Fixed Points, Tricriticality, and Order–Disorder Variables

The previous page constructed the Wilson–Fisher fixed point as a weakly coupled infrared fixed point in $D=4-\epsilon$ dimensions. That calculation matters, but the conceptual lesson is broader: a critical point is not just a place where a mass vanishes. It is a scale-invariant theory together with a list of perturbations. Some perturbations must be tuned away to stay critical; others disappear automatically as we look at longer distances.

This page makes that logic explicit. We first review fixed points and scaling fields. Then we explain why an ordinary Ising critical point and a tricritical Ising point are different fixed points, even though both have the same $\mathbb Z_2$ symmetry. Finally we return to the lattice Ising model and introduce the two variables that will dominate the next part of the course: the order variable $\sigma$ and the disorder variable $\mu$.

The common theme is that the long-distance theory knows about operators, not microscopic details. A polynomial Landau action gives a useful coordinate system, but the true critical theory is characterized by scaling dimensions, operator products, and defect insertions.

Required background. The epsilon expansion and one-loop scaling supplies the Wilson–Fisher fixed point, the critical surface, and the distinction between relevant and irrelevant flow directions.

Helpful background. The Ising graphical expansions supplies the closed-loop expansion, while Kramers–Wannier duality explains why a high-temperature variable can become an ordered variable in the dual model.

Scaling fields near a fixed point

Long-distance RG eigenvalues

$D$ denotes the Euclidean dimension of the statistical system. The long-distance RG time is

$$\ell=\log b, \qquad b>1,$$

so increasing $\ell$ means looking at longer distances. A coupling $g_a$ near a fixed point obeys

$${dg_a\over d\ell}=\beta_a(g).$$

A diagonalized perturbation, called a scaling field, obeys

$${du_a\over d\ell}=y_a u_a+O(u^2).$$

If the associated operator has scaling dimension $\Delta_a$, then

$$y_a=D-\Delta_a.$$

Thus $y_a>0$ is relevant, $y_a<0$ is irrelevant, and $y_a=0$ is marginal before higher-order beta functions decide its fate.

Let $S_*$ be a fixed-point action. A nearby action can be written schematically as

$$S=S_*+\sum_a g_a\int d^D x\,O_a(x),$$

where the $O_a$ are local operators. The RG acts as a flow on the couplings:

$${dg_a\over d\ell}=\beta_a(g).$$

A fixed point satisfies

$$\beta_a(g^*)=0$$

for every $a$. At such a point, after relevant perturbations have been tuned away and in infinite volume, the theory has no intrinsic correlation length. Correlation functions become power laws rather than exponentials. For the unitary, short-range critical systems considered later—especially in two dimensions—scale invariance is enhanced to conformal invariance under the standard assumptions, but the RG statement is enough here.

Near the fixed point, write

$$g_a=g_a^*+\delta g_a.$$

Linearizing the beta functions gives

$${d(\delta g_a)\over d\ell} =\sum_b M_{ab}\,\delta g_b+\cdots, \qquad M_{ab}=\left.{\partial\beta_a\over\partial g_b}\right|_{g=g^*}.$$

Diagonalizing $M$ gives scaling fields $u_a$:

$${du_a\over d\ell}=y_a u_a+\cdots.$$

The sign of $y_a$ determines the fate of the perturbation:

• $y_a>0$: $u_a$ grows toward long distances, so the perturbation is relevant;
• $y_a<0$: $u_a$ dies toward long distances, so the perturbation is irrelevant;
• $y_a=0$: $u_a$ is marginal at linear order, and nonlinear terms decide whether it is marginally relevant, marginally irrelevant, or exactly marginal.

This is the cleanest meaning of universality. Microscopic Hamiltonians contain infinitely many symmetry-allowed couplings, but most of their scaling fields are irrelevant near a stable critical fixed point. Long-distance physics remembers only a small number of relevant perturbations and a small amount of discrete information: dimension, symmetry, locality, and sometimes topology.

