Abstract Strong-disorder renormalization group (SDRG), despite being a relatively simple real-space renormalization procedure, provides in principle exact results on the critical properties at the infinite-randomness fixed point of random quantum spin chains. Numerically, SDRG can be efficiently implemented as a renormalization of matrix product operators (MPO-RG). By considering larger blocks than SDRG, MPO-RG was recently used to compute non-critical quantities of finite chains that are inaccessible to SDRG. In this work, the accuracy of this approach is studied and two simple and fast improvements are proposed. The accuracy on the ground state energy is improved by a factor at least equal to four for the random Ising chain in a transverse field. Finally, the proposed algorithms are shown to yield Binder cumulants of the three-color random Ashkin–Teller chain that are compatible with a second-order phase transition while a first-order one is predicted by the original MPO-RG algorithm.
The N -color quantum Ashkin–Teller spin chain is a prototypical model for the study of strong-randomness phenomena at first-order and continuous quantum phase transitions. In this paper, we first review the existing strong-disorder renormalization group approaches to the random quantum Ashkin–Teller chain in the weak-coupling as well as the strong-coupling regimes. We then introduce a novel general variable transformation that unifies the treatment of the strong-coupling regime. This allows us to determine the phase diagram for all color numbers N , and the critical behavior for all . In the case of two colors, N = 2, a partially ordered product phase separates the paramagnetic and ferromagnetic phases in the strong-coupling regime. This phase is absent for all , i.e., there is a direct phase boundary between the paramagnetic and ferromagnetic phases. In agreement with the quantum version of the Aizenman–Wehr theorem, all phase transitions are continuous, even if their clean counterparts are of first order. We also discuss the various critical and multicritical points. They are all of infinite-randomness type, but depending on the coupling strength, they belong to different universality classes.
We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak intercolor coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization group calculations. We also find evidence for unconventional critical behavior in the case of strong intercolor coupling, even though an unequivocal determination of the universality class is beyond our numerical capabilities. We compare our results to earlier simulations, and we discuss implications for the classification of phase transitions in the presence of disorder.
Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents
beta and
nu vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to
128times128 spins, each of which is represented by three colors taking values
pm1, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents,
nu and
beta, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound
2/Dlenu, where
D is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in
(1+1) dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.
CORRECTION NOTICE (v5, 2026-09-06). The central result of v1-v4 is WITHDRAWN. This record has not been peer reviewed. Issued 2026-09-06 by the author (v5 of this record). This notice was added to the archived record; the manuscript itself is unaltered. This record has not been peer reviewed. What is wrong The force exponent reported here as alpha = 3.92 +/- 0.16 is an artifact of the fitting procedure, not a property of the data. The fit was unweighted least squares in linear space applied to a steeply falling profile, which places 95.8 percent of its objective on the single point at the shortest retained separation. Re-fitting the same data with nothing discarded, changing only the error model, gives 2.82 (relative-error weighting) or 2.89 (log-space regression). Across fifteen combinations of standard estimator and fit window the exponent spans 2.53 to 3.92, and the value published here is the largest of them. What the data show instead, and how firmly The retraction is certain. The replacement is not. On the open 24-site chain the same data give alpha = 2.6 +/- 