Theta Angle and Instanton Corrections

The previous page constructed finite-action instantons in the two-dimensional $O(3)$ nonlinear sigma model. Those saddles are maps $S^2\to S^2$, their topological charge $Q$ is an integer, and the charge-one saddle has action

$$S_{\rm inst}={4\pi\over\alpha}.$$

This page asks what these sectors do in the quantum theory. The answer starts with the theta angle. The Euclidean path integral is not a single integral over one connected space of fields; it is a coherent sum over disconnected topological sectors,

$$Z(\theta)=\sum_{Q\in\mathbb Z}e^{i\theta Q}Z_Q.$$

The factor $e^{i\theta Q}$ is invisible in perturbation theory around the trivial sector, but it is visible to instantons. A single instanton contributes a factor $e^{-4\pi/\alpha+i\theta}$, while a single anti-instanton contributes $e^{-4\pi/\alpha-i\theta}$. Thus theta dependence enters through nonperturbative terms such as

$$2e^{-4\pi/\alpha}\cos\theta.$$

These terms are tiny at weak coupling, but they can decide the infrared physics when different long-distance behaviors compete. The angle $\theta=\pi$ is especially delicate: it is CP invariant, but odd topological sectors acquire a minus sign.

Required background. Sigma-model instantons and topological charge supplies the integer charge, Bogomolny bound, and holomorphic saddles used here.

Helpful background. Spin chains and theta terms explains why integer and half-integer antiferromagnets realize $\theta=0$ and $\theta=\pi$, respectively.

Theta sectors and periodicity

Normalization. We use the same normalization as the previous sigma-model pages:

$$S_0[\mathbf n] ={1\over2\alpha}\int d^2x\,\partial_\mu\mathbf n\cdot\partial_\mu\mathbf n, \qquad \mathbf n^2=1,$$

and

$$Q[\mathbf n] ={1\over8\pi}\int d^2x\,\epsilon_{\mu\nu}\, \mathbf n\cdot(\partial_\mu\mathbf n\times\partial_\nu\mathbf n)\in\mathbb Z, \qquad \epsilon_{12}=+1.$$

The Euclidean theta term is included as

$$S_E[\mathbf n;\theta]=S_0[\mathbf n]-i\theta Q[\mathbf n],$$

so a charge-$Q$ sector is weighted by $e^{i\theta Q}$. With $\beta(\alpha)=\mu d\alpha/d\mu$, the perturbative one-loop $O(3)$ beta function is

$$\beta_{\rm pert}(\alpha)=-{\alpha^2\over2\pi}+O(\alpha^3).$$

Scope of semiclassics. Instantons reliably show the transseries structure and theta phases of the weak-coupling theory. They do not by themselves solve the full infrared $O(3)$ model, because the size modulus samples large instantons where the running coupling is strong. Whenever this page uses a dilute-gas formula, read it as a controlled short-distance/semiclassical guide unless an additional infrared cutoff or deformation is specified.

The decomposition into topological sectors is

$$Z(\theta)=\sum_{Q\in\mathbb Z}e^{i\theta Q} \int_Q\mathcal D\mathbf n\,e^{-S_0[\mathbf n]}.$$

Since $Q$ is integer-valued,

$$Z(\theta+2\pi)=Z(\theta).$$

The theta angle is therefore an angular parameter. The theories at $\theta$ and $\theta+2\pi$ are the same theory.

At weak coupling, the lowest nonzero sectors are $Q=\pm1$. Their leading semiclassical weights are

$$e^{-4\pi/\alpha+i\theta}, \qquad e^{-4\pi/\alpha-i\theta}.$$

For a CP-even quantity, instantons and anti-instantons combine into

$$e^{-4\pi/\alpha+i\theta}+e^{-4\pi/\alpha-i\theta} =2e^{-4\pi/\alpha}\cos\theta.$$

This is the first basic selection rule: perturbative diagrams give powers and logarithms of $\alpha$, while instantons give exponentials in $1/\alpha$ multiplied by theta phases.