Near a fixed point, the RG flow can be diagonalized into scaling fields. Relevant perturbations flow away and must be tuned to reach the critical theory. Irrelevant perturbations flow into the fixed point and account for universality.

The set of theories that flow into the fixed point is the critical surface or stable manifold. Within a specified symmetry sector, its codimension is the number of independent relevant scaling fields that must be tuned. For the ordinary Ising universality class, if the magnetic field is forbidden by the $\mathbb Z_2$ symmetry, one tunes only a temperature-like parameter. If the magnetic field is allowed, it is another relevant perturbation.

The ordinary Ising fixed point therefore has the local form

$$S=S_*+t\int d^D x\,\varepsilon(x)-h\int d^D x\,\sigma(x) +\sum_i u_i\int d^D x\,O_i(x),$$

where $t$ is temperature-like, $h$ is the magnetic field, and the $u_i$ multiply irrelevant operators. The fields $\varepsilon$ and $\sigma$ are the continuum energy and spin operators. Their dimensions are not generally their Gaussian dimensions; they are dynamical data of the fixed point.

Gaussian power counting as a first map

Before the exact scaling dimensions are known, the Gaussian fixed point gives a useful first map. Consider a real scalar field with $\mathbb Z_2$ symmetry,

$$\phi\mapsto-\phi,$$

and Euclidean action

$$S[\phi] =\int d^D x\, \left[ {1\over2}(\partial\phi)^2 +{1\over2}t\phi^2 +{u\over4}\phi^4 +{v\over6}\phi^6 +\cdots \right].$$

At the Gaussian fixed point, the kinetic term fixes the engineering dimension of the field:

$$[\phi]_{\rm G}={D-2\over2}.$$

The operator $\phi^{2n}$ has Gaussian dimension

$$\Delta_{2n}^{\rm G}=2n[\phi]_{\rm G}=n(D-2),$$

so its coupling has RG eigenvalue

$$y_{2n}^{\rm G}=D-n(D-2)=2n-(n-1)D.$$

For the first few even operators,

$$\begin{aligned} \phi^2 &: \qquad y_2=2,\\ \phi^4 &: \qquad y_4=4-D,\\ \phi^6 &: \qquad y_6=6-2D=2(3-D). \end{aligned}$$

Thus the mass term is always relevant, the quartic interaction is marginal at $D=4$, and the sextic interaction is marginal at $D=3$.

Gaussian power counting predicts the upper critical dimension of each interaction. Ordinary Ising criticality is controlled by $\phi^4$ near $D=4$, while tricritical Ising behavior is controlled by $\phi^6$ near $D=3$.

This is why the previous page expanded around $D=4-\epsilon$: the quartic interaction is weakly relevant and can balance loop effects at the Wilson–Fisher fixed point. For tricriticality, the analogous expansion is around

$$D=3-\epsilon,$$

because the stabilizing $\phi^6$ interaction is weakly relevant there.

Power counting is only the start. At an interacting fixed point, scaling dimensions are shifted by anomalous dimensions. For the spin field,

$$\Delta_\sigma={D-2+\eta\over2}$$

in a scalar-field normalization, so the critical two-point function behaves as

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

In two-dimensional Ising theory, for example,

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

so $\eta=1/4$. Landau theory would have given $\eta=0$. The missing quarter power is not a cosmetic correction; it is the signal of a genuinely non-Gaussian fixed point.

Dimensional estimates from the interaction

The same classification can be seen without formal RG language. At a length scale $L$, the free critical field has

$$\phi^2\sim {1\over L^{D-2}}.$$

The quartic interaction in a region of volume $L^D$ has size

$$S_{\rm int}(L) \sim u L^D\phi^4 \sim uL^D{1\over L^{2D-4}} =uL^{4-D}.$$

Equivalently the dimensionless quartic coupling at scale $L$ is

$$u_{\rm eff}(L)\sim uL^{4-D}.$$

For $D>4$ this goes to zero at long distances, so the Gaussian fixed point is stable. For $D<4$ it grows, so the Gaussian fixed point cannot describe the ordinary Ising critical theory. For $D=4$ it is marginal and logarithms decide the flow.

