude ratio

$U_0$:

0.536 +/- 0.002 †

comment: [1], equation (55) and Table III.

$U_2$:

4.713 +/- 0.007 †

comment: [1], equation (31) and Table III.

$U_\xi$:

1.939 +/- 0.005 †

comment: [1], equation (33) and Table III.

$Q_\xi^-$:

1.020 +/- 0.005 †

comment: [1], equation (40) and Table III. The uncertainty includes the estimated effect of using a finite-distance effective correlation length to approximate the exponential correlation length.

$Q_+$:

0.01899 +/- 0.00010 †

comment: [1], equation (64) and Table III.

$Q_-$:

0.00487 +/- 0.00002 †

comment: [1], equation (65) and Table III.

$u^*$:

14.08 +/- 0.05 †

comment: [1], equation (38) and Table III.

$Q_c$:

0.3293 +/- 0.0002 †

comment: [1], equation (36) and Table III.

Definition

The universal amplitude ratios of the three-dimensional Ising universality class are dimensionless combinations of bulk critical amplitudes that are independent of microscopic couplings and of thermal and magnetic scale factors [1].

Parameters

ratio

— universal critical-amplitude combination

Formulas

(1)

At $h=0$, $C_s\sim A_\pm |t|^{-\alpha}$, $\chi\sim C_\pm |t|^{-\gamma}$, $\xi_{\mathrm{2nd}}\sim f_{\mathrm{2nd},\pm}|t|^{-\nu}$ and $\xi_{\exp}\sim f_{\exp,\pm}|t|^{-\nu}$. The spontaneous magnetisation satisfies $M\sim B(-t)^\beta$ as $t\uparrow0$ [1].

(2)

$\displaystyle \xi_{\mathrm{2nd}}^2=\frac{\int_{\mathbb R^3}|x|^2G(x)\,d^3x}{6\int_{\mathbb R^3}G(x)\,d^3x}$. For a lattice, the integrals are replaced by sums in lattice units [3].

(3)

$\displaystyle U_0=\frac{A_+}{A_-},\quad U_2=\frac{C_+}{C_-},\quad U_\xi=\frac{f_{\mathrm{2nd},+}}{f_{\mathrm{2nd},-}},\quad Q_\xi^-=\frac{f_{\exp,-}}{f_{\mathrm{2nd},-}}$ [2].

(4)

$\displaystyle Q_\pm=\alpha A_\pm f_{\mathrm{2nd},\pm}^{\,3},\quad u^*=\frac{3C_-}{f_{\mathrm{2nd},-}^{\,3}B^2},\quad Q_c=\frac{f_{\mathrm{2nd},+}^{\,3}B^2}{C_+}$ [1].

(5)

$\displaystyle \frac{Q_+}{Q_-}=U_0 U_\xi^3,\qquad u^*Q_c=\frac{3U_\xi^3}{U_2}$ follow from (3) and (4). The relations $\alpha+3\nu=2$ and $2\beta+\gamma=3\nu$ cancel the powers of $|t|$ in the mixed combinations; the critical exponents satisfy hyperscaling in three dimensions.

Comments

(6)

The setting is an isotropic bulk system with short-range interactions and a scalar order parameter with $\mathbb Z_2$ symmetry, in three Euclidean dimensions. The thermodynamic limit is taken before approaching the critical point. The subscripts $+$ and $-$ denote the disordered high-temperature and ordered low-temperature phases.

(7)

Let $t=(T-T_c)/T_c$ and let $h$ be the dimensionless field conjugate to the magnetisation density $M$. Write $f_s(t,h)$ for the singular reduced free-energy density per unit volume, and define $C_s=-\partial_t^2 f_s$, $M=-\partial_h f_s$ and $\chi=\partial_h M$. Lengths and the volume in $f_s$ use the same length unit. The amplitudes are specified by (1). On the ordered side the limit $h\to0^+$ follows the thermodynamic limit, selecting a pure phase.

(8)

For the connected order-parameter correlation function $G(x)$ at zero field, the second-moment correlation length is defined by (2). The exponential correlation length $\xi_{\exp}$ is the inverse asymptotic exponential decay rate of $G$; it is not interchangeable with $\xi_{\mathrm{2nd}}$ [3].