The path integral is a sum over integer topological sectors. The sector with charge $Q$ carries the phase $e^{i\theta Q}$; hence the theory is periodic under $\theta\mapsto\theta+2\pi$, and the first instanton pair contributes through $2e^{-4\pi/\alpha}\cos\theta$.

The theta term is locally a total derivative. It does not change the local instanton equations, and the same holomorphic maps are the classical saddles. Its effect is global: it changes the interference among disconnected sectors.

CP and the special angle theta = pi

A parity or CP-type transformation reverses the orientation of Euclidean spacetime and sends

$$Q\mapsto -Q.$$

Therefore it maps the theory at $\theta$ to the theory at $-\theta$. Because $\theta$ is periodic, CP can be a symmetry at the fixed points of

$$\theta\equiv -\theta \pmod{2\pi}.$$

Thus the special CP-invariant angles are

$$\theta=0, \qquad \theta=\pi \pmod{2\pi}.$$

At $\theta=0$, all sectors have real positive theta phase. At $\theta=\pi$,

$$e^{i\pi Q}=(-1)^Q,$$

so every odd topological sector changes sign relative to $\theta=0$. This is not a small continuous perturbation. It is a different interference pattern protected by a discrete symmetry.

This is why $\theta=\pi$ must be handled separately. The semiclassical sign flip alone does not solve the infrared theory, but it warns us that the long-distance behavior at $\theta=\pi$ may differ qualitatively from that at $\theta=0$. Symmetry constraints make the distinction sharper: at $\theta=\pi$, a unique, trivially gapped vacuum preserving all the relevant symmetries is not an allowed infrared endpoint. A gapless theory or degenerate vacua with spontaneous CP breaking are possible ways to satisfy the constraint.

For the undeformed $O(3)$ model, the standard infrared identification is the gapless level-one $SU(2)$ Wess–Zumino–Witten theory, with logarithmic corrections from a marginally irrelevant interaction. This statement uses information beyond the dilute instanton calculation. In particular, the simple cosine vacuum energy derived below does not establish the $\theta=\pi$ endpoint; its large-instanton region is already strongly coupled. The spin-chain realization makes the same distinction concrete:

$$\theta=2\pi S,$$

so integer spin corresponds to $\theta=0\pmod{2\pi}$ and half-integer spin corresponds to $\theta=\pi\pmod{2\pi}$. The distinction is discrete, not perturbatively small.

The one-instanton measure and the scale modulus

Hold the value $\mathbf n_\infty$ at spatial infinity fixed. A convenient charge-one representative is then $w(z)=\rho e^{i\chi}/(z-X)$, with four real collective coordinates: a center $X\in\mathbb R^2$, a size $\rho>0$, and the residual target-space $U(1)$ angle $\chi$. If one instead integrates over the full global $O(3)$ orbit, additional orientation coordinates and the corresponding group volume must be treated consistently. With the fixed-boundary convention, the measure is schematically

$$Z_1(\theta) \sim \int d^2X\,{d\rho\over\rho^3}\,d\chi\, C(\alpha) \exp\left[-{4\pi\over\alpha}+i\theta\right],$$

where $C(\alpha)$ denotes determinant and zero-mode normalization factors. The precise prefactor is regulator-dependent; the important pieces are the collective-coordinate measure, the exponential $e^{-4\pi/\alpha}$, and the phase $e^{i\theta}$. The measure shown here is not the measure over all six real parameters of a general Möbius transformation: fixing the boundary value removes the two global target orientations that would move $\mathbf n_\infty$.