The same conclusion follows from the classical equation of motion. Schematic Euclidean $\phi^4$ theory gives

$$\partial^2\phi=u\phi^3.$$

Expanding a solution as

$$\phi=\phi_0+u\phi_1+\cdots,$$

one finds

$$\partial^2\phi_0=0, \qquad \partial^2\phi_1=\phi_0^3.$$

The inverse Laplacian contributes two powers of length, while $\phi_0^3$ contributes three powers of the field. The relative importance of the correction is again controlled by $uL^{4-D}$.

Solving the nonlinear field equation perturbatively produces tree diagrams. Loop corrections are a separate effect: they come from integrating over fluctuations around the classical solution.

Diagrams, equations of motion, and RG are just three languages for the same scale estimate.

Criticality without microscopic spin-flip symmetry

The Ising lattice model has an exact $\mathbb Z_2$ symmetry, but many physical systems in the Ising universality class do not. A liquid–gas critical point is the standard example. The coarse-grained order parameter may be taken to be a density fluctuation, and the most general local potential begins as

$$V(\phi)=h\phi+{r\over2}\phi^2+{g\over3!}\phi^3+{u\over4!}\phi^4+\cdots.$$

The field $h$ is conjugate to the order parameter. One may first shift the field so that the chosen equilibrium point is at $\phi=0$; its stationarity condition then removes the linear term there. Departing from that equilibrium reintroduces the ordering field $h$. A continuous critical point also requires the curvature to vanish. If the cubic term remains nonzero, the local potential is too asymmetric to produce the Ising critical singularity; generically the system is driven toward a first-order jump. Thus, in a two-parameter family one reaches an ordinary critical point by imposing

$$r(P_c,T_c)=0, \qquad g(P_c,T_c)=0, \qquad u(P_c,T_c)>0.$$

Here $P$ and $T$ are schematic control parameters; in a magnetic system they could be replaced by other microscopic couplings. The last inequality says that the quartic term stabilizes the critical potential. In modern terminology this is ordinary Ising criticality written in asymmetric variables, not tricriticality. The infrared fixed point has an emergent spin-flip symmetry even when the microscopic variables do not.

The equations $r=g=0$ are local Landau-coordinate conditions, not invariant definitions of two separate RG eigenfields. Field shifts and analytic mixing of $P-P_c$ with $T-T_c$ change their detailed form. The invariant statement is that the two relevant Ising scaling fields—the temperature-like and ordering-field directions—must be tuned.

In an exactly Ising-symmetric model at zero magnetic field, $g=0$ is automatic and only the temperature-like coupling $r$ must be tuned. Without microscopic $\mathbb Z_2$ symmetry, the leading asymmetric perturbation must also be tuned away. This is the Landau version of the fixed-point rule: tune all relevant deformations not allowed at the desired critical theory.

Ordinary criticality and tricriticality

The usual Ising critical point is reached by tuning the coefficient of $\phi^2$ to zero while keeping the stabilizing quartic coupling positive. In Landau language,

$$V(\phi)={t\over2}\phi^2+{u\over4}\phi^4, \qquad u>0.$$

At $h=0$, the transition is continuous at

$$t=0, \qquad u>0.$$

For $t>0$, the minimum is at $\phi=0$. For $t<0$, the minima are at

$$\phi=\pm\sqrt{-{t\over u}},$$

and the $\mathbb Z_2$ symmetry is spontaneously broken. The order parameter vanishes continuously as

$$|\phi|\sim (-t)^{1/2}.$$

That is the mean-field exponent $\beta_{\rm MF}=1/2$.

Now allow a sextic term and allow the quartic coefficient $u$ to pass through zero:

$$V(\phi) ={t\over2}\phi^2+{u\over4}\phi^4+{v\over6}\phi^6-h\phi, \qquad v>0.$$

For $u>0$, tuning $t$ gives the ordinary continuous transition. For $u<0$, the quartic term favors a jump, and the sextic term stabilizes the potential at large $|\phi|$. The transition becomes first order.

