# Quantum field theory 1, lecture 10

In this lecture we turn again to the $$O(N)$$-symmetric scalar theories that we have already introduced in the lecture 2.3. These models are simple enough to serve as good examples, and rich enough to show many interesting physical properties characteristic for quantum field theories. They serve as a ”working horse” for this lecture. With the formalism developed so far we can compute the propagator in the absence of interactions or in the limit of small interaction effects. We also discuss the setup for spontaneous symmetry breaking.

#### 5.3 Unified Scalar field theories

Euclidean space. Scalars play an important role in quantum field theory. Prominent examples are the Higgs scalar for the standard model of particle physics, scalar mesons for the strong interactions, or the inflaton for cosmology. The corresponding Lorentz invariant quantum field theory is formulated in Minkowski space. Analytic continuation from Minkowski to Euclidean space yields \begin{equation*} \eta ^{\mu \nu }\partial _\mu \partial _\nu \to \delta ^{\mu \nu } \partial _\mu \partial _\nu . \end{equation*} Another factor arises from $$dt = -id\tau .$$ In Euclidean space the action therefore reads \begin{equation*} S = \int _x \left \{ \frac{1}{2} \sum _a \partial ^{\mu } \chi _a \partial _\mu \chi _a + V(\rho ) \right \}, \end{equation*} where now $$\partial ^{\mu } = \delta ^{\mu \nu }\partial _\nu$$ and $$\int _x = \int \;dt\int d^3\vec{x}.$$ This is the four-dimensional $$O(N)$$-model introduced in lecture 2.3. The Euclidean action is also the one that appears for the $$T \to 0$$ limit of thermal equilibrium, while for $$T>0$$ the $$\tau$$-integration becomes periodic with period $$1/T$$.

In euclidean space, the Lorentz-symmetry $$SO(1,3)$$ gets replaced by the four dimensional rotations $$SO(4)$$. This symmetry is broken for $$T>0$$ since space and time are no longer treated equally. One should distinguish two different symmetries: The internal symmetry $$O(N)$$ acts on the internal degrees of freedom, while the symmetry $$SO(d)$$ corresponds to the Lorentz symmetry and acts as a space-time transformation, changing coordinates or momenta.

Unified description of scalar theories. The euclidean $$O(N)$$-models in arbitrary dimension $$d$$, admit a classical statistical probability distribution, with real action,

\begin{equation*} p = \frac{1}{T} e^{-S}, \quad \quad \quad Z= \int D\varphi e^{-S}. \end{equation*} They can be simulated on a computer.

We can classify important applications according to the dimension $$d$$ of euclidean space and number $$N$$ of real components of the scalar field:

 $$d=1,2,3$$ models of classical statistical systems in $$d$$-dimensions $$N=3$$ magnets, $$\langle\chi_a(x)\rangle$$ is magnetisation (order parameter) $$N=1$$ Ising type models $$N=2$$ $$d=2$$ two-dimensional x-y model with Kosterlitz-Thouless phase transition $$d=4$$ relativistic scalar theories in thermal equilibrium at $$T=0$$, or analytic continuation of quantum dynamics

If the euclidean model is solved, the $$n$$-point functions can be analytically continued to Minkowski space, using \begin{equation*} q_{0E} = q^0_E = -i q_{0M} = i q^{0}_{M}. \end{equation*}

Correlation functions or $$n$$-point functions. The task is the computation of $$n$$-point functions \begin{equation*} G^{(n)}_{ab\ldots f}(x_1\ldots x_n) = \langle \chi _a(x_1) \chi _b(x_2)\cdots \chi _f(x_n)\rangle , \end{equation*} with space-time argument $$x_i=x^\mu _i$$. Alternatively in Fourier space the $$n$$-point functions are \begin{equation*} G^{(n)}(p_1\ldots p_n), \end{equation*} where $$p_i=p^\mu _i$$. As an example take the two point function or propagator \begin{equation*} G_{ab}(p_1,p_2) =\langle \chi _a(p_1) \chi _b(-p_2) \rangle - \langle \chi _a(p_1)\rangle \langle \chi _b(-p_2)\rangle =G(p_1) \delta (p_1-p_2)\delta _{ab}. \end{equation*} It can only depend on one momentum by virtue of $$d$$-dimensional translation symmetry. Invariance under $$SO(d)$$-rotations implies that $$G$$ can only depend on \begin{equation*} p^2 = p_\mu p_\nu \delta ^{\mu \nu }, \end{equation*} or, in other words, $$G(p^\mu ) = G(p^2)$$. Analytic continuation does not change $$G(p^2)$$, one only has to switch to $$p^2 = p_\mu p_\nu \eta ^{\mu \nu }$$ in momentum space.