Classically the instanton has no preferred size. Quantum mechanically, an instanton of size $\rho$ probes momenta of order

$$\mu\sim {1\over\rho}.$$

The classical exponential should therefore be RG-improved by using the running coupling $\alpha(1/\rho)$. Let $\alpha_0=\alpha(\Lambda_{\rm UV})$ at a weakly coupled ultraviolet reference scale. The one-loop $O(3)$ running gives

$${1\over\alpha(\mu)}={1\over\alpha_0}+{1\over2\pi}\log {\mu\over\Lambda_{\rm UV}}.$$

Setting $\mu=1/\rho$,

$${1\over\alpha(1/\rho)} ={1\over\alpha_0}-{1\over2\pi}\log(\rho\Lambda_{\rm UV}),$$

and hence

$$\exp\left[-{4\pi\over\alpha(1/\rho)}\right] = \exp\left[-{4\pi\over\alpha_0}\right](\rho\Lambda_{\rm UV})^2.$$

Thus the running coupling turns the naive size dependence into

$${d\rho\over\rho^3} \exp\left[-{4\pi\over\alpha(1/\rho)}\right] \sim \Lambda_{\rm UV}^2e^{-4\pi/\alpha_0}{d\rho\over\rho}.$$

The logarithmic measure $d\rho/\rho$ is the quantum trace of the classical scale modulus. Instantons in the window $\Lambda_{\rm UV}^{-1}\lesssim \rho\ll M^{-1}$ are controlled by asymptotic freedom at the scale $1/\rho$. At $\rho\sim M^{-1}$ the running coupling becomes order one, so the one-loop instanton density loses control. Treating that scale as a sharp upper cutoff is an estimate, not a derivation from the semiclassical measure:

$$\rho\lesssim M^{-1}.$$

An instanton of size $\rho$ is naturally evaluated at the momentum scale $\mu\sim1/\rho$. In the $O(3)$ model, RG improvement changes the naive size measure into the logarithmic form $d\rho/\rho$. The mark at $\rho\sim M^{-1}$ denotes loss of semiclassical control, not a derived sharp cutoff.

This is why the instanton expansion in the pure $O(3)$ model is instructive but not completely harmless. It cleanly reveals topology and theta dependence at short distances, but its size integral reaches the same strongly coupled region responsible for the mass gap.

Dilute gas and vacuum energy

If instantons and anti-instantons are dilute and approximately independent, their sum becomes a grand canonical gas. Mixed instanton–anti-instanton configurations are not exact holomorphic multi-instanton saddles; here they are approximate, widely separated constituents whose interactions are neglected at leading order. Let $\zeta$ be the integrated fugacity of a single instanton, including its determinant and collective-coordinate integral after an explicitly chosen ultraviolet prescription and infrared regulator. A configuration with $N_+$ instantons and $N_-$ anti-instantons has charge

$$Q=N_+-N_-,$$

and contributes

$${(V\zeta)^{N_+}\over N_+!} {(V\zeta)^{N_-}\over N_-!} e^{i\theta(N_+-N_-)}.$$

Summing over $N_+$ and $N_-$ gives

$${Z(\theta)\over Z_0} \approx \sum_{N_+,N_-\ge0} {(V\zeta e^{i\theta})^{N_+}\over N_+!} {(V\zeta e^{-i\theta})^{N_-}\over N_-!} = \exp\left[2V\zeta\cos\theta\right].$$

The vacuum free-energy density is

$$f(\theta)=-\lim_{V\to\infty}{1\over V}\log Z(\theta).$$

After subtracting the constant at $\theta=0$,

$$f(\theta)-f(0)=2\zeta(1-\cos\theta).$$

The topological susceptibility is therefore

$$\chi_t={\partial^2 f\over\partial\theta^2}\bigg|_{\theta=0}=2\zeta.$$

In the dilute-gas model, instantons and anti-instantons exponentiate independently. The result is a leading vacuum-energy shift proportional to $1-\cos\theta$ and a topological susceptibility $\chi_t=2\zeta$.

The dilute-gas formula is a model of the theta dependence, not a complete solution of the $O(3)$ model. The exact function $f(\theta)$ can be more complicated, especially near $\theta=\pi$. But periodicity, CP symmetry, and the appearance of $\cos\theta$ at one-instanton order are robust.