To find the first-order coexistence line at $h=0$, set

$$s=\phi^2.$$

Nonzero stationary points obey

$$t+us+vs^2=0.$$

At such a stationary point,

$$V(s)={t\over2}s+{u\over4}s^2+{v\over6}s^3.$$

Using $t=-us-vs^2$ gives

$$V(s)=-{u\over4}s^2-{v\over3}s^3.$$

Coexistence with the $\phi=0$ minimum requires $V(s)=0$ with $s\ne0$, hence

$$s=-{3u\over4v}.$$

This is positive only for $u<0$. Substituting back gives

$$\boxed{ t={3u^2\over16v} \qquad (u<0). }$$

The first-order line and the ordinary continuous line meet at

$$t=0, \qquad u=0, \qquad h=0.$$

This meeting point is the tricritical point.

The curve above is the coexistence line, found by requiring degenerate minima. It should not be confused with the spinodal curves, where a metastable minimum disappears; those follow from the additional condition $V''(\phi)=0$ and lie elsewhere.

The Landau potential with $v>0$ contains both ordinary criticality and tricriticality. For $u>0$, tuning $t$ gives a continuous Ising transition. For $u<0$, the transition is first order. The two meet at $t=u=0$, where the leading stabilizing interaction is $\phi^6$.

Historically, “tricritical” denotes a point where a line of continuous transitions meets a line of first-order transitions. In the symmetric Landau picture, the disordered region and the two symmetry-related ordered vacua all come together there. Within the $\mathbb Z_2$-symmetric scalar sector, RG language says that one extra relevant even perturbation must be tuned. To reach the ordinary Ising critical point at $h=0$, tune $t$. To reach the tricritical point at $h=0$, tune both $t$ and $u$.

Mean-field tricritical exponents

At the tricritical point the quartic term has also been tuned away, so the leading even potential is $\phi^6$. Set $u=0$ and $h=0$:

$$V(\phi)={t\over2}\phi^2+{v\over6}\phi^6.$$

For $t<0$, the nonzero minima obey

$$t+v\phi^4=0,$$

so

$$|\phi|=\left({-t\over v}\right)^{1/4}.$$

Therefore

$$\beta_{\rm tri,MF}={1\over4}.$$

At $t=0$ and $u=0$, include a magnetic field:

$$V(\phi)={v\over6}\phi^6-h\phi.$$

The equation of state is

$$h={dV\over d\phi}=v\phi^5,$$

so

$$\phi\sim h^{1/5}, \qquad \delta_{\rm tri,MF}=5.$$

In the symmetric phase $t>0$, the susceptibility is

$$\chi=\left({d^2V\over d\phi^2}\bigg|_{\phi=0}\right)^{-1}={1\over t},$$

so

$$\gamma_{\rm tri,MF}=1.$$

The correlation length is controlled by the mass term in the Landau–Ginzburg action,

$$\xi\sim t^{-1/2}, \qquad \nu_{\rm tri,MF}={1\over2}.$$

The singular free energy below the transition is obtained by substituting the minimum. Since $\phi^4=-t/v$,

$$V_{\min}=-{1\over3}{(-t)^{3/2}\over v^{1/2}},$$

so

$$f_{\rm sing}\sim |t|^{3/2}.$$

Using $f_{\rm sing}\sim |t|^{2-\alpha}$ gives

$$\alpha_{\rm tri,MF}={1\over2}.$$

These values are mean-field exponents. Because $\phi^6$ is marginal at $D=3$, three dimensions is the upper critical dimension of tricritical Ising theory, so logarithmic corrections appear there. Below three dimensions, the tricritical point is an interacting fixed point. In two dimensions it becomes a conformal field theory with a finite list of primary fields, but that belongs to the CFT part of the course.