0.2, where the interval is the spread of a plateau in the local logarithmic slope over six consecutive separations, not a statistical error (the bootstrap error on the published fit to the same data is 0.63); the plateau value was reproduced at 2.5921 by a separately written implementation run on the same recorded state (SIM-2026-073). This is a single geometry at a single size. On a 24-site periodic ring, analysed with chord distance as pre-registered, two probes that are both energy-density-like (Delta = 1) give 1.83 and 2.37 against a CFT two-point value of 2 (SIM-2026-074): one size, not converged, disagreeing with each other by 29 percent, and not attributable to a single operator channel. No corrected exponent is established. The exchange channel is an open question. Treat 2.6 +/- 0.2 as the provisional open-chain value, not as a result. Also withdrawn or voided alpha = 3.92 +/- 0.16 (24 sites) and alpha = 3.97 +/- 0.15 (16 sites, exact) from stress-tensor exchange. Was: headline result and its first corroboration. Now: withdrawn. Why: the same estimator artifact: both move in lockstep under it (3.92 to 2.56, 3.97 to 2.75 on dropping the contact point), because they share one observable, one fit window and one fitting routine. scalar exchange (alpha = 2) excluded. Was: stated as excluded at about 12 sigma. Now: withdrawn. Why: the 12 sigma used the covariance error 0.16; at the bootstrap error 0.63 computed before release the exclusion is about 2.2 sigma, and under any window starting at separation 3 the point estimate itself falls to about 2.56. alpha = 4 is universal; alpha = 4.00 at Ising, tricritical Ising and Potts. Was: confirmed at three classes. Now: withdrawn; the operator towers at all three classes stand. Why: the exponent was measured only at Ising and inherited by the other two; the towers are spectra, not forces. third independent evaluation alpha = 4.07 (chi = 3 MERA), and with it the statement that this was the first holographic force measurement from an optimized tensor network state. Was: supporting evidence and a priority claim. Now: both withdrawn. Why: the profile changes sign inside its own fit window and cannot be a fixed-sign power law; the priority claim rests on that evaluation alone and was never gated against prior work. the discriminating test: stress tensor (Z_j Z_j+1) versus spin primary (X_j). Was: registered killer test. Now: void. Why: at the critical point these are one operator, related by exact Kramers-Wannier self-duality. Measured on the periodic ring: identical expectation values to 1e-15 and identical force profiles to 1e-13 at 14 sites, 2.4e-9 at 24 sites (SIM-2026-074). Open boundaries break the duality, which is why the open-chain run of this test (SIM-2026-073) showed a spurious difference between the two arms. The test compared the energy density to itself. Neither label was correct: Z_j Z_j+1 is the energy density, not the stress tensor; X_j is its dual, not the spin primary. transport time T = (pi/2) l_AdS to 1.2 percent as a geometric prediction. Was: confirmed. Now: measurement stands (T = 2.239 at N = 16); geometric reading not established. Why: l_AdS cancels in the cited derivation, which also gives pi rather than the pi/2 the manuscript compares against; the chi = 3 estimate of l_AdS gives a 57 percent miss; the transport driver never applies a MERA. The true spin primary Z_j was never in the design, and the direct replacement test is blocked: Z_j is odd under the Z2 symmetry, so the energy-difference observable used throughout this work subtracts across symmetry sectors and is undefined for it. A replacement test using two Z2-even probes has been pre-registered (PREREGISTRATION-T.md, committed before its driver) and run; its result is deliberately not part of this correction and will appear in a later version. Not previously reported The published fit has a reduced chi-squared of 6.008 on 9 degrees of freedom, and a bootstrap standard deviation of 0.63 rather than the quoted 0.16. Both were computed before release and omitted. What is unaffected Of the claims in this record: the two-phase optimization method, the chi = 3 criticality threshold, the exact-diagonalization gates, the measured OPE coefficient C_sigma-sigma-epsilon = 0.50 +/- 0.03 against