Scaling Dimensions, Correlators, and Critical Exponents

The previous page ended with the simplest continuum correlators of the critical Ising theory: free chiral Majorana fermions and the energy bilinear. This page steps back and explains the general logic behind those formulas. At a continuous phase transition, the correlation length is infinite, so the long-distance theory has no preferred scale. Local observables reorganize into scaling fields, and their correlation functions become homogeneous functions of the separations.

The central number attached to a scaling field is its scaling dimension. For the two-dimensional Ising spin field,

$$\langle \sigma(z_1,\bar z_1)\sigma(z_2,\bar z_2)\rangle ={C_\sigma\over |z_1-z_2|^{1/4}},$$

so the spin field has

$$\Delta_\sigma={1\over8}.$$

That single exponent already knows about the anomalous dimension, the magnetization exponent, the susceptibility exponent, and the critical isotherm. The energy operator similarly controls the correlation length and the singular specific heat. The goal of this page is to make this operator-to-exponent dictionary precise.

Required background. Lesson 11 supplies the critical Ising spin, energy, and Majorana correlators used as the main examples.

Helpful background. Lesson 4 introduces critical power laws, and Lesson 5 develops RG eigenvalues and anomalous dimensions perturbatively.

Scaling fields

Scaling action and exponent notation

We work in Euclidean $d$ dimensions unless stated otherwise. In two dimensions,

$$z=x+iy, \qquad \bar z=x-iy, \qquad |z|^2=z\bar z.$$

A scalar scaling field $O(x)$ of dimension $\Delta_O$ is written in the convention

$$\mathcal D_\lambda O(x)\mathcal D_\lambda^{-1} =\lambda^{\Delta_O}O(\lambda x).$$

At a scale-invariant vacuum, this is equivalent to the correlator homogeneity law

$$\left\langle \prod_{i=1}^n O_i(\lambda x_i)\right\rangle = \lambda^{-\sum_i \Delta_i} \left\langle \prod_{i=1}^n O_i(x_i)\right\rangle.$$

The critical exponent for the correlation length is also traditionally called $\nu$. To avoid confusion, this page uses $\Delta$ for operator dimensions and $\nu$ only for the correlation-length exponent.

A microscopic lattice observable is usually not a pure scaling field. Instead, near a critical point it expands into a sum of continuum scaling fields:

$$O_{\rm lat}(i) \sim \sum_k c_k a^{\Delta_k} O_k(x), \qquad x=ai,$$

where $a$ is the lattice spacing. At long distance the term with the smallest scaling dimension compatible with the symmetries dominates.

For the Ising spin variable,

$$\sigma_i \sim c_\sigma a^{\Delta_\sigma}\sigma(x)+\cdots.$$

For the nearest-neighbor energy density, one first subtracts the expectation value in order to remove the identity operator:

$$E_i-\langle E\rangle \sim c_\epsilon a^{\Delta_\epsilon}\epsilon(x)+\cdots.$$

The identity has dimension zero and contributes to one-point functions, but it is not the thermal perturbation that changes the critical theory. The energy field $\epsilon$ is the leading nontrivial scalar even under the Ising spin flip.

At the critical point, dilation acts diagonally on scaling fields:

$$O_i(x)\mapsto \lambda^{\Delta_i}O_i(\lambda x).$$

Away from the critical point the statement holds only inside the scaling window

$$a\ll |x|\ll \xi,$$

where $\xi$ is the correlation length.

Two-point functions and anomalous dimensions

Let $O$ be a scalar scaling field. Translation and rotation invariance imply

$$G_2(x)=\langle O(x)O(0)\rangle=G_2(r), \qquad r=|x|.$$

Scale covariance of the vacuum correlator gives

$$G_2(\lambda r)=\lambda^{-2\Delta_O}G_2(r).$$

The solution is

$$\boxed{ \langle O(x)O(0)\rangle={C_O\over |x|^{2\Delta_O}}. }$$

The normalization constant $C_O$ changes if we rescale the operator. The exponent $\Delta_O$ does not.

A scaling operator is defined by how it transforms under dilations. Homogeneity of the two-point function forces the critical correlator to be a power law, with exponent $2\Delta$.

