2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationcorrelation length and critical points

Statistical mechanics and phase transitions — probability weights, the length over which correlations reach, and the point where that length diverges

We call each single state of a system of many spins or particles a configuration $\sigma$, and its energy $H(\sigma)$. A system in thermal equilibrium is described by a probability distribution on configurations; the lower the energy of a configuration, the heavier it is, and the weight is exponential.

We call $\mu_\beta$ the Boltzmann distribution and $Z_\beta$ the partition function ($T$ is the temperature; below, Boltzmann's constant $k_B=1$, and $\beta$ is the inverse temperature). For $\beta\to0$ it becomes the uniform distribution, and for $\beta\to\infty$ it concentrates on the configurations of minimal energy. $\langle\cdot\rangle$ is the expectation under $\mu_\beta$ (1-08).

One reason for the exponential is the following variational principle Known. With the entropy $S(P)=-\sum_\sigma P\log P$,

holds, and the minimum is attained only by $P=\mu_\beta$ (the difference between the two sides is the relative entropy $\frac1\beta\sum_\sigma P\log(P/\mu_\beta)\ge0$). The left side is the free energy. It is a tug of war between the force that lowers the energy and the force that increases the number of configurations; at low temperature the former wins, at high temperature the latter. Precisely, in a finite system the minimum of (2) is always the single $\mu_\beta$. If we consider the minimisation of (2) per site in an infinite system, at low temperature the states attaining the minimum can be several rather than one. This "splitting from one into several" is one precise form of a phase transition (the coexistence of phases) (we see it in one-variable form in section 05).

Example. If $N$ non-interacting spins $\sigma_x=\pm1$ are in an external field $h$ ($H=-h\sum_x\sigma_x$), then $Z=(2\cosh\beta h)^N$, the spins are independent, and $\langle\sigma_x\rangle=\tanh\beta h$.

Differentiating $\log Z$ gives means and variances. In the language of probability, $\log Z_\beta$ is the cumulant generating function of $-H$ (plus a constant). The cumulant generating function of a random variable $X$ is $t\mapsto\log\mathbb{E}[e^{tX}]$; differentiating once in $t=0$ gives the mean, twice the variance.

Putting in the external field as $-hM$ with the total magnetisation $M=\sum_x\sigma_x$ gives the magnetisation and the susceptibility $\chi$ (the response to the external field).

The right-hand equality is the fluctuation–response relation: the response to a push is determined by the fluctuations without the push. Since variances are non-negative, $\log Z$ is convex both in $\beta$ and in $h$. The limit $f$ of $f_N=-\frac{1}{\beta N}\log Z_N$ per site as $N\to\infty$ is called the thermodynamic limit. For short-range interactions the limit is known to exist, and convexity survives Known (for lattice spin systems, the standard argument from the fact that when the system is divided into boxes, $\log Z$ differs from the sum over boxes only by the boundary contributions [subadditivity]; Ruelle 1969 and others).

Example. For independent spins, $\chi=\beta(1-\tanh^2\beta h)=\beta\operatorname{Var}(\sigma_x)$. For $h=0$, $\chi=\beta$, finite at every temperature. Without interactions nothing happens.

With interactions, distant points move together. The degree of this is the connected two-point correlation (covariance).

In a translation-invariant infinite system, $G(x,y)=G(x-y)$, and (4) is $\chi=\beta\sum_xG(x)$. For the susceptibility to diverge, the correlations must decrease only slowly. When correlations decrease exponentially, we call that length the correlation length $\xi$.

The logarithm is taken to allow polynomial prefactors. The correlation length of lattice gauge theory is defined in the same form, except that instead of spins it uses correlations of the variables of small square loops (plaquettes) (5-01, Chatterjee 2018 §5, 5-09) Known. There are two examples.

The Ising model: at each point of $\mathbb{Z}^d$, $\sigma_x=\pm1$, $H=-J\sum_{\langle xy\rangle}\sigma_x\sigma_y-h\sum_x\sigma_x$ ($\langle xy\rangle$ is a neighbouring pair, and $J\gt0$ is a ferromagnetic coupling that rewards alignment; a worked example is in 2-09).