Ordinary Ising versus tricritical Ising

It is tempting to say that the tricritical theory is just ordinary Ising theory with a small quartic coupling. That is too casual. The sign and flow of the quartic perturbation decide the destination.

For the ordinary Ising transition,

$$S_{\rm Ising} =\int d^D x\, \left[ {1\over2}(\partial\phi)^2 +{1\over2}t\phi^2 +{u\over4}\phi^4 +\cdots \right], \qquad u>0.$$

At $D<4$, the quartic interaction grows away from the Gaussian point and flows toward the Wilson–Fisher fixed point. Positive microscopic $u$ is not a fine tuning; it is part of the basin of attraction.

For tricritical Ising behavior,

$$S_{\rm tri} =\int d^D x\, \left[ {1\over2}(\partial\phi)^2 +{1\over2}t\phi^2 +{u\over4}\phi^4 +{v\over6}\phi^6 +\cdots \right], \qquad v>0,$$

and one must tune

$$t=0, \qquad u=0, \qquad h=0.$$

The even perturbation $u\phi^4$ is relevant at the tricritical point. After the temperature direction is retuned, positive $u$ sends the system toward ordinary Ising critical behavior; negative $u$ sends it toward a first-order transition.

This is the first example in the course where the same symmetry admits more than one interesting fixed point. Symmetry tells us which operators are allowed. It does not by itself tell us which fixed point controls the infrared.

Landau theory and exact criticality

Landau theory is invaluable because it organizes phases and perturbations. It is not the exact theory of critical exponents below the upper critical dimension. The ordinary two-dimensional Ising transition is the warning sign.

For ordinary Ising mean-field theory,

$$\beta_{\rm MF}={1\over2}, \qquad \gamma_{\rm MF}=1, \qquad \nu_{\rm MF}={1\over2}, \qquad \eta_{\rm MF}=0.$$

The exact two-dimensional Ising fixed point instead has

$$\beta={1\over8}, \qquad \gamma={7\over4}, \qquad \nu=1, \qquad \eta={1\over4},$$

with a logarithmic singularity in the specific heat. These exponents are not obtained by minimizing a polynomial potential. They are properties of the fixed point.

The right attitude is therefore:

$$\text{Landau theory gives the relevant variables and phase topology,}$$

while

$$\text{the fixed point gives the scaling dimensions.}$$

The next step is to learn how to describe the operator content of the two-dimensional fixed point. For that we need a second kind of Ising variable.

Order variables

In the lattice Ising model, the microscopic order variable is the spin

$$\sigma_i=\pm1.$$

The magnetization

$$M={1\over N}\sum_i\sigma_i$$

is an order parameter for the broken $\mathbb Z_2$ symmetry. In the ordered phase,

$$\lim_{|i-j|\to\infty}\langle\sigma_i\sigma_j\rangle\ne0.$$

In the disordered phase, the connected correlator decays exponentially:

$$\langle\sigma_i\sigma_j\rangle_c\sim e^{-|i-j|/\xi}.$$

The high-temperature expansion gives a graphical interpretation of this statement. For nearest-neighbor coupling $K=\beta J$,

$$e^{K\sigma_i\sigma_j} =\cosh K\left(1+\sigma_i\sigma_j\tanh K\right).$$

Let

$$w=\tanh K.$$

Then

$$Z=(\cosh K)^{N_b}\sum_{\{\sigma\}} \prod_{\langle ij\rangle} \left(1+w\sigma_i\sigma_j\right),$$

where $N_b$ is the number of bonds. Expanding the product selects a set of bonds $\Gamma$. A spin sum survives only when every site is incident on an even number of selected bonds. Thus

$$Z=2^N(\cosh K)^{N_b}\sum_{\Gamma\,\mathrm{closed}}w^{|\Gamma|}.$$

For the two-point function, insert $\sigma_a\sigma_b$:

$$\langle\sigma_a\sigma_b\rangle ={1\over Z}(\cosh K)^{N_b} \sum_{\{\sigma\}}\sigma_a\sigma_b \prod_{\langle ij\rangle} \left(1+w\sigma_i\sigma_j\right).$$

Now the surviving graphs have odd degree at $a$ and $b$, and even degree at every other site. Therefore they contain open paths connecting $a$ and $b$, possibly dressed by closed loops. The order-field correlator is a sum over graphs whose endpoints are order insertions.

