Critical Phenomena and RG Bridge

A continuous critical point is a route from a regulated many-body model to a continuum field theory. If $a$ is a microscopic spacing and $\xi$ is the correlation length of the critical sector under study, the critical regime contains a scaling window

$$a \ll \lvert x\rvert \ll \xi.$$

At the critical point, $\xi/a$ diverges in the thermodynamic limit. The microscopic spacing then becomes invisible to long-distance correlators after fields, operators, and parameters are matched appropriately.

The central QFT translation is

$$\mathcal S = \mathcal S_\star + \sum_i \lambda_i \int d^D x\, \mathcal O_i(x).$$

Here $\mathcal S_\star$ is a fixed-point theory, $\mathcal O_i$ are its scaling operators, and the $\lambda_i$ are deformations. Under a scale change $x\mapsto bx$,

$$\lambda_i \longmapsto b^{y_i}\lambda_i, \qquad y_i = D-\Delta_i,$$

where $\Delta_i$ is the scaling dimension of $\mathcal O_i$. This one relation connects the many-body language of relevant perturbations and critical exponents to the QFT language of operators, anomalous dimensions, masses, and renormalized correlation functions.

Helpful background. Statistical field theory defines the regulated action; renormalization group explains coarse graining and linearized flow. Here $D$ initially denotes an isotropic Euclidean dimension; quantum anisotropic scaling is treated separately.

Critical Scaling and Continuum Operators

The same structure receives different names in neighboring communities.

Many-body or statistical language	Continuum-QFT language
microscopic spacing $a$	ultraviolet regulator $\Lambda\sim a^{-1}$
block or collective variable	coarse field or renormalized operator
reduced temperature or tuning control	relevant coupling
diverging correlation length $\xi$	vanishing renormalized mass $m_{\mathrm{corr}}$
critical surface	continuum-limit tuning surface
correction to scaling	irrelevant-operator contribution
universality class	fixed point plus operator and symmetry data
finite size $L$	infrared regulator
susceptibility	integrated connected two-point response
lattice anisotropy	symmetry-allowed deformation

This dictionary is not an identification of bare symbols. Every row requires a regulator, normalization, and observable-matching prescription.

From a Lattice Model to a Continuum Theory

Let a regulated model have lattice spacing $a$ and cutoff

$$\Lambda \sim \frac{c_\Lambda}{a},$$

where $c_\Lambda$ depends on the regulator convention. A continuum scaling limit requires

$$\frac{\xi_{\mathrm{phys}}}{a} \longrightarrow \infty.$$

The ratio, not merely the numerical size of $\xi$ in one set of units, is the essential statement.

Critical and massive scaling limits

At the critical point,

$$\xi_{\mathrm{phys}} \longrightarrow \infty, \qquad m_{\mathrm{corr}} := \xi_{\mathrm{phys}}^{-1} \longrightarrow 0.$$

The limiting QFT is massless in this correlation-length sense.

One can also take a massive scaling limit. Send $a\to0$ while tuning the bare controls toward the critical surface so that

$$m_{\mathrm{corr}} = \xi_{\mathrm{phys}}^{-1}$$

remains finite. In lattice units,

$$m_{\mathrm{corr}}a = \frac{a}{\xi_{\mathrm{phys}}} \longrightarrow 0.$$

The resulting continuum theory is the fixed-point theory deformed by one or more relevant operators.

A continuum limit is a tuned family

A single lattice Hamiltonian at a single spacing is not yet a continuum limit. One needs a family of regulated models whose bare couplings depend on $a$ so that selected renormalized observables remain fixed.

The data defining that family include:

1. regulator and boundary conditions;
2. fields and source normalizations;
3. controls tuned toward the critical surface;
4. physical quantities held fixed;
5. operator definitions and subtraction conventions;
6. the order of infinite-volume and continuum limits.

Keeping every bare coupling fixed while sending $a\to0$ generally does not accomplish this.

The thermodynamic limit still matters

When the bulk correlation length becomes comparable to or exceeds the system size, finite-size rounding replaces bulk asymptotic behavior. A bulk or fitted correlation length can exceed $L$; the issue is loss of a region in which its infinite-volume decay can be measured. A clean bulk scaling limit separates

$$a \ll \xi \ll L$$

when studying a massive theory, or takes $L/a\to\infty$ together with $\xi/a\to\infty$ at criticality. Aspect ratios and boundary conditions remain part of universal finite-size statements.