For the critical Ising spin field,

$$\langle \sigma(x)\sigma(0)\rangle\sim {1\over |x|^{1/4}},$$

hence

$$2\Delta_\sigma={1\over4}, \qquad \Delta_\sigma={1\over8}.$$

In the language of critical phenomena one often writes the order-parameter correlator as

$$\langle \sigma(x)\sigma(0)\rangle\sim {1\over |x|^{d-2+\eta}}.$$

Comparing the two forms gives

$$\boxed{ 2\Delta_\sigma=d-2+\eta. }$$

Thus $\eta$ is simply another way of measuring the anomalous part of the scaling dimension of the order parameter. For the two-dimensional Ising model,

$$d=2, \qquad \Delta_\sigma={1\over8}, \qquad \eta={1\over4}.$$

Ising operators in the continuum

The critical Ising theory has a small set of basic fields that already explain many of the exact exponents. The spin field and the disorder field have the same scaling dimension,

$$\Delta_\sigma=\Delta_\mu={1\over8}.$$

The Majorana fermions are chiral fields with weights

$$(h_\psi,\bar h_\psi)=\left({1\over2},0\right), \qquad (h_{\bar\psi},\bar h_{\bar\psi})=\left(0,{1\over2}\right),$$

so $\Delta_\psi=\Delta_{\bar\psi}=1/2$. The energy field is the mass operator,

$$\epsilon(z,\bar z)\sim i\psi(z)\bar\psi(\bar z),$$

up to normalization and a convention-dependent phase. With

$$\langle \psi(z)\psi(w)\rangle={1\over z-w}, \qquad \langle \bar\psi(\bar z)\bar\psi(\bar w)\rangle={1\over \bar z-\bar w},$$

Wick contraction gives

$$\langle \epsilon(z,\bar z)\epsilon(0,0)\rangle \propto {1\over z}{1\over \bar z} ={1\over |z|^2}.$$

Therefore

$$\boxed{\Delta_\epsilon=1.}$$

Long-distance lattice observables expand into continuum scaling fields. The spin and disorder fields have dimension $1/8$, the energy field has dimension $1$, and the chiral Majorana fields have weights $(1/2,0)$ and $(0,1/2)$.

The spin field $\sigma$ is not a local polynomial in $\psi$ and $\bar\psi$. It is a twist field for the Majorana fermion: taking a fermion around a spin insertion changes the fermion boundary condition. This is the continuum version of the order–disorder branch-cut construction from the previous pages.

Multi-point functions and shape data

For scaling fields $O_i$,

$$G_n(x_1,\ldots,x_n)=\langle O_1(x_1)\cdots O_n(x_n)\rangle$$

obeys the homogeneity law

$$G_n(\lambda x_1,\ldots,\lambda x_n) = \lambda^{-\sum_i \Delta_i}G_n(x_1,\ldots,x_n).$$

This is powerful, but it does not determine all multi-point functions. Translation invariance removes one vector, rotation invariance removes an overall orientation, and scale invariance removes one length. For three or more points there can still be dimensionless shape data.

For example, scale invariance alone permits a three-point function of scalar fields to be written schematically as

$$\langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle = {|x_{12}|^{-\Delta_1-\Delta_2-\Delta_3}} F_{123}\left({|x_{13}|\over |x_{12}|},{|x_{23}|\over |x_{12}|},\text{angles}\right).$$

Scale invariance fixes the total degree of homogeneity. It does not fix the function $F_{123}$.

Full conformal invariance is stronger. For scalar primary operators it fixes the three-point function up to one coefficient:

$$\langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle ={C_{123}\over |x_{12}|^{\Delta_1+\Delta_2-\Delta_3} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1} |x_{31}|^{\Delta_3+\Delta_1-\Delta_2}} .$$

This formula will be derived later from conformal transformations. The important distinction is that similarity transformations leave arbitrary triangle-shape dependence, whereas special conformal transformations remove it from scalar three-point functions. For four points, even full conformal invariance leaves nontrivial functions of cross ratios. In two-dimensional complex coordinates, the basic complex cross ratio is

$$\eta={z_{12}z_{34}\over z_{13}z_{24}}.$$

Similarity transformations remove position, orientation, and one overall length but leave dimensionless shape data. The full global conformal group reduces the four-point dependence to the complex cross ratio $\eta$ and its conjugate $\bar\eta$.

The remaining function of $\eta$ and $\bar\eta$ contains dynamical information: operator product coefficients, exchanged scaling fields, and the consistency constraints that later become crossing symmetry.

