FR37 · Conformal bootstrap

Quantum Critical Ising Chain: The Same Pattern at Every Scale

Why does a critical spin chain look the same on a ring of 16 spins and a ring of 256? Because at the critical field nothing sets a length. Spin correlations fall as a power of distance, so after rescaling by a fixed power of the ring size, rings of 16 and 256 spins show the same pattern. The energy levels, scaled by the ring size, settle on the numbers of the Ising conformal field theory. Away from the critical field a correlation length appears and the rings no longer match.

Compare spin correlations on Ising rings of 16 to 256 sites. At the critical field every ring shows the same rescaled pattern and the Ising CFT spectrum.

Correlation at the probe $C(16)=0.325$

Local decay exponent (Ising:

1/4

) $\eta_{\mathrm{eff}}=0.250$

Rescaled spin gap (Ising:

1/8

) $\Delta_\sigma^{(N)}=0.125$

Central charge (Ising:

1/2

) $c_{\mathrm{eff}}=0.500$

Tune to the critical point

Transverse field

1

h/J

Ring for the readouts

N=128

N=2^k

Probe separation

r/N=0.125

r/N

Drag the probe around the rings, then move the field away from one.

Model, method and sources

$H=-J\sum_i\sigma^x_i\sigma^x_{i+1}-h\sum_i\sigma^z_i,\qquad E_0\simeq Ne_\infty-\frac{\pi v c}{6N}$

Each ring is solved as free fermions: correlations are determinants of Majorana contractions, levels are sums of quasiparticle energies. Gaps are rescaled with the critical velocity 2J at every field. The local exponent compares the correlation at r and 2r in chord distance. The bulk energy per site is integrated numerically.

Mbeng, Russomanno & Santoro (2020), Eqs. 55, 247–249 · Cardy, Les Houches lectures (2008), Sec. 5.1, Eq. 19 · Simmons-Duffin, TASI lectures (2016), §10

Read why scale invariance matters

Model assumptions

Five periodic transverse-field Ising rings, from 16 to 256 spins, are solved exactly as free fermions. Each site is coloured by its correlation with the top spin, multiplied by the fourth root of the ring size. At the critical field this product depends only on the fraction of the ring between the two spins, so every ring shows the same colours. The rescaled energy gaps approach the Ising CFT dimensions one eighth, one and nine eighths, and the ground-state energy gives the central charge one half. Off the critical field the rings disagree. All results are computed; no measured data are shown.

Every ring on one curve at the critical point

Energy levels land on Ising CFT dimensions

More numbers and how they are computed

Ring used for the readouts: $0$
Probe separation in sites: $0$
First even level, rescaled (Ising: 1): $0$
Levels shown below 2.4: $0$

The dashed curve in the first chart is the conformal shape, normalised once to the 256-site critical ring at half the ring; it does not change with the field. Energy gaps are divided by the critical velocity 2J at every field, so off the critical point the rescaled numbers are not dimensions.

Start with the essentials

Focus question: Why does a critical spin chain look the same on a ring of 16 spins and a ring of 256?
One-sentence intuition: At the critical field, correlations fall as a power of distance, so multiplying by a fixed power of the ring size makes every ring identical. The energy levels, rescaled by the ring size, settle on the numbers of the Ising conformal field theory.

Core mathematical model

Power law on a ring

\langle\sigma^x_0\sigma^x_r\rangle\propto\Bigl[\tfrac{N}{\pi}\sin\tfrac{\pi r}{N}\Bigr]^{-2\Delta_\sigma},\qquad 2\Delta_\sigma=\eta=\tfrac14

On a ring the distance that enters is the chord. Multiplying by $N^{1/4}$ leaves a function of $r/N$ alone.

Levels are scaling dimensions

E_n-E_0\simeq\frac{2\pi v}{N}\,\Delta_n,\qquad \Delta_n\in\{\tfrac18,\,1,\,\tfrac98,\,2,\dots\}

A critical ring of $N$ sites is a CFT on a cylinder, so each gap is a scaling dimension times $2\pi v/N$ , as derived in Cardy's lectures; here $v=2J$ .

Central charge from the ground energy

E_0=Ne_\infty-\frac{\pi v c}{6N}+\dots,\qquad c=\tfrac12

The same cylinder formula puts the central charge into the $1/N$ term of the ground energy. The exact ring energies give $c=0.500002$ at $N=256$ .

Common difficulties

Ordered rings also look alike

Typical misconceptionIf correlations look similar at every size, the chain must be critical.

Better mental modelIn the ordered phase correlations approach a constant, so the raw values also agree. The test is the rescaled pattern: with the factor of the ring size included, ordered rings grow brighter with size.

Run the experiment

01
Compare the rings
Start from the critical preset and follow the probe around.
What to observe: At every angle the five rings have the same colour; the curves in the first chart lie on top of each other.
02
Leave the critical point
Move the field to 0.8, then to 1.2.
What to observe: The rings separate: brighter outer rings when ordered, fading colour when disordered. The central charge drops to zero.
03
Read the spectrum
Return to the critical field and look at the second chart.
What to observe: Rescaled levels sit on the dashed lines at $\tfrac18,\ 1,\ \tfrac98$ and $2$ .

Quantum Critical Ising Chain: The Same Pattern at Every Scale

A critical chain has no preferred length

Core mathematical model

Power law on a ring

Levels are scaling dimensions

Central charge from the ground energy

Common difficulties

Ordered rings also look alike

Run the experiment

Compare the rings

Leave the critical point

Read the spectrum

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