Microscopic Observables Become QFT Operators

A microscopic observable is not automatically equal to a continuum field. At long distance, a local lattice observable $A_{\mathrm{lat}}$ has an operator expansion of the schematic form

$$\begin{aligned} A_{\mathrm{lat}}(n) &= \sum_i C_i(a)\, \mathcal O_i(x) + \text{descendants} + \text{contact terms}, \\ x&=an. \end{aligned}$$

The coefficients $C_i(a)$ contain powers of $a$, normalization constants, and mixing information. Symmetry restricts which $\mathcal O_i$ can appear.

For example, an Ising-odd microscopic spin can overlap with the leading odd scaling operator $\sigma$:

$$S_n = C_\sigma(a)\, \sigma(x) + \cdots.$$

Its long-distance two-point function is controlled by the lowest-dimension allowed operator with nonzero overlap. A different microscopic observable can have the same leading exponent but a different amplitude.

Renormalized correlators

A continuum two-point function can be defined schematically by

$$G_R(x) = \lim_{a\to0} Z_A^2(a) \times \left\langle A_{\mathrm{lat}}(x/a) A_{\mathrm{lat}}(0) \right\rangle_{\mathrm c}.$$

The factor $Z_A$ is chosen by a renormalization or normalization condition. Composite operators can mix, and additive contact terms may be needed at coincident points. Long-distance agreement does not make microscopic operators identical at the cutoff.

Sources must be matched too

If a microscopic source $h_{\mathrm{lat}}$ couples to $A_{\mathrm{lat}}$, its continuum image couples to every operator allowed in the expansion:

$$\delta\mathcal S = -\int d^D x\, \sum_i j_i(x)\mathcal O_i(x).$$

Source matching is what turns continuum correlators into predictions for susceptibilities, scattering intensities, and response coefficients.

The Fixed Point Is the Critical Field Theory

A fixed-point theory reproduces itself after coarse-graining and rescaling. For a scalar scaling operator,

$$\mathcal O(bx) \sim b^{-\Delta_{\mathcal O}} \mathcal O(x).$$

Translation, rotation, and scale invariance then constrain its separated-point two-point function to

$$\left\langle \mathcal O(x) \mathcal O(0) \right\rangle = \frac{C_{\mathcal O}}{ \lvert x\rvert^{2\Delta_{\mathcal O}} },$$

up to tensor structures, operator mixing, boundaries, and normalization.

For an order-parameter field $\phi$ in an isotropic $D$-dimensional critical theory,

$$\Delta_\phi = \frac{ D-2+\eta }{2}.$$

The anomalous exponent $\eta$ is therefore part of the field’s scaling dimension, not a correction appended after the QFT has been defined.

Beyond exponents

Critical exponents are only a small part of fixed-point data. A continuum description can also organize:

• the spectrum of scaling operators;
• their internal-symmetry representations and spins;
• normalized two- and three-point coefficients;
• operator mixing and descendants;
• boundary, defect, and topological sectors;
• conserved currents and stress tensors.

When a conformal description is valid, the short-distance product of local operators takes the form

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k C_{ij}^{\ k} \times \lvert x\rvert^{ \Delta_k-\Delta_i-\Delta_j } \times \mathcal O_k(0) + \cdots.$$

The operator dimensions and operator-product coefficients provide a much sharper universality fingerprint than one fitted exponent.

Operators and RG Eigen-Directions

Perturb the fixed point by

$$\delta\mathcal S = \lambda_i \int d^D x\, \mathcal O_i(x).$$

If $\mathcal O_i$ is an eigenoperator of the linearized RG, then

$$\lambda_i(b) = b^{y_i}\lambda_i, \qquad y_i = D-\Delta_i.$$

Thus:

• $\Delta_i<D$ gives a relevant deformation;
• $\Delta_i>D$ gives an irrelevant deformation;
• $\Delta_i=D$ is marginal at linear order.

The word eigenoperator matters. Bare monomials such as $\phi^2$, $\phi^4$, and $(\partial\phi)^2$ can mix under renormalization. The simple classification applies after the mixing matrix has been diagonalized near the fixed point.