We start from the action for a free field \begin{equation*} S= \int _x\left \{\frac{1}{2}\partial ^\mu \chi _a\partial _\mu \chi _a + \frac{1}{2} M^2 \chi _a \chi _a \right \}. \end{equation*} This is a sum of independent pieces. Each particle with associated field can be treated separately. Consider for simplicity a single complex field \begin{equation*} S=\int _x \left \{\partial ^\mu \chi ^{*}\partial _\mu \chi + M^2 \chi ^{*}\chi \right \}, \end{equation*} and transform to Fourier space \begin{equation*} S= \int _q (q^2 + M^2) \chi ^{*} (q) \chi (q),\quad \quad \quad \int _q =\int \frac{d^d q}{(2\pi )^d}. \end{equation*} The propagator in Fourier space is given by \begin{equation*} G(p,q) = \langle \chi (p) \chi ^{*}(q)\rangle - \langle \chi (p) \rangle \langle \chi ^{*}(q)\rangle . \end{equation*}

We want to compute this propagator. For this purpose we use a torus with discrete modes and take the volume to infinity at the end. For \begin{equation*} S = \sum _q (q^2 +M^2) \chi ^{*}(q) \chi (q) \end{equation*} the expectation value obeys \begin{equation*} \langle \chi (p) \rangle = \frac{1}{Z} \int D\chi \;e^{-S} \; \chi (p) = 0. \end{equation*} This is a simple consequence of the invariance of $$S$$ and $$\int D\chi$$ under the reflection $$\chi \to -\chi$$. Similarly, for $$p \neq q$$, one finds \begin{equation*} \langle \chi (p) \chi ^{*}(q) \rangle = \frac{1}{Z} \int D\chi e^{-S} \chi (p) \chi ^{*}(q) =0. \end{equation*}

Only for equal momenta $$p=q$$ the two point function differs from zero, \begin{equation*} \begin{split} \langle \chi (q) \chi ^{*}(q)\rangle &= \frac{1}{Z}\int D\chi e^{-S}\chi (q) \chi ^{*}(q)\\ &= \frac{\int d\chi (q) e^{-(q^2+M^2)\chi ^{*}(q)\chi (q)}\chi ^{*}(q)\chi (q)}{\int d\chi (q) e^{-(q^2+M^2)\chi ^{*}(q)\chi (q)}}. \end{split} \nonumber \end{equation*} For the second identity we use the fact that for all $$q' \neq q$$ the same factor appears in the numerator and denominator.

We first compute the Gaussian integral \begin{equation*} \tilde{Z}(M^2) =\int d\chi (q) e^{-(q^2+M^2)\chi ^{*}(q)\chi (q)}, \end{equation*} and then take the derivative with respect to $$M^2,$$ \begin{equation*} \langle \chi (q) \chi ^{*}(q) \rangle = -\frac{\partial }{\partial M^2} \ln (\tilde{Z}(M^2)). \end{equation*} The Gaussian integral has the solution \begin{equation*} \tilde{Z}(M^2) = \frac{\pi }{q^2+M^2}, \end{equation*} \begin{equation*} -\ln (\tilde{Z}) = \ln (q^2+M^2) - \ln (\pi ), \end{equation*} \begin{equation*} -\frac{\partial }{\partial M^2} \ln (\tilde{Z}) = \frac{1}{q^2+M^2}. \end{equation*} We can summarise for the free propagator \begin{equation*} G(q,p) = \frac{1}{q^2+M^2}\delta (q-p). \end{equation*} For the last identity we have performed the infinite volume limit for which the Kronecker delta $$\delta _{p,q}$$ becomes the distribution $$\delta (p-q) = (2\pi )^{-d} \delta ^d(p_\mu -q_\mu )$$, which plays in our conventions the role of the unit matrix in momentum space.