Two robust derivatives of $f(\theta)$ are often more meaningful than the absolute normalization. The first derivative gives the expectation value of topological charge density,

$${\partial f\over\partial\theta}=-{i\over V}\langle Q\rangle_\theta,$$

in Euclidean conventions. The second derivative at $\theta=0$ is the topological susceptibility. The dilute gas gives $\chi_t=2\zeta$, but the definition of $\chi_t$ is nonperturbative and does not rely on the dilute approximation.

Instanton corrections to running couplings

Perturbation theory in the $Q=0$ sector gives a beta function independent of $\theta$:

$$\beta_{\rm pert}(\alpha) =-{\alpha^2\over2\pi}+O(\alpha^3).$$

Instantons can add nonanalytic corrections. Since the charge-$\pm1$ sectors carry phases $e^{\pm i\theta}$, a CP-even Wilsonian flow equation can contain terms of the schematic form

$$\delta\beta(\alpha,\theta) \propto e^{-4\pi/\alpha}(e^{i\theta}+e^{-i\theta}) =2e^{-4\pi/\alpha}\cos\theta.$$

Thus one may write schematically

$$\beta(\alpha,\theta) =-{\alpha^2\over2\pi}+O(\alpha^3) -C(\alpha)e^{-4\pi/\alpha}\cos\theta +O(e^{-8\pi/\alpha}),$$

where the prefactor $C(\alpha)$ is scheme-dependent. In a coupling convention where the leading perturbative coefficient is scaled to one, the same idea is often compressed to

$$\beta(\alpha,\theta) =-\alpha^2 -c e^{-4\pi/\alpha}(e^{i\theta}+e^{-i\theta})+ \cdots.$$

The numerical coefficient $c$ is not the invariant statement. The invariant statement is the transseries structure: exponentially small, theta-dependent corrections supplement the perturbative power series.

Perturbative running is blind to $\theta$. The first instanton correction carries the factor $e^{-4\pi/\alpha}\cos\theta$, so its sign flips between $\theta=0$ and $\theta=\pi$. The drawing is schematic; the prefactor depends on the definition of the running coupling.

At very weak coupling this correction is incredibly small. But the RG flow drives the theory toward larger $\alpha$ in the infrared. Once the flow reaches strong coupling, the distinction between $\theta=0$ and $\theta=\pi$ is no longer parametrically tiny.

This is the main reader-facing lesson of the page. Perturbation theory says that all theta angles look identical near the ultraviolet fixed point. Instantons explain how the theta angle re-enters through effects that are invisible order by order in $\alpha$. Strong infrared dynamics can then amplify those initially tiny differences.

Plasma language and the mass gap

The holomorphic multi-instantons of the $O(3)$ model can be written as rational maps,

$$w(z)=C\prod_{j=1}^k{z-a_j\over z-b_j}.$$

The points $a_j$ and $b_j$ are zeros and poles of the stereographic coordinate $w$. They are collective coordinates of a degree-$k$ map, not literal instantons and anti-instantons. Nevertheless, the collective-coordinate measure contains logarithmic interactions among these moduli, so it resembles a two-dimensional Coulomb plasma.

In an ordinary two-dimensional Coulomb plasma, screening replaces scale-free correlations by a finite screening length. The rational-map measure suggests the analogous mechanism here. In field-theory language, such a finite correlation length would be a mass gap:

$$\xi<\infty, \qquad M\sim \xi^{-1}>0.$$

The rational-map moduli of multi-instantons resemble a logarithmically interacting two-dimensional plasma. This is an auxiliary collective-coordinate picture: zeros and poles are moduli of $w(z)$, not separate topological charges. The screening analogy suggests, but does not by itself derive, a finite correlation length.

This idea prepares the next step in the course. In compact three-dimensional gauge theory, monopole-instantons form a genuine Coulomb gas, and its screening mechanism leads directly to confinement. The sigma-model discussion is the two-dimensional cousin: topology plus logarithmic interactions naturally produces finite-length physics.