Marginal does not mean constant

When $y_i=0$, nonlinear terms decide the flow:

$$\frac{d\lambda}{d\ell} = A\lambda^2 + \cdots.$$

The deformation can be marginally relevant, marginally irrelevant, or exactly marginal. Logarithmic scaling, exponentially generated scales, and fixed lines all arise from this distinction.

Redundant directions

Some apparent deformations are generated by field redefinitions, equations of motion, or total derivatives. They can change coordinates on theory space without changing bulk separated-point physics. A total derivative can nevertheless contribute at a boundary or in a nontrivial topological sector; those conditions must be fixed before calling it redundant. Counting relevant controls requires physical eigenoperators after such redundancies and exact symmetries are handled.

Relevant Deformations Generate Scales

Let $u_t$ be the leading temperature-like scaling field:

$$u_t(b) = b^{y_t}u_t, \qquad y_t>0.$$

The flow remains near the fixed point until

$$b_\star^{y_t} \lvert u_t\rvert \sim 1.$$

If departure in this direction produces a finite correlation length and no additional singular scale controls it, that correlation length is then

$$\xi \sim a b_\star \sim a \lvert u_t\rvert^{-1/y_t}.$$

Therefore

$$\nu = \frac{1}{y_t}, \qquad m_{\mathrm{corr}}a \sim \lvert u_t\rvert^\nu.$$

The many-body statement “the correlation length diverges” and the QFT statement “the mass scale vanishes” are two descriptions of this same departure from a critical fixed point.

Critical surfaces and tuning

Near a fixed point, write

$$\mathcal S = \mathcal S_\star + \sum_{a=1}^{n_{\mathrm{rel}}} u_a \int d^D x\, \mathcal O_a + \cdots.$$

Under standard regularity assumptions, the critical surface has codimension $n_{\mathrm{rel}}$. Reaching the fixed point requires tuning every symmetry-allowed physical relevant $u_a$ to zero. The identity operator only shifts the bulk free-energy normalization and is excluded from this count. Marginally relevant directions require separate analysis beyond this hyperbolic linearized picture.

At an Ising-symmetric critical point, the thermal deformation and magnetic source are both relevant. Exact $\mathbb Z_2$ symmetry sets the odd magnetic source to zero, leaving one experimental tuning control. At a multicritical point, more relevant even operators can remain and more controls must be tuned.

The sign selects the infrared phase

Opposite signs of one relevant deformation can flow into distinct phases. The characteristic crossover length can be finite even when the resulting phase contains Goldstone modes or another gapless sector; a relevant perturbation does not always make the entire theory massive. The fixed-point theory governs the common short-to-intermediate scaling regime, while the sign and magnitude of the deformation determine the eventual infrared endpoint.

Universality Requires an Operator Dictionary

Suppose two microscopic models flow toward the same fixed point:

$$\mathcal S_A \longrightarrow \mathcal S_\star, \qquad \mathcal S_B \longrightarrow \mathcal S_\star.$$

This explains shared scaling dimensions and normalized scaling functions, but predictions still require maps

$$\begin{aligned} A_{\mathrm{lat}} &\longleftrightarrow \sum_i C_{A i}\mathcal O_i, \\ B_{\mathrm{lat}} &\longleftrightarrow \sum_i C_{B i}\mathcal O_i. \end{aligned}$$

The coefficients are generally nonuniversal. A fixed point without an operator dictionary does not tell an experimenter which probe measures which continuum correlation function.

What can be universal

With geometry and normalization specified, universal information can include:

• scaling dimensions and critical exponents;
• operator-product data;
• normalized scaling functions;
• selected amplitude ratios;
• fixed-point values of dimensionless observables;
• the number and symmetry of relevant directions.

What usually is not

The following are normally regulator or realization dependent:

• the critical temperature or microscopic coupling;
• the fixed-point coordinates in a chosen coupling chart;
• field-normalization constants;
• raw correlation and susceptibility amplitudes;
• the width of the observable scaling window;
• amplitudes of irrelevant corrections.

Universal data survive smooth changes of scheme. Bare and fixed-point coupling values generally do not.