The Gaussian chain: put a real number $\phi_x$ at each point of $\mathbb{Z}$, and with $\beta=1$, $m\gt0$ (the mass), take

Restricting to an interval of length $L$ with $\phi=0$ outside, and taking $\phi$ as a column vector, we can write $H=\frac12\phi^{\mathsf T}K\phi$ ($K$ is the $L\times L$ matrix with $m^2+2$ on the diagonal and $-1$ next to it); $e^{-H}$ is a finite-dimensional Gaussian measure with covariance $K^{-1}$ (1-08). As $L\to\infty$, the $G$ at the centre of the interval converges to (8) below. (8) is obtained by putting $G=Ae^{-\kappa|x|}$ for $x\ne0$ into the equation $\sum_yK_{xy}G(y)=\delta_{x0}$ of the infinite chain to get $m^2+2=e^\kappa+e^{-\kappa}$, and fixing $A$ by the equation at $x=0$.

The smaller the mass, the longer the correlation length, and it diverges at the rate $1/m$ as $m\to0$.

ComputedFor $m=1,\ 0.2,\ 0.05$, $\xi=1.03904,\ 5.00831,\ 20.00208$ ($\xi m\to1$). The $G(20)$ obtained by numerically inverting $K$ at the centre of the interval (1601 to 4001 points) agreed with (8) to 13 digits.

This chain is the imaginary-time harmonic oscillator of 2-07 placed on a lattice in time. Distributing $m^2\phi_x^2$ half and half to the couplings on either side, $e^{-H}$ becomes the product of weights of neighbouring pairs

We call the integral operator $(Tf)(\phi)=\int T(\phi,\phi')f(\phi')\,d\phi'$ with this $T$ as kernel the transfer matrix (the $\sum$ of a matrix product replaced by an integral). For a periodically joined chain of $L$ points, $Z=\int T(\phi_1,\phi_2)T(\phi_2,\phi_3)\cdots T(\phi_L,\phi_1)\,d\phi_1\cdots d\phi_L$, that is, the trace of $T^L$. The ratio of the two largest eigenvalues of $T$ is $e^{-\kappa}$, and we call this $\kappa$ the gap of the transfer matrix. This is the simplest example in which the inverse of the correlation length is the gap Known (the general form is in 4-11). ComputedThe ratio of eigenvalues of the matrix discretising $T$ with 1500 points agreed with $e^{-\kappa}$ to 10 digits at $m=1,\ 0.5,\ 0.2$.

We call it a phase transition when a slight change in temperature or the like changes the behaviour completely. In mathematical language, the $f(\beta,h)$ of the thermodynamic limit is not analytic at that point (cannot be represented by a Taylor expansion around the point). It does not occur in finite systems. Precisely, $Z_N$ is a finite sum of exponentials $e^{-\beta H(\sigma)}$ of $\beta,h$, positive for real $\beta,h$. So $\log Z_N$ can be differentiated any number of times and its Taylor expansion converges (this is called being real-analytic).

For the Ising model the following is known Known. For $d=1$ there is no phase transition (Ising 1925, 2-09). For $d=2$, $h=0$,

a phase transition occurs. Assuming there is just one transition, Kramers–Wannier 1941 located it by duality, and Onsager 1944 found the free energy exactly and showed the $-\log|T-T_c|$ divergence of the specific heat. The argument showing that the alignment survives at low temperature is said to begin with Peierls 1936 (put into shape as a proof in later years). What states this precisely is spontaneous magnetisation.

Example. The order of limits matters. Taking $h\to0$ first, the $H$ of the finite system is invariant under the global flip $\sigma\to-\sigma$, so $\langle\sigma_0\rangle=0$. Going to infinity first and then switching off the external field, in two dimensions $\bar m_0\gt0$ remains for $\beta\gt\beta_c$ (going to 0 like $(\beta-\beta_c)^{1/8}$ as $\beta\downarrow\beta_c$; Yang 1952). That the answer changes with the order of limits is a recurring issue in Yang–Mills too (5-09, 7-10).

The view of phase transitions through the zeros of the partition function is treated in the detour after section 05.