Temperature as an energy perturbation

Moving the Ising model away from criticality introduces a relevant perturbation. In continuum notation,

$$S=S_*+\tau\int d^d x\,\epsilon(x)+\cdots,$$

where $\tau$ is proportional to $T-T_c$ or, equivalently, to the deviation of the lattice coupling from its critical value. Since the action is dimensionless, the coupling $\tau$ has RG eigenvalue

$$y_t=d-\Delta_\epsilon.$$

The correlation length is the scale at which the effective dimensionless perturbation becomes order one:

$$\tau \xi^{y_t}\sim 1.$$

Therefore

$$\boxed{ \xi\sim |\tau|^{-\nu}, \qquad \nu={1\over d-\Delta_\epsilon}. }$$

For the two-dimensional Ising model,

$$d=2, \qquad \Delta_\epsilon=1, \qquad \nu=1.$$

The physical mass gap is

$$m_{\rm phys}=\xi^{-1}.$$

In the Majorana description this is exactly the fermion mass scale. Its sign distinguishes the ordered and disordered phases, while its magnitude sets the inverse correlation length.

Specific heat from the energy correlator

The singular specific heat is controlled by energy fluctuations. Differentiating the free energy twice with respect to the temperature-like coupling inserts two energy operators:

$$C_{\rm sing} \sim \int d^d x\,\langle \epsilon(x)\epsilon(0)\rangle_c.$$

The connected correlator is used because the identity contribution has been subtracted. Near the fixed point,

$$\langle \epsilon(x)\epsilon(0)\rangle_c\sim {1\over |x|^{2\Delta_\epsilon}} \qquad (a\ll |x|\ll \xi).$$

Thus

$$C_{\rm sing} \sim \int_a^\xi dr\,r^{d-1-2\Delta_\epsilon}.$$

The singular specific heat is the integrated connected energy–energy correlator. The ultraviolet cutoff is the microscopic spacing $a$, while the infrared cutoff is the correlation length $\xi$.

Writing $p=d-2\Delta_\epsilon$, the regulated radial integral gives

$$\int_a^\xi dr\,r^{p-1} \sim \begin{cases} \xi^p/p, & p>0,\\ \log(\xi/a), & p=0,\\ C_{\rm UV}-\xi^p/|p|, & p<0. \end{cases}$$

For $p<0$, the constant $C_{\rm UV}$ belongs to the cutoff-dependent analytic background. The remaining $\xi^p$ term is finite but nonanalytic in the temperature-like coupling; depending on its exponent it appears as a cusp or as a singularity in a higher derivative. Thus a UV-dominated integral does not mean that all critical nonanalyticity has disappeared.

For the two-dimensional Ising model,

$$d=2, \qquad \Delta_\epsilon=1,$$

so the energy integral is logarithmic:

$$\boxed{ C_{\rm sing}\sim \log {\xi\over a}\sim -\log |\tau|. }$$

This is why the specific-heat exponent is quoted as $\alpha=0$ but the singularity is still present.

The same conclusion is consistent with hyperscaling, provided the fixed point is below its upper critical dimension and no dangerously irrelevant coupling changes the free-energy scaling. The singular free-energy density then scales as one correlation volume per unit volume,

$$f_{\rm sing}\sim \xi^{-d}\sim |\tau|^{d\nu}.$$

If $C_{\rm sing}\sim |\tau|^{-\alpha}$, then

$$\boxed{\alpha=2-d\nu.}$$

For two-dimensional Ising, $d=2$ and $\nu=1$, so $\alpha=0$.

Scaling forms away from criticality

At the fixed point, a two-point function is a pure power. Away from criticality, the correlation length can appear. If $O$ has a nonzero one-point function, the clean massive scaling law applies to the connected correlator:

$$\boxed{ \langle O(r)O(0)\rangle_{\tau,c} ={1\over r^{2\Delta_O}} \Phi_{O,c}\left({r\over \xi}\right). }$$

For $r\ll \xi$, the scaling function tends to a constant and the critical power law is recovered. For $r\gg \xi$, the connected correlator in a massive phase decays exponentially, up to powers of $r/\xi$. When $\langle O\rangle=0$, the full and connected correlators coincide.

Near criticality, the connected spin correlator behaves as $r^{-2\Delta_\sigma}$ for $r\ll\xi$ and crosses over to massive decay when $r$ becomes comparable to the correlation length.