This is the first hint that local operators are not merely functions in an action. They create allowed endpoints, twists, or singularities in the graphical expansion.

Disorder variables

The disorder variable is more subtle. It does not multiply the spin configuration by a local function of the spins. Instead it changes the couplings along a line.

Place a dual lattice site $p^*$ at the center of each plaquette of the square lattice. Given two dual sites $p^*$ and $q^*$, choose a path $\Gamma$ on the dual lattice connecting them. Define signs on original-lattice bonds by

$$\eta_{ij}^{(\Gamma)}= \begin{cases} -1, & \text{if the bond }\langle ij\rangle\text{ is crossed by }\Gamma,\\ +1, & \text{otherwise.} \end{cases}$$

Let $Z_\Gamma(K)$ denote the partition function with these flipped bonds. The disorder two-point function is defined up to a local normalization $C_\mu(K)$ by

$$\boxed{ \langle\mu(p^*)\mu(q^*)\rangle_K = C_\mu(K){Z_\Gamma(K)\over Z(K)}. }$$

Equivalently, the insertion $\mu(p^*)\mu(q^*)$ flips the sign of the Ising coupling on every bond crossed by $\Gamma$. The factor $C_\mu(K)$ fixes the normalization of the local endpoint operator; it does not affect path independence, phases, or critical exponents.

A pair of disorder variables is represented by a defect line $\Gamma$ on the dual lattice. Bonds crossed by $\Gamma$ have their coupling sign reversed. Moving the line without moving its endpoints is a change of variables, so the endpoints are the physical insertions.

At first this looks path-dependent. A deformation of $\Gamma$ through empty lattice plaquettes can be undone by flipping the spins in the swept region, so only the endpoints are physical. The next lesson proves this statement and explains what changes when the line crosses an order insertion.

For the present page, the phase interpretation is enough. In the high-temperature phase, changing the signs of a long string of weak bonds costs little free energy, and the disorder field has long-range order. In the low-temperature phase, the insertion forces an energetically costly domain wall between its endpoints, so the disorder correlator decays. This is the opposite behavior from the spin order parameter.

The Kramers–Wannier duality relation

$$e^{-2K^*}=\tanh K$$

exchanges these two descriptions. Schematically,

$$\sigma\quad\longleftrightarrow\quad\mu,$$

while high temperature in one model becomes low temperature in the dual model. At the self-dual critical point, order and disorder are placed on equal footing.

Why both variables belong to the fixed point

The order variable $\sigma$ is local in the original spin variables, while the disorder variable $\mu$ is local in the dual description. At the self-dual critical point neither can be discarded: each has a power-law correlator and creates a legitimate local scaling field in its own operator algebra. Their mutual branch-cut structure and the resulting fermionic variables belong to the next two lessons, where the necessary line prescription is made explicit.

The broader lesson is that the operator content of a critical theory is richer than the polynomial action suggests. The action may begin with $\phi^4$ or $\phi^6$, but the fixed point contains order fields, disorder fields, energy fields, descendants, and eventually a stress tensor. This operator viewpoint is the natural entrance into conformal field theory.

Summary

A fixed point is a scale-invariant theory in coupling space. Perturbations around it are classified by RG eigenvalues $y=D-\Delta$. Relevant perturbations grow and must be tuned; irrelevant perturbations decay and explain universality.

The ordinary Ising critical point and the tricritical Ising point are different fixed points with the same $\mathbb Z_2$ symmetry. Ordinary Ising criticality is reached by tuning the temperature-like variable $t$ at positive quartic coupling. Tricriticality requires tuning both $t$ and the quartic coupling $u$ to zero, leaving the sextic interaction as the leading stabilizer.