The Curie–Weiss model $H_N=-\frac{J}{2N}M^2-hM$ ($M=\sum_{x=1}^N\sigma_x$, $J\gt0$), in which every spin is coupled with the same strength, is the smallest example in which a phase transition can be followed by hand. Grouping by the number of up spins $k$ gives $Z_N=\sum_k\binom Nk e^{-\beta H_N}$. Let $\bar m=M/N$ be the magnetisation per spin (with a bar, to distinguish it from the mass $m$ of the Gaussian chain). Using $\binom{N}{N(1+\bar m)/2}=e^{Ns(\bar m)+o(N)}$, $s(\bar m)=-\frac{1+\bar m}2\log\frac{1+\bar m}2-\frac{1-\bar m}2\log\frac{1-\bar m}2$ from Stirling's formula, the sum is determined by its largest term (Laplace's method).

The tug of war of (2) has shrunk to one variable (the first term is the energy, the last the entropy). $\varphi'(\bar m)=0$ is

For $h=0$: if $\beta J\le1$, the only solution is $\bar m=0$; for $\beta J\gt1$, $\pm\bar m^*\ne0$ are the minima (critical temperature $T_c=J$). From $\tanh u\approx u-u^3/3$, $\bar m^*\approx\sqrt{3(\beta J-1)}$, and differentiating (12) in $h$ gives $\chi=\beta/(1-\beta J)$ for $T\gt T_c$ (the Curie–Weiss law).

ComputedThe figure computes (11)(12) ($J=1$) on the spot. The $\bar m^*$ at $h=0$ is $0.52543,\ 0.71041,\ 0.90733$ at $T/T_c=0.9,\ 0.8,\ 0.6$ (agreeing with python3).

ComputedTable 1 $\sqrt{\langle\bar m^2\rangle}$ ($h=0$; for finite $N$ the sum over $\binom Nk$ is computed exactly) and the $\bar m^*$ of $N=\infty$.

$\beta J=0.8$ goes to 0 like $N^{-1/2}$ ($\sqrt N$ times it is 2.2358, against the prediction 2.2361 from (4) and the Curie–Weiss law), $\beta J=1.25$ goes to $\bar m^*$, and the critical $\beta J=1$ goes to 0 like $N^{-1/4}$ (about 1.08 when multiplied by $N^{1/4}$). For finite $N$ each is a smooth function of $\beta$, and the kink first appears as $N\to\infty$.

Detour: zeros of the partition function

We call the point of a transition in which the spontaneous magnetisation rises continuously from 0 a critical point. Approaching it, the correlation length diverges, and many quantities behave as powers. Their exponents are the critical exponents. With $t=(T-T_c)/T_c$,

we write ($\sim$ means the ratio approaches a constant. The exponent $\beta$ is a different thing from the inverse temperature. $\eta$ is written $\eta_s$ in later chapters when it must be distinguished from the dynamical exponent: 8-03 §07. The relaxation time and the dynamical exponent $z$ are in 2-10). From the scaling hypothesis the following relations among the exponents follow (the renormalisation group explains them; equations (1.38)–(1.40) of Pelissetto–Vicari 2002) Physics, and the right two are held to hold for $d\lt4$.

$\alpha$ is the exponent of the specific heat. Putting in the $\nu=1,\ \eta=1/4$ of two-dimensional Ising gives back the exact values $\gamma=7/4,\ \beta=1/8,\ \alpha=0$ Computed.

ComputedTable 2 Critical exponents. Mean-field and two-dimensional values are from Pelissetto–Vicari (Table 13 and elsewhere), the three-dimensional $\nu,\eta$ are numerical values from the conformal bootstrap (Kos et al. 2016), and * is computed from (14) (errors move it by about ±1 in the last digit).

Example. In the Gaussian chain, the squared mass $m^2$ (section 03) plays the role of $t$, and $\nu=1/2$ follows from $\xi\approx(m^2)^{-1/2}$. Together with the $\beta=1/2,\ \gamma=1$ of section 05, this is the same as the mean-field column. $\eta$ measures the departure from Gaussian behaviour (Pelissetto–Vicari), and for three-dimensional Ising it is small but not 0.