For the Ising spin field, it is also useful to retain the full correlator:

$$\langle \sigma(r)\sigma(0)\rangle_\tau ={1\over r^{1/4}} \Phi_\pm\left({r\over \xi}\right),$$

where the two functions correspond to the two sides of the transition. In the disordered phase $\Phi_+$ decays exponentially. In the ordered phase, cluster decomposition requires the full spin correlator to approach $M^2$; equivalently $\Phi_-(s)\sim B_-s^{2\Delta_\sigma}$ as $s\to\infty$. The scaling form then implies

$$M\sim \xi^{-\Delta_\sigma}\sim |\tau|^{\nu\Delta_\sigma}.$$

Therefore the magnetization exponent is

$$\boxed{\beta_{\rm mag}=\nu\Delta_\sigma.}$$

The subscript keeps this exponent distinct from the inverse temperature often denoted by $\beta$.

Critical exponent dictionary

The two relevant Ising perturbations are the thermal field and the magnetic field:

$$S=S_*+\tau\int d^d x\,\epsilon(x)-h\int d^d x\,\sigma(x)+\cdots.$$

Their RG eigenvalues are

$$y_t=d-\Delta_\epsilon, \qquad y_h=d-\Delta_\sigma.$$

Two representative derivations make the dictionary transparent. First, the zero-field susceptibility is the integrated connected spin correlator,

$$\chi\sim \int^{\xi}d^d x\, \langle\sigma(x)\sigma(0)\rangle_c \sim \xi^{d-2\Delta_\sigma},$$

which gives $\gamma=\nu(d-2\Delta_\sigma)$. Second, the scaling form of the singular free energy,

$$f_{\rm sing}(\tau,h) =b^{-d}f_{\rm sing}(\tau b^{y_t},h b^{y_h}),$$

at $\tau=0$ and $b=|h|^{-1/y_h}$ gives $M\sim |h|^{(d-y_h)/y_h}$. Hence $\delta=y_h/(d-y_h)=(d-\Delta_\sigma)/\Delta_\sigma$.

Assuming hyperscaling and no dangerously irrelevant variable, the standard exponents are

$$\nu={1\over d-\Delta_\epsilon},$$ $$\eta=2\Delta_\sigma-d+2,$$ $$\alpha=2-d\nu,$$ $$\beta_{\rm mag}=\nu\Delta_\sigma,$$ $$\gamma=\nu(d-2\Delta_\sigma),$$

and

$$\delta={d-\Delta_\sigma\over \Delta_\sigma}.$$

For the two-dimensional Ising dimensions,

$$d=2, \qquad \Delta_\sigma={1\over8}, \qquad \Delta_\epsilon=1,$$

we obtain

$$\boxed{ \nu=1, \qquad \eta={1\over4}, \qquad \alpha=0\text{ with a logarithm}, \qquad \beta_{\rm mag}={1\over8}, \qquad \gamma={7\over4}, \qquad \delta=15. }$$

This is the operator-dimension dictionary in action. The thermodynamic exponents are not independent mysteries; they are consequences of the scaling dimensions of the relevant operators.

Summary

At a critical point, long-distance observables organize into scaling fields. Their two-point functions obey

$$\langle O(x)O(0)\rangle\sim |x|^{-2\Delta_O}.$$

For the two-dimensional Ising fixed point,

$$\Delta_\sigma=\Delta_\mu={1\over8}, \qquad \Delta_\epsilon=1, \qquad \Delta_\psi={1\over2}.$$

The energy dimension gives the correlation-length exponent and the specific-heat singularity. The spin dimension gives the anomalous-dimension exponent, magnetization exponent, susceptibility exponent, and critical isotherm. Multi-point functions are homogeneous, but their remaining dependence on dimensionless shapes is where the operator algebra begins to appear. The next pages sharpen this by deriving conformal symmetry and then the OPE.

Common pitfalls

Scaling dimensions need not be engineering dimensions. They coincide at a suitable Gaussian fixed point, but interactions generally add anomalous dimensions and shift the powers in correlation functions.

Do not confuse the two uses of $\beta$. The exponent $\beta_{\rm mag}$ is the magnetization exponent, not the inverse temperature. This page uses $\tau$ for the reduced temperature to avoid overloading $\beta$.

The value $\alpha=0$ does not specify the singularity by itself. In the two-dimensional Ising model the specific heat is logarithmic because the energy–energy integral is marginal.

Scale invariance fixes homogeneity, not all shape dependence. Conformal invariance is the stronger statement that fixes scalar three-point functions and reduces four-point functions to functions of conformal cross ratios.

Use connected correlators when the one-point function is nonzero. In the ordered phase, $\langle\sigma(r)\sigma(0)\rangle\to M^2$ rather than zero. The exponentially decaying object is $\langle\sigma(r)\sigma(0)\rangle_c$.

