eps-geo: A Diagnostic for Emergent Geometry from Mutual Information — What It Can and Cannot Do

A numerical investigation (free-fermion / small exact-diagonalization) of emergent geometry from entanglement (the Van Raamsdonk / Cao–Carroll–Michalakis program), probed by a single quantity: the non-Euclidean obstruction of the mutual-information distance, ε_geo. The aim is to delimit precisely what this diagnostic can and cannot tell us.

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Summary

From the mutual-information matrix I_ij we build a distance d_ij, form the classical-MDS Gram matrix B = -½ J D² J (with J the centering matrix), and take its negative-eigenvalue mass

ε_geo = Σ_{λ<0} |λ| / Σ |λ|,

read as "how far the MI-distance is from embeddable into a metric geometry" — i.e. the degree of geometry failure.

In one line: ε_geo is a robust detector of geometric phases, but it is not a meter of critical exponents.

Established claims:

1. Geometry failure is governed by the spatial structure of mutual information (whether MI is local or spread out isotropically), not by the area-law/volume-law scaling of entanglement entropy.
2. ε_geo detects a genuine quantum phase transition (the Aubry–André localization transition), and that detection is robust to the choice of distance functional.
3. By contrast, the static critical exponent extracted from ε_geo is construction-dependent (it ranges over 1.7–15 across functionals) and is therefore meaningless.
4. The geometry-destruction front is carried by coherent mutual information: ballistic for free-fermion unitary dynamics (z ≈ 1), dynamically stable in localized phases, and the diffusive/scrambling regimes require interacting or non-local dynamics that lie beyond the reach of these methods.

Along the way, three plausible-looking claims were falsified or withdrawn (see the discipline table below).

Unified synthesis: see INTERPRETATION.md — what ε_geo does and does not measure across all models, the validity guards, and a one-line verdict per model.

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Background and definition

• Emergent geometry. Degrees of freedom that share more mutual information are treated as "closer," defining a distance and reconstructing a spatial geometry from a state's entanglement structure (Van Raamsdonk 2010; Cao–Carroll–Michalakis 2017). Cao–Carroll et al. restrict to area-law "redundancy-constrained" states and extract an emergent dimension via MDS.
• Questions asked here. Is the area law really the essential condition? Can we quantify the class of states for which geometry fails to emerge? Is that boundary a phase transition? How does geometry break dynamically?
• ε_geo. Distance d_ij = -log(I_ij / I_max) by default. For free-fermion (Gaussian) states, I_ij is computed exactly from the 2×2 sub-blocks of the single-particle correlation matrix C. To isolate non-Euclidean-ness we also tried an effective dimension d_eff (participation ratio of positive eigenvalues); it was confounded by the negative-eigenvalue mass and discarded. ε_geo is the clean discriminator.

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Results

(c) Non-Hermitian spectral pathology is unrelated to geometry breakdown ✗ (killed)

Hypothesis. Exceptional points / non-normality of a non-Hermitian generator (measured by the eigenvector-matrix condition number cond(V)) coincide with the rise of ε_geo — the intuition being that both detect "loss of a good basis."

Test. A Hatano–Nelson chain where cond(V) spans ~7 decades (1 → 1.2×10⁷); the non-equilibrium steady state of a stable quadratic Lindblad with this coherent part is correlated against ε_geo. Result:

Spearman(log cond V, ε_geo) = -0.69   (-0.62 excluding low-MI noise)  -- WRONG SIGN
Pearson(total MI, ε_geo)    = +0.23   -- weak: ε_geo is independent of total MI

Stronger non-normality (skin effect → one-sided correlations) makes geometry simpler (lower ε_geo), not broken. Hypothesis (c) is false.

(a) Geometry failure is controlled by MI spatial structure, not entropy scaling ✓

Four free-fermion state classes (N = 16):

class	ε_geo	MI-range R	vol-coeff
gapped GS	0.245	1.04	+0.000 (area law)
critical GS	0.308	1.61	+0.026
thermal	0.247	1.02	+0.448 (most volume-law)
random / scrambled	0.368	5.76	+0.313

The decisive row is thermal: the most volume-law state of all, yet its ε_geo matches the gapped (area-law) state — geometry is fine, because its MI is spatially local.

Control (thermal temperature sweep). Holding R ≈ 1.0 fixed while raising the volume-law coefficient +0.09 → +0.67, ε_geo decreases 0.30 → 0.20:

Spearman(vol-coeff, ε_geo) = -1.00

Isolating the entropy-scaling axis reverses the naive correlation. The controlling variable is the MI-range R (spatial structure). Area law is not even necessary; the right condition is spatial locality of MI. This extends Cao–Carroll–Michalakis beyond their area-law restriction.