Summary

The theta angle weights integer topological sectors by

$$e^{i\theta Q}.$$

Consequently the theory is $2\pi$ periodic in $\theta$, and CP maps $\theta$ to $-\theta$. The special CP-invariant angles are $\theta=0$ and $\theta=\pi$.

Instantons generate nonperturbative factors

$$e^{-4\pi/\alpha+i\theta},$$

while anti-instantons generate

$$e^{-4\pi/\alpha-i\theta}.$$

For CP-even observables the leading theta dependence is therefore

$$2e^{-4\pi/\alpha}\cos\theta.$$

The one-instanton size modulus links semiclassics to RG flow. Running-coupling improvement changes the naive size measure into a logarithmic integral,

$${d\rho\over\rho^3}e^{-4\pi/\alpha(1/\rho)} \sim {d\rho\over\rho},$$

up to overall scale and determinant factors. Large instantons probe the strongly coupled infrared. The scale $\rho\sim M^{-1}$ marks where the one-loop instanton density ceases to be reliable; using it as a cutoff only estimates the uncontrolled region.

A dilute gas gives the model formula

$$f(\theta)-f(0)=2\zeta(1-\cos\theta), \qquad \chi_t=2\zeta.$$

More generally, instantons can correct Wilsonian flow equations by terms such as

$$\delta\beta\sim e^{-4\pi/\alpha}\cos\theta.$$

The sign flip at $\theta=\pi$ is the semiclassical shadow of the special infrared role of that angle.

Common pitfalls

Discarding the theta term because it is locally a total derivative. Its integral labels disconnected sectors, and the relative phase between sectors affects the quantum theory.

Expecting perturbation theory to know about $\theta$. Perturbation theory around a trivial vacuum lives in $Q=0$, so its beta function is theta independent.

Treating the $O(3)$ instanton gas as uniformly controlled. The size modulus reaches the strongly coupled infrared. The scale $M^{-1}$ marks loss of control, rather than providing a sharp cutoff derived within semiclassics.

Assigning a universal coefficient to $e^{-4\pi/\alpha}\cos\theta$ in a beta function. The coefficient depends on the definition of the running coupling. The theta structure and exponential order are the robust facts.

Calling zeros and poles separate instantons and anti-instantons. They are moduli of one holomorphic multi-instanton map, although their measure has a useful plasma-like interpretation.

Exercises

Exercise 1: Periodicity from integer charge

Show that the theta-dependent partition function is $2\pi$ periodic if $Q\in\mathbb Z$.

Solution

Using

$$Z(\theta)=\sum_{Q\in\mathbb Z}e^{i\theta Q}Z_Q,$$

we find

$$Z(\theta+2\pi) =\sum_{Q\in\mathbb Z}e^{i\theta Q}e^{i2\pi Q}Z_Q.$$

Since $Q$ is an integer, $e^{i2\pi Q}=1$. Hence

$$Z(\theta+2\pi)=Z(\theta).$$

Exercise 2: CP-invariant angles and distinct infrared physics

Assume CP maps $Q\mapsto -Q$. Show that the CP-invariant theta angles are $\theta=0$ and $\theta=\pi$ modulo $2\pi$. Explain why this shared symmetry does not require the two theories to have the same infrared behavior.

Solution

The theta factor transforms as

$$e^{i\theta Q}\mapsto e^{-i\theta Q}=e^{i(-\theta)Q}.$$

Thus CP maps the theory at $\theta$ to the theory at $-\theta$. CP can be a symmetry only if

$$-\theta\equiv\theta\pmod{2\pi}.$$

This gives

$$2\theta=2\pi k, \qquad k\in\mathbb Z,$$

so

$$\theta=0,\pi\pmod{2\pi}.$$

Symmetry invariance only says that CP maps each of these theories to itself. Their sector weights are different: at $\theta=0$ every sector has phase one, whereas at $\theta=\pi$ odd sectors have phase $-1$. The resulting interference can produce different infrared physics. Indeed, the standard $O(3)$ model is massive at $\theta=0$ and is identified with a gapless $SU(2)_1$ theory at $\theta=\pi$.

