2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationthe Ising model and transfer matrices

Detour: the gap and the correlation length in the Ising model — the ratio of eigenvalues of the transfer matrix sets the length over which correlations reach

The mass gap was the difference between the two lowest eigenvalues of the Hamiltonian (2-05), and the correlation length was the length over which the joint movement of two distant points disappears (2-08). We check in the simplest model that the two are a single number.

The key is the transfer matrix. If the eigenvalues of the matrix $T$ that advances the lattice one column at a time are $\lambda_0\gt\lambda_1\ge\cdots$, $E_n=-\log\lambda_n$, then

(the conditions are in section 04). The left side is a statistical quantity, the right side is the gap of the "Hamiltonian" $H=-\log T$, and reading one tick as one step of imaginary time, this is the same content as "the rate of decay is the gap" stated by (8) of 2-05.

Example. On a two-dimensional strip of width $L$, $E_1-E_0$ settles to a positive value on the high-temperature side, is proportional to $1/L$ at the critical point, and closes exponentially on the low-temperature side (sections 05–08).

Arrange $N$ spins $\sigma_i=\pm1$ on a ring ($\sigma_{N+1}=\sigma_1$), coupling neighbours. With $K=\beta J$, $B=\beta h$, and distributing the external-field term half and half to neighbouring pairs, the weight is a product of factors for each neighbouring pair.

Each factor is determined by $\sigma_i$ and $\sigma_{i+1}$ alone, so it becomes a $2\times2$ matrix (rows are $\sigma_i$, columns $\sigma_{i+1}$, $+1$ is the first row), and since the sum goes once round the ring and returns, it is a trace.

$T$ is real symmetric, so its eigenvalues are real (1-01), and

For large $N$, $Z_N\approx\lambda_0^N$. The method of finding the partition function from the largest eigenvalue of a matrix was carried out by Kramers–Wannier 1941 for the one-dimensional chain, and taken on to two dimensions Known.

Example. ComputedFor $N=14$, $K=0.7$, $B=0.2$, the $Z_N$ obtained by summing all $2^{14}$ configurations, and the correlations, agreed with the trace formula (3) within rounding error (differences in correlations below $10^{-13}$).

Write the orthonormal eigenvectors of $T$ as $v_0,v_1$ and the spin as $S=\operatorname{diag}(1,-1)$. In $N\to\infty$ the rest of the ring is replaced by $\lambda_0^{N-r}v_0v_0^{\mathsf T}$, and expanding the $T^r$ in between in eigenvalues gives the correlation at distance $r$.

The last equality follows because putting $v_0=(\cos\varphi,\sin\varphi)$, $v_1=(-\sin\varphi,\cos\varphi)$ gives $m=\cos2\varphi$, $v_0^{\mathsf T}Sv_1=-\sin2\varphi$. The correlation length is as in (1). For $B=0$, $\lambda_0=2\cosh K$, $\lambda_1=2\sinh K$, and

ComputedTable 1 Eigenvalues and correlation lengths for $B=0$.

For every $K\lt\infty$, $\lambda_0\gt\lambda_1$, so $\xi$ is finite, and switching off the external field gives $m=\sinh B/\sqrt{\sinh^2B+e^{-4K}}\to0$. The quantity under the square root in (4) is positive, so $\lambda_0$ does not coincide with $\lambda_1$, and the free energy per site $f$ is $\beta f=-\lim_{N\to\infty}\frac1N\log Z_N=-\log\lambda_0$, an analytic function of $K$ and $B$. This is the reason, in terms of the transfer matrix, why there is no phase transition in one dimension (Ising 1925; for the definition of a phase transition see section 04 of 2-08) Known. Even for a general square matrix with all entries positive, the largest eigenvalue does not coincide with another; this is called the Perron–Frobenius theorem (here we give only the name). Intuitively, the fraction of neighbours that disagree (the fraction of walls) $p=\frac{1-\tanh K}{2}\approx e^{-2K}$ does not become 0 however low the temperature, and the correlation changes sign each time it crosses a wall, so $\xi\approx1/(2p)$.

Computed$G(r)$ and the levels are computed on the spot from (4)(5); the chain is a sample of the Markov chain built from $v_0$ (random numbers with a fixed seed). The readout agrees with python3 ($\xi=2.9563$ at $K=1$, $B=0.1$).

The $T$ of $B=0$ is the sum of $e^K$ times the identity matrix and $e^{-K}$ times $\sigma^x=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$ (1-04). Comparing with $e^{K^*\sigma^x}=\cosh K^*+\sigma^x\sinh K^*$, it can be written as

We call $K^*$ the dual coupling (the quantity of Kramers–Wannier duality). $T=e^{-H}$, $H=-K^*\sigma^x+\text{const.}$ is a single spin in a transverse magnetic field, that is, a two-level system (2-05), and its gap is $E_1-E_0=2K^*=\log\coth K=1/\xi$. The direction of the chain is imaginary time, and one tick is a step of 1 (2-07). The stronger the coupling (the lower the temperature), the smaller $K^*$, the closer the two levels, and the longer the correlations persist.

