When logarithmic entanglement lies about universality

Key claim

Logarithmic entanglement does not, by itself, imply conformal criticality.

An MPS can reproduce the logarithm and still misidentify the physics that generated it.

Theme

The XXZ ferromagnetic endpoint: when entanglement scaling lies about universality.

Guiding question

In the spin-$1/2$ XXZ chain, can a CFT entropy formula and finite-entanglement scaling distinguish a Luttinger liquid from the exceptional point $\mathrm{\Delta }=-1$?

The collision joins three subjects:

• integrable spin chains;
• conformal field theory;
• matrix product states.

The trap is unusually clean. The endpoint has logarithmic ground-state entanglement, but its low-energy dynamics are not relativistic.

Setup

Take an even periodic chain,

For $-1<\mathrm{\Delta }\le 1$, the thermodynamic low-energy theory is a $c=1$ compact boson, or Luttinger liquid.

The endpoint $\mathrm{\Delta }=-1$ is different. Apply a staggered $\pi$ rotation about $z$:

Because every bond contains one even site,

Thus the endpoint is unitarily equivalent to the isotropic Heisenberg ferromagnet.

Its ground-state manifold has maximal total spin $S=N/2$. Fixing $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^z=0$ selects the half-filled Dicke state, up to the staggered onsite rotation.

That rotation factorizes as $U_A\otimes U_{\overline{A}}$ across any spatial cut. It therefore leaves the entanglement spectrum unchanged.

For a block of $L$ sites,

with

The Schmidt probabilities form a hypergeometric distribution. This gives an exact route to the logarithm without invoking CFT.

Analysis

Derivation — solve the endpoint in its native language

Set $L=N/2$. The hypergeometric distribution has

Its width is therefore

Near its peak, $p_k$ approaches a discrete Gaussian. Its Shannon entropy is

Hence

The logarithm records the number of macroscopically plausible ways to distribute a fixed total magnetization between the two halves.

It does not come from a tower of relativistic local modes.

The distinction is visible in the excitation spectrum. For the rotated ferromagnet, a one-magnon excitation has

Thus

whereas an ordinary $(1+1)$-dimensional CFT requires linear low-energy dispersion and $z=1$.

Scope and assumptions — attack “logarithm means central charge”

For a periodic conformal chain, the von Neumann entropy of an interval of length $L$ is

Here $a$ is a short-distance cutoff. At $L=N/2$, the coefficient of $\log N$ is $c/3$.

If the Dicke result were inserted blindly, one would infer

This number is an effective fit coefficient, not the central charge of an endpoint CFT. No relativistic CFT governs the endpoint.

The same functional form has arisen from a different mechanism: a collective conserved-charge fluctuation inside a highly degenerate ferromagnetic ground-state manifold.

This is not only a conceptual counterexample. Near $\mathrm{\Delta }=-1$, finite systems show a long crossover between the endpoint coefficient and the neighboring $c=1$ regime.

Entropy fits can therefore return plausible but scale-dependent central charges.

The correct question is not merely

It is

Physical interpretation — what an MPS actually knows

First separate two geometries.

For an open-boundary MPS cut across one virtual bond of dimension $\chi$,

For an interval in a periodic MPS, the bipartition crosses two virtual bonds:

These are exact algebraic bounds. They do not diagnose a universality class.

Now consider an optimized uniform MPS for a genuine one-dimensional conformal ground state. Finite $\chi$ induces an effective correlation length ${\xi }_{\chi }$.

For the entropy across one cut,

The standard finite-entanglement theory predicts

under its CFT assumptions.

This theory is more than the inequality $S\le \log \chi$. It uses the universal distribution of entanglement-spectrum eigenvalues at a conformal critical point.

Finite $\chi$ is often described as introducing a mass-generating relevant perturbation. That description is a useful analogy, not an operator identity.

A 2026 analysis finds that the effective perturbations selected by optimal tensor-network approximations can differ from those guessed from the most relevant CFT operator alone.

The geometry of the variational manifold also matters.

False claim to diagnose

If $S_{\chi }$ is linear in $\log {\xi }_{\chi }$ and the fitted slope gives $c\approx 1$, the XXZ state must be a Luttinger liquid.

The claim is tempting because this is a standard finite-entanglement diagnostic.

But the fit tests one relation over a finite range. It does not independently establish $z=1$, a conformal tower, or freedom from crossover contamination.

The $\mathrm{\Delta }=-1$ state supplies the counterexample:

A defensible diagnosis requires mutually consistent evidence:

• entropy scaling;
• low-momentum dispersion or finite-size energy gaps;
• MPS transfer-matrix spectra;
• correlation-function exponents;
• stability of fitted parameters as $N$ and $\chi$ increase.