Nature of the boundary (finite-size scaling): free-fermion ground states are robustly geometric

• All-to-all GUE interpolation (gapped → GUE): R̂(α) curves collapse with no rescaling and the maximum slope is N-independent → crossover, not a transition.
• Power-law hopping exponent p: R̂ is independent of p and → 0 with N (the translation-invariant free-fermion eigenvectors are plane waves regardless of p; the Fermi sea is unchanged — Fermi-surface universality).

Aubry–André: ε_geo detects a genuine transition ✓

Aubry–André model (Fibonacci β, PBC, phase-averaged, N = 34…377), localization transition at λ_c = 2:

• ε_geo ≈ 0.37 (high, N-independent) in the extended phase (λ<2), dropping in the localized phase (λ>2);
• the steepest descent |dε/dλ|_max grows with N (0.28 → … → 0.73) and its location λ* converges to λ_c = 2 (2.27 → 2.02);
• R̂ collapses across λ = 2, confirming the localization transition itself.

So geometric phase = localized (short-range MI), non-geometric phase = extended (long-range MI) — fully consistent with (a).

The static critical exponent is construction-dependent ✗ (withdrawn)

We initially claimed ε_geo had its own exponent ν_eff ≈ 3.5 at the AA transition (true ν = 1). Re-measuring with only the distance functional changed:

functional	ν_eff
-log x	5.9
1/x - 1	1.7
1 - x	13.9
√(-log x)	15.1

ν_eff ranges over 1.7–15 (and even varies with grid resolution for a fixed functional). Meanwhile the transition location λ_c ≈ 2 is detected by every functional. So ε_geo's static exponent is not a meaningful quantity. The ν_eff ≈ 3.5 claim is withdrawn. (A cross-transition check on the 3D Anderson model was inconclusive: the ground-state ε_geo is blind to a mobility-edge transition, and the energy-resolved version did not converge at accessible sizes.)

(b) Dynamics: geometry destruction is ballistic; in detector mode the exponent is robust ✓

Quench from a localized (geometric) SSH ground state under uniform hopping (ballistic):

• Two timescales. Onset (30%-rise) is N-independent (z ≈ 0); completion (70%-rise) gives z ≈ 0.87 (ballistic z = 1 with finite-size corrections).
• The completion z is the same (~0.87) for the log and inverse functionals — unlike the static ν_eff, the dynamical z is robust, because t* tracks a physical ballistic front rather than a fragile spectral shape.
• Ballistic vs diffusive. Dephasing (L_i = √γ n_i) does not produce a diffusive front: it destroys MI (total MI → 0 on timescale 1/γ). The apparent rise under weak dephasing is ε_geo reading a vanishing MI matrix (a noise value ~0.4) — ε_geo must be guarded by a "total MI alive" condition.
• Stability. Quenching under a localized Hamiltonian (AA λ = 6) keeps ε_geo low (~0.19 vs ballistic ~0.43): the geometric phase is dynamically stable.

Interacting (non-Gaussian) ED: the front exponent is beyond ED reach

XXZ + next-nearest-neighbour ZZ spin chain (Jordan–Wigner ↔ t–V fermions), quenched from a singlet product, N = 10…14:

• ε_geo rises then relaxes; total MI stays alive (~5–7), so ε_geo is reliable here (unlike the dephasing case). Late-time geometry is set by the steady-state (GGE / thermal) MI structure — the dynamical echo of result (a).
• Front. The 70%/60%-rise is onset-dominated (N-independent at these sizes). The peak time grows with N (free 10 → 12; chaotic 7 → 21, more steeply), hinting that interactions delay full destruction — but N ≤ 14 is far too small to resolve whether the front is ballistic, diffusive, or scrambling.

→ The diffusive end (z = 2) needs MPS/TEBD at N ~ 50–100; the scrambling end (log N) needs all-to-all models. Both are beyond exact diagonalization here.