Wilsonian Flow and Renormalized QFT

The Wilsonian and renormalized descriptions answer related questions with different scale conventions.

Wilsonian convention

Let $k$ be a sliding infrared cutoff below a fixed ultraviolet cutoff $\Lambda$. Integrating modes above $k$ produces an action

$$\mathcal S_k[\phi].$$

With

$$\ell := \ln \frac{\Lambda}{k},$$

increasing $\ell$ means flowing toward the infrared. A coupling satisfies

$$\frac{d g_i}{d\ell} = \beta_i^{\mathrm{IR}}(\boldsymbol g).$$

Renormalization-scale convention

Perturbative QFT often uses a subtraction scale $\mu$ and defines

$$\beta_\mu(g) := \mu \frac{dg}{d\mu}.$$

If $\mu$ tracks $k$, then

$$\frac{dg}{d\ell} = -\beta_\mu(g).$$

A statement that a coupling “grows” is incomplete until the scale direction is declared.

Callan–Symanzik equation

For a critical renormalized $n$-point function in a one-coupling theory, one common convention is

$$0 = \Bigg[ \mu\frac{\partial}{\partial\mu} + \beta_\mu(g) \frac{\partial}{\partial g} + n\gamma_\phi(g) \Bigg] G_R^{(n)}.$$

Define the field renormalization by $\phi_{\mathrm B}=Z_\phi^{1/2}\phi_R$, with bare parameters held fixed in the scale derivative. Then

$$\gamma_\phi := \frac12 \mu \frac{d\ln Z_\phi}{d\mu}.$$

Masses, composite insertions, and several couplings add terms. At a fixed point,

$$\beta_\mu(g_\star) = 0,$$

and the anomalous dimension contributes to the scaling law. In this convention,

$$\eta = 2\gamma_\phi(g_\star).$$

The Callan–Symanzik equation expresses independence from the arbitrary subtraction scale after bare data are held fixed. It is not a physical-time evolution equation.

Counterterms and coarse graining are compatible

Counterterms define finite renormalized parameters and operators in a chosen scheme. Wilsonian integration explains why every allowed operator is generated and why long-distance predictions depend on only a few relevant coordinates near a fixed point. These are complementary formulations, not competing explanations.

Wilson–Fisher Fixed Point Preview

The $O(N)$ scalar model in

$$D = 4-\epsilon$$

dimensions gives a perturbative bridge for $0<\epsilon\ll1$, positive quartic coupling, and integer $N\ge1$. The noninteger dimension is an analytic continuation used for the expansion. Use the convention

$$\mathcal S = \int d^D x\, \Bigg[ \frac12 (\partial_\mu\phi_a)^2 + \frac12 r\phi_a^2 + \frac{ \mu^\epsilon g }{4!} (\phi_a\phi_a)^2 \Bigg].$$

The mass parameter must be tuned to its critical value $r_c(g)$. At $D=4$, $g$ is marginal by engineering power counting. For $D=4-\epsilon$, it is relevant at the Gaussian fixed point.

In minimal-subtraction-type conventions, the one-loop beta function is

$$\beta_\mu(g) = -\epsilon g + \frac{N+8}{ 3(4\pi)^2 } g^2 + O(g^3).$$

Besides the Gaussian fixed point, there is an interacting fixed point

$$g_\star = \frac{ 3(4\pi)^2 }{N+8} \epsilon + O(\epsilon^2).$$

Its coordinate depends on the normalization and scheme. The linearized slope

$$\beta_\mu'(g_\star) = \epsilon + O(\epsilon^2)$$

means that deviations in the quartic coupling decay as $\mu\to0$: this direction is infrared attractive.

The thermal eigenvalue and field anomalous dimension are

$$\begin{aligned} y_t &= 2 - \frac{N+2}{N+8} \epsilon + O(\epsilon^2), \\ \eta &= O(\epsilon^2). \end{aligned}$$

Therefore

$$\nu = \frac12 + \frac{N+2}{ 4(N+8) } \epsilon + O(\epsilon^2).$$

This small calculation displays the bridge:

1. a Landau–Ginzburg action supplies a regulated field theory;
2. loop renormalization supplies a beta function and anomalous dimensions;
3. a non-Gaussian fixed point replaces mean-field scaling;
4. stability eigenvalues become critical exponents;
5. the continuum QFT describes many microscopic realizations.