Exercises

Exercise 1: Two-point homogeneity

Let $O$ be a scalar scaling field of dimension $\Delta$ in a translation- and rotation-invariant critical theory. Show that

$$\langle O(x)O(0)\rangle={C\over |x|^{2\Delta}}.$$

Solution

Let $F(r)=\langle O(x)O(0)\rangle$, where $r=|x|$. Scale covariance gives

$$F(\lambda r)=\lambda^{-2\Delta}F(r).$$

Set $r=1$ and choose $\lambda=r$. Then

$$F(r)=F(\lambda\cdot 1)=\lambda^{-2\Delta}F(1)=C r^{-2\Delta}.$$

Thus

$$F(r)={C\over r^{2\Delta}}.$$

Exercise 2: Energy fluctuations and specific heat

Use the energy correlator

$$\langle \epsilon(x)\epsilon(0)\rangle_c\sim {|x|}^{-2\Delta_\epsilon}$$

to determine the singular scaling of

$$C_{\rm sing}\sim \int_a^\xi dr\,r^{d-1-2\Delta_\epsilon}.$$

Apply the result to the two-dimensional Ising model.

Solution

If $d-2\Delta_\epsilon\ne0$, then

$$\int_a^\xi dr\,r^{d-1-2\Delta_\epsilon} ={\xi^{d-2\Delta_\epsilon}-a^{d-2\Delta_\epsilon}\over d-2\Delta_\epsilon}.$$

For $d>2\Delta_\epsilon$, the $\xi$ term dominates. For $d<2\Delta_\epsilon$, the first term vanishes as $\xi\to\infty$ while the cutoff term contributes to the analytic background. The remaining power $\xi^{d-2\Delta_\epsilon}$ is a finite nonanalytic correction; it can produce a cusp or a singularity only in a higher derivative. For $d=2\Delta_\epsilon$,

$$\int_a^\xi {dr\over r}=\log(\xi/a).$$

In the two-dimensional Ising model, $d=2$ and $\Delta_\epsilon=1$, so

$$C_{\rm sing}\sim \log \xi\sim -\log |\tau|.$$

Exercise 3: The two-dimensional Ising exponent dictionary

Using $d=2$, $\Delta_\sigma=1/8$, and $\Delta_\epsilon=1$, compute $\nu$, $\eta$, $\alpha$, $\beta_{\rm mag}$, $\gamma$, and $\delta$.

Solution

The correlation-length exponent is

$$\nu={1\over d-\Delta_\epsilon}={1\over2-1}=1.$$

The anomalous-dimension exponent is

$$\eta=2\Delta_\sigma-d+2={1\over4}.$$

The specific-heat exponent is

$$\alpha=2-d\nu=2-2=0,$$

with a logarithmic singularity. The magnetization exponent is

$$\beta_{\rm mag}=\nu\Delta_\sigma={1\over8}.$$

The susceptibility exponent is

$$\gamma=\nu(d-2\Delta_\sigma)=2-{1\over4}={7\over4}.$$

Finally,

$$\delta={d-\Delta_\sigma\over \Delta_\sigma} ={2-1/8\over 1/8}=15.$$

Exercise 4: Energy dimension from Majorana fields

Assume

$$\langle \psi(z)\psi(0)\rangle={1\over z}, \qquad \langle \bar\psi(\bar z)\bar\psi(0)\rangle={1\over \bar z}.$$

If $\epsilon\sim i\psi\bar\psi$, show that $\Delta_\epsilon=1$.

Solution

Wick contraction gives

$$\langle \epsilon(z,\bar z)\epsilon(0,0)\rangle \propto \langle \psi(z)\psi(0)\rangle \langle \bar\psi(\bar z)\bar\psi(0)\rangle ={1\over z\bar z} ={1\over |z|^2}.$$

A scalar operator with dimension $\Delta$ has two-point function $|z|^{-2\Delta}$. Therefore $2\Delta_\epsilon=2$, and

$$\Delta_\epsilon=1.$$

Further reading

• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996). A compact route from scaling hypotheses to critical exponents.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer (1997). See the two-dimensional treatment of Ising correlators and operator dimensions.
• B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Harvard University Press (1973). The exact lattice solution and correlation-function results.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987). See the discussions of statistical systems, duality, and conformal field theory.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., Oxford University Press (2002). See the field-theoretic treatment of scaling, RG, and critical exponents.