In general, if $T$ is a positive symmetric matrix (in infinite dimensions, a positive self-adjoint operator) with eigenvectors $|n\rangle$, the correlation of an observable $A$ is, by the same computation,

Known. Example. A level whose coefficient $\langle0|A|n\rangle$ is 0 by symmetry does not appear in the correlation of that $A$. The gap is "the slowest decrease, valid for every observable", and individual correlations decrease at least that fast (the remark after Table 2 of 2-05). That the transfer matrix of lattice gauge theory is positive self-adjoint follows from reflection positivity Known (said to be Osterwalder–Seiler 1978; details in 5-03).

Put the two-dimensional Ising model on the square lattice ($B=0$, $K$ both vertically and horizontally) on a strip of width $L$. The width direction is joined periodically (a cylinder), and the length direction is read as imaginary time. One column has $2^L$ states; writing the couplings within a column as the diagonal matrix $V_1=\exp(K\sum_j\sigma_j\sigma_{j+1})$ and the couplings to the next column as $V_2$, using (7) at each point gives

Here $\sigma^x_j$ is the matrix of order $2^L$ that flips only the $j$-th spin, and $\sigma^z_j$ is the diagonal matrix that multiplies by the sign $\pm1$ of the $j$-th spin; neither does anything to the other spins. The exponent of $V_1$ is the sum of $\sigma^z_j\sigma^z_{j+1}$, and the exponent of $V_2$ the sum of $\sigma^x_j$. Putting half of $V_1$ on each side makes $T$ a symmetric matrix, so (8) of section 04 can be used. $T$ has the form of one step of imaginary time for a system of quantum spins in a row (the transverse-field Ising chain). Two-dimensional classical statistical mechanics has become a one-dimensional quantum system.

$T$ commutes with the global flip $\sigma\to-\sigma$, so the eigenvectors split into even and odd; the largest even one is $\lambda_0$, the largest odd one $\lambda_1$, and the correlation length of the strip is $\xi_L=1/(E_1-E_0)$ (equation (1) of de Queiroz 2011). The critical point $\sinh2K_c=1$ ((9) of 2-08) is the self-dual point where, by (7), $K^*=K$.

ComputedTable 2 Level differences on strips of width $L$ ($E_2$ is the third level). The bottom row gives the values of the formulas for $L\to\infty$ (sections 06–08).

$L=20$ (order $2^{20}\approx10^6$) too was found without building the matrix, only by applying the actions of $V_1$ and $V_2$ in turn. At the three temperatures the behaviour is completely different.

For $T\gt T_c$ ($K\lt K^*$), $E_1-E_0$ settles to a positive value even as $L$ is widened. The eigenvalues of the strip have a closed form (Onsager 1944; summarised in the review of Domb 1960, with the form of the formulas in equations (6)–(8) of de Queiroz 2011), and its building block is the following $\gamma(\omega)$. The right-hand form follows from $\sinh2K\sinh2K^*=1$.

As $\gamma(0)=2|K^*-K|$, the $E_1-E_0$ of Table 2 approaches $2(K^*-K)$ Computed (difference $4\times10^{-6}$ at $T/T_c=1.5$, $L=16$). $\gamma(\omega)$ can be read as the energy of a single particle of momentum $\omega$ (the free-fermion view; Schultz–Mattis–Lieb 1964). On the strip the momentum is quantised to $2\pi n/L$, and the second odd level is close to $\gamma(2\pi/L)$, agreeing better the further from $T_c$ (at $L=16$: 0.763461 and 0.763457 for $T/T_c=1.5$, 0.495433 and 0.494596 for 1.2) Computed. As $L\to\infty$ these come down to $\gamma(0)$, and above the gap there is a continuous band (a real example of section 06 of 2-05).

Example. Near the critical point, when $\gamma,\omega$ is small, expanding the right-hand form of (10) to second order gives

which has the relativistic form $E^2=m^2+p^2$ (speed of light 1, 2-04). Precisely, (11) is an approximation only when $\gamma(0)$ and $\omega$ are small (for $T/T_c=1.05$, $\omega=0.1$: left side 0.017234, right side 0.017263) Computed.

At $T=T_c$, $\gamma(0)=0$, and the gap of an infinitely wide strip is 0. On a finite strip $\xi_L$ grows in proportion to $L$, and the constant of proportionality is a universal quantity independent of the details of the model. The first formula of (12) below (a general relation) is a prediction from conformal invariance (the property, at a critical point, of not changing form even under transformations that preserve angles; here we give only the name) Physics (Cardy 1984; the form is equations (3)(5) of de Queiroz 2011). For the two-dimensional Ising model, the same value $4/\pi$ comes out of the closed form of the eigenvalues of the strip in section 06 Known.