Gaussian power counting gives the upper critical dimensions: $D_c=4$ for $\phi^4$ ordinary criticality and $D_c=3$ for $\phi^6$ tricriticality. Below those dimensions, interactions change the scaling dimensions.

The Ising spin $\sigma$ is an order variable. Its correlator is represented in the high-temperature expansion by open graphs ending at the inserted spins. The disorder variable $\mu$ is defined by flipping couplings along a dual-lattice line. Its endpoint is a twist operator. Kramers–Wannier duality exchanges order and disorder.

Common pitfalls

Letting symmetry choose the fixed point. Ordinary and tricritical Ising theories both have $\mathbb Z_2$ symmetry, but their relevant spectra and critical exponents differ. Symmetry determines which operators are allowed, not which fixed point controls the infrared.

Calling any small quartic coupling tricritical. The quartic perturbation is relevant at the tricritical point and must be tuned to zero. The sextic term must remain positive to stabilize the Landau potential.

Promoting mean-field exponents below the upper critical dimension. Landau theory correctly identifies phases and tuning parameters, but anomalous dimensions and critical exponents are fixed-point data. In two dimensions the exact Ising exponents differ sharply from their mean-field values.

Treating $\mu$ as a local polynomial in the original spins. A disorder pair is defined by modifying Boltzmann weights along a line. The line is deformable, while its endpoints are the operator insertions; the next lesson proves the corresponding path-independence statement.

Exercises

Exercise 1: Upper critical dimensions

At the Gaussian fixed point in $D$ dimensions, compute the engineering dimension of the coupling $g_{2n}$ multiplying $\phi^{2n}$:

$$S\supset g_{2n}\int d^D x\,\phi^{2n}.$$

For which dimension is $g_{2n}$ marginal?

Solution

The kinetic term gives

$$[\phi]={D-2\over2}.$$

Therefore

$$[\phi^{2n}]=n(D-2).$$

Since the action is dimensionless,

$$[g_{2n}]+[-D]+n(D-2)=0.$$

Thus

$$[g_{2n}]=D-n(D-2)=2n-(n-1)D.$$

The coupling is marginal when

$$D-n(D-2)=0,$$

so

$$\boxed{D_c={2n\over n-1}.}$$

For $n=2$, $D_c=4$ for $\phi^4$. For $n=3$, $D_c=3$ for $\phi^6$.

Exercise 2: The tricritical coexistence line

For

$$V(\phi)={t\over2}\phi^2+{u\over4}\phi^4+{v\over6}\phi^6, \qquad v>0,$$

show that for $u<0$ the first-order coexistence line between $\phi=0$ and $\phi\ne0$ is

$$t={3u^2\over16v}$$

in the normalization used on this page.

Solution

Let

$$s=\phi^2.$$

A nonzero stationary point obeys

$$0={dV\over d\phi}=\phi(t+us+vs^2),$$

so

$$t=-us-vs^2.$$

At the stationary point,

$$V(s)={t\over2}s+{u\over4}s^2+{v\over6}s^3.$$

Substitute $t=-us-vs^2$:

$$V(s)=-{u\over2}s^2-{v\over2}s^3+{u\over4}s^2+{v\over6}s^3 =-{u\over4}s^2-{v\over3}s^3.$$

Coexistence with the $\phi=0$ minimum requires $V(s)=0$ with $s\ne0$:

$$-{u\over4}-{v\over3}s=0.$$

Thus

$$s=-{3u\over4v}.$$

This is positive only when $u<0$. Then

$$t=-us-vs^2 ={3u^2\over4v}-{9u^2\over16v} ={3u^2\over16v}.$$

Exercise 3: Mean-field tricritical exponents

At the tricritical mean-field point $u=0$, derive $\beta_{\rm tri,MF}=1/4$ and $\delta_{\rm tri,MF}=5$.