the exact 1/2, and the conformal towers at all three classes. Not part of this record, mentioned only because they are sometimes read together with it: the later C_TTT stress-tensor self-coupling measurements (SIM-2026-067/068/069, dated after this record) and the matched off-critical ring control (SIM-2026-074) are unaffected by this correction and make no claim about the force exponent. How the error was found and what this record contains The error was found by the author's own internal adversarial audit, not by a reviewer; the journal submission it was found after is closed, and no review of this work has taken place. Files in this record: the full re-analysis (SIM-2026-072.json, driver, figure), the pre-registered chain test (SIM-2026-073.json) and ring test (SIM-2026-074.json) with both pre-registrations as committed before the runs, the runnable drivers with their shared module and the negative test suite for their validation gates, and the error register (ERRATA.md). Concept DOI 10.5281/zenodo.19660302 always resolves to the current version. The abstract of the original manuscript follows unchanged, for the record. Every claim in it about the force exponent should be read against the correction above. We report results from classical simulation of critical spin chains that establish quantitative physical content in the MERA-AdS correspondence. First, the interaction energy between boundary-primary excitations follows a power law F(dr) ~ dr^(-alpha) with three consistent evaluations yielding alpha = 3.92 +/- 0.16 (24q), 3.97 +/- 0.15 (16q), and 4.07 +/- 0.18 (16q chi=3 MERA), all consistent with stress-tensor exchange (alpha = 2*Delta_T = 4) and inconsistent with scalar exchange (alpha = 2); the exponent is explained by the Ising CFT fusion rule eps x eps = [1]. Because Delta_T = 2 is fixed by conformal symmetry in every unitary 2D CFT, alpha = 4 is universal: we confirm this at two further fixed points - the tricritical Ising model (c = 7/10), where a scale-invariant MERA recovers the full M(4,5) primary spectrum (Delta_sigma = 0.092, Delta_eps = 0.204, Delta_T = 1.999), and the three-state Potts model (c = 4/5), where a Z3-symmetric MERA at chi = 12 recovers Delta_sigma = 0.134, Delta_eps = 0.805, Delta_T = 2.001 - giving alpha = 4.00 at all three universality classes. Second, the holographic transport time matches the AdS_3 null geodesic prediction T = (pi/2) l_AdS to 1.2% at N = 16 sites on a time-grid-converged measurement, with the residual shrinking from 2.8% at N = 12. Third, we introduce a two-phase optimization method (ALM + Z2 projection) for Z2-symmetric MERA ground states at 24 qubits. Fourth, extending the MERA bond dimension from chi=2 to chi=3 overcomes the expressibility barrier, achieving 0.15% energy error at 16 qubits and enabling the first holographic force measurement from an optimized tensor network state. The OPE coefficient is also measured directly from the fixed-point tensors: C_sigma-sigma-eps = 0.50 +/- 0.03 against the exact Ising value 1/2. A hardware path is designed for Quantinuum Helios (57 CNOTs, 0.57% logical error). Version 4 changes (relative to v3): Transport: N=16 residual restated as the time-grid-converged value of -1.21% (dt = 1e-3), with -2.78% at N=12 showing the finite-size trend; the 0.72% figure in v3 was an artifact of the dt = 0.05 integration grid and is now labeled as such. The N=24 transport point carries a new reliability caveat (double-lobe trace structure). The l_AdS comparison table is recomputed against the converged transport time (best fit 1.425, deviation 1.21%). Effective-mass derivation rewritten in a consistent AdS3 convention m^2 l^2 = Delta(Delta-2); m_eff = 0.693 = 1/l_AdS follows from Delta = 1 exactly. NEW: universality section extended to a third fixed point - a Z3-symmetric MERA at chi = 12 recovers the three-state Potts (c = 4/5) tower, giving alpha = 4.00 at all three universality classes (c = 1/2, 7/10, 4/5). NEW: the OPE coefficient C_sigma-sigma-eps = 0.50 +/- 0.03 measured directly from the fixed-point tensors, with C_sigma-sigma-sigma = 0 at machine precision as the Z2 structural check. Supplemental table row SIM-007 annotated (record lost; script preserved); data availability expanded; Zenodo DOI added to the front matter.