$x$ is the scaling dimension of the spin, with $G(r)\sim r^{-2x}$ at the critical point; in two dimensions it is half of $\eta=1/4$ of 2-08. Example. ComputedExtrapolating from the $\xi_L/L$ of $L=18,\ 20$, assuming the difference vanishes like $c/L^2$, gives 1.273248, differing from $4/\pi$ by $8\times10^{-6}$.

ComputedThe numbers in the figure were computed on this machine ($L=4$–14, $T/T_c=0.72$–1.5 in steps of 0.04 (0.02 for the last only); python3 values embedded to 4 significant figures). The dashed and dotted lines on the right compute the values of (10)(13) on the spot.

The lines on the left gather almost at one point at $T_c$ even for different widths, and open downwards on the high-temperature side and upwards on the low-temperature side, the larger $L$.

For $T\lt T_c$, $E_1-E_0$ becomes exponentially small with $L$. On the strip there are two states, "almost all $+$" and "almost all $-$", and the largest even and odd eigenvectors are their sum and difference. To move from one to the other needs a wall of length $L$ across the strip, whose weight is of order $e^{-\sigma L}$. $\sigma$ is the free energy per unit length of the wall (the interface tension), and for the two-dimensional Ising model

is held to hold (the free energy of the boundary between orders of opposite direction was found by Onsager 1944). (13) becomes 0 at $T_c$ (the self-dual point $K=K^*$), and as $K\to\infty$ it approaches $2K$, one disagreeing bond per unit length of wall (the weight $e^K$ becomes $e^{-K}$). The relation (7) also holds with $K$ and $K^*$ exchanged ($\tanh K=e^{-2K^*}$), so an equivalent form is $\sigma=2K+\log\tanh K$. This is the same picture as the double well of section 07 of 2-05. Precisely, what corresponds to the height of the hill is the free energy of the wall $\sigma L$, which grows in proportion to the width.

On the other hand, the difference from the third level $E_2-E_0$ does not close, and is held to approach two particles' worth of (10), $2\gamma(0)=4(K-K^*)$; the extrapolation of Table 3 does not contradict this Physics. It is two particles' worth because, in one column joined periodically in the width direction, a disagreement of direction returns to the original after going once round, so only an even number of them fit, and one disagreement corresponds to one particle of (10).

ComputedTable 3 $T/T_c=0.8$. The slope is $\frac12\log\frac{(E_1-E_0)_{L-2}}{(E_1-E_0)_L}$; the correction assumes the prefactor $L^{-1/2}$; the extrapolation is from the $E_2-E_0$ of $L-2$ and $L$ (Table 2), assuming the difference vanishes like $c/L^2$.

The slope moves towards $\sigma$, and the extrapolation towards $4(K-K^*)$. The prefactor and the form of the correction are assumptions, so this is a check at the level of not contradicting (13). Conversely, fixing $\sigma$ at the value of (13), taking the prefactor as $L^{-p}$, and solving back for $p$ from two neighbouring $L$, we get 0.587, 0.564, 0.551 at $L=12,\ 16,\ 20$, approaching $1/2$ Computed. Finite-size effects of first-order transitions on a finite strip (cylinder) are said to have been discussed by Privman–Fisher 1983. Lowering $T/T_c$ below 1 on the right of Figure 2, you can see the ● line up parallel to the dotted line of slope $\sigma$.

This is an important reading. If the width $L$ is finite, every entry of $T$ is positive (the entries of $V_2$ are products of $e^{\pm K}$), so by the Perron–Frobenius theorem named in section 03, $\lambda_0\gt\lambda_1$, that is, $E_1-E_0\gt0$ (as in section 03, there is no phase transition). $E_1-E_0\to0$ is the sign that in the limit where the width $L$ is widened to infinity there are two ordered states, $+$ and $-$, and above each a positive gap (the limit of $E_2-E_0$ above) is held to remain Physics. The gap that closes is a sign that "there are two vacua", not a sign that "there is no mass gap". When discussing the mass gap, one checks separately that the vacuum is unique (4-11).

We list the correspondences of this chapter. All are reinterpretations at the level of physics and show nothing about Yang–Mills Physics. The $\beta$ in the table and in the paragraph below is the constant multiplying the action of lattice gauge theory (proportional to the inverse of $g^2$; 2-08 §08), a different quantity from the inverse temperature $\beta$ of this chapter.

There are also differences. The critical point of four-dimensional non-abelian gauge theory is believed to be at $\beta=\infty$, and how it is approached is determined by asymptotic freedom (4-06). Nor is there a closed-form solution like that of the two-dimensional Ising model.

Detour: Z₂ gauge theory and the Ising model

What this chapter can say

Sources and reproduction

Next chapter: 3-01 The gauge principle. The main treatment of the definition of the gap is in 4-11, and the transfer matrix of the lattice in 5-03.

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