Setting $\epsilon=1$ at first order is only a rough estimate for three dimensions. Precision work needs higher orders, resummation, simulation, bootstrap, or other nonperturbative information. The displayed result is a preview of method, not a precision table.

Worked Example: Approach to the Wilson–Fisher Fixed Point

Within the displayed one-loop truncation, define $x=g/g_\star$ and $s=\epsilon\ell$, where $\ell=\ln(\Lambda/k)$ grows toward the infrared. Then

$$\frac{dx}{ds}=x(1-x), \qquad x(s)=\frac{1}{1+(x_0^{-1}-1)e^{-s}}, \qquad x_0>0.$$

This solution follows by separating $dx/[x(1-x)]=ds$. Initial positive couplings above or below $x=1$ approach the same fixed point. For example, $x_0=1/2$ reaches $x=0.9$ at $s=\ln9$. The scale ratio is $b=e^\ell=9^{1/\epsilon}$: for small $\epsilon$, a substantial change in scale may be required before the asymptotic regime appears.

This attraction in the quartic direction does not remove thermal tuning. A small thermal field still grows as $u_t(\ell)=u_t(0)e^{y_t\ell}$ near the fixed point. The flow can remain critical only if that physical relevant field is tuned to zero. At $N=1$, $\epsilon=0.2$, the displayed first-order result gives $y_t=2-\epsilon/3\simeq1.933$.

The curves solve the one-loop flow exactly within that truncation, with $x_0=0.25,0.6,1.5,2$. They approach $g/g_\star=1$ from both sides as $s=\epsilon\ln(\Lambda/k)$ increases. The thermal relevant coupling is assumed tuned; it would otherwise carry the system away from criticality.

Classical, Quantum, and Dynamic Criticality

The dimension entering a scaling relation must match the theory being scaled.

Thermal equilibrium criticality

For a static thermal transition in $d$ spatial dimensions, the long-distance statistical field theory is usually $D=d$ dimensional after massive temporal and microscopic modes have been integrated out.

Quantum criticality

At a zero-temperature quantum critical point,

$$\mathbf x \longmapsto b\mathbf x, \qquad \tau \longmapsto b^z\tau.$$

The integration measure scales with

$$D_{\mathrm{sc}} = d+z.$$

An eigenoperator can then obey

$$y_i = d+z-\Delta_i$$

when $\Delta_i$ is defined in this anisotropic convention. The notation $d+z$ is a scaling count, not a guarantee of an isotropic relativistic QFT.

Finite temperature gives an imaginary-time extent

$$L_\tau = \beta\hbar.$$

It cuts off the zero-temperature quantum scaling at a thermal scale. A bosonic static sector can continue flowing below that scale, including toward a finite-temperature transition; it does not stop every infrared flow.

Dynamics is additional structure

The same static fixed point can support different dynamic universality classes. Conservation laws, damping, reversible couplings, noise, and hydrodynamic modes determine $z$ and real-time response. A static Euclidean functional does not choose those ingredients.

Hydrodynamics and Effective Theory Preview develops that real-time slow-variable and fluctuation framework; this page retains the static continuum and RG construction.

Scale Invariance Is Not Automatically Conformal

At an RG fixed point, the theory is scale invariant in the sense defined by the RG transformation. Conformal invariance is stronger.

Many familiar local, short-range, equilibrium critical points do admit a conformal description. When they do, conformal symmetry sharply constrains correlators and organizes operators into primaries and descendants.

One should not infer conformal symmetry from the words “fixed point” alone. The conclusion can depend on:

• locality and reflection positivity or unitarity;
• spacetime dimension;
• discreteness of the operator spectrum;
• conserved virial currents;
• anisotropic or nonrelativistic scaling;
• disorder and broad distributions;
• boundaries, defects, or long-range interactions.

Likewise, emergent rotational or Lorentz symmetry must be demonstrated by the flow of anisotropies and velocities. It is not guaranteed by taking a long wavelength limit.

Where the Simple Fixed-Point Picture Needs Extension

The ordinary isolated-fixed-point picture is powerful but not universal.