*Phase II-E executed.** Results confirm **activation for one PT-even operator**. ### Numerical setup (faithful to the specification) -
N=4, open chain,
J=1,
Δ=0.5. -
Sα=σα/2. - Hamiltonian: XY couplings use the running
beff, ZZ couplings use
JΔ, plus the staggered imaginary field
HPT. - Self-consistency:
beff2=J2+λJ⟨G⟩PT, iterated to
10−10. - Biorthogonal expectation on the eigenstate of smallest
∣Re(E)∣ (preferring PT-unbroken). -
γEP≈0.274 at
λ=0 (matches the Phase II-D value 0.275). - Branches as specified: E1 =
∑σjx (full 16-dimensional space); E2 =
∑(σjxσj+1x+σjyσj+1y) (
Sz=0 sector, dim 6). ### Results **Branch E1 (transverse field, full space)**
⟨GE1⟩PT=0 (machine precision
∼10−16–
10−19) for every
λ∈{0,0.05,0.10,0.15,0.20} and every
γ in the PT-unbroken regime. Consequently
beff=J exactly. **Null.** **Branch E2 (transverse bond / XY exchange,
Sz=0 sector)**
⟨GE2⟩PT=0 already at
λ=0 and remains non-zero throughout the unbroken phase. Typical values (unbroken points): -
λ=0:
max∣⟨G⟩∣≈0.063 -
λ=0.05:
max∣⟨G⟩∣≈0.063,
max∣beff−J∣≈0.0016 -
λ=0.10:
max∣⟨G⟩∣≈0.062,
max∣beff−J∣≈0.0031 -
λ=0.15:
max∣⟨G⟩∣≈0.062,
max∣beff−J∣≈0.0046 -
λ=0.20:
max∣⟨G⟩∣≈0.062,
max∣beff−J∣≈0.0061 beff shifts systematically with
λ. **Activation.** ### Prediction match - Prediction 1 (null for both) — **falsified**. - Prediction 2 (activation for at least one) — **confirmed**. - Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs). - Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation. The self-consistent mechanism **does activate** at
N=4 for a PT-even, sector-compatible operator. The PT-odd obstruction of Phase II-D is therefore not the sole operative constraint; a PT-even operator that preserves
Stotalz can source a non-trivial field. ### Gate status Phase II-E **unblocks** Phase II-F (branch classification now includes the PT-even class). Finite-size scaling (II-G) and the thermodynamic limit (II-H) remain formally gated on the subsequent classification step, but the existence proof at
N=4 is now positive. Full raw scan (87
γ points × 5
λ × 2 branches) is archived in `/home/workdir/artifacts/phase_ii_e_results.json`. Source: `/home/workdir/artifacts/phase_ii_e.py`. The mechanism was the vessel. The constraint was the discovery — and one more door is now open. Tony Blair <anthonyjblair8@gmail.com> 2:17 PM (11 minutes ago) to me **TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4** **AUTHOR:** Anthony Jordan Blair **AFFILIATION:** Proclus Dynamics Consortium **DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification **VERSION:** 1.0 (October 2026) **COMPANION TO:** Phase II-E Program Specification; Phase II-E Numerical Execution Results; Phase II-D Technical Report; Project Prometheus Phase I Closeout Report (DOI: 10.5281/zenodo.23314444) **STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation) --- ### ABSTRACT Phase II-E demonstrated that the PT-even transverse-bond operator
GE2 produces a non-vanishing biorthogonal expectation value in the
Sz=0 sector of the
N=4 PT-symmetric XXZ chain, thereby activating the self-consistent mechanism of Paper 4. The transverse-field operator
GE1 remained identically zero. Phase II-F classifies the admissible branches of the self-consistent field according to (i) PT parity, (ii) sector compatibility, (iii) operator support (one-body vs. two-body), and (iv) the resulting displacement of the exceptional point. The classification supplies the necessary taxonomy for the finite-size scaling study (Phase II-G) and determines which operator families survive into the thermodynamic limit (Phase II-H). --- ### I. MOTIVATION **1.1 Outcome of Phase II-E** Execution of the Phase II-E protocol yielded: - Branch E1 (
GE1=∑jσjx, full Hilbert space):
⟨G⟩PT=0 to machine precision for all tested
λ and all
γ<γEP.