Operator front vs conserved-Sz transport (MPS/TEBD): ballistic front, anomalous transport ✓ (exp09)

MPS / TEBD at N ~ 40 (beyond the ED reach of exp06/exp07), separating two dynamical exponents. After a quench, two independent measures:

1. MI / operator front — connected correlators from a Néel quench; front position r*(t) = furthest r above a threshold. Result: BALLISTIC (d log r* / d log t ~ 1) for both an integrable (TFIM) and a non-integrable (mixed-field Ising) chain. The geometry-destruction front that ε_geo tracks is a Lieb-Robinson (operator-spreading) quantity → ballistic regardless of integrability. Resolves the exp07 open question: interactions do NOT make the ε_geo front diffusive. Caveat: this "front" is the connected ⟨Z_c Z_{c+r}⟩ light cone used as a Lieb-Robinson proxy — ε_geo(t) itself was not computed at N = 40. The integrable value is 1/z ≈ 1.0, but the non-integrable value is only ≈ 0.8 over a short, χ-limited window; the robust evidence for ballistic is the constant front velocity, not the log-log slope.

2. Conserved-Sz transport — domain-wall melting in clean XXZ; transferred magnetization dM(t) ~ t^(1/z). It carries its own distinct exponent: ballistic at Δ = 0.5 (1/z ~ 0.9) but KPZ-superdiffusive at the isotropic point Δ = 1.0 (1/z ~ 2/3, z ~ 3/2). So the conserved-density exponent is decoupled from the ballistic operator front; ε_geo follows the front, not transport.

True diffusion z = 2 is not reachable from a pure-state quench (a sharp domain wall is pinned for Δ > 1); it requires the infinite-temperature purification protocol — addressed in exp10, which finds behaviour consistent with z = 2 (with the caveats noted there). Note also that the domain-wall transferred-magnetization exponent used here is a far-from-equilibrium probe, not identical to the linear-response transport z.

MPS/TEBD purification: conserved transport approaches diffusion z=2 (exp10)

The z = 2 left open by exp09 is probed with an infinite-temperature purification (each physical spin maximally entangled with an ancilla so ρ_phys ∝ I; physical XXZ gates applied across the ancilla by a swap network). The conserved-Sz autocorrelation second moment σ²(t) = Σ_r r² C_r(t) / Σ_r C_r(t) (sum rule Σ_r C_r = 1/4 conserved to 4 digits) shows the full anisotropy crossover: ballistic at Δ = 0.5 (σ² ∝ t²), KPZ-superdiffusive at the isotropic point Δ = 1.0, and behaviour consistent with diffusion at Δ = 1.5, where σ²(t) is approximately linear in t (D ≈ 0.6).

A dedicated convergence study (exp10b, second figure below) removes the boundary and truncation objections and sharpens the verdict. At N = 18 the spreading packet stays far from the edges (edge/centre weight ≤ 10⁻² through t = 6), and σ²(t) is triple-χ converged — χ = 100, 160, 256 agree to < 1 % through t ≈ 4.5, with an exact sum rule at χ = 256. In that converged, boundary-free window the local exponent d log σ² / d log t = 2/z descends monotonically 1.85 → 1.17 (i.e. z falling from the early ballistic transient toward z = 2) and is still descending, while the ballistic Δ = 0.5 case stays pinned at ≈ 2.0 (z = 1). The approach to z = 2 is therefore χ-converged and boundary-free, not a short-window artefact.

It does not reach z = 2 exactly: the trustworthy floor is 2/z ≈ 1.17 (z ≈ 1.7). The slope falls to 1 only at much longer times (t ~ 15–30), which at infinite temperature need χ ~ 10²–10³ because operator entanglement grows linearly — beyond brute-force TEBD here (χ = 256 already reaches only t ≈ 4 in budget, so raising χ confirms convergence but cannot extend the window). Honest verdict: a χ-converged, boundary-free monotonic approach to z = 2 (floor z ≈ 1.7), cleanly separated from ballistic; the asymptote is consistent and expected but reachable only with long-time methods (e.g. dissipation-assisted operator evolution). The σ² second moment is clean in one respect: C_r(t) was verified non-negative (no sign cancellation).

Together with exp09 this rounds out the dynamical picture: the ε_geo geometry-destruction front is ballistic (Lieb-Robinson, z = 1) regardless of integrability, while the conserved-Sz transport is ballistic → KPZ → diffusive (up to z ≈ 2) depending on anisotropy. ε_geo tracks the operator front, not the (slower) conserved-density transport. Two honest limits: ε_geo(t) is measured directly only for free fermions (exp06, ballistic) — for interacting systems the "front is ballistic" statement is inferred from the correlator light cone plus Lieb-Robinson. The exp09 front exponents are single-size estimates; the exp10 diffusive measurement is χ-converged (χ = 100/160/256) and checked at N = 12/16/18, though without a full finite-size-scaling analysis.