First-order transitions

An ordinary first-order transition need not make its transition-associated correlation length diverge. An ordered phase can nevertheless retain Goldstone modes or other gapless spectators, and a special critical endpoint can have divergent critical fluctuations. It does not generally define a scale-invariant continuum QFT by tuning to coexistence alone.

Berezinskii–Kosterlitz–Thouless transitions

Marginal flows and a fixed line produce an essential singularity rather than ordinary power-law tuning by one positive eigenvalue.

Fermi surfaces

Low-energy modes live near an extended momentum-space surface. Scaling is patch dependent, and four-fermion interactions require a channel-sensitive classification.

Disorder and infinite-randomness flow

The running object can be a probability distribution rather than a finite set of couplings. Typical and average observables can scale differently.

Boundaries and defects

A bulk fixed point can coexist with independent boundary or defect operators, relevant perturbations, and universality classes.

Several soft sectors

Order parameters coupled to gauge fields, Goldstone modes, fermions, or hydrodynamic densities can generate nonlocal kernels and additional relevant directions. Integrating out a gapless field does not generally leave a local polynomial action.

Connections

• Statistical Field Theory Preview supplies the regulated measure whose long-distance flow is analyzed here.
• Renormalization Group Preview develops the Wilsonian transformation and flow geometry in depth.
• Critical Exponents and Scaling turns the eigenvalues introduced here into thermodynamic, correlation, and finite-size predictions.
• Universality explains how symmetry, dimension, interactions, and observables distinguish classes.
• Landau–Ginzburg Theory Preview supplies the scalar functional used in the Wilson–Fisher example.
• Quantum Phase Transitions develops anisotropic scaling and quantum-critical crossover.
• Finite-Temperature QFT Bridge develops the compact thermal direction, Matsubara scales, screening, and the conditions for a static lower-dimensional effective theory.
• Connected Correlation Functions explains clustering, connected subtraction, and correlation-length definitions.
• Sources to Generating Functionals develops source differentiation and connected functionals.
• Further field-theory study connects these calculations to renormalization, composite operators, and effective field theory.

Exercises

1. Tune a Massive Continuum Limit

Suppose a lattice correlation length obeys

$$\xi_{\mathrm{lat}}(t) = A \lvert t\rvert^{-\nu},$$

where $\xi_{\mathrm{lat}}$ is measured in lattice spacings. Find how $t$ must depend on $a$ so that the physical correlation length $\xi_{\mathrm{phys}}=a\xi_{\mathrm{lat}}$ approaches a fixed nonzero value $\Xi$ as $a\to0$.

Solution

Require

$$\Xi = aA \lvert t(a)\rvert^{-\nu}.$$

Solving gives

$$\lvert t(a)\rvert = \left( \frac{Aa}{\Xi} \right)^{1/\nu}.$$

Thus $t(a)\to0$ as the cutoff is removed. Meanwhile,

$$\frac{\xi_{\mathrm{phys}}}{a} = \frac{\Xi}{a} \longrightarrow \infty.$$

The continuum theory is massive because $m_{\mathrm{corr}}=\Xi^{-1}$ remains finite, but its bare lattice control is tuned toward the critical point.

2. Operators and Coupling Eigenvalues

In an isotropic $D=3$ fixed-point theory, four scalar eigenoperators have dimensions

$$\Delta \in \{1.04,\ 2.40,\ 3.00,\ 4.20\}.$$

Classify their integrated couplings at linear order and give each RG eigenvalue.

Solution

For an integrated scalar deformation,

$$y = D-\Delta = 3-\Delta.$$

Therefore:

$$\begin{aligned} \Delta=1.04,\quad y=1.96 &: \text{relevant}, \\ \Delta=2.40,\quad y=0.60 &: \text{relevant}, \\ \Delta=3.00,\quad y=0 &: \text{marginal}, \\ \Delta=4.20,\quad y=-1.20 &: \text{irrelevant}. \end{aligned}$$

The $\Delta=3$ operator requires the nonlinear beta function. Symmetry can forbid a relevant operator, and redundant operators must be removed before counting physical tuning directions.

3. Correlation Length as an RG Stopping Scale

Let one relevant field obey

$$u(b) = b^y u, \qquad y>0.$$

Assume this is the only generated long length and that the infrared phase has a finite correlation length in this sector. Derive its exponent and inverse-length mass scaling.