Solution

Set $h=0$ first. The potential is

$$V(\phi)={t\over2}\phi^2+{v\over6}\phi^6.$$

For $t<0$, nonzero minima obey

$$0={dV\over d\phi}=\phi(t+v\phi^4),$$

so

$$\phi^4=-{t\over v}.$$

Therefore

$$|\phi|\sim(-t)^{1/4},$$

and

$$\beta_{\rm tri,MF}={1\over4}.$$

At $t=0$ and $u=0$, include the magnetic field:

$$V(\phi)={v\over6}\phi^6-h\phi.$$

The equation of state is

$$h={d\over d\phi}{v\over6}\phi^6=v\phi^5.$$

Thus

$$\phi\sim h^{1/5},$$

so

$$\delta_{\rm tri,MF}=5.$$

Exercise 4: Open graphs from order insertions

Use the high-temperature expansion to show that the numerator of $\langle\sigma_a\sigma_b\rangle$ is a sum over graphs with odd degree at $a$ and $b$, and even degree at every other site.

Solution

Start from

$$e^{K\sigma_i\sigma_j}=\cosh K(1+w\sigma_i\sigma_j), \qquad w=\tanh K.$$

The numerator is proportional to

$$\sum_{\{\sigma\}}\sigma_a\sigma_b \prod_{\langle ij\rangle}(1+w\sigma_i\sigma_j).$$

Expanding the product selects a set of bonds $\Gamma$. The spin dependence of a selected graph is

$$\sigma_a\sigma_b\prod_{\langle ij\rangle\in\Gamma}\sigma_i\sigma_j.$$

At a site $k\ne a,b$, the power of $\sigma_k$ is the number of selected bonds incident on $k$. The spin sum

$$\sum_{\sigma_k=\pm1}\sigma_k^m$$

vanishes unless $m$ is even. At $a$ and $b$, the inserted factor contributes one extra power of the spin, so the selected-bond degree must be odd there. Therefore the surviving graphs have odd degree at $a$ and $b$ and even degree elsewhere. Such graphs contain open paths from $a$ to $b$, possibly together with closed loops.

Exercise 5: Classical corrections and the quartic scale

For the critical classical equation

$$\partial^2\phi=u\phi^3,$$

write $\phi=\phi_0+u\phi_1+\cdots$. If a free critical configuration varying on scale $L$ has $\phi_0\sim L^{-(D-2)/2}$, show that

$${u\phi_1\over\phi_0}\sim uL^{4-D}.$$

Solution

At successive orders in $u$,

$$\partial^2\phi_0=0, \qquad \partial^2\phi_1=\phi_0^3.$$

On a configuration varying over distance $L$, the inverse Laplacian contributes a factor of order $L^2$. Therefore

$$\phi_1\sim L^2\phi_0^3.$$

Dividing by $\phi_0$ and using $\phi_0^2\sim L^{-(D-2)}$ gives

$${u\phi_1\over\phi_0} \sim uL^2\phi_0^2 \sim uL^{4-D}.$$

Thus the classical perturbation is irrelevant for $D>4$, relevant for $D<4$, and marginal by power counting at $D=4$, in agreement with the RG analysis.

References

• L. P. Kadanoff and H. Ceva, “Determination of an Operator Algebra for the Two-Dimensional Ising Model,” Physical Review B 3 (1971), 3918–3939, doi:10.1103/PhysRevB.3.3918.
• K. G. Wilson and J. Kogut, “The Renormalization Group and the $\epsilon$ Expansion,” Physics Reports 12 (1974), 75–199, doi:10.1016/0370-1573(74)90023-4.

Further reading

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer, New York, 1997.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Course of Theoretical Physics, vol. 5, Butterworth–Heinemann, Oxford, 1980.
• I. D. Lawrie and S. Sarbach, “Theory of Tricritical Points,” in C. Domb and J. L. Lebowitz (eds.), Phase Transitions and Critical Phenomena, vol. 9, Academic Press, London, 1984, 1–161.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., International Series of Monographs on Physics, vol. 113, Oxford University Press, Oxford, 2002.