Propagator in Minkowski space. The analytic continuation of the free euclidean propagator is straightforward in momentum space, \begin{equation*} \begin{split} G(p,q) &= \frac{1}{(q^2+M^2)}\delta (p-q)\\ &= \frac{1}{-q^2_0+ \vec{q}^2+M^2}\delta (p-q). \end{split} \nonumber \end{equation*} This propagator has poles at \begin{equation*} q_0 = \pm \sqrt{\vec{q}^2 +M^2}. \end{equation*} These two poles correspond to a particle and its antiparticle.

The solutions of the free field equations are \begin{equation*} \chi _{+} = e^{-i\sqrt{\vec q^2+M^2}t} \end{equation*} and \begin{equation*} \chi _{-} = e^{+i\sqrt{\vec q^2+M^2}t} = e^{-i\sqrt{\vec q^2+M^2}\tilde{t}},\quad \quad \quad \tilde{t}= -t. \end{equation*} Antiparticles appear formally as particles propagating “backwards in time”. The oscillatory behaviour in time is also visible in the Fourier transform of the propagator.This contrasts with the behaviour in euclidean space. There the Fourier transform becomes a function of $$r=|\vec x - \vec y|$$. For $$d=3$$ the result is a Yukawa potential $$G(r)$$ proportional $$\exp{(-M r)}/r$$. The propagator vanishes rapidly for large separations $$r \gg 1/M$$.

Adding an interaction with strength $$\lambda$$, as specified by a potential \begin{equation*} V=M^2 \rho + \lambda \rho ^2 /2, \end{equation*} will modify the propagator through the effects of fluctuations. For small $$\lambda$$ the leading effects are a shift of $$M^2$$ and a multiplicative constant for the terms in the action which are quadratic in $$\chi$$. These effects can be absorbed by a multiplicative rescaling of fields and an additive ”renormalisation” of $$M^2$$. Modifications of the momentum dependence of the propagator occur in the order $$\lambda ^2$$ or higher. The free propagator remains often a very good approximation.

#### 5.5 Magnetisation in classical statistics

In the next part we link our formalism to a first set of physical questions. We discuss magnetisation and the notion of spontaneous symmetry breaking. This is done in the view of a later treatment of the Higgs mechanism for the electroweak interactions in particle physics.

Action. We investigate the thermal equilibrium state for classical statistics of magnets. We employ microscopic fields $$\sigma _a(x)$$ which represent elementary magnets averaged over small volumes. The Hamiltonian with next neighbour interaction reads in this continuum description \begin{equation*} H= \int _x \left \{  K\, \partial _i \sigma _a (x) \partial _i\sigma _\alpha (x) + c\, \sigma _a(x)\sigma _a(x) + d\, (\sigma _a(x)\sigma _a(x))^2 -B_a\sigma _a(x) \right \}. \end{equation*} We take $$K>0$$, which tends to align magnets at neighbouring points. The homogeneous magnetic field $$B$$ breaks the $$O(N)$$-symmetry. Typical isotropic magnets in three dimensions correspond to $$N=3$$. The internal symmetry $$O(3)$$ reflects independent spin rotations that are decoupled from rotations in space. One can also consider asymmetric magnets with $$N=2$$ (xy-models) or $$N=1$$ (Ising-type models). Magnets in lower dimensions are also highly interesting, with $$d=2$$ corresponding to physics dominated by layered structures as for materials leading to high temperature superconductivity. At this level there is no longer any difference between ferromagnets and antiferromagnets. The internal symmetry is the same.

The partition function in classical statistical thermal equilibrium obeys as usual \begin{equation*} Z= \int D\sigma \;e^{-\beta H}=\int D\sigma e^{-S}, \end{equation*} where the classical action is \begin{equation*} S=\beta H. \end{equation*}

Rescaled fields. By a rescaling of fields \begin{equation*} \sigma _a(x) = \sqrt{\frac{1}{2 \beta K}}\, \chi _a(x). \end{equation*} we can bring the action to the standard form for $$O)N)$$-models \begin{equation*} S=\int _x \left \{ \frac{1}{2} \partial _i \chi _a(x) \partial _i \chi _a(x) +\frac{c}{2 K} \chi _a(x)\chi _a(x) + \frac{d}{4 \beta K^2}(\chi _a(x)\chi _a(x))^2 - \frac{B_a\sqrt{\beta }}{\sqrt{2K}} \chi _a(x) \right \}, \end{equation*} or with other naming conventions for the couplings \begin{equation*} S = \int _x \left \{ \frac{1}{2} \partial _i \chi _a(x) \partial _i\chi _a(x) + \frac{m^2}{2}\chi _a(x)\chi _a(x) + \frac{\lambda }{8}(\chi _a(x)\chi _a(x))^2 - J_a \chi _a(x) \right \}. \end{equation*} This relates the standard couplings $$m^2$$, $$\lambda$$ and the source $$J$$ to the microscopic model parameters. The parameter $$m^2$$ can be positive or negative. It is often called a ”mass term”, in analogy to the mass term for a relativistic particle.