The entropy coefficient is evidence. It is not a verdict.

What follows — and what does not

The practical conflict is controlled by three competing lengths:

The endpoint crossover scale ${\xi }_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ grows as $\mathrm{\Delta }\to -1^{+}$.

If $N$ or ${\xi }_{\chi }$ remains below that scale, a calculation performed inside the $c=1$ phase can still look endpoint-like.

If the accessible scale exceeds it at fixed $\mathrm{\Delta }>-1$, the asymptotic Luttinger-liquid behavior can emerge.

Writing only

hides this competition. One must specify the path through scale space.

An MPS does not classify universality directly. It provides a compressed state whose entropy, transfer matrix, correlations, symmetry sector, and convergence must be interpreted together.

Exercise

Take the half-filled Dicke state $|D_N^{N/2}⟩$ and divide an even chain into two equal halves.

1. Derive its Schmidt probabilities $p_k$.
2. Approximate $p_k$ by a Gaussian and obtain $S_A=\frac{1}{2}\log N+O(1)$.
3. For a one-cut Schmidt truncation, find the rank needed to retain a fixed fraction $0<f<1$ of the total weight.
4. Explain why none of these results establishes conformal invariance.

Hint 1

Use the entropy of a discrete Gaussian:

$$H\simeq \frac{1}{2}\log \left(2\pi \mathrm{e}{\sigma }^2\right).$$Copy LaTeX

Hint 2

A fixed fraction of a Gaussian lies within a window of width $O(\sigma )$ around its mean.

Oral check 1. Why is $S\le \log \chi$ exact for one MPS cut, while ${\xi }_{\chi }\propto {\chi }^{\kappa }$ is not?

Oral check 2. Which most directly separates the endpoint from the Luttinger liquid: the entropy coefficient, $⟨S^z⟩$, or the low-$q$ dispersion?

Solution

Counting configurations with $k$ up spins in the left half gives

$$p_k=\frac{(\genfrac{}{}{0ex}{}{N/2}{k})(\genfrac{}{}{0ex}{}{N/2}{N/2-k})}{(\genfrac{}{}{0ex}{}{N}{N/2})}.$$Copy LaTeX

This distribution has

$${\sigma }^2=\frac{N^2}{16(N-1)}\sim \frac{N}{16}.$$Copy LaTeX

Therefore,

$$S_A\simeq \frac{1}{2}\log \left(2\pi \mathrm{e}{\sigma }^2\right)=\frac{1}{2}\log N+O(1).$$Copy LaTeX

To retain any fixed fraction $0<f<1$, one keeps a central window containing $O(\sigma )$ Schmidt values.

Thus a one-cut truncation needs

$${\chi }_f\sim \sigma \sim \sqrt{N}.$$Copy LaTeX

Exact representation across the half-chain cut requires the full Schmidt rank

$${\chi }_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=\frac{N}{2}+1.$$Copy LaTeX

For a periodic MPS interval, two virtual bonds cross the bipartition. Its geometric rank bound is ${\chi }^2$, not $\chi$.

The calculation reveals the width of a collective number distribution. It says nothing by itself about $z$, local conformal symmetry, or a Virasoro spectrum.

The low-$q$ dispersion is the cleanest oral-check answer:

$$\epsilon (q)\propto |q|\text{for a Luttinger liquid},\epsilon (q)\propto q^2\text{at the endpoint}.$$Copy LaTeX

Check your understanding

Is entanglement scaling a property of the infrared theory, the selected state sector, or the variational representation?

Name one measurement that separates those three layers in this example.

You may also reply with “deeper,” “too easy,” “too hard,” or your attempted derivation.

Further Reading

• Entanglement entropy scaling of the XXZ chain
• Permutation operators, entanglement entropy, and the XXZ limit $\mathrm{\Delta }\to -1^{+}$
• Theory of finite-entanglement scaling at one-dimensional quantum critical points
• On the origin of finite entanglement scaling

Connections and next step

• Pressure point: a logarithmic entropy is not a unique fingerprint of CFT.
• Exact layer: the Dicke Schmidt spectrum and MPS rank bounds.
• Dynamical layer: $z=2$ at the ferromagnetic endpoint versus $z=1$ in the Luttinger liquid.
• Numerical layer: $N$, ${\xi }_{\chi }$, and ${\xi }_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ compete near $\mathrm{\Delta }=-1$.
• Revisit: extract $c$, $z$, and the Luttinger parameter from independent observables in three runs.