This work investigates whether a quantum phase transition can be located using record distinguishability measured only within a small subsystem. Using the massive Schwinger model at θ = π, the study defines a regional record susceptibility from the fidelity distinguishability of reduced states at neighboring fermion masses and compares it directly with the conventional global ground-state fidelity susceptibility. The principal result is an exact agreement of the critical-point locators at the tested lattice sizes. On a mass grid with spacing 0.025, the global susceptibility and the regional susceptibilities of both six-site and four-site regions peak at the same pseudo-critical mass: 0.2125 for N = 12 and 0.2375 for N = 16. The peaks sharpen with increasing lattice size, while the pseudo-critical location moves upward toward the known continuum critical value m_c/g ≈ 0.3335. A half-step refinement around the N = 16 maximum moves the estimate to 0.24375, consistent with the expected finite-size drift from below. A particularly strong locality result is that an observer holding only four sites together with the Wilson-line rotor identifies the same transition location as the full many-body state. The matter sector provides an additional test. Even after the rotor is traced out, the matter-only regional susceptibility peaks at the same pseudo-critical point. Thus, although the order record itself migrates from matter into the gauge-field zero mode across the transition, the local matter subsystem retains enough sensitivity to locate where the transition occurs. The results distinguish two different informational questions: where an order record is physically stored and where the system is maximally sensitive to a change in coupling. These need not be carried by the same degrees of freedom. This work demonstrates that, at the studied finite sizes, quantum criticality is locally record-detectable using fidelity-based reduced-state measurements. The analysis is limited to two lattice sizes and one coupling, so no continuum extrapolation or critical exponent determination is claimed.
This work investigates whether a quantum phase transition can be located using record distinguishability measured only within a small subsystem. Using the massive Schwinger model at θ = π, the study defines a regional record susceptibility from the fidelity distinguishability of reduced states at neighboring fermion masses and compares it directly with the conventional global ground-state fidelity susceptibility. The principal result is an exact agreement of the critical-point locators at the tested lattice sizes. On a mass grid with spacing 0.025, the global susceptibility and the regional susceptibilities of both six-site and four-site regions peak at the same pseudo-critical mass: 0.2125 for N = 12 and 0.2375 for N = 16. The peaks sharpen with increasing lattice size, while the pseudo-critical location moves upward toward the known continuum critical value m_c/g ≈ 0.3335. A half-step refinement around the N = 16 maximum moves the estimate to 0.24375, consistent with the expected finite-size drift from below. A particularly strong locality result is that an observer holding only four sites together with the Wilson-line rotor identifies the same transition location as the full many-body state. The matter sector provides an additional test. Even after the rotor is traced out, the matter-only regional susceptibility peaks at the same pseudo-critical point. Thus, although the order record itself migrates from matter into the gauge-field zero mode across the transition, the local matter subsystem retains enough sensitivity to locate where the transition occurs. The results distinguish two different informational questions: where an order record is physically stored and where the system is maximally sensitive to a change in coupling. These need not be carried by the same degrees of freedom. This work demonstrates that, at the studied finite sizes, quantum criticality is locally record-detectable using fidelity-based reduced-state measurements. The analysis is limited to two lattice sizes and one coupling, so no continuum extrapolation or critical exponent determination is claimed.
This work extends the record-based study of the θ = π quantum phase transition in the massive lattice Schwinger model to a third lattice size, N = 20, with ground-state Hilbert-space dimension 1,662,804. The addition of the third size allows the previously observed critical-point diagnostics to be tested as finite-size scaling quantities. Using uniform three-point parabolic interpolation across N = 12, 16, and 20, the global fidelity-susceptibility peaks move as 0.2125 → 0.2364 → 0.2485, showing a monotonic drift toward the known continuum critical mass. The coherence-minimum locator identified through the local-unitary floor also moves upward with system size while remaining below the susceptibility peak. Its separation from the peak decreases, indicating that the two independent record-based locators converge toward the same critical region from below. At N = 20, several independent diagnostics agree at sub-grid resolution: the global ground-state susceptibility, six-site and four-site regional record susceptibilities, and an alternative fidelity convention all locate the same transition region. The susceptibility peak also sharpens with increasing system size, with peak heights 1.80 → 3.04 → 4.56. No critical exponent is inferred from the available three-size dataset. The information structure of the critical record evolves with size as well. The minimum local-unitary floor increases from approximately 0.019 → 0.043 → 0.061, so the critical record remains strongly coherence-dominated at N = 20, with approximately 93.9% of its Fisher information carried by the coherence sector. Finally, the identity between the local-unitary floor and the classical-to-quantum Fisher-information ratio remains valid at Hilbert-space dimension 1.66 × 10^6 to approximately 3–6 × 10^−4. The results provide three-size finite-size scaling evidence that local and global record-based observables consistently locate the θ = π transition in the lattice Schwinger model, while also revealing how the coherence content of the critical record changes with system size. The study is limited to three finite lattice sizes and a relatively coarse mass grid. No critical exponents or infinite-volume extrapolation are claimed.