(9)

The symbols $U_0,U_2,U_\xi,Q_\xi^-$ follow Table 7 of [2]; $Q_\pm,u^*,Q_c$ use the normalisation of [1]. In particular, $Q_\pm$ are the combinations $\alpha A_\pm f_{\mathrm{2nd},\pm}^{\,3}$, not their cube roots. The critical exponents $\alpha,\beta,\gamma,\nu$ are listed in the Ising entries of the three-dimensional exponent table.

(10)

The computational thermal coordinate in [1] is $\beta_c-\beta_{\mathrm{inv}}$, where $\beta_{\mathrm{inv}}$ is inverse temperature. Defining the heat-capacity amplitude through the second derivative of the reduced free energy keeps $Q_\pm$ unchanged when this coordinate is rescaled to $t$. The symbol $\beta$ in (1) denotes the magnetisation exponent.

(11)

Each entry is one published Monte Carlo estimate, with its quoted uncertainty retained. The radii include the source's assessment of statistical and systematic errors; they are not proven containment bounds, and the eight intervals do not specify a simultaneous confidence region.

(12)

The quoted intervals for $U_2,U_\xi,Q_+,Q_-,Q_c$ do not overlap the corresponding series estimates in Table 1 of [2]. The interval for $u^*$ does not overlap the earlier Monte Carlo estimate in [3]. Each row gives the estimate from [1] with its published uncertainty; the earlier determinations are cited for comparison.

References

[1]

M. Hasenbusch, Universal amplitude ratios in the three-dimensional Ising universality class, Physical Review B 82 (2010), 174434. (arXiv) (doi)

[2]

M. Campostrini, A. Pelissetto, P. Rossi and E. Vicari, 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice, Physical Review E 65 (2002), 066127. (arXiv) (doi)

[3]

M. Caselle and M. Hasenbusch, Universal amplitude ratios in the three-dimensional Ising model, Journal of Physics A: Mathematical and General 30 (1997), 4963–4982. (arXiv) (doi)

Links

[4]

Author manuscript with the measured amplitude ratios

Similar tables

Critical exponents of the three-dimensional universality classes — the three-dimensional critical exponents that specify the power laws whose amplitudes enter these ratios

Critical couplings of the Ising model on lattices — the lattice-dependent critical points whose bulk neighbourhood is described by these universal ratios

Critical exponents of the two-dimensional universality classes — critical-exponent data for the two-dimensional counterparts, including the two-dimensional Ising class

Data properties

Entries are of type: real number

How well the digits are known: measured

Table is complete: no (it holds all eight ratios in the “here” row of Table III of [1], covering the specific heat, susceptibility, correlation lengths and their standard mixed combinations; critical-isotherm, interface and correction-to-scaling amplitude ratios are not included)

Sources of data: [1]

How they were obtained:

The entries transcribe the author's Monte Carlo estimates for the improved Blume-Capel model, as printed in Table III of the version 3 author manuscript [4]. Each entry identifies its equation. Parenthesised errors are converted exactly to centre-and-radius notation; for example, $0.01899(10)$ becomes $0.01899\mathbin{+/-}0.00010$. No simulation is rerun, and no precision beyond the source is claimed.

The attached checks compare every centre and radius with a separate transcription of the printed table. Exact rational arithmetic gives a common point in all eight error boxes satisfying both identities in (5). Positive and deliberately inconsistent controls test the parser, containment and identity checks. A rational control also tests invariance under rescaling the thermal coordinate, magnetic field and length unit.

Independent comparisons use Table 1 of [2] for six entries, and equations (45)–(46) of [3] for $Q_\xi^-$ and $u^*$. Six pairs have non-overlapping quoted intervals, as recorded in (12). Every separation is less than three times the square root of the sum of squared quoted errors. This is a descriptive comparison, not a confidence-level test or an independent certification of the digits. The cause of the shifts between determinations is not established by these checks. All transcription, identity and comparison checks are repeated on the stored entries, alongside the generator's full verification.