Exercise 3: RG improvement of the size integral

Using

$${1\over\alpha(\mu)}={1\over\alpha_0}+{1\over2\pi}\log {\mu\over\Lambda_{\rm UV}},$$

show that

$${d\rho\over\rho^3} \exp\left[-{4\pi\over\alpha(1/\rho)}\right]$$

is proportional to $d\rho/\rho$ up to a $\rho$-independent factor. What is the meaning of the scale $\rho\sim M^{-1}$ at the upper end of a semiclassical estimate?

Solution

Set $\mu=1/\rho$:

$${1\over\alpha(1/\rho)} ={1\over\alpha_0}+{1\over2\pi}\log {1\over\rho\Lambda_{\rm UV}} ={1\over\alpha_0}-{1\over2\pi}\log(\rho\Lambda_{\rm UV}).$$

Therefore

$$-{4\pi\over\alpha(1/\rho)} =-{4\pi\over\alpha_0}+2\log(\rho\Lambda_{\rm UV}),$$

so

$$\exp\left[-{4\pi\over\alpha(1/\rho)}\right] =e^{-4\pi/\alpha_0}(\rho\Lambda_{\rm UV})^2.$$

Multiplying by $d\rho/\rho^3$ gives

$${d\rho\over\rho^3} \exp\left[-{4\pi\over\alpha(1/\rho)}\right] = \Lambda_{\rm UV}^2e^{-4\pi/\alpha_0}{d\rho\over\rho}.$$

The prefactor is independent of $\rho$, so the size dependence is logarithmic. At $\rho\sim M^{-1}$, the scale $1/\rho$ reaches the strong-coupling region and $\alpha(1/\rho)$ is no longer small. This marks the breakdown of the one-loop density; it does not derive a literal step-function cutoff.

Exercise 4: Dilute-gas exponentiation

In a dilute gas with instanton fugacity $\zeta$, derive

$${Z(\theta)\over Z_0}=\exp(2V\zeta\cos\theta)$$

and compute $\chi_t$.

Solution

A configuration with $N_+$ instantons and $N_-$ anti-instantons has topological charge

$$Q=N_+-N_-.$$

The dilute-gas sum is

$${Z(\theta)\over Z_0} = \sum_{N_+,N_-\ge0} {(V\zeta e^{i\theta})^{N_+}\over N_+!} {(V\zeta e^{-i\theta})^{N_-}\over N_-!}.$$

The two sums exponentiate independently:

$${Z(\theta)\over Z_0} =e^{V\zeta e^{i\theta}}e^{V\zeta e^{-i\theta}} =e^{2V\zeta\cos\theta}.$$

Thus

$$f(\theta)-f(0)=2\zeta(1-\cos\theta),$$

and

$$\chi_t={\partial^2 f\over\partial\theta^2}\bigg|_{\theta=0}=2\zeta.$$

References

• I. Affleck, “Exact Critical Exponents for Quantum Spin Chains, Non-Linear Sigma Models at $\theta=\pi$ and the Quantum Hall Effect,” Nuclear Physics B 265 (1986), 409–447, doi:10.1016/0550-3213(86)90167-7.
• A. A. Belavin and A. M. Polyakov, “Metastable States of Two-Dimensional Isotropic Ferromagnets,” JETP Letters 22 (1975), 245–248.
• F. D. M. Haldane, “Continuum Dynamics of the 1-D Heisenberg Antiferromagnet: Identification with the $O(3)$ Nonlinear Sigma Model,” Physics Letters A 93 (1983), 464–468, doi:10.1016/0375-9601(83)90631-X.
• F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State,” Physical Review Letters 50 (1983), 1153–1156, doi:10.1103/PhysRevLett.50.1153.

Further reading

• S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, 1985, especially “The Uses of Instantons.”
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, vol. 3, Harwood Academic Publishers, Chur, 1987.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press, Oxford, 2021.