This work develops a quantitative theory of how much information about a quantum parameter can and cannot be erased by unitary operations, and applies it to the θ = π quantum critical point of the massive lattice Schwinger model. For two nearby parameter-dependent quantum states, the distinguishability stored in eigenvalue populations cannot be changed by unitary rotations, whereas information stored in eigenbasis orientation can be moved by such operations. This motivates the erasability floor: the minimum fidelity-based distinguishability remaining after optimal unitary erasure. For smooth full-rank state families with simple spectra, the paper proves E_spec / E → F_cl / F_Q, where F_Q is the quantum Fisher information and F_cl is its eigenvalue-population contribution. Thus the locally uneraseable fraction of parameter information is exactly the population share of the quantum Fisher information, while the erasable fraction is its coherence contribution. The result is further extended to eigenvalue crossings and degenerate subspaces. A transversal crossing produces an explicit correction to the floor that is parabolic in the crossing position within the sampling window. Degenerate-point slopes are determined by the eigenvalues of the projected derivative within the degenerate subspace rather than by basis-dependent diagonal entries. Symmetric crossings can drive the erasability floor exactly to zero. The theoretical results are verified numerically on analytic and engineered state families, including crossing, degeneracy, additivity, and random-family tests. The framework is then applied to the massive lattice Schwinger model at θ = π with the Wilson-line zero mode retained as a dynamical rotor. Matrix-free GPU Krylov calculations extend the finite-size tower to N = 30 sites, corresponding to a Hilbert-space dimension of approximately 1.4 × 10^9. The global fidelity-susceptibility peak grows approximately as N^s with s = 1.87–1.89 over N = 22–30, corresponding to ν = 1.06(1), consistent with the two-dimensional Ising universality class. The pseudo-critical peak locations extrapolate to a lattice critical mass near 0.278 at a = 1/2, with a fit-form range extending to 0.296. The central value lies within 0.007 of the independently predicted value obtained by transporting the established continuum critical mass through the known additive lattice mass shift. The erasability floor reveals the information structure of the transition. Near criticality, only about 6–8% of the mass-parameter information carried by a gauge-invariant six-site region is population-borne, implying that more than 92% is coherence-borne and therefore movable by local unitary rotations. Away from the transition, the population contribution rises substantially. A regional susceptibility constructed from the same reduced states locates the critical point to within approximately 0.009 of the global susceptibility across all five large system sizes, demonstrating that the critical point remains locally detectable even as the total Hilbert space grows by more than two orders of magnitude. The study also reports unresolved finite-grid questions rather than forcing conclusions: monotonic scaling of the floor minimum and of the dip–peak separation cannot be established on the present mass grid and require finer sampling. Together, the results connect quantum Fisher information, unitary erasability, eigenvalue-crossing geometry, and quantum criticality within a single reproducible framework. The accompanying package includes theorem-verification scripts, the matrix-free lattice solver, validation gates, analysis code, figure-generation code, and raw large-N data.
This work uses the floor theorem as a quantitative probe of how information is distributed across a quantum phase transition. The system is the massive Schwinger model at θ = π, where an earlier record-susceptibility analysis located the finite-size transition. Here, neighboring ground states are analyzed through the local-unitary floor and its exact Fisher-information decomposition. Three main results are obtained. First, the floor theorem remains quantitatively valid throughout the transition. Across sixteen mass intervals, the directly measured local-unitary floor agrees with the ratio F_cl/F_Q at every point, with a maximum deviation of 0.0019. Thus the theorem functions as an accurate diagnostic even in the rapidly changing critical region. Second, the critical record is overwhelmingly coherence-borne. The population fraction of the quantum Fisher information drops to approximately 0.043 near the transition, implying that about 95.65% of the critical sensitivity is carried by eigenvector rotation rather than eigenvalue populations. Away from criticality, the information becomes progressively more population-borne, with the floor rising to approximately 0.39 by the end of the sweep. The coherence minimum does not coincide exactly with the conventional fidelity-susceptibility peak. It lies below the peak at both studied lattice sizes, with the separation decreasing as system size increases. This identifies the coherence extremum as a distinct finite-size critical locator rather than merely another representation of the susceptibility maximum. Third, the symmetry-broken order record itself is found to be almost completely local-unitary erasable. The two broken-symmetry vacua have nearly identical regional spectra, giving a unitary floor of approximately 0.014. Their distinction is therefore carried predominantly by a symmetry rotation. This does not contradict the substantial protection previously found under field-preserving channels. The same order record has different fates under different allowed channel classes: it is erasable by the broken-symmetry rotation while remaining protected when the field populations carrying the order parameter are explicitly preserved. The results provide an information-theoretic anatomy of criticality in which quantum Fisher information is separated into population and coherence components, revealing that the transition is dominated by coherence rather than population rearrangement. The analysis is limited to two lattice sizes and one coupling. The finite-size scaling of the offset between the coherence minimum and the susceptibility maximum remains open.
Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility χF can detect a QCP provided that the correlation length exponent satisfies ν< 2. We then show that χF can be used to locate a QCP even if ν 2 if we introduce boundary conditions labeled by a twist angle Nθ ,w hereN is the system size. If the QCP lies at g = 0, we find that if N is kept constant, χF has a scaling form given by χF ∼ θ −2/ν f(g/θ 1/ν )i fθ � 2π/N. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which ν = q, and in a XY spin-1/2 chain in which ν = 2. Finally we show that when q is very large, the model has two additional QCPs at h =± 2 which cannot be detected by studying the energy spectrum but are clearly detected by χF. The peak value and width of χF seem to scale as nontrivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
This work synthesizes the gauge-theory branch of the quantum-record program and derives a quantitative relation between record irreversibility and Fisher information. The paper first corrects an interpretation from an earlier study of theta records. At a finite angular separation, the fraction of the record that could not be removed by local unitaries was approximately one half in both the massless and massive Schwinger models. The present tangent-space analysis shows that this apparent mass independence was a finite-angle coincidence. In the small-parameter-separation limit, the local-unitary floor is instead governed by a Fisher-information law: E = d²F_Q/8, E_spec = d²F_cl/8, and therefore E_spec/E = F_cl/F_Q, where F_Q is the quantum Fisher information and F_cl is its population, or diagonal, contribution in the eigenbasis of the reduced state. The relation is verified numerically to four decimal places at two mass values, two expansion centers, and two step sizes. The resulting floor fractions are approximately 0.639 in the massless theory and 0.495 in the massive theory. Operationally, the result states that the fraction of a parameter record that no local unitary can erase is precisely the fraction of its Fisher information encoded in eigenvalue populations rather than eigenvector rotations. The paper also consolidates eleven companion studies into a single gauge-record classification. Twenty-nine previously frozen numerical results spanning continuum field theory, interacting gauge theory, boundary conditions, theta physics, symmetry breaking, criticality, and non-Abelian center charge are audited against the general record principle. All twenty-nine are found to be mutually consistent. The resulting classification separates protected carriers—including Gauss flux, zero modes, energy, emergent conserved structures, and center charge—from erasable information classes such as arrangements, reference-frame orientation, and finite-size screening remnants. A final SU(3) diagnostic further localizes a previously observed quark–antiquark residual asymmetry. The effect appears only in the screened regime and only for one string convention, strongly indicating that it originates in the geometry of the screening cloud on the string side, although an analytic derivation remains open. The Fisher relation is demonstrated numerically in the present system rather than proven as a general operator theorem. Establishing that general theorem is the natural next theoretical step.
This note removes the open-boundary constraint from the lattice Schwinger model and studies the periodic theory with its physical Wilson-line zero mode. The goal is to determine which features of the previous lattice results belong to the gauge theory itself and which were caused by the boundary. Using exact diagonalization of the periodic lattice model with fermions and flux degrees of freedom, the study reveals four main results. The transition is restored: the mass-gap collapse and the critical response reappear, with the mass-direction evidence developing an interior peak near the transition region. The θ-lock is recovered at zero fermion mass, with the sector energy sensitivity strongly reduced compared with the open-boundary case, while the mass-induced unlock follows the expected perturbative behavior and becomes critically enhanced near the transition. The transition-amplifier law also returns: sector evidence grows monotonically with the mass deformation, showing that the open-boundary inversion was a boundary artifact caused by pinning both sectors to a single vacuum. The central result is a record-migration law. In the massless theory, the θ-record is stored mainly in matter correlations and can be accessed through charged endpoints of spatial cuts. In the massive broken phase, the record moves into the topological flux sector: the competing vacua differ by a Wilson-loop flux, making the information invisible to purely matter-based local observers. This establishes how the location of quantum information changes across phases of an interacting gauge theory: symmetry breaking does not only create records, it determines who can access them. The results are obtained within the finite periodic lattice framework, with bounded system sizes, rotor truncation checks, and no continuum extrapolation.