Solution

The flow leaves the fixed-point neighborhood when

$$b_\star^y \lvert u\rvert \sim 1.$$

Hence

$$b_\star \sim \lvert u\rvert^{-1/y}.$$

The only generated long length is

$$\xi \sim a b_\star \sim a \lvert u\rvert^{-1/y}.$$

Comparing with $\xi\sim\lvert u\rvert^{-\nu}$ gives

$$\nu = \frac{1}{y}.$$

The inverse-correlation-length mass behaves as

$$m_{\mathrm{corr}}a \sim \frac{a}{\xi} \sim \lvert u\rvert^{1/y} = \lvert u\rvert^\nu.$$

4. Leading Operator in a Lattice Correlator

A $\mathbb Z_2$-odd lattice observable has the continuum expansion

$$A_{\mathrm{lat}} = c_1\sigma + c_2\mathcal O_{\mathrm{odd}} + \cdots,$$

with

$$\Delta_\sigma < \Delta_{\mathrm{odd}}.$$

What controls the leading long-distance two-point function? What changes if $c_1=0$?

Solution

When $c_1\ne0$, the smallest-dimension allowed operator dominates:

$$\left\langle A_{\mathrm{lat}}(x) A_{\mathrm{lat}}(0) \right\rangle_{\mathrm c} \sim c_1^2 \times \frac{C_\sigma}{ \lvert x\rvert^{2\Delta_\sigma} }.$$

The amplitude $c_1^2C_\sigma$ depends on microscopic normalization, while the power is fixed by the continuum operator dimension.

If symmetry, an improved-observable construction, or an accidental cancellation gives $c_1=0$, then the next operator with nonzero coefficient controls the asymptotic power:

$$\lvert x\rvert^{-2\Delta_{\mathrm{odd}}}.$$

This is why an operator dictionary needs overlap coefficients as well as a list of allowed symmetries.

5. One-Loop Wilson–Fisher Data

Use

$$\beta_\mu(g) = -\epsilon g + A_N g^2, \qquad A_N = \frac{N+8}{ 3(4\pi)^2 }.$$

Find the interacting fixed point and its slope. Then use the first-order formula for $\nu$ to estimate the $N=1$, $D=3$ value.

Solution

The nonzero root is

$$g_\star = \frac{\epsilon}{A_N} = \frac{ 3(4\pi)^2 }{N+8} \epsilon.$$

The slope is

$$\begin{aligned} \beta_\mu'(g_\star) &= -\epsilon + 2A_Ng_\star \\ &= \epsilon. \end{aligned}$$

Thus a small deviation obeys

$$\delta g(\mu) \propto \mu^\epsilon$$

and decays toward the infrared when $\epsilon>0$.

For $N=1$ and $\epsilon=1$,

$$\begin{aligned} \nu &\simeq \frac12 + \frac{3}{36} \\ &= \frac{7}{12} \simeq 0.583. \end{aligned}$$

This is a first-order estimate, not the precision three-dimensional Ising value. Comparison with a precise value tests this low-order estimate; it is not an internally determined error bar. Higher orders and controlled resummation or nonperturbative methods are needed for precision.

6. Translate the Beta-Function Sign

Let

$$\ell = \ln \frac{\Lambda}{\mu}.$$

Show how a QFT beta function

$$\beta_\mu(g) = \mu\frac{dg}{d\mu}$$

is related to the infrared-length convention

$$\beta_{\mathrm{IR}}(g) := \frac{dg}{d\ell}.$$

Solution

Because

$$\frac{d\ell}{d\ln\mu} = -1,$$

the chain rule gives

$$\begin{aligned} \beta_{\mathrm{IR}}(g) &= \frac{dg}{d\ell} \\ &= -\frac{dg}{d\ln\mu} \\ &= -\beta_\mu(g). \end{aligned}$$

The zeros are the same, but arrows on a flow diagram reverse. Relevance and stability statements must therefore name the convention.

7. Count the Required Tunings

A candidate fixed point has three relevant eigenoperators:

• an even thermal operator $\varepsilon$;
• an odd magnetic operator $\sigma$;
• an even anisotropy operator $\mathcal A$.