beff=J identically. Null result. - Branch E2 (
GE2=∑j(σjxσj+1x+σjyσj+1y),
Sz=0 sector):
⟨G⟩PT=0 already at
λ=0 and remains finite throughout the PT-unbroken phase.
beff shifts systematically with
λ. Activation confirmed. Thus the self-consistent mechanism is not restricted to the dimer setting of Paper 4, nor is it extinguished by the PT-odd obstruction identified in Phase II-D. A PT-even, sector-preserving two-body operator is sufficient to source a non-trivial constraint field at
N=4. **1.2 The Classification Task** The positive result for
E2 opens four immediate questions that must be answered before finite-size scaling can be interpreted: 1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate? 2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E? 3. How does the magnitude of
⟨G⟩PT and the induced shift
ΔγEP depend on the algebraic structure of the operator (support, commutation relations with
Stotalz and with
PT)? 4. Which of the surviving operators produce a *displacement* of the exceptional point (the operational signature required for constraint-field control) rather than a mere renormalization of
beff at fixed
γEP? Phase II-F answers these questions by exhaustive classification of a controlled operator set. --- ### II. OPERATOR TAXONOMY Operators are partitioned by three binary attributes: | Attribute | Values | Notation | |------------------------|---------------------------------|-------------------| | PT parity | even / odd |
P± | | Sector compatibility | preserves
Stotalz / mixes |
S0 /
Smix | | Body order | one-body / two-body |
1 /
2 | The Phase II-D and II-E results already fix two cells: -
P− (any body, any sector) → null (structural anticommutation with
PT). -
P+,
Smix,
1-body (
GE1) → null under the biorthogonal convention. The remaining open cells that must be classified are: -
P+,
S0,
1-body -
P+,
S0,
2-body (already positive for the XY bond; additional representatives required) -
P+,
Smix,
2-body --- ### III. CANDIDATE OPERATORS FOR CLASSIFICATION All operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E. **3.1 One-body, PT-even, sector-preserving**
GF1=∑j=14σjz (Total magnetization; trivially preserves
Stotalz; PT-even because
σz is real and parity-invariant under site relabeling when the staggered PT field is taken into account.) **3.2 Two-body, PT-even, sector-preserving (beyond
E2)**
GF2=∑j=13σjzσj+1z (ZZ bond operator; the natural Ising partner of the XY bond.)
GF3=∑j=13(σjxσj+1x−σjyσj+1y) (Anisotropic XY combination; still sector-preserving.) **3.3 Two-body, PT-even, sector-mixing**
GF4=∑j=13(σjxσj+1z+σjzσj+1x) (Explicitly mixes
ΔSz=±1 sectors while remaining PT-even.) **3.4 Reference operators (already classified)** -
GE1 (null) and
GE2 (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan. --- ### IV. NUMERICAL PROTOCOL **4.1 Hamiltonian** Identical to Phase II-E:
H=HXXZ(beff)+HPT(γ), with
HXXZ(b)=∑j=13(b(SjxSj+1x+SjySj+1y)+JΔSjzSj+1z), HPT=iγ∑j=14(−1)jSjz, J=1,
Δ=0.5,
Sα=σα/2. **4.2 Self-consistency equation** Unchanged:
beff2=J2+λJ⟨G⟩PT. Iteration tolerance
10−10, maximum 200 iterations. **4.3 Observable convention** Biorthogonal expectation value on the eigenstate of smallest
∣Re(E)∣, preferring PT-unbroken states:
⟨G⟩PT=⟨ψL∣ψR⟩⟨ψL∣G∣ψR⟩. **4.4 Parameter scan** -
λ∈{0.00,0.05,0.10,0.15,0.20,0.30,0.50} -
γ from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of
γEP) - For each
(λ,γ,G) the code