Magnetisation. For $$m^2>0$$ the microscopic magnets have for $$J=0$$ a preferred value $$\chi _a = 0$$. For $$m^2<0$$ the preferred value differs from zero for $$J=0$$. The minimum of the potential \begin{equation*} V_0(\rho ) = m^2 \rho +\frac{\lambda }{2}\rho ^2,\quad \quad \quad \rho = \frac{1}{2}\varphi _a\varphi _a, \end{equation*} obeys \begin{equation*} \frac{\partial V_0}{\partial \rho } = m^2+\lambda \rho = 0. \end{equation*} For $$m^2<0$$ it occurs at $$\rho _0 = -\frac{m^2}{\lambda }$$. A non-vanishing magnetic field $$J_a$$ singles out a certain direction. The minimum of $$V= m^2\rho + \frac{\lambda }{2} \rho ^2 - J_a \varphi _a$$ defines the microscopic magnetisation.

We want to compute the macroscopic magnetisation $$\langle \chi (x) \rangle$$ as a function of the magnetic field $$J_a$$. For this problem fluctuations play an important role. We concentrate on $$m^2 < 0$$ where things are most interesting. The factor $$e^{-S}$$ is maximal if $$S$$ is minimal. One may first look for the minimum of S and expand around it. This procedure is called the ”saddle point approximation”. The minimum of $$S$$ is given by the microscopic magnetisation. Without loss of generality we choose $$J = (J_1, 0, 0)$$. The configuration with constant $$\chi$$, $$\chi _a(x) = \chi _{a,0}$$ minimises the kinetic term. The minimum of the action is then given by the minimum of $$V$$. It occurs in the direction $$\chi _1$$, for which the potential reduces to \begin{equation*} V = \frac{1}{2} m^2\chi ^2_1+\frac{\lambda }{8}\chi ^4_1-J\chi _1. \end{equation*} The minimum of $$V$$ is determined by the homogeneous field equation \begin{equation*} \frac{\partial V}{\partial \chi } = m^2 \chi _1 +\frac{\lambda }{2}\chi ^3_1 -J =0. \end{equation*}

If we take $$J>0$$ a positive $$\chi _{1,0}$$ is preferred, being the absolute minimum of $$V$$. The absolute minimum flips sign if we change the sign of $$J$$. At $$J=0$$ one observes two degenerate minima. Such a behaviour is characteristic for a first order phase transition as a function of the magnetic field, as observed in ferromagnets or antiferromagnets.

In the limit of small $$J>0$$ one has \begin{equation*} \frac{\lambda }{2}\chi ^2_{10} = -m^2,\quad \quad \quad \chi _{10}= \sqrt{-\frac{2m^2}{\lambda }}. \end{equation*}

Fluctuations tend to wash out the microscopic magnetisation. If we want to know how strong is this effect, we have to compute the partition function $$Z(J)$$ as a function of the source $$J$$. Then the magnetisation $$\tilde{M}$$ in appropriate units is determined by \begin{equation*} \frac{\partial \ln Z}{\partial J} = \left \langle \int _x \chi _1 \right \rangle = \Omega \langle \chi _1 \rangle =\tilde{M}, \end{equation*} where $$\Omega$$ the volume. We are interested here in small $$J\to 0$$.

To do thermodynamics we start from the free energy \begin{equation*} F= -T \ln Z = -\frac{1}{\beta }\ln Z. \end{equation*} As well known in thermodynamics the magnetisation is determined by the minimum of the free energy.