How many controls must be tuned with and without exact $\mathbb Z_2$ symmetry?

Solution

Without an exact symmetry, all three relevant couplings are allowed. Reaching the fixed point generically requires tuning three independent controls.

Exact $\mathbb Z_2$ symmetry forbids the odd source coupled to $\sigma$, so that coordinate is fixed to zero by symmetry. The even thermal and anisotropy couplings remain allowed and relevant. Two controls must still be tuned.

Symmetry reduces the accessible coupling space; it does not make every remaining relevant operator vanish.

8. Susceptibility from a Fixed-Point Correlator

At criticality in $D$ dimensions, suppose

$$\left\langle \phi(x)\phi(0) \right\rangle_{\mathrm c} \sim \frac{1}{ \lvert x\rvert^{D-2+\eta} }.$$

Estimate the finite-size scaling of the zero-momentum susceptibility in a box of linear size $L$.

Solution

Ignoring geometry-dependent constants and the ultraviolet contact region,

$$\begin{aligned} \chi_L &\sim \int_a^L dr\, r^{D-1} r^{-(D-2+\eta)} \\ &= \int_a^L dr\, r^{1-\eta}. \end{aligned}$$

For $2-\eta>0$, the upper limit dominates:

$$\chi_L \sim L^{2-\eta}.$$

This estimate assumes that integrating the stated correlator to the box scale captures the leading finite-size contribution. Dangerous irrelevant variables and a separately enhanced uniform mode, especially above an upper critical dimension with periodic boundaries, can invalidate that assumption; the infinite-volume power alone then does not determine the full finite-box susceptibility.

The cutoff-dependent lower-limit contribution is analytic background. The leading infrared power is fixed by the continuum scaling dimension

$$2\Delta_\phi = D-2+\eta.$$

References

• C. G. Callan, Jr., “Broken Scale Invariance in Scalar Field Theory”, Physical Review D 2, 1541–1547 (1970) – renormalization-scale equations for field-theory amplitudes.
• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996) – scaling operators, universality, and continuum limits.
• M. E. Fisher, “The Renormalization Group in the Theory of Critical Behavior”, Reviews of Modern Physics 46, 597–616 (1974) – scaling fields and universal critical behavior.
• M. E. Fisher, “Renormalization Group Theory: Its Basis and Formulation in Statistical Physics”, Reviews of Modern Physics 70, 653–681 (1998) – conceptual foundations and statistical applications.
• J. A. Hertz, “Quantum Critical Phenomena”, Physical Review B 14, 1165–1184 (1976) – field theory for quantum critical systems.
• P. C. Hohenberg and B. I. Halperin, “Theory of Dynamic Critical Phenomena”, Reviews of Modern Physics 49, 435–479 (1977) – the distinction between static and dynamic universality.
• L. P. Kadanoff, “Scaling Laws for Ising Models near $T_c$”, Physics Physique Fizika 2, 263–272 (1966) – block variables and scaling.
• A. Pelissetto and E. Vicari, “Critical Phenomena and Renormalization-Group Theory”, Physics Reports 368, 549–727 (2002) – high-precision critical field theory and universality.
• J. Polchinski, “Renormalization and Effective Lagrangians”, Nuclear Physics B 231, 269–295 (1984) – Wilsonian flow and perturbative renormalizability.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011) – quantum critical scaling and continuum field theories.
• F. J. Wegner, “Corrections to Scaling Laws”, Physical Review B 5, 4529–4536 (1972) – nonlinear scaling fields and irrelevant corrections.
• K. G. Wilson, “Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture”, Physical Review B 4, 3174–3183 (1971) – fixed points and the Kadanoff picture.
• K. G. Wilson, “Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior”, Physical Review B 4, 3184–3205 (1971) – critical behavior from repeated coarse graining.
• K. G. Wilson and M. E. Fisher, “Critical Exponents in 3.99 Dimensions”, Physical Review Letters 28, 240–243 (1972) – the original $\epsilon$-expansion fixed point.
• K. G. Wilson and J. Kogut, “The Renormalization Group and the $\epsilon$ Expansion”, Physics Reports 12, 75–199 (1974) – the statistical-field and QFT synthesis.