Classical OU / Langevin: a weak signal under a finite-sample noise floor △ (exp08)

A classical analogue of the Aubry–André experiment — node fields under a network Ornstein–Uhlenbeck (linear Langevin) flow whose normal modes localize at λ_c = 2. With exact (noise-free) MI, ε_geo carries a weak but functional-robust, correctly-directed signal of the transition (extended ≈0.146 → localized ≈0.128, same direction as the quantum case; Spearman(log, inverse) = +0.70; tracks the soft-mode participation ratio and R at +0.72). The contrast is small (~10%) because a classical steady state reshapes the MI-distance geometry far less than a quantum ground state.

With estimated MI from a finite ensemble, per-pair estimation noise inflates ε_geo toward a floor (~0.3–0.4) that buries the signal and destabilizes the functional-robustness check; the level drifts back toward the true value only as n_real → ∞. Practical lesson: on estimated MI, verify the ε_geo level is not just the n_real noise floor and that functional robustness is stable under increasing ensemble size. The quantum experiments above use exact analytic MI and are free of this effect.

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Overall conclusion

ε_geo (the MDS negative-eigenvalue mass of the MI-distance matrix) is a robust detector of geometric phases:

1. governed by MI spatial structure, not entropy area/volume scaling (thermal decoupling -1.00);
2. it locates phases, transitions (AA λ_c = 2), and geometry-destruction times, robustly across distance functionals;
3. it does not yield reliable static critical exponents (construction-dependent, 1.7–15);
4. the geometry-destruction front is carried by coherent mutual information: ballistic for free fermions (z ≈ 1), stable in localized phases, and requiring interacting / non-local dynamics for the diffusive and scrambling regimes.

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Discipline log: falsified / withdrawn claims

claim	verdict	evidence
non-Hermitian EP / non-normality ↔ geometry breakdown	killed	Spearman(log cond V, ε_geo) = -0.69 (wrong sign)
ε_geo has an intrinsic critical exponent ν_eff ≈ 3.5	withdrawn	ν_eff spans 1.7–15 across distance functionals
dephasing reveals diffusive geometry destruction (z = 2)	killed	dephasing destroys MI; the apparent rise is noise

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Limitations and outlook

• Mostly free-fermion (Gaussian), 1D, -log distance. Absolute values are construction-dependent; only signs, orderings, detected locations, and time-scalings are robust.
• The AA exponent used 5 sizes and a derived observable — it is not a rigorous field-theory ν.
• Mobility-edge transitions are invisible to the ground-state ε_geo (energy-resolved versions were inconclusive here).
• Next steps: the front is ballistic even with interactions (exp09) and conserved transport shows a χ-converged, boundary-free monotonic approach to diffusion z = 2 (exp10 / exp10b; local exponent 1.85 → 1.17, floor z ≈ 1.7). Reaching the asymptote needs long-time methods (e.g. dissipation-assisted operator evolution); the scrambling end (log N) with all-to-all / non-local models also remains open.

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Reproducing

pip install -r requirements.txt
cd scripts
python exp01_nonhermitian.py      # (c) killed
python exp02_state_classes.py     # (a) MI structure controls geometry
python exp03_fss_crossover.py     # FSS: crossover, no transition
python exp04_aubry_andre.py       # AA transition detected
python exp05_exponent_artifact.py # exponent is construction-dependent
python exp06_dynamics.py          # ballistic destruction; dephasing kills MI
python exp07_interacting_ed.py    # interacting: rise-relax
python exp08_classical_ou.py      # classical OU: weak signal under a finite-sample noise floor
python exp09_mps_dynamics.py      # ballistic front vs anomalous conserved-Sz transport
python exp10_purification_z2.py   # purification: conserved transport reaches diffusion z=2 (needs quimb, ~6-8 min)

Each script prints its key numbers and writes its figure to figures/. Some scripts use modest sizes for speed; comments note the larger N used for the paper figures. See scripts/README.md for details.

References

1. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42, 2323 (2010).
2. C. Cao, S. M. Carroll, S. Michalakis, Space from Hilbert space: recovering geometry from bulk entanglement, Phys. Rev. D 95, 024031 (2017).
3. J. Maldacena, L. Susskind, Cool horizons for entangled black holes (ER=EPR), Fortschr. Phys. 61, 781 (2013).
4. S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
5. B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86, 065007 (2012).
6. G. Franzmann, S. Jovancic, M. Lawson, On the relative distance of entangled systems in emergent spacetime scenarios, arXiv:2210.14875 (2022).
7. S. Aubry, G. André, Analyticity breaking and Anderson localization in incommensurate lattices (1980).
8. N. Hatano, D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).