Spontaneous symmetry breaking. Spontaneous symmetry breaking occurs if the magnetisation remains different from zero in the limit of vanishing magnetic field, $$\tilde{M} \neq 0$$ for $$J\to 0$$. The magnetisation $$\tilde M_a$$ is proportional to the expectation value \begin{equation*} \varphi _a = \langle \chi _a \rangle . \end{equation*} For $$J=0$$ the $$O(N)$$-symmetry is not violated. Any direction for $$\varphi _a$$ in internal space is equivalent. Nevertheless, the state $$\varphi _a =0$$, which corresponds to vanishing magnetisation, is not a minimum of the free energy, but rather a local maximum. The minimum occurs for $$\rho _0= (\varphi _a \varphi _a) /2$$ different from zero, and the system has to choose ”spontaneously” a direction of the magnetisation. Once this direction is chosen, the symmetry of the ground state is less than the symmetry of the action. This explains the name ”spontaneous symmetry breaking”. For the example of an $$O(3)$$-symmetry of the action the ground state only exhibits the symmetry $$O(2)$$ of rotations in the plane perpendicular to the vector $$\vec \varphi$$. In practice, the direction of $$\vec \varphi$$ is often determined by tiny amounts of symmetry breaking or a tiny effective source $$J$$. Nevertheless, a discussion of the simple situation $$J=0$$ covers the relevant physics.

We will discuss this issue here in terms of the classical action. In view of the importance of fluctuation effects this may not seem to be a good idea at first sight. We will see later, however, that the main effect of the fluctuations is to replace the microscopic potential $$V(\chi )$$ by an ”effective potential” $$U(\varphi )$$. Here $$\varphi _a$$ are macroscopic fields. The symmetry of the ”effective action” that includes the fluctuation effects is the same as for the microscopic or classical action $$S$$. Also the general form has often only small modifications, such that the dominant effect of the fluctuations is a change of parameters. The microscopic parameters $$m^2$$ and $$\lambda$$ are replaced by macroscopic parameters of ”renormalised couplings” $$m^2_R$$ and $$\lambda _R$$. Since we do not fix the parameters we can discuss many aspects in terms of the microscopic action $$S$$, keeping the later replacements in mind.

Goldstone bosons. One of the characteristic signs of spontaneous breaking of a global continuous symmetry ( as $$O(3)$$ in our case ) is the presence of massless “Goldstone bosons”. They correspond to excitations perpendicular to $$\chi _{a,0}$$. For $$J=0$$ the potential has the same height for arbitrary directions of $$\chi _{a,0}$$. A change of the direction will correspond to massless excitations, the Goldstone bosons.

“Massive” or “gapped” excitations correspond to a propagator $$G$$ porportional $$1/(q^2 + M^2)$$, whereas for “massless” or “gapless” excitations one has $$M=0$$ and therefor a propagator $$G$$ which is proportional to $$1/q^2$$. For our translation invariant setting the propagator in momentum space is a matrix in internal space, $$G_{ab}(q^2)$$. In order to see the massless or massive excitations we have to diagonalise the propagator matrix.

Spontaneous symmetry breaking occurs for $$m^2_R < 0$$, or in our ”classical setting” for $$m^2 < 0$$. In this case it is useful to write the potential in the form \begin{equation*} V = \frac{\lambda }{2}(\rho -\rho _0)^2, \quad \quad \quad \rho _0 =-\frac{m^2}{\lambda }. \end{equation*} We concentrate for simplicity on a single complex field, $$N=2$$, \begin{equation*} \rho = \chi ^{*}\chi = \frac{1}{2}(\chi ^2_1 + \chi ^2_2). \end{equation*} For the magnetisation in absence of a magnetic field, $$J=0$$, we choose without loss of generality \begin{equation*} \chi _{1,0} \neq 0, \quad \quad \chi _{2,0}=0, \quad \rho _0 =\frac{1}{2} \chi ^2_{10} . \end{equation*} We expand around $$\chi _{1,0}$$, with \begin{equation*} \chi _1 = \chi _{10}+ \delta \chi _1, \end{equation*} \begin{equation*} \frac{1}{2} \chi ^2_1 = \rho _0 + \chi _{10} \delta \chi _1 +\frac{1}{2}\delta \chi ^2_1, \end{equation*} \begin{equation*} \rho -\rho _0 = \chi _{10}\delta \chi _{1} + \frac{1}{2}\delta \chi ^2_{1} + \frac{1}{2}\chi ^2_{2}. \end{equation*}

For the extraction of the propagator it is sufficient to keep only terms quadratic in the fields $$\delta \chi _1$$ and $$\chi _2$$. A proof in terms of the effective potential will be given in later lectures. In quadratic approximation the potential reads \begin{equation*} \frac{\lambda }{2}(\rho -\rho _0)^2 = \frac{\lambda }{2} \chi ^2_{10} \delta \chi ^2_1 = \lambda \rho _0 \delta \chi ^2_1. \end{equation*} In this approximation the potential does not depend on $$\chi _2$$. The field $$\chi _2$$ corresponds to a ”flat direction of the potential” and will be associated with the Goldstone boson.