The exponents of Table 2 are thought not to depend on the shape of the lattice or the details of the couplings, but only on the dimension, the symmetry and the range of the interaction. A collection of systems with the same exponents is a universality class Physics. The gas–liquid critical point of simple fluids and uniaxial antiferromagnets are held to belong to the three-dimensional Ising class (Pelissetto–Vicari §3.1), so that liquids and magnets have the same exponents.

Near a critical point $\xi$ becomes much longer than the lattice spacing. Kadanoff 1966 regrouped spins into blocks larger than the lattice spacing and smaller than the correlation length and looked again, and Wilson 1971 turned the repetition of this operation into the equations of the renormalisation group. Details fade, and systems that flow into the same fixed point have the same exponents. Precisely, universality is a principle of physics confirmed in many examples, not a general theorem (6-05, 4-04).

As the smallest example, we add to (7) a coupling $c\ge0$ to the next-but-one neighbour.

By the same computation as section 03, $G$ becomes a sum of terms of the form $e^{-\kappa|x|}$, and the $\kappa$ of the most slowly decreasing term is the smallest positive root of $m^2=2(\cosh\kappa-1)+2c(\cosh2\kappa-1)$. This is the quadratic equation $4cw^2+2w-(2+4c+m^2)=0$ in $w=\cosh\kappa$, so the root can be written by a formula. When $\kappa$ is small the right side is about $(1+4c)\kappa^2$, so $\xi\approx\sqrt{1+4c}/m$.

ComputedTable 3 $\xi m$ for the chain with $c$ added. The limit is $\sqrt{1+4c}=1,\ 1.41421,\ 2.23607$ (also agreeing with the inverse matrix to 10 digits).

ComputedThe figure computes the root of the quadratic equation above on the spot. At $c=0,\ 1/4,\ 1$ it agrees with Table 3 (and with python3's root search and inverse matrix).

The exponent ($\xi\propto m^{-1}$) does not depend on $c$, but the coefficient $\sqrt{1+4c}$ does. For a slowly varying field $(\phi_{x+2}-\phi_x)^2\approx4(\phi_{x+1}-\phi_x)^2$, so the next-but-one coupling looks merely like the neighbour coupling multiplied by $1+4c$, and disappears when the ruler of length is changed.

Introduce the lattice spacing $a$ and read the point $x$ as the position $r=xa$. If the correlation length in lattice units is $\xi$ (the $1/\kappa$ of section 03), the correlation length in physical units is $a\xi$. Keeping $a\xi$ fixed and letting $a\to0$, $\xi$ has no choice but to go to infinity. Taking the continuum limit is approaching a critical point (Chatterjee 2018 §5) Physics. For the Gaussian chain everything is explicit: fixing $M\gt0$ and putting $m=Ma$, from $\kappa/a\to M$, $a/(2\sinh\kappa)\to1/(2M)$ the following holds Known.

The right side is the kernel of the inverse of $-d^2/dr^2+M^2$ (the propagator, 1-11), and $M$ is the mass in physical units.

ComputedThe figure computes (8)(16) on the spot. Table 4 $\xi$ (in lattice units) and $aG$ (python3) as $a$ is shrunk with $M=1$. The bottom row is the right side of (16).

For Yang–Mills, in many non-abelian lattice gauge theories the critical point is believed to be at $\beta=\infty$ (the $\beta$ here is the constant multiplying the action of lattice gauge theory, proportional to the inverse square of the coupling constant $g$, and plays the role of inverse temperature; 5-01), and the inverse of the correlation length corresponds to the mass of the lightest glueball (a particle made of gluons alone, 7-01) (Chatterjee 2018 §5) Physics. One of the questions on the lattice side is "that correlations decrease exponentially for each $\beta\gt0$ ($\xi(\beta)$ is determined in $(0,\infty)$), and that $\xi(\beta)\to\infty$ as $\beta\to\infty$" (Problem 5.1 of the same section, 5-09), and whether the mass stays positive in the continuum limit is a further question (5-05, 4-11). Weights, the correlation length, the critical point and universality correspond to the path integral of field theory (4-02), the inverse of the mass, the continuum limit and renormalisation, and for the Gaussian chain this is exact.

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Next chapter: 2-09 Detour: the gap and the correlation length in the Ising model. For the relation to the mass gap see 4-11; for the relaxation time and the dynamical exponent, 2-10.

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