Why a critical chain looks the same at every scale

Correlation length, and what happens when it disappears

Pick one spin and ask how strongly a spin $r$ sites away follows it. Away from a critical point this correlation dies off exponentially, and the decay length $\xi$ is the size of a typical correlated patch. Measure distances in units of $\xi$ and every such system looks alike.

At the critical point $\xi$ grows without limit. What remains is a power law. Doubling the distance always multiplies the correlation by the same factor $2^{-\eta}$ , wherever you start, so no length is singled out.

C(r)\sim e^{-r/\xi}\ \ (\text{off critical}),\qquad C(r)\propto r^{-\eta}\ \ (\text{critical})

The exponent

\eta

is a property of the critical point, not of the material details. For the two-dimensional Ising class it is

1/4

Source: S. Rychkov

A chain we can solve exactly

The transverse-field Ising chain has spins coupled to their neighbours with strength $J$ and a field $h$ that tries to flip them. For $h<J$ the spins order; for $h>J$ they do not. At $h=J$ the chain is critical and belongs to the two-dimensional Ising class.

Mapped to free fermions, the ground state of a ring of $N$ spins is known exactly, and the correlation is a determinant of size $r$ . On a ring the distance that enters the power law is the chord $(N/\pi)\sin(\pi r/N)$ , which equals $r$ for short separations.

H=-J\sum_i\sigma^x_i\sigma^x_{i+1}-h\sum_i\sigma^z_i,\qquad C(r)=\langle\sigma^x_0\sigma^x_r\rangle\propto\Bigl[\tfrac{N}{\pi}\sin\tfrac{\pi r}{N}\Bigr]^{-1/4}

The second formula holds at the critical field for separations much larger than one site.

A ring of

N=16

spins. At the critical field the correlation between two spins depends on their chord distance

(N/\pi)\sin(\pi r/N)

through a single power, so rings of any size collapse after rescaling by

N^{1/4}

. Schematic.

Source: G. B. Mbeng · D. Simmons-Duffin

The test: rescale and compare

If $C$ depends on $r$ and $N$ only through the chord to the power $-1/4$ , then $N^{1/4}C(r)$ depends only on the fraction $r/N$ . Rings of every size should fall on one curve. That is a sharp, falsifiable statement.

In the ordered phase the correlation tends to a constant, so $N^{1/4}C$ grows with the ring. In the disordered phase it vanishes beyond a few sites. Only at the critical field do the curves collapse.

Source: G. B. Mbeng

Worked example: rings of 16 to 256 spins at the critical field

At $h=J$ and separation $r=N/8$ the exact correlations are $0.5439$ , $0.3859$ and $0.2729$ for $N=16$ , $64$ and $256$ . They drop by a factor of two over this range, yet $N^{1/4}C$ is $1.0878$ , $1.0915$ and $1.0918$ . The small ring differs by 0.4 percent; the two large ones agree to four digits.

Between $r=N/4$ and $r=N/2$ on the 256-site ring the chord grows by $\sqrt2$ , and the correlation falls by $2^{-1/8}\approx0.917$ . That ratio gives $\eta=0.2500$ to four digits.

At $h=0.8J$ the 256-site correlation at half the ring is $0.7746$ , equal to $(1-h^2/J^2)^{1/4}$ to six digits: a finite plateau, the square of the magnetisation. Rescaling by $N^{1/4}$ then makes larger rings look brighter.

Try it in the Lab

Collapse the rings

Start from the critical preset. Predict: will the five rings show the same colours once rescaled?
Follow the probe around. Check that the curves in the first chart lie on top of each other.
Move the field to 0.9 and then 1.1. Describe how the rings separate in each phase.

The same chain, short and out of equilibrium

Choose “Quench without a local flip” and view “Connected correlation”. The Hamiltonian is the same, on an open chain of four to eight spins.
Predict before you look: can a chain this short show a clean power law? Compare its length with the separations used on the critical ring.

Exercises with explained answers

With $C\propto r^{-1/4}$ , by what factor does the correlation fall when the separation grows from 10 to 160 sites?
Show the answer
The distance grows sixteenfold, so the correlation falls by $16^{-1/4}=1/2$ .
Why does an exponential decay single out a length, while a power law does not?
Show the answer
For $e^{-r/\xi}$ , moving out by $\xi$ always costs a factor $e$ , so $\xi$ is a natural unit. For $r^{-\eta}$ only ratios of distances matter: doubling costs $2^{-\eta}$ at any scale.
A ring of 128 spins at the critical field gives $C(16)=0.3246$ . Estimate $C(32)$ on a ring of 256 spins.
Show the answer
Both points sit at $r/N=1/8$ , so $N^{1/4}C$ is the same: $C\approx0.3246\times(128/256)^{1/4}=0.2729$ , the exact value to four digits.

Sources and what we recomputed

D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)
Version 1. Eqs. (157)-(158): the four-point function as a sum over conformal multiplets, one conformal block each. Eqs. (164)-(168): the Casimir equation for blocks and their closed form in two dimensions, built from the one-variable function k. Eqs. (194)-(199), Sections 10.5-10.7: the crossing equation for identical scalars as a sum of vectors, the derivative functionals and the assumptions behind the scalar-gap bound. Fig. 26: 2D crossing vectors. Fig. 27 and Table 1: 2D bounds for derivative cutoffs 6 to 28 with the kink at the Ising point. Fig. 28: the 3D bound with its kink.
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, arXiv:1601.05000v2 (2016)
Version 2. Section 1.1: fixed points and scale invariance at a critical point. Section 1.3.4: how primary operators transform. Section 2.2: two- and three-point functions of primary scalars. Sections 3.1.3-3.1.4: the state-operator correspondence and the cylinder picture. Section 3.3: the operator product expansion. Section 4.3.1, Eqs. (4.14)-(4.16): the minimal models and the central charge and spin and energy dimensions of the 2D Ising model.
G. B. Mbeng, A. Russomanno and G. E. Santoro, The quantum Ising chain for beginners, arXiv:2009.09208v2 (2020)
Version 2. Sections 1-3: the Jordan-Wigner mapping of the chain to free fermions. Eq. (55): the quasiparticle energies of the uniform chain, of which the transverse-field Ising ring is the fully anisotropic case. Eqs. (247)-(249): the ground-state correlation functions as determinants. The course uses these for the periodic transverse-field ring and checks them against exact diagonalisation of eight sites.