The kinetic term adds to the action in momentum space a piece $$q^2(\delta \chi _1(q)\delta \chi _1(-q)+\chi _2(q)\chi _2(-q))$$. In the quadratic approximation we therefore end with a free theory, for which we have already computed the propagator. We conclude that the excitation $$\delta \chi _1$$ behaves as massive field, with $$M^2 = 2\lambda \rho _0$$, and propagator \begin{equation*} G = \frac{1}{q^2 + 2\lambda \rho _0}. \end{equation*} On the other hand, only the kinetic term contributes to the propagator of the excitation $$\chi _2$$, which behaves as massless field with propagator \begin{equation*} G=\frac{1}{q^2.} \end{equation*} This massless field is called a Goldstone boson.

We may add a small source $$J$$, which breaks the symmetry explicitly. This modifies the potential, \begin{equation*} \begin{split} V &=\frac{\lambda }{2}(\rho -\rho _0)^2 - J \chi _1\\ &= \lambda \rho _0 \delta \chi ^2_1 -J\chi _{1,0}-J\delta \chi _1. \end{split} \end{equation*} The action takes the form \begin{equation*} S = S_0 + \Delta S, \end{equation*} \begin{equation*} S_0 = -\Omega J \chi _{1,0}, \end{equation*} \begin{equation*} \Delta S = \int _x \frac{1}{2} \delta \chi _1(x) (-\Delta + 2\lambda \rho _0) \delta \chi _1(x) -J\;\delta \chi _1(x) + \frac{1}{2} \chi _2(x)(-\Delta ) \chi _2(x). \end{equation*} Correspondingly, one obtains for the partition function in lowest order \begin{equation*} Z_0 = e^{-S_0} = \text{exp}(\Omega J \chi _{1,0}), \end{equation*} \begin{equation*} \ln Z_0 = \Omega J \chi _{1,0}, \end{equation*} \begin{equation*} \tilde{M} = \frac{\partial \ln Z_0}{\partial J} = \Omega \chi _{1,0}. \end{equation*}

Phase transitions and fluctuations. What remains is a computation of the fluctuation effects that relate the ”microscopic parameters” $$m^2$$ and $$\lambda$$ to the ”macroscopic parameters” or ”renormalised couplings” $$m^2_R$$ and $$\lambda _R$$. If $$m^2_R$$ turns out positive, the symmetry is not spontaneously broken and one speaks about the ”symmetric phase”. In contrast, for the range of $$(m^2,\lambda )$$ for which $$m^2_R$$ is negative one has spontaneous symmetry breaking. One speaks about the ”ordered phase” or ”SSB phase”. If both possibilities can be realised for suitable $$(m^2,\lambda )$$, and $$m^2_R$$ depends continuously on these parameters, there must be a transition where $$m^2_R = 0$$. This is a phase transition. There is a ”critical surface” in the space of microscopic parameters for which the phase transition occurs. For two parameters this is a critical line, determined by the condition $$m^2_R(m^2,\lambda )=0$$. Both $$m^2$$ and $$\lambda$$ depend on the temperature $$T$$. For given functions $$m^2(T)$$ and $$\lambda (T)$$ one has $$m^2_R (T)$$. The critical temperature $$T_c$$ for the phase transition is determined by the condition $$m^2_R (T_c) =0$$.

Not all models admit a phase transition. For the example $$d=1$$, $$N>3$$, or for $$d=2$$, $$N>2$$, one can show that a true phase transition is not possible. For these models one finds $$m^2_R > 0$$ for all possible values of $$m^2$$ and $$\lambda$$. This is the content of the Mermin-Wagner theorem. An interesting boundary case is $$d=2, N=2$$. In this case one encounters a ”Kosterlitz-Thouless phase transition”, which can be connected to vortices.

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